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 Permanent Link:
 https://ufdc.ufl.edu/UF00089539/00001
Material Information
 Title:
 A molecular thermodynamic model for aqueous solutions of nonpolar compounds and micelle formation
 Series Title:
 A molecular thermodynamic model for aqueous solutions of nonpolar compounds and micelle formation
 Creator:
 Brugman, Robert James
 Publisher:
 Robert James Brugman
 Publication Date:
 1979
 Language:
 English
Subjects
 Subjects / Keywords:
 Enthalpy ( jstor )
Entropy ( jstor ) Free energy ( jstor ) Hydrocarbons ( jstor ) Liquids ( jstor ) Micelles ( jstor ) Monomers ( jstor ) Solutes ( jstor ) Surfactants ( jstor ) Water temperature ( jstor )
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 University of Florida
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 University of Florida
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 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 05581592 ( OCLC )
023017921 ( AlephBibNum )

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Full Text 
A MOLECULAR THERMODYNAMIC MODEL FOR AQUEOUS SOLUTIONS
OF NONPOLAR COMPOUNDS AND MICELLE FORMATION
By
ROBERT J. BRUGMAN
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1979
ACKNOWLEDGMENTS
I wish to express my deepest appreciation to Dr. John O'Connell
for his interest and enthusiastic guidance throughout my graduate studies.
Working with him has been a truly valuable and enjoyable experience.
I also wish to thank Drs. J. C. Biery, G. Y. Onoda and F. A.
Vilallonga' and. Prof. R. D. Walker for serving on my supervisory committee.
It is a pleasure to thank the faculty and students of the Depart
ment of Chemical Engineering for providing an enjoyable, pressurefree
environment in which I could pursue my research.
I am extremely grateful to Mrs. Thomas Larrick for her excellent
typing despite the author's poor handwriting and the highly technical
nature of the task.
Finally, I am grateful to the Department of Energy and the members
of the Enhanced Oil Recovery Consortium at the University of Florida
who provided financial support for this work.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS . . . . . . . . . . ..... ii
LIST OF TABLES . . . . . . . . . . . . . vi
LIST OF FIGURES . . . . . . . . . . . .... ix
KEY TO SYMBOLS . . . . . . . . . . . . . x
ABSTRACT . . . . . . . . . . . . . . xiv
CHAPTER
1 INTRODUCTION . . . . . . . . . . . 1
2 THERMODYNAMICS OF MICELLIZATION . . . . . . 4
Introduction . . . . . . . . . . . 4
Thermodynamic Formalism of Micellization . . . . 4
Effect of Added Salt on Micellization . . . . . 12
Some Theories for Free Energy Changes Upon Micellization 15
Contributions to Thermodynamic Properties of Micellization
from Various Species . . . . . . . . 20
A Thermodynamic Process for Micelle Formation . . . 25
3 SCALED PARTICLE THEORY MODIFIED FOR AQUEOUS SOLUTIONS . 29
Introduction . . . . . . . . . . . 29
Basis of Scaled Particle Theory . . . . . .. 29
Scaled Particle Theory and Aqueous Solutions . . . 37
Curvature Dependence of Surface Tension . . . . 43
Determination of the Radial Distribution Function and
the Direct Correlation Function for Liquid Water
from XRay Diffraction.. . . . . . . . 45
4 MODELING OF SPHERICAL GAS SOLUBILITY . . . . . 62
Introduction . . . . . . . .. . .. . 62
Thermodynamic Properties of Solution from
Experimental Data . . . . . . . . . 63
Application of ScaledParticle Theory to Aqueous
Solubility . . . . . . . . . . . 66
Contributions to the Thermodynamic Properties
of Solution from Cavity Formation . . . . . 72
TABLE OF CONTENTS (Continued)
CHAPTER Page
4 (Continued)
Contributions to the Thermodynamic Properties
of Solution from Intermolecular Forces . . . . 78
Analysis of Spreading Pressure of a Solute Occupying
a Cavity . . . . . . . . . . .... 81
Discussion of Results and Suggestions for Future
Research . . . . . . . . . . . 84
5 AQUEOUS SOLUBILITY OF ALIPHATIC HYDROCARBONS . . . 92
Introduction . . . . . . . . . . . 92
Calculation of Thermodynamic Properties of Cavity
Formation forAliphatic Hydrocarbons . . . . 93
Free Energy of Interaction Between a Spherocylindrical
Solute and Spherical Solvent . . . . . . 94
Changes in Rotational and Vibrational Degrees of Freedom
of Aliphatic Hydrocarbons Upon Solution . . . 104
Results of the Model for Aqueous Solubility of
Aliphatic Hydrocarbons . . . . . . . . 110
Comparison with Infinite Dilution Properties
of Surfactants . . . . . . . . . . 129
Suggestions for Future Work . . . . . . . 129
6 MODELING OF THE THERMODYNAMIC PROPERTIES OF MICELLIZATION 132
Introduction and Review of Thermodynamic Process
for Micelle Formation . . . . . . . . 132
Derivation and Application of an ArbitraryShape
Hard Body Equation of State . . . . . . 134
Contributions to a Model for the Thermodynamics
of Micellization . . . . . . . . . 139
Discussion and Suggestions for Future Research . . 149
7 EXPERIMENTAL INVESTIGATION OF PHASE BEHAVIOR AND
TRANSITIONS FOR CONCENTRATED SURFACTANT SOLUTIONS . . 156
Introduction . . . . . . . . . . . 156
Experimental Objectives . . . . . . . . 156
Description of Experimental Apparatus . . . . . 156
Operating Procedures . . . . . . . . . 160
Pertinent Calculations for Dissolved Gas Experiments . 162
Results and Suggestions for Future Work . . . . 166
8 SUMMARY AND CONCLUSIONS . . . . . . . . 168
TABLE OF CONTENTS (Continued)
APPENDIX
A PROGRAM FOR CORRELATION OF SPHERICAL GAS
SOLUBILITY PROPERTIES . . . . . . .
B HELMHOLTZ FREE ENERGY OF INTERACTION BETWEEN
A SPHEROCYLINDRICAL SOLUTE AND SPHERICAL SOLVENT
Integrations of the Components of the Helmholtz
Free Energy of Interaction . . . . .
Correlation of the Helmholtz Free Energy of
Interaction with s ,L and Temperature . .
s
C PROGRAMS FOR GAS AND LIQUID HYDROCARBON
SOLUBILITY PROPERTIES . . . . . . .
D PROGRAM FOR CALCULATION OF THERMODYNAMIC
PROPERTIES OF MICELLIZATION . . . . . .
BIBLIOGRAPHY . . . . . . . . . . . .
BIOGRAPHICAL SKETCH . . . . . . . . . .
Page
. . 172
. . 184
. . 194
. . 201
. . 209
. . 258
. . 272
. . 280
LIST OF TABLES
Table Page
31 Surface Tension and Curvature Parameter Calculated
for Liquid Water at Its Saturated Vapor Pressure
Using the Pierotti Approximation . . . . . 39
32 Radial Distribution Function for Liquid Water . . . 47
33 Direct Correlation Function for Liquid Water . . 50
34 Direct Correlation Function for Liquid Water . . . 53
35 Reduced Direct Correlation Function for Liquid Water . 58
4la Solution Properties from Pierotti's Model . . . . 70
4lb Enthalpy and Heat Capacity Contributions . . . . 71
42 Universal Correlation for the Reduced Hard
Sphere Diameter . . . . . . . . . 76
43 Characteristic Parameters . . . . . . . . 77
44 Intermolecular Potential Energy Parameter . . . . 80
45a Contributions to Free Energy and Entropy of Solution . 85
45b Contributions to Enthalpy and Heat Capacity of Solution 88
51 c Parameter Values for Aliphatic Hydrocarbons . . . 107
52 Properties Required to Analyze Liquid Hydrocarbon
Solubility . . . . . . . . . . . 109
53a Contributions to Free Energy of Solution of
Gaseous Hydrocarbons . . . . . . . . 111
53b Contributions to Enthalpy of Solution of
Gaseous Hydrocarbons . . . . . . . . 113
53c Contributions to Entropy of Solution of
Gaseous Hydrocarbons . . . . . . . . 115
53d Contributions to Heat Capacity of Solution
of Gaseous Hydrocarbons . . . . . . . 117
vi
LIST OF TABLES (Continued)
Table Page
54 Energy Parameter Values and Length Function . . . 119
55a Contributions to Free Energy and Entropy of
Solution of Liquid Hydrocarbons . . . . . 121
55b Contributions to Enthalpy and Heat Capacity of
Solution of Liquid Hydrocarbons . . . . . 125
56 Infinite Dilution Heat Capacity of Surfactants
in Water at 298.150K . . . . . . . . 130
61 Comparison of Properties of Hard Spheres with Those
of Some NonSpherical Particles . . . . . 137
62a Contributions to Gibbs Free Energy of Micellization . 141
62b Contributions to Enthalpy of Micellization . . . 142
62c Contributions to Entropy of Micellization . . . . 143
63 Parameter Values for Micellization Model . . . . 145
71 Temperature Dependence of Two Phase Region . . . 164
72 Pressure Dependence of Two Phase Region . . . . 165
73 Effect of Dissolved Methane . . . . . . . 165
Ala Contributions to Free Energy and Entropy of Solution . 187
Alb Contributions to Enthalpy and Heat Capacity
of Solution . . . . . . . . . . 190
Bla Parameters for Temperature Dependence of o
Interaction Correlation Coefficients (a = 3.40 A) . 202
s
Blb Parameters for Temperature Dependence of o
Interaction Correlation Coefficients (as = 3.60 A) . 204
B1c Parameters for Temperature Dependence of o
Interaction Correlation Coefficients (as = 3.80 A) . 206
Cla Contributions to Free Energy of Solution of
Gaseous Hydrocarbons . . . . . . . . 226
Clb Contributions to Enthalpy of Solution of
Gaseous Hydrocarbons . . . . . . . . 228
vii
LIST OF TABLES (Continued)
Table Page
Clc Contributions to Entropy of Solution of
Gaseous Hydrocarbons . . . . . . . . 230
Cld Contributions to Heat Capacity of Solution of
Gaseous Hydrocarbons . . . . . . . . 232
C2a Contributions to Free Energy and Entropy of
Solution of Liquid Hydrocarbons . . . . . 248
C2b Contributions to Enthalpy and Heat Capacity of
Solution of Liquid Hydrocarbons . . . . . 252
C3 Energy Parameter Values and Length Function . . . 256
viii
LIST OF FIGURES
Figure Page
21 Contributions of Species to Property Changes
of Micellization . . . . . . . . . . 22
22 A Thermodynamic Process for Micelle Formation . . . 26
31 Contact Correlation Function; Comparison of
Different Models . . . . ... . . . . 42
32 Reduced Direct Correlation Functions . . . ... . 57
5la Fixed Potential at y = 0 Interacting with Molecular
Centers in 0 < y 5 L and 0 < z < .. . . . . . 95
5lb Fixed Potential at y = 0 Interacting with Molecular
Centers in y < 0 and y > L . . . . . . . 95
5ic Distributed Potential Along Spherocylinder Axis
from y = 0 to y = L Interactint with Molecular
Centers in 0 S y S L and 0 z
5ld Distributed Potential Along Spherocylinder Axis
from y = 0 to y = L Interacting with Molecular
Centers in y < 0 and y > L . . . . . .. 96
61 A Thermodynamic Process for Micelle Formation . . . 133
71 High Pressure Experimental Apparatus . . . . . 158
KEY TO SYMBOLS
A = Helmholtz free energy
A = Helmholtz free energy
a = activity
a = cavity surface area
c(r) = direct correlation function
Cdis = dispersion coefficient in intermolecular potential
Cp = heat capacity
d = hard sphere diameter
f = fugacity
g(r) = radial distribution function
G = Gibbs free energy
G(r) = contact correlation function
H = enthalpy
H (s) = scattering structure function
m
J = arithmetic mean curvature
K = Henry's constant
L = spherocylinder length
LCH2 = segmental length
N = average micelle aggregation number
P = pressure
P = probability of an empty cavity
Q = canonical partition function
R = gas constant
R = separation between molecules
S = entropy
T = temperature
T = characteristic temperature
U = interaction energy
V = volume
V = characteristic volume
W(r) = work of cavity formation
X = mole fraction
Y = reduced solvent density
Greek Letters
a = fraction of counterions bound to micelle
a = solvent coefficient of thermal expansion
P
a2 = solute polarizability
S = I/KT
y = surface tension
y = planar surface tension
F. = relative adsorption of i to water
l,w
6 = curvature dependence parameter for surface tension
A = denotes a property change
= interaction energy parameter
p = chemical potential
p = number density
a = potential distance parameter, hard sphere diameter
= intermolecular pair potential
xi
.Subscripts
0 = overall mole fraction of surfactant
1 = monomer or solvent property
2 = solute property
c = cavity property
ca = counterion adsorption
ci = counterion cavity property
cal = total calculated property
exp = experimental property
HS = hard sphere property
hs = hard sphere property
i = interaction property
m = micelle property
mic = micelle property
mmi = monomermonomer interaction property
0 = absence of added salt
ref = refers to reference solute
s = solvent property
w = water
ws = watersolute property
Superscripts
o = standard state
+ = property value at CMC
AQ = aqueous solution
diss = dissolved state
g = gas phase
el = electrostatic quantity
hs = hard sphere quantity
L = liquid phase
ref = refers to reference solute
V = vapor phase
Overline
= denotes partial molar, average, or reduced property
xiii
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor Philosophy
A MOLECULAR THERMODYNAMIC MODEL FOR AQUEOUS SOLUTIONS
OF NONPOLAR COMPOUNDS AND MICELLE FORMATION
By
Robert J. Brugman
June 1979
Chairman: John P. O'Connell
Major Department: Chemical Engineering
A thermodynamic process for micellization has been developed
which provides a basis for better understanding of molecular mechanisms
important in the formation of micelles as well as other processes of
aqueous solution. The foundation of a model of the micelle formation
process is a unified molecular thermodynamic theory of aqueous solutions.
For spherical gases, a modification of scaled particle theory
has been used for the excluded volume contribution while a mean field
theory approach has been used for intermolecular interactions. Very
good correlation of experimental thermodynamic properties was achieved
over a considerable temperature range (2770K 3580K).
Extension of the model to aliphatic hydrocarbon solutes required
development of an expression for the total interaction energy between
a spherocylindrical solute and a spherical solvent. The radial distri
bution function was considered to be a function of distance from the
spherocylinder surface and the intermolecular potential was distributed
along the spherocylinder axis. The thermodynamic properties of solution
for gaseous hydrocarbons (C1 C4) are well correlated and predicted
trends in solution properties for the liquid hydrocarbons were in
reasonable accord with the few experimental data available. The entropy
contribution from intermolecular interactions was unexpectedly large,
actually dominating the cavity (excluded volume) contribution at higher
temperatures.
A primitive model for the thermodynamic properties of ionic
micellization was tested using the aqueous solubility model and electro
static theories. Quantitative agreement with experiment was not found
due to contributions to the thermodynamic process which were omitted
as well as model inadequacies. Suggestions for improvement have been
made.
A limited experimental investigation was conducted into the
effect of temperature, pressure and dissolved gas on the isotropic
anisotropic transition for lyotropic liquid crystals. Tentative results
indicate a two phase region between 40 and 80C and little or no effect
of pressure or dissolved gas on the isotropicanisotropic transition,
except for transients.
CHAPTER 1
INTRODUCTION
The objective of this investigation is development of a unified
molecular thermodynamic theory of aqueous solutions with particular inter
est in the aggregation of surfactant monomers to form micelles. This
development is based upon building stepwise from a treatment of aqueous
solubility of spherical nonpolar gases to consideration of the numerous
molecular effects important for a rigorous model of the thermodynamics of
micellization.
Chapter 2 progresses from a discussion of the thermodynamic
formalism of micellization through a review of previous models to develop
ment of a thermodynamic process for micellization. Previous models
(Tanford, 1974b) considered micelle formation as a result of a balance
between favorable (AGm <0) "hydrophobic forces" arising from minimization
of waterhydrocarbon contact area and unfavorable (AG > 0) electrostatic
m
forces between surfactant head groups in the micelle. While these models
provide reasonable correlating expressions, they present an overly sim
plistic picture of the important molecular effects involved. The present
process analyzes the situation in considerably more detail, accounting
for several other significant effects such as changes in intermolecular
interactions upon aggregation, entropy and enthalpy changes upon restric
tion of surfactant head groups to the micelle surface and most importantly
1
excluded volume effects associated with creating and destroying the appro
priate sized cavities in water.
Interest in this excluded volume effect leads to a discussion
in Chapter 3 of the application of scaled particle theory to aqueous
solutions of nonpolar gases as originally done by Pierotti (1965), with
modifications suggested by Stillinger (1973) to account for the special
effect of the hydrogen bond structure of water. Fundamental difficulties
arise when considering the appropriate matching of the microscopic expres
sion of Stillinger for the work of cavity formation with the macroscopic
expression resulting from the relation of Koenig (1950) for the curvature
dependence of the surface tension. Rather than address this controversy
directly, we have chosen to use argon as a reference solute and use only
the macroscopic expressions to obtain properties for the other solutes
from those of argon.
Application of the model to aliphatic hydrocarbons is presented
in Chapter 5. To model the linear hydrocarbons as spherocylinders, an
expression is derived from perturbation theory for the total interaction
energy between a spherocylindrical solute and a spherical solvent. This
derivation considers the radial distribution function to be a function
of distance from the spherocylinder surface rather than a centertocenter
distance (spherical case) and distributed the intermolecular potential
along the spherocylinder axis.
The correlation for the gaseous hydrocarbons is quite good and
predicted trends in liquid hydrocarbon solubility properties seem reason
able. Sensitivity of the results to chain segment length is examined.
Attempts at development of a model for the thermodynamic process
of micellization are discussed in Chapter 6. Some contributions such as
electrostatic repulsions between head groups in the micelle surface and
watermicelle interactions are omitted from the model and other contribu
tions are likely to be inadequate. Comparison of the contributions of
all effects is discussed and suggestions for improvement are given.
Chapter 7 presents the results of a limited experimental investi
gation not directly related to the theoretical study. The objective of
the experiment was to determine the effect of temperature, pressure and
dissolved gas on the isotropicanisotropic phase transition in lyotropic
liquid crystals. A study of a single system containing potassium oleate
showed a twophase region between about 40 800C and no reproducible
effect of pressure or dissolved gas on the phase behavior.
Chapter 8 provides a summary of conclusions and suggestions
for future research as discussed in detail in the earlier chapters.
CHAPTER 2
THERMODYNAMICS OF MICELLIZATION
Introduction
The purpose of this chapter is to discuss the fundamental
thermodynamics with which a theory of micellization must be consistent.
In the first section the basic formalism is developed, along with dis
cussion of the effects of the distribution of micelle aggregation number.
Consideration is then given to the effect of added electrolyte.
Previous theories for the free energy change upon micellization are
critically reviewed, leading to a reconsideration of contributions to
thermodynamic properties of micellization from the various species
involved and development of a novel thermodynamic process for micellization.
Thermodynamic Formalism of Micellization
For nonionic amphiphiles, Hall and Pethica (1970) show a relation
derived from Hill's small systems thermodynamics (1963) for the Gibbs'
free energy of formation, AG of a system of micelles of average aggre
gation number, N, in a standard state of infinite dilution, from monomeric
amphiphiles, also in a standard state of infinite dilution, in terms of
the mole fraction of the monomers, xI, and of the micelles, xm, when
ideal solution is assumed
AGO 1 0
m m 1 1 I
m = n xI _x n x (21)
NRT NRT RT N
At the critical micelle concentration (CMC), relatively abrupt
changes in properties are observed because the concentration of monomeric
species begins to change very little with the mole fraction of added
amphiphile, x. micelless are being replicated)
x = xI + N x (22)
o 1 m
Around the CMC the value of 8xl/@x TP falls rapidly from near
unity to near zero. The CMC definition of Phillips (1955), explored by
Hall (1972), is
83
lim c = 0 (23)
\ + @x3
o o T,P
where x is the CMC and c isan "ideal colligative property" which depends
only on the number of solute species monomerss and micelles). As noted
by Chung and Heilweil (1970), this definition is difficult to interpret
unambiguously in terms of the experiments done with micellar solutions.
As an alternative, the expression of Hall and Pethica (1970) can
be used
8(x +x )
lim 1 m = 0.5 (24)
+ xo
x +x
o o T,P
which is essentially equivalent to equation (23) for sharp CMC points.
This expression can be used to obtain (Hall and Pethica (1970))
x = (N2 2N)x (25)
1 m
From the definition of x in equation (22)
x = x /(N N) (26)
m o
+ + 2 2 
xl = xo(N 2N)/(N N) (27)
which then yields
AG + 1 + 2
 n x = n x+ n N + n(l N/N
NRT o N o
+ Yn [(12N/N 2)/(1N/N2) (28)
To assume that polydispersity is unimportant the righthand
side of equation (28) should be small enough to neglect, which means its
+
value is approximately 0.05 for 5% error in x (typical experimental
0
uncertainty). By taking
1 >> 1/N = N/N2 (29)
equation (28) becomes, in first order approximation
AG n x + 2%n N 1
n x = (210)
NRT N
The righthand side of equation (210) is always positive so
+ 0
Zn x represents a lower algebraic bound to AG /NRT. For the effect to
o m
be less than 5%, N should be greater than about 300 to 600. However, it
ranges upward from 10, so polydispersity is normally important and thermo
dynamic theories for AG/NRT should give an equation which can yield
m
2 0
values of N and N. Further, data analysis to obLain values of AC from
In
CMC values must allow for these terms.
The division of the free energy change of micellization into
enthalpy and entropy components is accomplished by the relations
AH 0 AGo/RT AG
m _1 m m nN (211)
D1/T 31/T
NR N P,n NRT P,n
Al0 + +
AH 0Zn x 1 n + n x 0
SN ( ZnN nx +3 nN (212)
NRT nT P,n N 0Pn
AS0 AH0 AG
m m m (213)
NR NRT
As0 + +T n x
m + o 1 o
n x  + T
NR o0 ZnT P,n N P,n
2 [T n N nx + (214)
T P,n P,nJ
The last term in equation (212) is small when the righthand side
of equation (210) is, so the standard state enthalpy change is normally
close to the temperature derivative of the CMC. As determined by Desnoyers
et al. (1978) from direct calorimetry, at lower temperatures (T 5 250C)
the micellization process is entropy driven (TAS <0, AHo >0), whereas at
m m
higher temperatures it is enthalpy driven (TAS0 >0, AH <0). This con
m m
siderable variation of AHo with temperature is reflected in a large nega
m
tive heat capacity change (ACp ) upon micellization. These trends agree
m
qualitatively with those determined using equations (212) and (214) by
Moroi et al. (1975) for sodium alkyl sulfates. Thus theories which
attempt to correlate the data must exhibit considerable flexibility in
their temperature dependence.
The above analysis was developed by Hall and Pethica (1970) for
nonionic species. The small systems analysis has not been applied to
ionic systems where the effect of counterions on the thermodynamics must
be included. This may be due to the extreme complexity of such an effort.
At present, the thermodynamics of micellization for ionic systems
is divided into two formalisms based on the mass action approach, These
are reviewed by Mijnlieff (1970). In the first, such workers as Stigter
(1964, 1974ab, 1975ab), Emerson and Holtzer (1965, 1967ab), and Mukerjee
(1969) focus on the changes associated with the amphiphilic ions forming
an aggregate. Thus, for a singly charged anionic monomer, the reaction is
N
NM M (215)
1 m
and the thermodynamic formulation is
o el
0i = Pi + P + RT Zn a (216)
o el
m = p + p + RT kn a (217)
m m m m
+ + o o el el +  +
P Np = 0 = (m Npo) + (,p Np ) + RT(Zn a N n a ) (218)
m 1 m 1 m 1 m 1
where p is the standard state (infinite dilution) chemical potential of
the uncharged monomer, p is the standard state chemical potential of the
m
el el
uncharged micellar aggregate, and ( N p ) is the chemical potential
m 1
difference associated with changing the charge on the micelle and the
monomers from zero to full value while it is in the presence of the ionic
atmosphere of the counterions. Such a change involves the response of the
counterions and is sensitive to the detailed molecular structure assumed,
as the calculations of Stigter (1975ab) show. It is particularly sensi
tive to the fraction of ions assumed bound to the micelle in the Stern
layer as related to electrophoretic and electrical conductance measure
ments (Stigter, 1964). This fraction is apparently of the order of one
half to threefourths when the micelle is fully charged but how this value
depends upon the charging process is unclear.
el
The value of Pi is determined from some expression such as that
1
of DebyeHuckel theory leading to
iel = i + RT An yl (219)
el
The relationship of the counterions to el is one of equilibrium between
m
those in bulk solution and those in the Stern (bound) layer and the Gouy
Chapman (diffuse) layer (Stigter, 1964)
p (solution) = Po(s) + iel + RT in a (s) (220)
C C JC
el
o 1m + RT n a (m) (221)
p (solution) = V micellee) = p (m) +  c
c c c N
or el
el
( o m P = RT Zn(a (s)/a (m)) (222)
P (m) (s) + 1 c c
C C N
el el
Substituting for NIp in equation (18) and combining the standard
m 1
state chemical potentials yield
+ + + N
a a (m)
N(p + ) + RT kn = 0 (223)
S1(aa +(s)] N
+ + +
Assuming that we can replace am and a (m) by unity micelless) and a1
and a (s) by mole fraction (solution), using the definition of AG0 from
c m
equation (21) where all the species are uncharged gives
AG
m n x_ x+ (224)
NRT
+ +
which for no added salt (x x ) is
c 1
0
AGm +
 = 2 Pn x 1 (2 25)
NRT
This relation also appears in the work of Shinoda and Hutchinson (1962).
It is important to note that all standard state chemical potentials and
activities given above are for neutral species.
The second approach to the thermodynamic relationships for ionic
amphiphiles (Sexsmith and White, 1959ab; White, 1970; Phillips, 1955;
Molyneaux and Rhodes, 1972; Kaneshima et al., 1974) writes the reaction as
+ N(1a)
NM + NCG+ M( (226)
1 m
where a is the apparent fraction of amphiphiles whose charge is neutral
ized by bound counterions. The chemical potential relation is then
+ 0+ o 
pm NIl Na4+ = 0 = u Nl Nap0
+ + +0 a
+ RT[Yn a N n a (a ) ] (227)
m 1 c
where the standard state is the charged species at unit activity. Again
+ + +
assuming that we can replace a by unity and a and a by mole fractions,
m 1 c
using the definition of AG0 from equation (21) gives
m
AG
m0 + + ,
= n x1(x) (228)
NRT c
In these relations, the chemical potentials are for ionic species, a
concept which is tenuous since in the definition
u = PG/n i) (229)
j#i
charge neutrality prevents holding all n. constant while ni is varied if
species i is charged. For the case of no added salt equation (228) yields
AG0 +
m +
= (l+a) kn x (230)
NRT
In order for equation (230) to yield equation (225), the value of a
must be unity.
This conclusion has two important consequences for theoretical
analysis. Since a has not been assumed to be unity in the semiempirical
expression and data analysis of Phillips (1955), Molyneaux and Rhodes
(1972) and others, it is not clear what interpretation should be placed
on their results.
The second consequence involves the work of Sexsmith and White
(1959ab) which, when assuming a < 1, gives a maximum in the monomeric
amphiphile concentration. Using equations (22) and (227) with mole
fractions for activities at all concentrations plus
xc = x C(x xl) (231)
yields the relation
1/N
x =1 (232)
X J K[x t(x x )]
where K E exp [AGo/NRT]. (233)
m
At small values of x, x1 = x but at larger values of x x> two
limiting cases appear
x 1/x= K(lc)o a < 1 (234)
1 o
1/2N 1/2
x = (x /N) /K a = 1. (235)
1 o
Equation (234) is chosen by Sexsmith and White (1959b) which indicates
a rapidly decreasing monomeric concentration with total amphiphile while
equation (235) gives a slowly increasing monomer concentration. Experi
mental evidence is mixed (Hall and Pethica, 1970).
A possible resolution of this conflict between the two approaches
to the thermodynamics of ionic amphiphiles lies in the new specific ion
surfactant electrode technique (Evans, 1978). When combined with ion
specific electrodes for the counterion, measurements on long chain sur
factants permit the fractional charge on the micelle to be determined.
Present estimates yield values of a = 0.8, whereas inferred results gave
values closer to 0.5.
Effect of Added Salt on Micellization
An extremely important aspect of the thermodynamics of micellization
is concerned with the effect of added salt on the CMC of ionic amphiphiles.
Examination of the better data reported by Mukerjee and Mysels (1971) for
systems such as alkyl ammonium chlorides and bromines, sodium alkyl sulfates
and alkyl sulfonates with added salts such as the sodium halides, potassium
nitrate and potassium bromide up to IM, confirm the relation first given by
Corrin and Harkins (1947), Hobbs (1951) and by Shinoda (1953) and described
in detail by Mijnlieff (1970) and Lin and Somasundaran (1971). The data can
be reproduced to within the estimated experimental error with
11 L+
En xi/x = K' in +o (236)
1
+o +
where x is the CMC without added salt, x is the value with added salt
of mole fraction x2 and K' is a constant independent of the salt whose
value is 0.66 .03 for anionic amphiphiles and 0.58 .03: for
cationics. The data are insufficient and probably of inadequate accuracy
to properly test this correlation for multiply charged salts. Mijnlieff
writes the reaction for the neutral species (M E Amphiphilic Salt,
S E Added Salt, MMQ Micelle)
N M + Q S2 MQ (237)
and the mass action relation for amphiphile (1) and salt (2) as
N P' + Q 2 Q MQ (238)
Now for an ideal solution where the added salt has a common ion with the
amphiphilic salt
PMQ = Q (239)
MQ MQ
o
u1 = + RT in XlXc (240)
P2 = 02 + RT Zn x2xc (241.)
where xc = xl + x2 is the mole fraction of counterion in the system from
both a 11 amphiphilic salt and a 11 added salt. The relations for other
salts would be similar in form but more complex in detail. The equilib
rium relation is then
AG0 p N P Q P 0
m =MQ 2 + nx/N x+ (1+Q/N))
NRT NRT in l (x + x2) (242)
NRT NRT L )
In the limit x2 = 0, Q = 0
o o
MO 1_ +0
WNRT 2 n (243)
where N is the micelle number in the absence of added salt and the
standard state chemical potential of the micelle without salt P0 may
0
differ from that with salt, MQ Again, these are neutral species, not
charged.
Finally, this may be rearranged to give
o o
L+ I IMQ 4 Q 1o
+ +o X + NN N
Zn(x /x ) = in + NxN
+ 0+ RT
Zn [x (x + x2)]. (244)
N
For the correlation of equation (236) to hold, the form of
the standard state chemical potential must be
o o
Q 2 
N N N = nx + + 1 + K' In (x + x2)
RT N L N
(1 + K') in xl (245)
Mijnlieff shows that the reciprocity relation
( 246)
Dn2 n1
T,P,n1 T,P,n2
leads to
Q (1 + K') < 0. (247)
N 2 + (IK')x /x
In the limit x /x 1
2 1
(1 + K')x
K)x2 (248)
N (1 K')
This equals zero when x = 0. In the limit x2/x >>1, Q/N = 0.16 for
anionic amphiphiles and 0.21 for cationics. The fact that it is
constant, but different for the charge types must be of significance.
Finally, after some rearrangement
o o_
___ MO Q o
v+
N N (I+K') 1 XK) n
RT+ x 2
RT 2 + (lK')x_/x2 2
+ ( + (IK')x /x2) n (1 + xn/x2 Pn x (249)
Thus, theories for the standard state Gibbs' free energy change should be
of the above form. When x /x+ >> 1 or high salt concentration equation
(249) becomes
S0O
Q 10 + (1 + K')p2
2
SN +o(250)
RT (1 + K') Zn xl = constant. (250)
Some Theories for Free Energy Changes
Upon Micellization
Before proceeding to describe the theories for calculating AGo,
consideration should be given to a significant phenomenological observa
tion previously discussed by Lin and Somasundaran (1971). From tables
of Mukerjee and Mysels (1971), the critical micelle concentration for
amphiphiles with paraffinic tails varies with the number of carbons in the
following way:
a) for all ionics such as sulfates, sulfonates and alkanoates, each
+
additional carbon changes Zn x within experimental error by a value of
0.69( 0.02) with negligible effect of temperature and added salt concen
tration (Lin and Somasundaran (1971) cited values essentially the same);
b) for nonionics such as oxyethylene 3 alcohols, 6 alcohols, and n and
cbetaines the variation is from 1.09 to 1.28 although it is constant
for each compound;
c) by contrast, Tanford (1973) quotes the results of McAuliffe (1966)
for each carbon group changing the alkane solubility, n xw, in water at
25C by 1.49( 0.02). It is not surprising that the nonionics should
show some difference of carbon number effect with head groups and/or
perhaps mean aggregation number. However, it is quite surprising that
these effects do not appear for the ionics.
One explanation which can be advanced is that, except for small
differences in potential energy and in conformational entropy of the hydro
carbon tails in bulk alkanes compared to micelles of amphiphiles (and
even these should probably vary proportionally to the carbon member), the
carbon number dependence of micelles and alkane solubility should be the
same since it is caused solely by the "hydrophobic" effect on the monomers.
The variation to be described is
A(AGm/N(l + a)RT)
An = An (251)
c c
where a = 0 for alkanes and nonionics and a = 1 (?) for ionics and
+
x is x for micelles and xW for alkane solubility. As noted above, the
value for ionics (0.69) is slightly less than onehalf that for alkane
solubility (1.49). While it is unclear why the nonionics do not have
a value equal to twice that for the ionics, it is possible that the volume
excluded to water by nonionic micelles varies with the nature of the com
pound due to differences of penetration of water around the head groups
which causes differences in the "hydrophobic" effect.
The above observation has been considered by Tanford (1972,1974ab)
in which a theory of Tartar (1955) is expanded for micelle formation and
size distribution. Tanford separates AG /NRT into a portion linear in the
carbon number, nc, a portion which depends upon the area of the hydro
carbon core in the micelle, AHM, plus a portion dependent only on the area
per head group ARM. Tanford's empirical expression for an ideal solution is
AG0
1 + + 2n x m
m 1 
N N NRT
= [klk2nc + k3AM] + Z6i/ARM (252)
where the constants k. are positive, the 6. are constants and there may be
1 1
as many as three different terms in the 6. sum. The first group
1
of terms on the righthand side is the same as (p /N 1 ) in equation (18)
el el
while the summation is apparently (Pi /N pe ). No distinction is made
m 1
by Tanford between enthalpy and entropy contributions. However, the second
term is likely to be enthalpy dominated, whereas the first is mixed imply
ing certain temperature variations of the k. and 6.. Tanford identified
1 1
ARM with that of an ellipsoid whose minor axis is that of the flexible
hydrocarbon chain length plus 3A. (An extensive discussion of the role of
geometric constraints in micellization can be found in Israelachvili,
Mitchell and Ninham (1976)..) Tanford's values for k1 and k2 are apparently
derived empirically for micelles since they are not the same as those for
alkane solubility.
A preferable approach is the concept of BenNaim (1971), Tenne and
BenNaim (1977) and Pratt and Chandler (1977), who indicate that the
"hydrophobic" effect arises from aggregation of the volume (cavity) occu
pied by the hydrocarbon which is excluded from occupancy by the water.
As calculated using scaled particle theory (the application of scaled
particle theory and subsequent modifications to aqueous solutions is dis
cussed in considerable detail in Chapter 3), this effect is essentially
entropic at low temperature (T = 100C) and becomes increasingly enthalpic
with increasing temperature as reflected in a large heat capacity (Pierotti,
1965). As previously noted, the thermodynamic properties of micellization
follow this same temperature trend.
Early approaches beginning with Debye (1949) tended to view micelli
zation as an enthalpic rather than entropic process. Debye ascribed micelle
formation to the opposition between hydrocarbonchain attraction and ionic
repulsion. The more complex approach of Poland and Scheraga (1965, 1966)
attributed the solvent contribution to micelle formation to the theory of
hydrophobic bonding of Nemethy and Scheraga (1962ab). This theory attrib
utes hydrophobic bonding to an increase in: the amount of hydrogen bonding
of the water near a solute over its average value in pure water and is
similar to the concept of "iceberg formation" around nonpolar solutes of
Frank and Evans (1945). It should be noted that there are both strong
entropy and enthalpy contributions in this approach. The calculated stan
dard free.energies, enthalpies and entropies of solution for alkanes and
benzene homologs agree well with experimental values within the temperature
range of 0 to 700C.
The principal difference between this approach and that of Ben
Naim (1971) is that scaled particle theory is applicable to any solvent
and does not rely explicitly on microscopic details concerning change in
water structure or conformation upon inclusion of the solute. These effects
may well be implicit in the details of scaled particle theory (particularly
with the modifications discussed in Chapter 3).
The hydrophobic free energy change Ap HS/RT associated with a sphere
of diameter C being inserted into a solvent of diameter as is given by
HS = An 3y) + 9  { 3 9 f
RT 2 (ly)2 (1y) (ly)2
l y (1y) (y)
+ (__1 18y )2 (253)
where y = I 03 and p is the solvent number density (a very small term
6 s
which varies as the pressure has been ignored). At constant temperature
this means that ApHS/RT is a quadratic in the ratio of solute to solvent
diameter.
Aup /RT = a + b(o/C ) +c(o/ )2 (254)
HS S S
To create a micelle of diameter a from N solutes of diameter a ,
m 1
the free energy change per solute will vary as
0 _/ 2 2
AG A Ap ( b(a IN ) ( I/N a )
mHS m m 1 m 1
NRT a 1 + c +2 (255)
NRT NRT N1S HS i s a
HSs H Ss
where a is the solvent diameter.
s
For this to coincide with Tanford's concept the first bracketed
terms must vary linearly with the amphiphile carbon number nd and there must
be a direct correspondence between the hydrocarbon water contact area
2
AHM and a0. The first is precisely what is appropriate for an amphiphile
monomer cylinder of constant radius r whose length Z is proportional to
2
n modelled as a sphere of equal area. (The term in 0 in equation (253) is
the most important.) Thus
2
A sphere = = 2r = Acylinder (256)
sphere 1 cylinder
with
k = c + c nc (257)
50 2
so = c' + c' n (258)
1 1 2 c
Second, the ellipsoidal micelle geometry is close to spherical so AHM
should be proportional to 0 .
m
To add further evidence to this assertation, the results of
el el
Stigter (1975ab) who has developed a very detailed theory for [p /N p ]
m 1
should be examined. When these calculated contributions are subtracted
+
from experimental kn x values, a correlation is found with the amphiphile
0
water contact areas of monomer and micelle of the form
Sel e
Gm N J (259)
 = k' k' A + k' A /N
NRT RT 1 2 1 3 m
= [a' + b'n ] + c' O2/N (260)
c m
where the constants are all positive. The second form is again entirely
consistent with the "hydrophobic" concept. Thus the form of the hydrophbic
effect for micelle formation is accounted for by rigid body effects.
Thus, calculations of the nonelectrostatic contributions to
micellization from rigidbody volumes excluded to the solvent (water)
appear to be consistent with present data and knowledge. In other solvents,
the effect will be significantly smaller due to a being larger; this may
s
explain why nonaqueous micellization is of considerable less importance
and the values of N are much smaller (Kitahara, 1970).
Contributions to Thermodynamic Properties
of Micellization from Various Species
Figure 21 describes a categorization of the thermodynamic contribu
tions which must be taken into account in development of theories describing
micellization. The breakdown is into overall energetic effects in various
regions of the system, entropic contributions due to changes in molecular
conformation and excluded volume effects. The last is chosen as a sepa
rate category (which has enthalpy and entropy contributions) because it
has been modeled in several theories as the "hydrophobic" effect. Electro
static effects are indicated which would be restricted to ionic amphi
philes. Finally, a distinction is made between those contributions which
are associated with aspects internal to a micelle, including conforma
tional constraints and those which occur across the interface between the
micelle and the aqueous environment.
Effects arising from the properties of salt species are restricted
to ionic amphiphiles. Important enthalpic effects are possible changes
in ion hydration (uncertain at present) upon binding to the micelle sur
face as well as increased electrostatic interaction of the ions with the
much greater surface charge density of the micelle relative to the amphi
phile monomer. The latter effect has been extensively modeled; the most
recent and detailed being that of Stigter (1975ab). Unfortunately, due
to the great complexity of detail Stigter's model is not readily adapt
able to a general theory of micellization.
Two significant entropy effects are attributable to the salt
species, both resulting from binding of oppositely charged ions to the
micelle surface. The volume excluded to the solvent by the ions is
aggregated at the micelle surface resulting in an entropy increase as mod
eled by the scaled particle theory. Secondly, the ion configurational
entropy is considerably decreased as modeled by Stigter.
Fig. 21. Contributions of Species to Property Changes of Micellization
Enthalpic contributions due to properties of the amphiphile
species include repulsive interaction (AH > 0) between charged or polar
amphiphile head groups crowded at the micelle surface and attractive
(AH < 0) Van der Waals interactions between hydrocarbon chains in the
micelle interior. Theories of electrostatics such as Stigter's attempt
to model the repulsive interactions which have also been handled semi
empirically by Tanford. Unfortunately, the Van der Waals attractive
interactions are frequently considered as part of the "hydrophobic force"
(Tanford 1974a).
A significant entropy effect attributable to the amphiphile is
the change in constraints on the motion of the hydrocarbon chains in the
micelle compared to monomeric form. Considerable evidence indicates that,
at least near the polar head group, the hydrocarbon chains in micelles
exhibit a considerable degree of rigidity (Kalyanasundaram and Thomas,
1976; Roberts and Chachaty, 1973). Unfortunately, knowledge of the
configuration of the hydrocarbon chains in water is inconclusive.
A partial model of this effect is possible using a hardbody equation
of state (Gibbons, 1969; Boublik, 1975) to calculate the entropy change
associated with the change in hydrocarbon density from the monomer
solution to the micelle. More specific effects associated with rota
tional freedom of the hydrocarbon chains would best be modeled from
a lattice approach (Poland and Scheraga, 1965).
Enthalpic effects associated with the solvent include changes
in binding of the solvent to the amphiphile head groups and ions upon
micellization. A lack of knowledge concerning this effect is compounded
by considerable debate concerning the location of the micellesolvent
interface relative to the head group position (Tanford, 1972; Stigter,
1975ab) and thus whether the head group should be considered hydrated
upon micellization.
As noted in Figure 21, significant entropy and enthalpy
(relative importance is dependent on temperature as previously noted)
contributions to micellization are due to a change in the volume excluded
to the water molecules upon micelle formation. This effect can be readily
calculated using scaled particle theory, which will be discussed in
Chapter 3, both in the original form and modified for unusual structural
properties of water.
In principle, all of these effects should be considered in
development of a theory for micellization. However, such a develop
ment would lead to models containing too many parameters with exces
sively complex expressions. Empirically, it has been observed that
the entropy change upon micellization is large and positive at low tem
peratures and decreases, eventually becoming negative at higher temper
atures. The enthalpy change is positive at low temperatures, changes
sign at approximately 25C and becomes quite negative at higher temper
atures. Thus micellization switches from an entropy to an enthalpy
driven process with increasing temperature. This temperature behavior
is closely followed by both the water structure approach of Nemethy and
Scheraga (1962ab) and the excluded volume concept of BenNaim (1971).
As noted previously this latter approach may macroscopically utilize
some of the microscopic detail of the former.
For micellization, the excluded volume which is dispersed with
monomers is coalesced when micelles are present. While it may seem
unusual for coalescence to lead to increased entropy, it should be
remembered that the species whose entropy is increased is the solvent
water not the amphiphile. When the excluded volume is coalesced many
more configurations are available for the water molecules than when
it is dispersed.
A Thermodynamic Process for Micelle Formation
Development of a meaningful theory of micellization requires
integration of the various effects discussed in the previous section
into a thermodynamic process for micelle formation. Such a process is
illustrated in Figure 22. The calculation of the change in a thermody
namic property between two equilibrium states is independent of the
path followed between these states. Thus the process in Figure 22 can
be developed for conceptual expedience rather than physical reality.
Note that the process involves three parallel paths for the amphiphile,
counterions and solvent.
The initial step of the process involves removal of the
amphiphiles and an appropriate fraction of the counterions from their
cavities in solution at constant density. Since the density is main
tained constant, there is essentially no change in the entropy of the
monomers and counterions. However, enthalpy and entropy changes in the
solvent will occur due to removal of these species from their excluded
volume cavities. The enthalpy change will be positive due to elimination
of the intermolecular interactions between the amphiphile and water.
An entropy change (probably negative) will arise due to rearrangemnt of
local water structure around the cavities. Frank and Wen (1957) have
examined this effect for ions.
Step Ib
AS z a0
Step 3
AH >0 T
AS
Step 5b
AS 0
S Step la O Step 2 Step 4 9 Step 5a
AH =? AH<0 AH>O AH =?
G A =AS ? 0 (I) AS> 0
0. AS >0 AS
r G I G 9
Dispersed Monomers Water With Water With Micelle in Water
and Counterions in ispersed Cavities Micelle Ca With Bound Counterions
Watertep
AS z 0 AHS < 0
+
Dispersed
Counterions
Fig. 22. A Thermodynamic Process for Micelle Formation
The next step in the process involves collapsing the amphiphile
and counterion cavities. Scaled particle theory calculations yield
significant entropy and enthalpy contributions, whose relative magnitude
is temperature dependent.
The third step involves compression of the dispersed monomers to
micellar density, with restriction of the monomer head group to the
micelle surface. An appreciable entropy decrease will occur due to both
the compression and the restriction placed on head group location. The
enthalpy will increase since the electrostatic repulsion between the
head groups at the micelle surface will more than counteract attractive
Van der Waals interactions between the compressed hydrocarbon chains.
Step 4 is essentially the reverse of step 2 with creation of
a micelle cavity. This has small entropy and enthalpy changes because
of the surface area and curvature dependence of the excluded volume
effect. On a monomer basis the magnitude of the changes in step 4 is
much less than those of step 2.
The final step of this thermodynamic process for micelle
formation involves placement of the compressed monomers in the micelle
cavity and binding of the dispersed counterions to the micelle surface.
The enthalpy decrease accompanying this step results from reduction of
the head group repulsion at the micelle surface because of the counterion
binding and creation of attractive interaction between the head groups
and water.
Entropy changes are mixed with an entropy decrease due to binding
counterions to the micelle surface and a possible entropy increase due to
water structure rearrangement around the occupied cavity.
As previously noted, experimentally the overall process of
micelle formation is entropy driven at lower temperatures and become
enthalpy driven with increasing temperature. Since step 2 is the only
one with an appreciable entropy increase at lower temperatures (T 35C)
the driving force (at least at lower temperatures) for micellization
must be aggregation of the volume excluded to the solvent by the amphi
phile monomers. Since there are likely to be only weak temperature
variations in the other steps of the process, step 2 must also reflect
the change to an enthalpy driving force at elevated temperature. Since
there is a large heat capacity effect in it, agreement with this trend
is expected and observed. Further discussion and calculations involving
the more significant stages of this process are included in Chapter 6.
CHAPTER 3
SCALED PARTICLE THEORY MODIFIED
FOR AQUEOUS SOLUTIONS
Introduction
The initial section of this chapter provides a brief summary of
scaled particle theory with particular emphasis on aspects important in
extension of the theory to aqueous solutions. Previous efforts at such
an extension (Pierotti, 1965; Stillinger, 1973) are discussed critically,
particularly with regard to treatment of the contact correlation function
G(r) at macroscopic r values.
The expression of Koenig (1950) for the curvature dependence of
the surface tension is then utilized to derive an exact relation for G(r)
in the macroscopic region.
.Finally, structural aspects of liquid water are investigated
through calculation of the radial distribution function and direct corre
lation function from Xray diffraction data of Narten and Levy (1971).
Structural features are particularly apparent when comparing the direct
correlation function with that of liquid argon at similar density.
Basis of Scaled Particle Theory
The intent of this section is to provide a brief survey of scaled
particle theory with particular emphasis on the assumptions involved and
29
and applicability of the theory to aqueous solutions. Several more exten
sive reviews are available (Reiss, 1965, 1977).
Scaled particle theory was originally developed and used for the
study of hard sphere fluids (Reiss et al., 1959). Application was also
made to one and twodimensional systems [rods and rigid disks] (Helfand
et al., 1961; Cotter and Martire, 1970 ab; Cotter and Stillinger, 1972)
as well as to mixtures of disks and spheres (Lebowitz et al., 1965). An
attempt was also made to rigorously extend the formalism to real fluids
(Helfand et al., 1960) along with some application to simple fluids such
as the inert gases in the liquid state (Yosim and Owens, 1963, 1964).
More recently scaled particle theory has been applied to aqueous solutions
(Pierotti, 1965; Stillinger, 1973).
The starting point of scaled particle theory is consideration of
the work of creating a cavity at some fixed position in the fluid. In a
fluid consisting of hard spheres of diameter a, a cavity of radius r cen
tered at R is the same as a requirement that no centers of. the hard
spheres can be found in a sphere of radius r at R Thus, creation of a
0
cavity of radius r at R is equivalent to placing a hard sphere solute of
diameter b at R such that r = (a+b)/2. Hence the work required to create
such a cavity is also the work required to introduce a hard sphere solute
at R This work is computed by using a continuous process of "building up"
o
the solute in the solvent. Hence the name "scaled particle theory."
It is important to note that a cavity is considered "empty" in
scaled particle theory if no centers of particles are found in it. Also it
is worth noting that a hard sphere of zero diameter produces a cavity of
radius a/2 in the system, whereas a cavity of zero radius is equivalent to
placing a hard sphere of negative diameter b = a in the system.
The fundamental distribution function in scaled particle theory
is P (r), the probability that no molecule has its center within the
spherical region of radius r centered at some fixed R in the system.
This function was originally introduced by Hill (1958).
Let P (r+dr) be the probability that the centers of all molecules
0
are excluded from the sphere of radius r + dr. Now the probability that
the spherical shell of thickness dr and volume 4Tr 2dr contains a particle
center is 4Tr pG(r)dr where G(r) is defined so that pG(r) measures the con
centration of molecular centers just outside the sphere. Thus G(r) mea
sures the conditional probability that the center of a molecule will be
found within the spherical shell at r when the region enclosed by the
shell is known to be empty. The probability that the spherical shell is
free of molecular centers is
1 4r2 pG(r)dr (31)
For the volume of radius r + dr to be devoid of centers, it is
necessary that the volume of radius r, and the shell of thickness dr, be
simultaneously free of centers. Thus the probability P (r+dr) is given by
2
P (r+dr) = P (r)[14rr pG(r)dr]. (32)
o o
Expanding P (r+dr) to first order in dr yields
P
P (r+dr) = P (r) + Dro dr + ... (33)
Combining equations (32) and (33) yields
Sn P
D = 4Tr 2pG(r). (34)
Dr
Upon integration
(r
P (r) = exp [ 4Wr pG(r')dr'] (35)
o
Oo
where the initial condition P (0) = 1 has been applied (a cavity of zero
radius is always empty).
An important relationship can be derived between P (r) or G(r)
and the work of cavity formation W(r). This relation shall be derived
in the canonical (T,V,N) ensemble. The probability density of finding
a specific configuration RN = R1,...,R is given by Hill (1956) and
BenNaim (1974) as
N exp [ g U(RN]
P(R) = exp (R N (36)
SJ... exp [ U(R N)]dRN
where P = (kT)1 and U(R N) is the interaction energy among the N particles
at.the configuration R N. Thus, the probability of finding an empty spher
ical region of radius r, centered at R may be obtained from equation (36)
o
by integrating over all the region Vv(r) where v(r) denotes the spher
ical region of radius r.
P (r) = ...... P(RN)dRN (37)
The following relation exists between the Helmholtz free energy.
of a system and the corresponding partition function in the canonical
ensemble
exp [A(T,V,N)] = 3N ... exp [U(RN)dRN (38)
N!A V
where A3 is the momentum partition function, and no internal degrees of
freedom are ascribed to the particles.
Similarly the free energy of a system with a cavity of radius r
at R is given by
exp [3A(T,V,N;r)] = 3N ...... exp [U(RN)]dR (39)
N:A f Vv(r) f
Thus the ratio of (38) and (39) gives
exp {[A(T,V,N;r) A(T,V,N)]} =
SVV...... exp [BU(RN)]dRN
J VV(r) (310)
f ... exp [U(RN)]dRN
V
= P (r).
Equation (310) is an important connection between the work (at
given T,V,N) of creating a cavity of radius r, and the probability of
finding such a cavity in the system. This relation can be rewritten as
W(r) = A(r) A = kT in P (r)
r 2
= kTp r 4rr'2G(r)dr'. (311)
o
Since the work required to create a cavity of radius r is the
same as that required to place a hard sphere of diameter b = 2r a at
R the chemical potential of this added particle is equal to the work
o
plus the translational free energy
1b = W(r) + kT n pb A (312.)
(a+b)/2 2
W(r) = kTp 4Tr' G(r')dr'. (313)
fo
Here, pb = 1/V is the solutee" density, whereas p = N/V is the "solvent"
density.
An exact expression is available for P (r) at very small r (Hill,
1958). If the diameter of the hard sphere particles is a, then in a
sphere of radius r < a/2 there can be at most, one center of a particle
at any given time. Thus for such a small r, the probability of finding
the sphere occupied is 4rr 3p/3. Since the sphere may be occupied by
at most, one center of a hard sphere, the probability of finding it
empty is
P (r) = 1 p 4T for r < a/2. (314)
o 3
For spheres with a slightly larger radius, namely for r a//3
at most two centers of hard spheres can be accommodated. The correspond
ing expression for P (r) is
3 2 r r
P (r) = 1 3 P+ g(RI,R2)dRdR2 (315)
V(r)
where g(R1,R2) is the pair correlation function, and the integration is
carried out over the region defined by the sphere of radius r.
The probability that a cavity of radius r is empty, in the terms
of pair correlation functions g(n) for molecular centers in the pure
solvent is
00f
P(r) = 1 + E [(p)n/n!] dR1...dR g (R ...R). (316)
V(r)
The terms in this series will all vanish for n exceeding the maximum
number of solvent molecule centers that can be packed in a sphere of
radius r. Equation (315) represents the first three terms in this series.
When 0 5 r 5 a/2 all terms in equation (316)'beyond n=l vanish. In this
range equation (314) applies.
As r begins to exceed a/2, two solvent centers can fit into the
cavity, so the n = 2 term in series (316) begins to contribute. However,
P(r), W(r) and G(r) all remain continuous and differentiable at r = a/2.
As r m, W(r) becomes dominated by work against the external
pressure P and against the surface tension y of the cavitysolvent inter
face. Thus
W(r) = (47Tp/3)r3 + (4iry)r 2 (161Ty6)r + 0(1)* (317)
Here Y. is the surface tension in the planar interface limit, and 6 pro
vides the leading term in the curvature dependence of the surface tension
y (Buff, 1951)
y = y[l 26/r]. (318)
The integral relation (equation.311) between W(r) and G(r) results
in the following larger behavior for G(r)
2y 4y 6
G(r) = pkT p+ kTr 2 + (319)
PGT ) +pkTr2
Subsequent efforts (TullySmith and Reiss, 1970) showed that the coeffi
3
cient on the r term in the expansion for G(r) must be zero.
For small cavities, equations (34) and (314) yield
G(r) = (1 47r3p/3) for r a/2 (320)
and for W(r) from equation
W(r) = kT kn(l 4pr3p/3) for r a/2. (321)
For very large cavities, r  c, from equation 019)
P
G(r) = pkT (322)
Thus exact results exist for G(r) at very small and very large r.
Reiss et al. (1959) and Pierotti (1963) adopted the procedure of bridg
ing these two extremes with a smooth function of r. They assumed that
G(r) is a monotonic function of r in the entire range of r. They sug
gested the form of (319) with empirical parameters
Since this is a drastic linearization of the rigorous relation
of Koenig (1950), a more rigorous approach to the curvature dependence
of the surface tension is presented in a later section.
G(r) = A + B/r + C/r2. (323)
The coefficient A in equation (323) was determined from
equation (322). Expressions for B and C were determined by matching
values and derivatives of equations (319) and (320) when r equalled
a/2. If P is the experimental value the expressions lead to
S 3y kT 1 3 (324)
Y.= 7a2 Vy 2 ( 2 pkT2 (34
6 a + 3y (325)
8 2+y2(1y)2(P/pkT)
where y = rpa 3/6.
The lower solid curve in Figure 31 shows the resulting G(r)
0
function. Its most distinctive feature is the maximum at r z 2.0 A.
Similar maxima occur for other temperatures, but always at r = 46 in
max
the Pierotti approximation.
Integration of equation (311) with this expression for G(r)
yields
W(r) = Ko + K r + K2r2 + K3r3 (326)
where the coefficients are
K = kT[ kn(ly) + 4.5z2] 1 r Pa3 (327a)
o 6
K = (kT/a)(6z + 18z2) + T Pa2 (327b)
K = (kT/a 2)(12z + 18z 2) 2a Pa (327c)
K3 = 4ir P/3 (327d)
where z = y/(ly).
Thus, an approximate expression is obtained for the work required
to create a cavity of radius r in a hard sphere fluid of diameter a.
So far all relations are derived for a hard sphere fluid. The
application of this theory to real fluids proceeded along three lines.
One was by Yosim and Owens (1963) which involved using hard sphere
diameters for real fluids determined from experiment on one property
such as surface tension, isothermal compressibility or thermal expan
sivity in expressions for the other properties. The results were
reasonably good for nonpolar species but poor for polar substances, par
ticularly water. Another procedure was to predict entropy of phase
change by a clever thermodynamic cycle. The same experience was encoun
tered here as with the other procedure. Finally, Pierotti (1963) explored
the results of the theory for gas solubility in liquids. The process he
used consisted of creation of a cavity in the liquid, with the free energy
change calculated from the above expressions. The second step was to fill
the cavity with the solute, calculating the free energy effects as the
sum of contributions of pairwise intermolecular forces.
Scaled Particle Theory and Aqueous Solutions
Pierotti (1965) applied his scaled particle theory for solubility
to aqueous solutions of nonpolar gases.. Somewhat surprisingly, he found
it possible to predict enthalpies, entropies and heat capacities of solu
tion with the physical assumption that water molecules arrange themselves
spatially in the pure liquid as would hard spheres of an appropriate size
and several mathematical approximations. Considering the complexity of
water structure and interactions, this success seems quite fortuitous, since
the only explicit information required about the molecular structure of
0
water is a, the distance of closest approach which he set at 2.75 A.
Stillinger (1973) shows compelling evidence that the agreement
is not necessarily because of the correctness of the physical assump
tions. He lists values of y. and 6 for water using equations (324)
and (325) at several points along the saturation curve for water. He
also includes measured liquidvapor interfacial tension for comparison.
Table 31 shows they do not agree. Of particular significance is the
improper temperature dependence of the interfacial tension which may
contribute to error in the predicted entropy of solution. Stillinger
argues that the sign of is incorrect, although an analysis of the work
of Koenig (1950) shows that 6 is positive so long as the radius of curva
ture of the cavity r has a positive sign convention.
Stillinger developed a revised G(r) for water using the exper
imental liquidvapor interfacial tension, y and the radial distribu
tion function, g(r), as input data.
The most accurate determination to date of the oxygenoxygen
pair correlation function g (2)(r) in liquid water can be determined
from Fourier transformation of the structure function data as determined
from Xray diffraction by Narten and Levy (1969, 1971). Details of this
method will be discussed later in this chapter. Their results show that
0
essentially no pairs of oxygen nuclei occur closer than 2.40 A. There
0
fore, equation (316) will be correct for G(r) in the range 0
For larger r, at least the pair term in P(r), equation (315), should
contribute, and so the same would be true in G(r).
In ice, strong directional forces between neighbor molecules
produce characteristic isosceles triangles of oxygen nuclei. The apex
angle is the tetrahedral angle Ot = 1090; since the hydrogen bonds in
Table 31
Surface Tension and Curvature Parameter Calculated
for Liquid Water at Its Saturated Vapor Pressure
Using the Pierotti Approximation
Yv (expt.)
(dyne/cm)
75.07
72.01
67.93
63.49
58.78
37.81
14.39
yj[Eq.(324)]
(dyne/cm)
51.44
54.97
58.35
60.96
62.86
63.82
52.18
6[Eq. (325)]
0
A
0.5026
0.5022
0.5010
0.4992
0.4970
0.4845
0.4648
T
(K)
277.15
298.15
323.15
348.15
373.15
473.15
573.15
0
ice have length approximately 2.76 A, the smallest sphere which could
0
enclose these triangles would have radius r = 2.25 A. Thus, for ice
nothing beyond the pair (n=2) terms in P(r) and G(r) would be required,
0
provided r does not exceed 2.25 A.
Certainly the hydrogen bond pattern present in ice is severely
distorted upon melting. However, the coordination number in the liquid
remains low and thus it seems reasonable that the tendency toward tetra
hedral bonding persists in the liquid (Narten and Levy, 1969). Stillinger
assumed that for cold liquid water triplets of oxygen nuclei are seldom
distorted into a more compact configuration than would result from reduc
ing Ot to 90. The resulting triplet will fit into a sphere of radius
r = 1.95 A.
The repressions for G(r) are thus the following from equations
(34) and (314):
3]1 0
G(r) = [1 (47/3)pr ] (0 r 5 1.20 A) (328)
while from equations (34) and (315)
1 + {2r dt g (t)t (t2r)
r
G(r) = 2r (329)
4AE 3 + f2r (2) 21 3 2 8 3
pr + )2 (t)t ( t 2r t + r3)
0 o
(1.20 r 5 1.95 A).
0
In order to specify G(r) beyond r = 1.95 A in terms of correla
tion functions, knowledge of g (3), (4),... would be required. In the
absence of such knowledge Stillinger relied on the conventional series
expressions for G(r) (equation 319) truncated after the fourth term
2y G G
P + v 2 4 o
G(r) pkT + Pr +  (1.95 A < r < o). (330)
r r
3
The r term is missing in equation (330) as required by the
general theory (Stillinger and Cotter, 1971) so that W(r) does not
have contributions proportional to Rn r. G2 and G4 are adjustable
parameters. Matching the magnitude and first derivative of G(r) at
0
r = 1.95 A between the exact microscopic expression (329) and the macro
scopic series (330) can be used to fix their values. Series (310) can
be expanded and the further parameters obtained by matching higher order
derivatives.
The function G(r).as calculated using this procedure at 25C is
the upper solid curve in Figure 31. The present procedure tends to give
G(r) an appreciably larger maxima than the Pierotti hard sphere approach.
Also the Pierotti approximation is less temperature sensitive (it depends
essentially on the temperature dependence of the number density alone).
Stillinger (1973) postulated a relatively simple physical explan
ation for the larger G(r) maxima in his approach. Unlike the Pierotti
approximation, it accounts for the strong and directional hydrogenbonding
forces in water, not only through the pair correlation function g but
also in the selection of the r value at which triplets first contribute.
As the exclusion sphere expands, it is forced to stretch and tear the
hydrogenbond network in its neighborhood. While this process occurs,
the remaining hydrogen bonds probably reach around the exclusive sphere
f
in a tight net, which enhances G(4).
Although the Stillinger modification appears to be an appropriate
extension of the Pierotti concepts, it suffers a fundamental flaw which is
discussed in the next section. Therefore, it yields results for gas
solubility which are not significantly more meaningful.
3
2
G(r)
1
0 2 4 6 8 10
r (A)
Contact Correlation
Different Models
Function; Comparison of
Fig. 31.
Curvature Dependence of Surface Tension
The expression for the curvature dependence of the surface
tension (equation 318) is an approximation to the rigorous relation of
Tolman (1949) and Koenig (1950). From the Gibbs theory of surface
tension Koenig developed general equations for the change of surface
tension with curvature for systems having an arbitrary number of com
ponents. For the special case of a spherical surface Koenig's expres
sion reduces by an appropriate choice of Gibbs surfaces to a form iden
tical with that of Tolman for a onecomponent system.
n 2(l+6q+1 62q2
(6q) 12 2
T l+26q(l+6q + 3 6 q )
where q E 1/r. For 6q << 1 equation (331) reduces to equation (318).
6 is a measure of the distance from the surface of tension to a Gibbs
auxiliary surface measured from the spherical phase outward. It is
reasonable to assume that both the surface of tension and the auxiliary
surface lie within or very near to the interface layer and consequently
6 is of the order of magnitude of the thickness of the interface layer.
Kirkwood and Buff (1949) performed approximate calculations for liquid
argon based upon statistical mechanics which suggested that 6 is of the
0
order of 3 A.
However, Lovett (1966) in his analysis of fluctuations about the
mean position of the interface shows that 6 should be related to the
third moment of the fluctuations, whereas the interfacial thickness is
related to the second moment. Because of this uncertainty, we will view
6 as a parameter in the study of aqueous solutions.
Equation (331) can be rearranged to yield
Z Zn y 2[(1+ 6/r) 1] (332)
31 + 2[(l + 6/r)]
Equations (318) and (319) imply that G(r) in the macroscopic
region should be written as
G(r) = p + 2 (333)
pkT pkTr
Since the first term is negligible at atmospheric pressure
G(r) 2 (334)
pkTr
This leads to
D r G(r) 2 D (335)
Dr pkT Dr
2_ D n y (336)
pkTr 8D n r
Substituting equation (332)
D r G(r) r G(r) 2[(l1 + 6/r) 1](337)
Dr r Ll + 2( + 6r)3
Upon integration
Zn rG 2[(1 + 6/r) 1 dr (338)
(rG)o [1 + 2(1 + 6/r')3] r'
where (rG)o = 2 yo/pkT.
Substituting X = 6/r', equation (338) can be rewritten in
dimensionless form with finite limits
6/r
nn rG = 2[(1 + X) 1] dX (339)
n = dX. (339)
(rGo o X[l + 2(1+X) ]
An exact expression is now available for the contact correlation
function G(r) rather than the series approximation of equation (319).
From equation (334)
SG(r) = 2y + 2 Y (340)
@r pkTr2 pkTr Br
= 6y 3 (341)
pkTr I + 2(1 + 6/r)
From the form of equation (341) it is apparent that this exact
expression for the macroscopic G(r) cannot go through an extremum
(DG/Dr = 0) and match the microscopic G(r) on the lower side of the
extremum as does the series approximation. This behavior is illustrated
in the middle curve of Figure 31 where 6 = 0.80. Means of dealing with
this difficulty will be discussed in Chapter 4.
Determination of the Radial Distribution Function and the Direct
Correlation Function for Liquid Water from XRay Diffraction
The total scattered intensity in electron units per molecule,
I(s), obtainable from Xray diffraction measurements on liquid water
(Narten and Levy, 1971), is related to the orientationally averaged
radial distribution function g(r) through
I(s) = + 2 47r2p (g(r) 1) sin (sr) dr (342)
fo sr
from which g(r) can be obtained by Fourier transformation. The quantity
s is the scattering vector (magnitude s = (47/A) sin 0, with X the wave
length and 20 the scattering angle). The quantity < F2 >is the average
scattering from one independent molecule, depending only on the intra
2
molecular distribution of scattering density, whereas < F > describes
the average scattering from a molecule of random orientation with respect
to any other molecule taken as the origin. Narten and Levy obtained
and from a Self Consistent FieldMolecular Orbital
approximationwith the result that < F 2> for intermediate values
of s.
In practice the accessible range of scattering angles is limited
to finite values of the variable s 5 s Fourier transformation of
max
the structure function,
H (s) E [I(s) ]/2 (343)
yields a correlation function
gM(r) E 1 + (27T2 pr) max
0'
s H (s) sin (sr)ds
m
which becomes exactly equal to the function g(r) only if s * m. Also
max
since the Xray scattering center of a water molecule is so close to the
oxygen atom the gm(r) determined is essentially the oxygenoxygen atom
correlation function in liquid water.
A direct correlation function, as proposed by Ornstein and
Zernike (Reed and Gubbins, 1973), may be defined by the following equation
c(rl2) = (g(rl2) 1) p f c(rl3)(g(r23) l)dr3
where c(r) is the direct correlation function.
The direct correlation function can be obtained from the
function H (s) as follows (Fisher, 1964)
c (r) = (2 2pr)1 s H m(s) (l + H (s))I sin (sr)ds.
m J m m
(345)
structure
(346)
Tables 32 and 33 contain gm(r) and c (r) calculated from the
structure function H (s) data of Narten and Levy (1971) at several temper
m
atures. Table 34 contains an expansion of Table 33 in the region of prin
cipal structural features. Note the unusual local extremum in c (r)
m
o
at r= 2.9 A.
Gubbins and O'Connell (1974) present a remarkable correspondence
between the reduced isothermal compresibility for several molecules
(344)
Table 32
Radial Distribution Function for Liquid Water
O
gm(r)
Temperature ('C)
4
0.06
0.02
0.02
0.03
0.06
0.03
0.01
0.05
0.05
0.04
C. 14
0. 14
0.04
0.02
0.02
0.09
0.14
0.15
0.15
0.14
0.15
0.17
0.13
0.04
0.15
0.72
1.62
2.29
2.29
1.76
1.22
0.97
0.90
n.83
0.80
20
0.09
0.03
0.06
0.00
0.03
0.03
0.09
0.05
0.02
0.01
0.06
0.06
0.03
0.10
0.07
0.01
0.04
0.07
0.09
0.08
0.09
0.14
0.13
0.03
C.08
0.63
1.59
2.34
2.41
1 .92
1.40
1.10
0.93
0.80
0.78
25
0.30
0.03
0. 11
0.02
0.06
0.06
0. 11
0.02
0.01
0.10
0.20
0.16
0.07
0.07
0. 12
0.12
0. 10
C.12
0. 14
0.12
0. 12
0.14
0.05
0.11
0.03
0.57
1.51
2.22
2. 29
1.88f
1.41
1.13
0.99
0.89
0.85
50
C.30
0.07
0.16
0.04
0.02
0.16
0.22
0.16
0. 18
0.30
0.36
0.29
0.23
0.23
0.20
0.141
0. 12
0.10
0.02
0.16
0. 15
0.07
0. 14
0.29
0.09
0.69
1.66
2.22
2.22
1.90
1. 54
1.22
1.02
0.97
1.02
75
0.21
0.06
0.11
0.02
0.05
0.03
0.06
0.01
0.03
0.07
0.17
0.19
0. 17
0.16
0. 17
0.16
0.15
0.11
0.03
0.01
0.06
0.11
0.01
0. 12
0.05
0.66
1. 14
1.96
2.07
1.85
1.52
1.23
1.09
1.07
1.07
100
0. 15
0.00
0.04
0.04
0.09
0.06
0.04
0. 12
0.18
0. 12
0.03
0. 10
0. 08
0.05
0.11
0.19
0.21
0. 15
0.08
0.08
0.14
0. 15
0.09
0.06
0.26
0.76
1.37
1.78
1.86
1.70
1. 49
1.35
1.23
1.11
1.02
r(A)
0. 10
0. 20
0. 30
0.40
0.50
0. 60
0.70
0.80
0.90
1.00
1. 1C
1. 20
1.30
1. 40
1.50
1.60
1.70
1.80
1.90
2.00
2.10
2. 20
2. 30
2. 40
2.50
2. 60
2.70
2.80
2.90
3.00
3. 10
3.20
3. 30
3.40
3. 50
Table 32 (Continued)
r(A) gm (r)
Temperature (C)
4 20 25 50 75 100
3.60 0.84 0.87 0.89 1.04 1.03 0.97
3.70 0.92 0.94 0.95 0.99 0.98 0.93
3.80 0.94 0.95 0.96 0.95 0.97 0.91
3.90 0.92 0.94 0.94 0.95 0.97 0.91
4.00 0.96 0.98 0.94 0.96 0.97 0.95
4.10 1.03 1.03 0.99 0.97 0.97 0.99
4.20 1.10 1.07 1.05 0.99 0.99 1.02
4.30 1.12 1.09 1.08 1.02 1.00 1.03
4.40 1.13 1.10 1.10 1.03 0.99 1.02
4.50 1.15 1.11 1.12 1.03 0.98 1.02
4.60 1.17 1.13 1.13 1.06 1.02 1.01
4.70 1.17 1.15 1.13 1.11 1.05 1.02
4.80 1.14 1.14 1.10 1.11 1.04 1.02
4.90 1.09 1.07 1.07 1.06 1.01 1.01
5.00 1.05 1.00 1.04 1.00 1.00 1.00
5.10 1.01 0.97 1.00 0.98 1.00 0.99
5.20 0.96 0.96 0.96 0.96 0.98 0.99
5.30 0.91 0.92 0.93 0.93 0.95 0.98
5.40 0.88 0.88 0.91 0.91 0.95 0.96
5.50 0.86 0.87 0.89 0.91 0.96 0.95
5.60 0.86 0.89 0.87 0.92 0.95 0.94
5.70 0.86 0.89 0.86 0.92 0.94 0.92
5.80 0.86 0.87 0.87 0.92 0.94 0.92
5.90 0.89 0.89 0.93 0.95 0.95 0.94
6.00 0.92 0.93 0.93 0.95 0.96 0.98
6.10 0.95 0.96 0.96 0.95 0.96 1.01
6.20 0.97 0.98 0.98 0.96 0.97 1.02
6.30 0.99 1.01 0.99 0.98 1.00 1.03
6.40 1.02 1.04 1.01 1.00 1.01 1.03
6.50 1.04 1.05 1.04 1.02 1.02 1.03
6.60 1.06 1.05 1.06 1.04 1.03 1.02
6.70 1.07 1.06 1.06 1.06 1.04 1.01
6.80 1.07 1.07 1.05 1.07 1.04 1.02
6.90 1.06 1.07 1.05 1.06 1.04 1.03
7.00 1.06 1.05 1.06 1 05 1 nl 1 nI
. .. .
. .
Table 32 (Continued)
0
r(A)
4
1.05
1.04
1.03
1.02
1.01
1.00
0.98
0.98
0.98
0.98
0.98
0.97
0.99
0.99
0.99
0.99
1.00
1.01
1.01
1.00
1.00
1.01
1.00
0.99
0.99
1.00
0.99
0.99
0.99
1.00
20
1. 04
1.05
1 .04
1.01
0.99
0.98
0.98
0.98
0.98
0.99
0.98
0.97
0.98
1.00
0.99
0.98
0.99
1.01
1.01
1.00
1 .00
1.01
1.01
1.00
0.99
1.00
1 .00
1.00
1 .00
1.00
gm(r)
Temperature (C)
25
1.06
1.05
1.03
1.02
1.01
1.00
0.98
0.98
0.98
0.98
0.99
0. 99
0.98
0.99
0.99
0. 99
0.99
1.00
1.00
1.00
1.00
1.00
1.01
1.00
1.00
1.00
1.00
1.00
1.00
1.00
50
1.05
1.04
1.03
1.02
1.01
1.00
0.99
0.99
0.99
0.98
0.97
0.97
0.98
0.99
0. 99
0.99
1.00
1.00
1.00
1.00
1.00
1.0 1
1.01
1.00
1.00
1.00
1.00
1.00
1.01
1.01
75
1.04
1.04
1.03
1.02
1.02
1.00
0.98
0.98
0.98
0.99
0.98
0.98
0.98
0.99
0.99
1.00
1.00
0.99
1.00
1.00
1.01
1.00
1.00
1.00
1.01
1.00
1.00
1.00
1.00
1.00
7. 10
7.20
7. 30
7.40
7.50
7.60
7.70
7.80
7.90
8.00
8. 10
8.20
8. 30
8.40
8. 50
8.60
8.70
8.80
8.90
9.00
9. 10
9.20
9.30
9.40
9.50
9.60
9.70
9.80
9.90
10.00
100
1.03
1.04
1.03
1.02
1.00
0.99
0.98
0.98
0.98
0.98
0.99
0.99
0.99
0.99
0.99
0.99
0.99
0.99
1.00
1.01
1.01
1.02
1.02
1.00
0.99
0.99
1.00
1.00
1.00
1.00
Table 33
Direct Correlation Function for Liquid Water
0
r(A) cm(r)
Temperature (C)
4 20 25 50 75 100
0.10 12.53 15.61 13.18 13.21 11.17 9.74
0.20 12.07 15.05 12.45 12.45 10.64 9.41
0.30 11.54 14.42 11.84 11.83 10.23 9.12
0.40 11.05 13.84 11.41 11.37 9.94 8.90
0.50 10.56 13.26 10.90 10.80 9.57 8.64
0.60 10.06 12.63 10.29 10.13 9.10 8.30
0.70 9.81 12.05 9.81 9.58 8.70 7.97
0.80 9.29 11.61 9.50 9.20 8.43 7.74
0.90 8.95 11.24 9.17 8.77 8.12 7.51
1.00 8.53 10.79 8.70 8.25 7.71 7.17
1.10 8.10 10.29 8.26 7.82 7.29 6.76
1.20 7.79 9.85 7.98 7.54 6.97 6.42
1.30 7.57 9.51 7.74 7.27 6.70 6.19
1.40 7.33 9.15 7.42 6.94 6.41 5.96
1.50 6.99 8.69. 7.06 6.65 6.12 5.65
1.60 6.62 8.20 6.74 6.40 5.84 5.31
1.70 6.28 7.72 6.45 6.11 5.56 5.04
1.80 5.97 7.16 6.12 5.82 5.31 4.84
1.90 5.66 6.81 5.78 5.63 5.10 4.65
2.00 5.36 6.36 5.46 5.46 4.85 4.38
2.10 5.04 5.93 5.13 5.13 4.50 4.07
2.20 4.69 5.44 4.79 4.72 4.14 3.79
2.30 4.41 5.00 4.53 4.45 3.97 3.60
2.40 4.17 4.65 4.26 4.28 3.82 3.38
2.50 3.74 4.19 3.95 3.75 3.37 3.94
2.60 2.86 3.22 3.02 2.66 2.50 2.21
2.70 1.56 1.87 1.77 1.39 1.47 1.37
2.80 0.70 0.74 0.77 0.54 0.71 0.74
2.90 0.44 0.34 0.43 0.30 0.38 0.47
3.00 0.72 0.51 0.55 0.38 0.39 0.43
3.10 1.04 0.75 0.83 0.54 0.54 0.49
3.20 1.09 0.79 0.91 0.67 0.65 0.49
3.30 0.99 0.74 0.88 0.71 0.64 0.47
3.40 0.89 0.66 0.81 0.62 0.52 0.46
3.50 0.78 0.50 0.71 0.44 0.40 0.44
Table 33 (Continued)
c (r)
m
Temperature (0C)
20 25
3.60
3.70
3.80
3.90
4.00
4.10
4.20
4.30
4.40
4.50
4.60
4.70
4.80
4.90
5.00
5.10
5.20
5.30
5.40
5.50
5.60
5.70
5.80
5.90
6.00
6.10
6.20
6.30
6.40
6.50
6.60
6.70
6.80
6.90
7.00
4
0.60
0.42
0.30
0.22
0.11
0.04
0.16
0.22
0.27
0.32
0.36
0.37
0.34
0.30
0.25
0.19
0.12
0.04
0.02
0.05
0.05
0.06
0.05
0.02
0.01
0.03
0.03
0.04
0.05
0.07
0.07
0.07
0.06
0.05
0.05
50 76
0.26
0.05
0.07
0.17
0.29
0.41
0.49
0.55
0.58
0.60
0.62
0.64
0.61
0.51
0.40
0.32
0.26
0.17
0.08
0.03
0.02
0.01
0.04
0.04
0.01
0.00
0.01
0.00
0.01
0.00
0.03
0.03
0.03
0.03
0.06
0.54
0.37
0.26
0.19
0.11
0.01
0.12
0.20
0.25
0.30
0.33
0.34
0.32
0.29
0.24
0.18
0.11
0.05
0.01
0.05
0.09
0.10
0.09
0.06
0.03
0.01
0.00
0.00
0.01
0.04
0.05
0.04
0.03
0.02
0.02
r(A)
0.30
0.25
0.21
0.14
0.06
0.01
0.08
0.15
0.19
0.21
0.25
0.30
0.30
0.23
0.15
0.09
0.03
0.04
0.10
0.12
0.13
0.14
0.14
0.12
0.11
0.10
0.10
0.08
0.06
0.04
0.02
0.00
0.01
0.01
0.00
0.34
0.30
0.23
0.15
0.10
0.04
0.03
0.07
0.09
0.11
0.16
0.20
0.19
0.16
0.14
0.12
0.08
0.02
0.01
0.02
0.03
0.06
0.06
0.04
0.03
0.03
0.02
0.01
0.02
0.03
0.04
0.04
0.04
0.04
0.05
100
0.40
0.35
0.30
0.23
0.13
0.03
0.04
0.08
0.10
0.12
0.13
0.15
0.15
0.14
0.12
0.11
0.10
0.06
0.03
0.00
0.02
0.03
0.04
0.02
0.02
0.05
0.07
0.08
0.08
0.08
0.06
0.05
0.05
0.06
0.06
i_
Table 33 (Continued)
c Temperature (C)
Temperature (*C)
4 20
0.04
0.03
0.02
0.02
0.02
0.01
0.01
0.01
0.02
0.02
0.02
0.02
0.02
0.03
0.02
0.01
0.02
0.02
0.01
0.00
0.00
0.00
0.01
0.01
0.01
0.01
0.01
0.02
0.02
0.01
0.07
0.06
0.06
0.08
0.08
0.08
.0.07
0.06
0.04
0.03
0.03
0.03
0.02
0.00
0.01
0.02
0.01
0.00
0.01
0.00
0.00
0.01
0.01
0.00
0.00
0.01
0.01
0.01
0.01
0.02
25 50
0.02
0.01
0.00
0.00
0.01
0.01
0.01
0.01
0.00
0.02
0.03
0.03
0.03
0.03
0.03
0.03
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.00
0.00
0.00
0.00
0.00
0.01
0.02
0.03
0.04
0.04
0.04
0.05
0.06
0.06
0.07
0.07
0.07
0.07
0.06
0.06
0.05
0.05
0.04
0.04
0.03
0.02
0.02
0.02
0.03
0.02
r(A)
7.10.
7.20
7.30
7.40
7.50
7.60
7.70
7.80
7.90
8.00
8.10
8.20
8.30
8.40
8.50
8.60
8.70
8.80
8.90
9.00
9.10
9.20
9.30
9.40
9.50
9.60
9.70
9.80
9.90
10.00
75
0.05
0.04
0.04
0.04
0.03
0.02
0.01
0.01
0.02
0.02
0.02
0.02
0.02
0.03
0.03
0.03
0.02
0.02
0.01
0.02
0.01
0.01
0.00
0.00
0.00
0.00
0.01
0.01
0.00
0.01
100
0.06
0.06
0.06
0.04
0.03
0.02
0.01
0.00
0.00
0.01
0.02
0.02
0.01
0.01
0.00
0.00
0.01
0.01
0.01
0.00
0.00
0.00
0.00
0.01
0.03
0.03
0.02
0.02
0.02
0.02
53
Table 34
Direct Correlation Function for Liquid Water
0
r(A) c (r)
m
Temperature (oC)
4 20 25 50 75 100
2.50 3.74 4.19 3.95 3.75 3.37 2.94
2.53 3.53 3.96 3.73 3.48 3.15 2.74
2.56 3.27 3.67 3.46 3.15 2.89 2.53
2.59 2.97 3.34 3.14 2.79 2.60 2.29
2.62 2.63 2.97 2.78 2.40 2.29 2.04
2.65 2.26 2.56 2.41 2.01 1.98 1.78
2.68 1.89 2.15 2.02 1.63 1.67 1.53
2.71 1.53 1.74 1.65 1.28 1.38 1.30
2.74 1.20 1.35 1.31 0.98 1.12 1.08
2.77 0.92 1.02 1.01 0.73 0.89 0.90
2.80 0.70 0.74 0.77 0.54 0.71 0.74
2.83 0.54 0.54 0.60 0.41 0.56 0.63
2.86 0.46 0.41 0.49 0.34 0.46 0.54
2.89 0.43 0.35 0.44 0.30 0.40 0.49
2.92 0.47 0.34 0.44 0.30 0.36 0.46
2.95 0.54 0.38 0.47 0.32 0.36 0.44
2.98 0.65 0.45 0.54 0.36 0.37 0.45
3.01 0.76 0.54 0.62 0.40 0.40 0.46
3.04 0.87 0.62 0.70 0.45 0.44 0.47
3.07 0,97 0.69 0.77 0.49 0.49 0.48
3.10 1.04 0.75 0.83 0.54 0.54 0.49
3.13 1.08 0.78 0.88 0.58 0.58 0.49
3.16 1.10 0.80 0.90 0.62 0.62 0.49
3.19 1.10 0.80 0.91 0.66 0.65 0.49
3.22 1.08 0.79 0.91 0.69 0.66 0.48
3.25 1.04 0.77 0.90 0.71 '0.66 0.48
3.28 1.01 0.75 0.89 0.71 0.65 0.47
3.31 0.97 0.73 0.87 0.71 0.63 0.47
3.34 0.94 0.71 0.85 0.69 0.60 0.47
3.37 0.91 0.69 0.83 0.66 0.57 0.47
3.40 0.89 0.66 0.81 0.62 0.52 0.46
3.43 0.86 0.62 0.79 0.56 0.48 0.46
3.46 0.83 0.58 0.76 0.51 0.45 0.46
3.49 0.79 0.52 0.72 b,.45 0.41 0.45
3.52 0.75 0.46 0.68 0.40 0.39 0.44
Table 34 (Continued)
cm(r)
Temperature (oC)
3.55
3.58
3.61
3.64
3.67
3.70
3.73
3.76
3.79
3.82
3.85
3.88
3.91
3.94
3.97
4.00
4. 03
4.06
4.09
4.12
4. 15
4.18
4.21
4.24
4.27
4.30
4.33
4.36
4.39
4.42
4.45
4.48
4.54
4.57
4
0.70
0.64
0.58
0.52
0.47
0.42
0.37
0.33
0.30
0.28
0.26
0.23
0.21
0.18
0.15
0.11
0.07
0.02
0.02
0.06
0.10
0.14
0.16
0.19
0.21
0.22
0.24
0.25
0.26
0.28
0.29
0.31
0.32
0.33
0.35
r(A)
20
0.39
0.31
0.24
0. 17
0.10
0.05
0.00
0.03
0.06
0.09
0.12
0.15
0.1 8
0.21
0.25
0.29
0 .33
0.36
0.40
0.43
0.46
0.48
0.50
0.52
0.53
0.55
0.56
0.57
0.58
0.59
0.59
0.60
0.60
0.61
0.62
25
0.63
0.58
0. 52
0.47
0.42
0.37
0.33
0.29
0.27
0.24
0.22
0.21
0.19
0. 17
0. 14
0. 11
0. 08
0.04
0. 01
0.03
0.07
0.10
0.13
0. 16
0. 18
0.20
0.22
0.23
0.25
0.26
0.28
0.29
0.30
0.32
0.33
50
0.36
0.32
0.29
0.28
0.26
0.25
0.24
0.23
0.22
0.20
0. 1 8
0.15
0.13
0. 10
0.08
0.06
0. 04
0.02
0.00
0.02
0.04
0.06
0.08
0. 11
0. 13
0.15
0. 16
0.17
0.18
0.19
0.20
0.20
0.21
0.22
0.24
75
0.37
0.35
0.34
0.32
0.3 1
0.30
0.28
0.26
0.24
0.21
0.19
0. 17
0.15
0. 13
0. 11
0.10
0.08
0.07
0.05
0.03
0.0 1
0.01
0.03
0.05
0.06
0.07
0.08
0.09
0.09
0.09
0.10
0.1 1
0. 12
0.13
0.15
100
0.43
0.41
0.40
0.38
0.37
0.35
0.33
0.32
0.30
0.29
0.27
0.24
0.22
0. 19
0. 16
0.13
0. 1 0
0.07
0.04
0.02
0.01
0.03
0.04
0.06
0.07
0.08
0.09
0. 10
0.10
0. 11
0.11
0.12
0. 12
0.12
0. 13
Table 34 (Continued)
0
r(A)
4.60
4.63
4.66
4. 69
4.72
4.75
4.78
4.81
4. 84
4.87
4.90
4.93
4.96
4. 99
5.02
5.05
5.08
5.11
5.14
5.17
5.20
5.23
5.26
5.29
5.32
5.35
5.38
5.41
5.44
5.47
4
0.36
0.36
0.37
0.37
0.37
0.36
0.35
0.34
0.33
0.31
0.30
0.28
0.27
0.25
0.23
0.22
0.20
0.18
0.16
0.14
0.12
0.10
0.07
0.05
0.03
0.01
0.01
0.02
0.03
0.04
c (r)
Temperature (C)
20
0.62
0.63
0.64
0.64
0.64
0.63
0.62
0.60
0.58
0.54
0.51
0.47
0.44
0.41
0.38
0.36
0.34
0.32
0.30
0.28
0.26
0.24
0.21
0.18
0.15
0.12
0.09
0.07
0.05
0.04
25
0.33
0.34
0.34
0. 34
0.34
0.33
0.33
0.32
0.31
0.30
0.29
0. 27
0.26
0.24
0.23
0.21
0.19
0. 17
0.15
0. 13
0.11
0.09
0.07
0.05
0. 04
0.02
0.00
0.0 1
0.03
0.04
50
0.25
0.27
0.29
0.30
0.31
0.31
0.30
0.29
0.28
0.25
0.23
0.21
0.18
0.16
0.14
0.12
0.10
0.08
0.07
0.05
0.03
0.01
0.01
0.04
0.06
0.07
0.09
0.10
0.11
0.12
75
0.16
0.18
0.19
0.20
0.21
0.21
0.20
0.19
0.18
0. 1 7
0.16
0.15
0. 14
0.14
0.14
0.13
0.1 3
0.12
0.1 1
0.09
0.08
0.06
0.04
0.03
0.02
0.01
0.00
0.01
0.01
0.01
100
0.13
0.14
0.14
0.15
0.15
0.15
0.15
0. 15
0.15
0.14
0.14
0.13
0. 13
0.13
0. 12
0.12
0.11
0. 11
0.11
0.10
0. 10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
including water with diverse intermolecular forces and orientational
effects. The reduced isothermal compressibility can be related to the
direct correlation function by
1 DP/RT 2
pRTK = 1 4rp r c(r)dr. (347)
T IT
Figure 32 and Table 35 represent an attempt to extend this
investigation to a microscopic level using a reduced direct correlation
function. The reducing parameters obtained by Mathias (1978) by fitting
pure component compression data are utilized
Argon : V = 75.4 cc/gmole T = 150.70K
Water : V = 46.4 cc/gmole T = 438.70K.
The ratios of these values are similar to those found by Gubbins
and O'Connell (1974) although the temperature ratio is lower.
The reduced direct correlation function (integrand in equation
(347) 4pV*1/3 'r2c(r) as a function of reduced distance r* = r/V at
several temperatures is presented in Table 35. Figure 32 shows the
reduced direct correlation function for water compared with two states of
liquid argon whose reduced density bound that of water. (Argon data of
Yarnell et al., 1973, and Smelser, 1969).
Of particular interest in Figure 32 are the unusual extremum in
the reduced correlation function for water at r* 0.8 and the shift of
the maximum in the water results to considerably larger distances (r*=1.3)
than that of argon (r* = 0.9). While any interpretation of these results
is highly speculative, the dual nature of liquid water as proposed by
Narten and Levy (1969) may offer some assistance since it suggests the use
60
0 0.5 1.0 1.5
Fig. 32. Reduced Direct Correlation Functions
20 \
Ar(1080K)
40_
Ar(85K)
60
0 0.5 r 1.0 1.5
Fig. 32. Reduced Direct Correlation Functions
Table 35
Reduced Direct Correlation Function for Liquid Water
1/3 2
r* 4Tr p V* r c (r)
Temperature (C)
0.05
0. 10
0. 15
0.20
0. 25
0.30
0.35
0.40
0.45
0.50
0. 55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1. 15
1.20
1. 25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
4
0.59
2.19
4.54
7.45
10.90
14.36
18.28
22.49
25.89
29.11
32.04
33.98
35.60
34.75
18.80
5.45
12.93
16.76
15.37
12.05
6.90
3.96
2.06
6.40
9.47
12.16
11.43
8.74
4.53
0.56
2.46
2.74
0.83
1.53
2.33
20
0.74
2.74
5.70
9.30
13.67
18.20
23.00
27.97
31.96
35.36
38.11
39.60
40.04
38.85
21.25
4.62
9.21
12.27
11.30
5.41
1.02
5.32
11.34
15.45
18.25
20.96
20.05
14. 14
10.43
3.77
1.16
1.20
1.53
0. 11
0.32
25
0.61
2.25
4.66
7.57
11.14
14.60
18.67
22.63
26.34
29.77
32.55
34.52
36.71
36.59
20.08
5. 66
10.34
14.31
14.07
10.75
6.03
3.70
1.15
5.77
8.97
11.24
10.81
8.50
4.37
0.06
3.57
4. 89
2.80
0.28
0.10
50
0.61
2.22
4.57
7.32
10.56
13.69
17.42
21.02
24.76
28.07
32.10
33.92
35.79
34.17
15.72
3.87
6.53
10.85
10.32
5.95
4.82
2. 18
0.70
4.21
6.22
9.46
9.45
4.94
0.92
4.08
5.96
7. 11
6.24
5.74
4.86
75
0. 5 1
1.91
4. 01
6.57
9.65
12.61
15.86
19. 14
22.27
25.28
28.28
29.39
31.56
30.38
16.27
5. 17
6.38
10. 22
8.69
6.53
5.25
2.82
0.53
2. 12
3.23
6. 33
6.11
4.99
2.89
0. 17
1.25
2.98
1.91
1.44
0.19
100
0.44
1.68
3.58
5.92
8.78
11.51
14.37
17.50
19.88
22.63
25. 15
26.24
27.88
26.01
14.87
6.10
6.41
7.27
7.79
7.56
6.43
4.05
0. 18
2.28
3.41
4.56
5.02
4.37
3.64
1.37
1.45
1.84
0. 29
2.97
4.22
Table 35 (Continued)
*1/3 2
r* 4T pV r c (r)
m
Temperature (C)
4 20 25 50 75 100
1.80 4.02 0.27 1.85 2.72 1.82 4.88
1.85 4.64 2.18 3.13 0.56 2.76 3.26
1.90 4.16 1.79 1.68 0.69 2.94 3.65
1.95 3.31 4.33 1.73 0.20 3.40 4.37
2.00 2.56 4.65 1.18 0.24 3.42 4.60
2.05 1.50 5.81 0.25 0.20 2.80 3.80
2.10 1.25 7.11 0.60 0.35 2.54 1.69
2.15 0.77 6.25 0.96 1.66 0.60 0.59
2.20 2.10 3.63 0.53 3.29 1.65 0.23
2.25 1.88 2.84 2.78 3.78 2.10 1.67
2.30 1.99 2.65 3.06 5.48 2.11 1.12
2.35 2.53 0.18 3.40 6.91 2.98 0.69
2.40 1.41 2.29 3.04 7.77 2.98 0.54
2.45 1.94 0.31 2.72 8.07 1.89 1.25
2.50 0.15 0.00 2.30 6.74 1.87 0.57
2.55 0.10 0.56 2.03 6.51 1.26 0.34
2.60 1.37 0.76 2.05 5.29 0.22 0.53
2.65 1.66 0.13 1.10 3.84 0.33 3.62
2.70 1.77 1.64 1.66 2.94 0.91 2.45
2.75 2.93 1.86 1.32 3.75 0.63 2.62
2.80 1.00 2.63 1.54 2.94 1.14 1.88
2.85 1.87 2.49 1.55 1.17 0.95 1.53
2.90 1.96 2.41 2.36 1.10 0.95 2.22
2.95 1.73 1.32 1.38 0.55 1.28 1.78
3.00 2.43 2.74 1.62 0.08 1.27 2.73
3.05 2.25 1.52 0.55 1.40 0.88 0.74
3.10 1.11 1.08 0.85 2.37 0.53 0.01
3.15 0.85 1.29 1.56 2.83 1.35 0.65
3.20 0.82 1.26 1.40 2.99 1.70 0.07
3.25 1.68 1.12 1.20 2.91 2.09 0.10
3.30 2.05 0.02 0.26 3.68 1.04 0.06
3.35 0.63 0.12 0.28 1.63 0.77 0.49
3.40 1.12 0.62 1.77 1.36 0.48 2.64
3.45 0.18 0.65 1.75 3.16 0.24 1.48
3.50 1.48 0.58 2.83 1.61 0.45 1.33
Table 35 (Continued)
r* 4 p V*/3 r2 c (r)
m
Temperature (C)
4 20 25 50 75 100
3.55 1.94 0.18 2.87 2.06 0.54 0.57
3.60 0.95 1.96 1.33 2.30 0.70 0.09
3.65 1.77 0.43 2.53 1.80 0.58 2.15
3.70 1.62 1.10 1.68 1.62 0.55 1.07
3.75 1.51 0.37 1.61 1.42 0.49 1.63
3.80 0.54 0.75 1.06 1.70 0.11 0.55
3.85 1.79 1.71 0.67 1.05 1.13 0.56
3.90 2.01 1.59 0.50 1.51 0.27 1.12
3.95 0.45 1.57 0.14 0.60 0.47 0.20
4.00 0.60 0.85 0.14 0.98 0.44 0.11
4.05 0.55 1.47 0.55 1.39 0.07 0.64
4.10 0.04 0.75 0.37 0.75 1.30 0.01
4.15 0.58 1.64 1.09 1.63 0.63 0.66
4.20 0.35 0.83 1. 15 1.45 0.42 0.25
4.25 1.19 1.43 1.53 0.91 1.24 1.12
4.30 1.82 0.99 1.28 1.34 1.37 2.92
4.35 1.30 1.09 1.42 0.53 1.11 0.94
4.40 1.23 0.87 1.42 0.87 1.33 0.45
4.45 1.78 0.78 1.04 0.07 0.95 0.62
4.50 2.18 0.90 0.91 0.18 2.11 2.00
4.55 1.28 0.30 0.72 0.26 0.69 2.25
4.60 1.32 0.61 0.48 0.71 1.31 2.57
4.65 0.54 0.19 0.06 2.23 2.22 1.85
4.70 1.06 0.47 0.11 2.22 0.70 1.72
4.75 1.25 0.32 0.57 0.97 0.04 1.85
4.80 1.05 0.03 0.41 1.32 0.32 0.27
4.85 1.13 0.72 0.79 1.95 0.56 0.12
4.90 0.51 0.67 0.84 2.27 0.03 0.55
4.95 0.38 1.05 1.08 2.75 0.38 0.44
5.00 1.11 1.24 1.14 0.49 0.85 0.52
61
of a combination of two distinct correlation functions to model the
results of Figure 32.
The most remarkable fact about these results is that even though
the integrands do not scale, the integrals do over wide ranges of
conditions.
CHAPTER 4
MODELING OF SPHERICAL GAS SOLUBILITY
Introduction
The initial section of this chapter outlines the derivation of
the thermodynamic properties of solution from the experimentally deter
mined Henry's constant. Since several correlating equations yield equally
accurate fits of the Henry's constant but considerable variation in the
enthalpy, entropy and heat capacity changes upon solution, they provide
reasonable bounds for the experimental properties of solution.
The second section is concerned with a theory for the thermo
dynamic properties of aqueous solutions. Pierotti (1965) considered the
solution process to consist of two steps: creation of a solutesized
cavity in the solvent and introduction of an interacting solute molecule
into the cavity. Using scaledparticle theory for the first step and
a meanfield theory using the LennardJones potential for the second,
Pierotti obtained reasonable values for the properties.
Stillinger (1973) proposed an extended analysis for the first step.
However, reexamination of this method has led to the present use of a
reference solute to encompass the smallest scale details in the cavity
formation step. A straightforward model based on macroscopic properties
then correlates the work required to form a different solute cavity from
the reference cavity.
The second step is modelled similarly to Pierotti but using
a full reference radial distribution function and a more appropriate
pair potential for water. The energy parameters for unlike interactions
were based on the approach of Rigby et a]. (1969) but allowed to vary to
obtain a highly accurate fit of the Henry's constant at 298.150K. Care
has been taken to provide accurate models for the temperature dependent
hard sphere diameters and mixture radial distribution function..
The possibility of contributions to the properties of solution
arising from a change in the interfacial tension upon introduction of
a solute into a cavity is discussed. Such a possible effect is shown to
be insignificant.
The last section of this chapter discusses the results of the
modeling from the viewpoint of possible inadequacies and thus provides
a basis for suggestions of future research.
Thermodynamic Properties of Solution from Experimental Data
Appropriate derivatives of the Henry's constant with respect to
temperature yield the enthalpy and entropy of solution and the difference
in heat capacity of the gas between the liquid and gaseous phases. For
states at the same temperature, the activity of the gas in each phase
is related to the partial molal free energy, or chemical potential by
G2 = G + RT kn a2. (41)
If the standard state of the gas in the gaseous phase at any fixed
temperature is defined as that in which the gas has a fugacity of unit
pressure, the chemical potential for that phase is
G = G2' + RT Zn f (42)
64
For the dissolved gas, the standard state at any chosen temperature is
taken to be the hypothetical state found by extrapolation of the line
representing Henry's law on an f2 vs x2 graph to unit mole fraction for
the dissolved gas solute. The fugacity in this standard state is K, the
Henry's constant. This choice makes the activity, f /f 0= f /H, of the
dissolved gas approach the mole fraction of the dissolved gas in very
dilute solutions where Henry's law is valid. Thus, for dilute solutions,
equation (41) becomes
diss odissdiss RT (43)
2 (43)
For equilibrium
Gdiss = G2 (44)
2 2
and
AG = o 'diss G'g = RT(kn x Sn f ) (45)
or
AG = RT in K (46)
where Henry's law has been used.
The other standard state thermodynamic properties of solution
can be obtained from the appropriate temperature derivatives.
o 2 .(AG/T) A (AG)
AH DTT AS T IP
and ACp = 8AH) (47)
P
The sources of the Henry's constant data used in this work are listed
below.
Gas Henry's Constant Data Source
Helium, Neon, Argon, Krypton, Xenon Benson and Krause (1976)
Carbon Tetrafluoride, Sulfur Hexa Ashton et al. (1968)
fluoride
Methane Wilhelm et al. (1977)
Neopentane Wetlaufer et al. (1964),
Shoor et al. (1969)
Several equations for correlating in K as a function of temper
ature are available in the literature (Benson and Krause, 1976). Two
expressions are considered here.
kn (1/K) = ao + a1 n T + a2(kn T)2 (48)
Sn (1/K) = b + bl/T + b2 /T2 (49)
A standard leastsquares routine was utilized to determine the
parameter values and their standard deviations in equations (48) and
(49). Extreme values of the thermodynamic properties calculated at
one standard deviation of the parameters were used to provide an estimate
of the bounds on the true value. For purposes of subsequent modeling the
average value of the two extremes was used as the "true" experimental
property value. The "true" experimental values and error limits are
listed in Table 45.
Application of ScaledParticle Theory
to Aqueous Solubility
Pierotti (1963) considered the process of introducing a solute
molecule into a solvent as consisting of two steps. First a cavity is
created in the solvent of a suitable size to accommodate the solute
molecule. The reversible work or partial molar Gibbs free energy G
c
required to do this is identical with that required to introduce a hard
sphere of the same radius as the cavity into solution. The second step
is the introduction into the cavity of a solute molecule which inter
acts with the solvent according to a chosen intermolecular pair potential.
Associated with each step is a set of thermodynamic functions
with which the solution process can be described. Pierotti showed that
for extremely dilute solutions
kn K = G /RT + G./RT + kn (RT/V ) (410)
where K is the Henry's constant, G and G. are the partial molar Gibbs
c 1
free energy for cavity formation and interaction, respectively, and V1
is the solvent molar volume.
The molar enthalpy of solution is given by
o = n K 2
AH = ( RT Hc + H. RT + a RT2 (411)
where a is the coefficient of thermal expansion of the solvent.
The molar heat capacity change for the solution process is
given by
AC = = Cp+ Cp R + 2mt RT + RT2I (412)
A p T p Cc i p a aT )p
The partial molar volume of the solute is given by
V2 = V + V. + B RT (413)
2 c
where P is the isothermal compressibility of the solvent.
Pierotti calculated the partial molar Gibbs free energy of
creating a cavity in a fluid using the scaled particle theory approach
of Reiss et al. (1959) (Equations 326 and 327). Appropriate temper
ature derivatives yield H Cp and S the molar entropy of cavity
formation.
The interaction energy of a nonpolar solute with a polar solvent
can be described in terms of dispersion, induction and repulsive inter
actions. Pierotti approximated the dispersion and repulsive interactions
by a LennardJones pairwise additive potential while the inductive inter
action was described by an inverse sixth power law. The total interaction
energy per solute molecule was given by
G. = Cd. yz (r 6 6 r12)) C. Z r6 (414)
S dis p p 12 p ) nd p p
where r is the distance from the center of the solute to the center of
p
the pth solvent molecule and 012 is the distance at which the dispersion
and repulsive energies are equal.
6 6
Cdis = 4 12 V12 4(12) [(01 + 02)/216 (415)
where c1 and 2 are the energy parameters for the solvent and solute,
respectively, and 01 and 02 are the corresponding distance parameters in
the LennardJones potential.
Cnd = 2 2 (416)
Sis the solventd is the solute polarizability.
where p is the solvent dipole moment and o2 is the solute polarizability.
An alternative method of calculating Cdis discussed by Pierotti
is the KirkwoodMuller formula.
2 ait2
Cdis = 6 m c (/X) + (a2/X2) (417)
where m is the mass of an electron, c is the velocity of light and X and
X2 are the molecular susceptibilities of the solvent and solute.
In order to calculate G., Pierotti assumed the mean field
1
approach: the solvent is infinite in extent and uniformly distributed
according to its number density p around the solute molecule. The
number of molecules contained in a spherical shell a distance r from the
center of the solute molecule is then equal to 4Tp r 2dr where dr is the
shell thickness. Combining this with equation (414) and replacing the
summation by an integration gives
Gi 47rp Cdis +Cind Cdis 12 dr' (418)
kT L kT ,4 0 drl (418)
kT kT r4 r 'l0
where R is the distance from the center of the solute molecule to the
center of the nearest solvent molecule.
Pierotti obtained an estimate of the distance parameter for
water, 0 from a graphical extrapolation of K vs a2 and 02 vs a2 to
obtain values of K and 02 at a2 = 0. Since, from equations (416) (418),
G is proportional to o2
kn K = G /RT + Zn(RT/V ) at c2 = 0. (419)
Thus, a can be calculated given values of K and 0a Pierotti's value of
w 2
0 = 2.75 A was essentially independent of temperature.
Pierotti's final expression for Zn K from equations (410),
(414) (418) is
8 Cind Tp
n K + 3 G /RT in (RT/V) =
6 kT 012
13
(11.17p/T)(C /k) (e /k) a3 (420)
w 2 12
Pierotti determined E /k from the best linear fit of the left
w
hand side of equation (420) as a function of (E2/k) 12 A reasonably
straight line is obtained, insuring a good fit of the experimental K val
ues. However, the value of 6 /k (= 85.3) obtained seems unreasonably low
w
when compared to that resulting from fitting other thermodynamic property
data (e.g., Rigby et al. (1969)).
Table 41 presents the results of Pierotti's model at 298.15K
and 323.150K. Under the assumptions of his model AS. = 0 and ACpi = 0.
The experimental values as previously discussed are included for compar
ison. Note that terms arising from the term in (RT/V ) in equation (410)
have been lumped with the cavity terms in Table 41. Considering that
no fitting of solute parameters was done, the results are quite good
except for the heat capacities.
Table 4la
Solution Properties from Pierotti's Model
Solute Temperature c cal ex_ c exp
RT RT RT RT R R
Helium 298.15 12.20 0.67 11.53 11.86 12.35 12.15
323.15 12.18 0.61 11.57 11.84 11.62 11.02
Neon 298.15 12.63 1.65 10.98 11.71 12.72 13.25
323.15 12.60 1.52 11.08 11.77 11.90 11.81
argon 298.15 14.66 4.35 10.31 10.59 14.66 15.42
323.15 14.59 3.99 10.60 10.88 13.25 13.53
Krypton 298.15 15.33 5.80 9.53 10.00 14.97 16.29
323.15 15.29 5.24 10.05 10.41 13.25 14.25
Xenon 298.15 17.26 8.06 9.20 9.45 16.58 17.22
323.15 17.18 7.36 9.82 9.95 15.00 14.78
Methane 298.15 16.15 5.66 10.49 10.60 15.66 16.14
323.15 16.10 5.19 10.91 10.95 14.27 14.14
Table 4lb
Enthalpy and Heat Capacity Contributions
Solute Temperature R RT R Rexp
Helium 298.15 0.15 1.67 0.82 0.29 9.78 14.18
323.15 0.56 0.61 0.05 0.82 10.34 13.84
Neon 298.15 0.08 1.65 1.73 1.54 10.73 18.13
323.15 0.70 1.52 0.82 0.37 11.27 17.70
Argon 298.15 0.20 4.35 4.15 4.83 15.13 23.48
323.15 1.31 3.99 2.68 2.64 15.65 23.11
Krypton 298.15 0.29 5.80 5.51 6.28 16.69 25.60
323.15 1.53 5.24 3.71 3.84 17.24 25.08
Xenon 298.15 0.56 8.06 7.50 7.77 20.86 30.56
323.15 2.14 7.36 5.22 4.83 21.64 29.98
Methane 298.15 0.41 5.66 5.25 5.55 18.38 25.55
323.15 1.79 5.19 3.40 3.19 19.11 24.48
Contributions to the Thermodynamic Properties
of Solution from Cavity Formation
Consider the origin of the series approximation (equation 330)
to the contact correlation function G(r) when the cavity radius r is
large. As noted in Chapter 3, the exact expression for G(r) derived
from Koenig's (1950) expression for the curvature dependence of the
surface tension (339) should allow a more rigorous calculation of G(r)
than the series result of Stillinger. However, equation (341) shows
that G(r) cannot pass through an extremum (dG/dr = 0) for finite values
of 6. Thus, the extremum value of G(r) seems a natural dividing point
between microscopic and macroscopic regimes.
Now, previous expressions (329) for G(r) in the region where
r contains two water molecular centers can pass through an extremum and
be used to match equation (339). However, we have found that the assump
tion of constant 6 severely restricts the range of r values in which the
two functions can be matched. Coupled with a lack of knowledge concern
ing the value of r at which the triplet correlation function becomes sig
nificant, this casts considerable doubt on the rigor of the expressions.
While the contact correlation function must be continuous with r, the
form is truly unknown in the region of the maximum.
Considering these difficulties, it was decided to abandon
efforts at linking microscopic and macroscopic approaches and simply
use a reference solute. The experimental solution properties of the
reference solute encompass the microscopic detail and allow use of a
macroscopic expression for differences between the reference and other
solutes. Argon was chosen as the reference solute for this work.
For macroscopic properties, Melrose (1970) showed that for two
phases, a and 3, in contact
dU = TdS + E" dN. pdV P dV + yda + adJ (421)
where J is the arithmetic mean curvature.
1 1
R R
where R1 and R2 are the principal radii of curvature.
For an isothermal constant composition process, the Helmholtz
Free Energy (work) is
dA = PdVa PdV + yda + adJ. (422)
Since our cavity creation process is constant pressure with
dVO = dVB
dA = yda + j adJ (423)
Using the one term macroscopic approximation
y = Y (16 J) d = YO. (424)
The work of changing the cavity from that of the reference
solute (argon) is then
As A ref = y (aaref 6[(aJ)s (aJ) ref]
6(a J a J )
y (a a ) 1 s s ref ref (425)
asa ref (a a re)
Since for a sphere
a = 4rR2 and J = 2/R
A Aref = 47 y R2 R2 1 f[]26 (426)
s ref ref (R +Re )
sref)
Since dG = dA + d(PV), and our cavity formation process is at
constant pressure and constant overall volume (dVO = dv ), G Gref
s ref
As Aref
The free energy of creating the argon reference cavity is
obtained by difference between the experimental free energy of solution
and the interaction contribution discussed in the next section. The
other cavity contributions to the thermodynamic properties of solution
are obtained through the appropriate temperature derivative of equa
tion (426).
Calculation of the thermodynamic properties of cavity formation
requires a model for the hard sphere diameters of the solute and solvent
as a function of temperature. We chose the model of Mathias (1978).
From considerations involving the direct correlation function he
postulated that the reduced hard sphere diameter should be some universal
function of reduced temperature and reduced density. The functional
form was obtained by fitting the experimental reduced isothermal compres
sibility to that obtained for a hard sphere fluid from the Carnahan
Starling equation (Carnahan and Starling, 1969). The rather complicated
function is shown in Table 42. Table 43 shows the values of the char
acteristic parameters for water and solutes of interest in this chapter.
Slight adjustments in these parameters compared to those of Mathias were
sufficient (with helium and neon as notable exceptions) to provide a con
sistent fit of the solubility data. Mathias chose the particular form
because it has the following features:
a) At high reduced temperatures the hard sphere diameter is
a function of reduced temperature only.
b) At high densities the hard sphere diameter is a function of
reduced temperature only.
c) The first two exponential terms were chosen to represent the
minimum in the isothermal change of the hard sphere diameter with density.
The particular form was chosen since the position of the minimum seems
to shift linearly with temperature.
d) The last exponential term is used for changes in the hard
sphere diameter required to obtain an accurate representation in the
critical region.
Table 42
Universal Correlation for the Reduced Hard Sphere Diameter
Reduced Temperature: T = T/T*
Reduced Density: p = p/V*
a
T 2 0.73: fs = a7/T (427)
T < 0.73: fs = a14 exp [a15 T] (428)
r N d
3 aV 2
3V* = fs + a2/exp [a4(p+ aT) 2] 
a3/exp [a5(p+alT a6)2] + a9/exp [al0{(Ta13)2 +
all(p al2)2}] (429)
a1 = 0.54008832 a9 = 0.18874824
a2 = 1.2669802 al0 = 17.952388
a3 = 0.05132355 all = 0.48197123
a4 = 2.9107424 a12 = 0.76696099
a5 = 2.5167259 a13 = 0.76631363
a6 = 2.1595955 a14 = 0.809657804
a7 = 0.64269552 a15 = 0.24062863
a8 = 0.17565885
Table 43
Characteristic Parameters
Helium
Neon
Argon
Krypton
Xenon
Methane
Carbon Tetrafluoride
Sulfur Hexafluoride
Neopentane
Water
T*(oK)
39.0
45.2
150.8
209.4
289.7
190.6
227.6
318.7
433.8
438.7
(10.3)
V*(cc/g mol)
50.0 (37.5)
60.5 (40.3)
74.9
88.5 (91.2)
114.5 (118.0)
96.0 (99.0)
147.0
203.9 (198.0)
312.1 (303.0)
46.4
Values in parentheses are
from those utilized here.
those of Mathias (1978) if different
Contributions to the Thermodynamic Properties
of Solution from Intermolecular Forces
From perturbation theory (Reed and Gubbins, 1973) the configura
tion integral L for a mixture can be related to the intermolecular
potential by
ref 2ir r r P ref 2
in L = nL  E E pap V g r dr + ... (430)
kaT g=a 3=a j P
ref P
where L is the reference configuration integral, P8 is the differ
ence between the real pair.potential and the reference state pair poten
ref
tial, and gref is the reference mixture radial distribution function.
We have restricted the model to a firstorder perturbation theory.
A basic relation of statistical mechanics is
SkT n (431)
SiT,V,Ny .
With the hard sphere as the reference state, equations (430)
and (431) yield
y Phs 2 (432)
y = hs + 4 ws(r) gs (r) r2dr
where R is the distance from the center of the solute molecule to the
center of the nearest water molecule, kws(r) is the watersolute inter
molecular potential and gs (r) is the watersolute hard sphere radial
distribution function. Since G = P
R hs (r) r2dr. (433)
Gi = 4Tpw ws(r) gws (r) r dr. 433)
We have chosen to approximate ws(r) by a LennardJones form
ws
with 0.. E R.
13
(w(r) = 4c' (434)
ws wsr r
where C' includes both dispersion and induction interaction. An approx
ws
imate expression for c' from Ribgy et al. (1969)
ws
2
e' =e 1 + s w (435)
ws 4 3 (a + 2a )3
ws ws ws ws
utilizes StockmayerKihara potential parameters and 0 along with
solute polarizabilities and the dipole moment of water V .
w
Table 44 lists the values of c' calculated from equation (435)
ws
compared to those required to obtain an exact fit of the standard free
energy of solution AG at 298.150K. The two values are in close agree
ment in most cases.
In order to evaluate the integrals in equation (433), an
approximation must be formulated for the radial distribution function of
a hard sphere mixture. We have chosen the formulation of Mathias (1978).
hs
He chose to approximate g..(r) in terms of the value at contact
and an equivalent pure hard sphere distribution function. An average hard
sphere diameter can be defined as
n n
.+1 a
d = E x d / / x d (436)
av =1 i=1 i i
We used O = 3; equivalent to a volume fraction average.
hs
If ghs(d..) is the contact value of the radial distribution function of
hs P
the mixture and g (d ) is the radial distribution function of a pure
av
Table 44
Intermolecular Potential Energy Parameter
e' /k
ws
Helium 46.8
Neon 156.6
Argon 375.6 (375.6)
Krypton 467.2 (462.2)
Xenon 562.3 (547.5)
Methane 462.0 (495.4)
Carbon Tetrafluoride 509.7 (514.2)
Sulfur Hexafluoride 586.3
Neopentane 663.2 (707.8)
The water energy parameter c /k = 170 as obtained from Rigby et al.
W. .
(1969). Values in parentheses are from equation (435) whereas other
values result from solubility data fit at 298.150K.
Pure component parameter values for equation (435) are from
O'Connell (1967).
hard sphere fluid whose hard sphere diameter is d and is at a reduced
av
n
3
density of r = E p d., then our approximation is
i=l
hs (d 1
hs (r) ws ws 1 hsP
hs (r) s g P (r + d d ) 1] (437)
ws i hs P ( av ws
g ^av 
This form was deduced from the results of Throop and Bearman
(1965) who numerically evaluated the radial distribution function for
various hard sphere binaries using the PercusYevick equation.
We evaluated equation (433) for several R values over the
temperature range considered in this work and obtained the following
accurate correlation.
G. = ' (6.72024.954 x10 3/T + 6.548 x10 5/T 1.52R3.17R 2). (438)
1 ws
The other interaction contributions to the thermodynamic
properties H., S., Cp. can be obtained from the appropriate temperature
derivatives of equation (438).
Analysis of Spreading Pressure of
a Solute Occupying a Cavity
There is the possibility of a contribution to the thermodynamic
properties of solution due to a change in the interfacial tension upon
introduction of a solute into a cavity. This section will however illus
trate that the magnitude of such an effect is insignificant.
For an ideal dilute solution, the thermodynamic and monolayer
analysis of Chapters 7 and 12 of Defay et al. (1966), respectively, lead
to the same result
=Y Y. = r R T (439)
w 1,w i,w g
where 7T is the spreading pressure and P. is the relative adsorption
1,w
of i to water, independent of the choice of dividing surface.
From Defay et al. the relative adsorption can be written in terms
of the adsorption of each component
C' C"
r. = r. r 1 1 1 (440)
w z z C' Crr
w w
where denotes the bulk i phase and the bulk water phase. For the
present case C. = 0 = C'
I w
r. = rF + r c!/C" (441)
1,w i w i 1 w
z z
Also for any change in the dividing surface from z1 to z2,
F r. = (z2zl) (CC") (442)
1 z2 2 1 2 1 1
and
Cf
r. = r. (zz)C + (443)
1,w i 2 1 w Z1 C
If z is chosen so that r. = 0 and z1 such that F = 0,
Si 1 w
= (Z Z ) C (444)
2,w 1
and
(yY ) = (ZZI) C R T (445)
w i,w 2 1 I g
As noted by Reiss (1974) we must fix our frame of reference at
the center of mass of tihe i phase. Then for an unoccupied cavity (termed
an rcule by Reiss and TullySmith, 1971) of radius R. bounded by water,
1
the cavity boundary is R., the surface of tension is at R. + 6 and the
1 1
equimolecular water dividing surface is at R. + 6 + 6 .
1 1
The r = 0 dividing surface is at 0. Therefore Z2 Z =
(R. + 6 + 6 ) and
S rw = (R. + 6 + 6) C' R T. (446)
Yw r,w r g
Considering now the case of a solute molecule occupying the
cavity, the Fs = 0 dividing surface is again at 0. The location of the
r = 0 dividing surface may change upon addition of the solute to
w
R + 6' + 6', where 6' denotes the surface of tension in the presence of
the solute. Thus,
Y sw (R. + 6' + 6')C' R T. (447)
Since there is only one rcule or solute molecule in the volume
47T 3
3 Ri'
C' = C' = (448)
r s 4 N R3
3 o i
Subtracting equation (448) from (447) we obtain
[(66') + (61 61)]
Y Yrw =N R3 R T. (449)
r,w rw g
3 o 1
Since (66') and (616') are probably very small relative to
3
R., the change of interfacial tension upon addition of the solute should
i
0 0
be negligible. For example, if (66') = (616{) = 0.10 A and R. = 3 A,
2
y Y *' 1.3 dynes/cm .
s,w r,w
Discussion of Results and Suggestions
for Future Research
With accurate values for the characteristic volume and temper
ature and the interaction energy parameter as previously discussed, the
final fit to the experimental data involves fitting 6 to a temperature
dependence of the form
6 = A + B/T + C/T2 (Tin K) (450)
A minimum sum of squares fit resulted in A = 8.3194896
B = 2,605.2103
C = 189,930.69
Temperature (K) 6(A)
277.15 1.39
298.15 1.72
323.15 2.08
358.15 2.53
Experience has shown that a fit of similar accuracy to Table 45
can be obtained with a different set of interaction energy parameters
C and consequently 6 values. An example considerably different from that
of Table 45 can be found in Appendix A along with the computer program
which determines the coefficients in the 6 function (equation (450).
The surprisingly large magnitude of 6 relative to the radius of
the cavities involved may arise from several sources: (1) inadequacy in
using the surface tension of pure water for the calculations involving an
evacuated cavity, (2) indicative of strong structural changes in water
caused by cavity formation, and (3) covers other inadequacies in the model.
Table 45a
Contributions to Free Energy and Entropy of Solution
Solute T(K)
He 277.15
298.15
323.15
358. 15
Ne 277.15
298.15
323.15
358.15
Ar 277.15
298.15
323.15
Go o
V AG
c i
RT RT
.2.65 0.86
12.64 0.78
12.53 0.71
.2.24 0.61
14.70 3.15
14.59 2.88
4.36 2.60
.3.89 2.25
9.34 9. 13
9.01 8.42
8.49 7.60
.1
1
1
I
I
I
AG
cal
RT
11.79
11 .86
11.82
11 .63
11.55
11.71
11 .76
10.16
10.59
10.88
AG
exp
11.802
11 .863
11 .841
11 .691
11.543
11 .710
11 .773
11.689
10.159
10.588
10.883
AS
cR
R
+.001
+.001
+.002
+.005
+.001
+.001
+.002
+.035
+.003
+.002
+.009
13*17
11.87
10.51
8.70
14.02
12.42
10.73
8.51
15.78
13.52
11.09
AS
i
R
0. 13
0. 21
0.26
0.29
0. 48
0.72
0.89
0.98
1.31
1.89
2*27
AS
cal
R
13.30
12.08
10.77
a.s S
14.50
13.15
11.61
9.49
17 .0
15.41
13.37
AS0
exp
R
13.20 +.08
12.15 +.02
11.02 +.09
9.62 +_.41
14.59 +.11
13.25 +.03
11.81 +.13
10.01 +.57
17.13 +.08
15.42 +.30
13.53 .04
358.15 17.61 6.63 10.99 11.031 +.068
A
8.00 2.49 10*49 11.17 +.96

Full Text 
167
phase could be noted. No significant change in the visual appearance
of the sample resulted from methane introduction.
Table 73 concerns the influence of dissolved methane on the
phase behavior. No significant enhancement of the isotropic or aniso
tropic phase could be noted. No significant change in the visual appear
ance of the sample resulted from methane introduction.
Future efforts should attempt to reinforce or refute these
results with more detailed investigations and better temperature control.
Since interest is high in its use for tertiary recovery, investigations
into the effects of carbon dioxide should be undertaken. If future
results warrant, the effect of surfactant sample aging may be worth
investigating.
Surfactant formulations of more immediate importance to enhanced
oil recovery should be screened fairly rapidly to ascertain whether they
exhibit more sensitivity to pressure, temperature or dissolved gas. This
may provide more interesting candidates for detailed study.
54
Table 34 (Continued)
r(A) c (r)
m
Temperature (C)
4
20
25
50
75
100
3. 55
0.70
0.39
0. 63
0.36
0. 37
0.43
3.58
0.64
0 .3 1
0.58
0.32
0.35
0.4 1
3.61
 0.58
0. 24
0. 52
0.29
0.3 4
0.40
3.64
0.52
0.17
0.47
0.28
0.32
0. 38
3.67
0.47
0.10
0.42
0.26
0.3 1
0.37
3.70
0.42
0. 05
0. 37
0.25
0.30
0.35
3.73
0.37
0.00
0.33
0.24
0.28
0. 33
3. 76
0.33
0 03
0.29
0.23
0.26
0.32
3.79
0.30
0.06
0. 27
0.22
0. 24
0.30
3.82
0.28
0.09
0.24
0.20
0 .2 1
0.29
3.85
0.26
0. 12
0. 22
0. 1 8
0.19
0 .27
3.88
0.23
0.15
0.21
0.15
0. 1 7
0. 24
3. 91
0.21
0.18
0. 19
0.13
0.15
0.22
3.94
0.18
0.2 1
0. 1 7
0. 1 0
0. 1 3
0. 1 9
3.97
0.15
0 .25
0.14
0.08
0.11
0. 16
4. 00
0.11
0.29
0. 1 1
0.06
0.10
0.13
4.03
0.07
0 .33
0. 08
0. 04
0. 08
0. 1 0
4.06
0.02
0.36
0.04
0.02
0.0 7
0.07
4.09
0.02
0.40
0. 01
0. 00
0.05
0.04
4.12
0.06
0.43
0. 03
0.02
0.0 3
0. 0 2
4. 15
0. 1 0
0.46
0.07
0.04
0 .0 1
0.0 1
4. 18
0.14
0.48
0. 10
0. 06
0. 0 l
0. 03
4.21
0.16
0 .50
0.13
0.08
0.03
0.0 4
4.24
0.19
0 52
0. 16
0. 1 1
0.05
0.06
4.27
0.2 1
0.53
0.18
0. 1 3
0. 06
0.07
4.30
0.22
0.55
0.20
0.15
0.07
0.08
4.33
0.24
0. 56
0. 22
0. 1 6
0.08
0.09
4.36
0.25
0.57
0.23
0. 17
0.09
0. 1 0
4. 39
0.26
0.58
0.25
0.18
0.09
0.10
4. 42
0.28
0.59
0. 26
0. 19
0. 09
0. 1 1
4.45
0.29
0 .59
0.28
0.20
0.10
0.11
4.46
0.31
0.60
0. 29
0.2 0
0.1 L
0.12
4.51
0.32
0.60
0.30
0.21
0. 1 2
0. 1 2
4. 54
0. 33
0.61
0.32
0 .22
0.13
0.12
4.57
0.35
0.62
0. 33
0.24
0. 1 5
0. 13
AH
AH?
X
Solute
T(K)
c
RT
RT
C14H30
277.15
96.84
137.00
298.15
106.05
134.8
323.15
1 13.67
133.07
358.15
11970
 131.21
Table 55b (Continued)
AH
r,v
ABcal
ACp
c
ACp
i
ACp
r,v
aS4i
RT
RT
R
R
R
R
0.00
40.16
237.63
 104. 19
33. 53
166.97
2.07
26.76
217.18
109.37
26.66
134.47
3.37
15.54
192.08
 113. 19
24. 36
103.25
5. 84
5.67
159. 46
l14.00
23.93
69.44
128
161
care was taken to thoroughly remove all mercury to the left of HP10 to
prevent blockage of the gas sample from the cell. Once this was done
HP6 was closed and the system evacuated through HP5 and LP2.
With HP7 closed a sample of gas was introduced into the line
between the tank and HP10. Valve HP9 was closed and the system evacu
ated to remove all gas except in the measuring line between HP9 and
HP10. Valve HP8 was closed, HP7 opened and mercury forced from the
intensifier into the evacuated region. HP9 was then opened and the pres
sure increased to compress the gas into a very small volume near HP10.
With HP10 closed, HP11 was opened to allow expansion of trapped
air in the cell to force a small quantity of mercury into the line from
HP11 to HP12. HP10 was opened and the system pressurized to force
mercury in the connection from HP10 into the cell along with perhaps a
portion of the gas sample. HP10 was again closed and the material
trapped between HP11 and HP12 drained through HP12. A small sample of
the cell contents was again drained into the HP11HP12 line and the
cell repressurized. This process was repeated until one could be reason
ably certain that the entire gas sample was in contact with the liquid
sample in the cell. Adequate pressure was then used to insure that the
gas was dissolved in solution (note calculations in next section) and
desired conditions were investigated. Introduction of more gas into the
sample was accomplished by repeating this procedure.
66
Application of ScaledParticle Theory
to Aqueous Solubility
Pierotti (1963) considered the process of introducing a solute
molecule into a solvent as consisting of two steps. First a cavity is
created in the solvent of a suitable size to accommodate the solute
molecule. The reversible work or partial molar Gibbs free energy
required to do this is identical with that required to introduce a hard
sphere of the same radius as the cavity into solution. The second step
is the introduction into the cavity of a solute molecule which inter
acts with the solvent according to a chosen intermolecular pair potential.
Associated with each step is a set of thermodynamic functions
with which the solution process can be described. Pierotti showed that
for extremely dilute solutions
Â£n K = G /RT + G./RT + Â£n (RT/Vj (410)
c i 1
where K is the Henry's constant, G^ and G^ are the partial molar Gibbs
free energy for cavity formation and interaction, respectively, and Vp
is the solvent molar volume.
The molar enthalpy of solution is given by
Al (Hts), \ + . RT + ap RT2 (411)
where is the coefficient of thermal expansion of the solvent.
The molar heat capacity change for the solution process is
given by
/3 AW0^ o / 9a A
ACP = i9T~)p= Cpc + Cpi R + % RT + RT' (4"12)
The partial molar volume of the solute is given by
V = V + V. + 3 RT
2 c i
(413)
35
W(r) =(4Trp/3)r3 + (47TYJr2 (liy^)r + 0(1) (317)
Here Yra is the surface tension in the planar interface limit, and 6 pro
vides the leading term in the curvature dependence of the surface tension
Y (Buff, 1951)
Y = yjl 26/rJ. (318)
The integral relation (equation.311) between W(r) and G(r) results
in the following larger behavior for G(r)
p X J
G(r) ikx + Sri ^2 + (319)
Subsequent efforts (TullySmith and Reiss, 1970) showed that the coeffi
(320)
(321)
(322)
Thus exact results exist for G(r) at very small and very large r.
Reiss et al. (1959) and Pierotti (1963) adopted the procedure of bridg
ing these two extremes with a smooth function of r. They assumed that
G(r) is a monotonic function of r in the entire range of r. They sug
gested the form of (319) with empirical parameters
Since this is a drastic linearization of the rigorous relation
of Koenig (1950), a more rigorous approach to the curvature dependence
of the surface tension is presented in a later section.
3
cient on the r term in the expansion for G(r) must be zero.
For small cavities, equations (34) and (314) yield
for r ^ a/2
and for W(r) from equation
G(r) = (1 47Tr3p/3) 1
W(r) = kT Â£n(l 4pr~p/3) for r < a/2.
For very large cavities, r , from equation 019)
P
G(r) =
pkT '
102
C
Upon numerical integration of equation (521), was fit to
a function of L with temperature dependent coefficients.
M=Â£ws(C19L + C20l2 + C21L3 + C22L4) h<3'6A
n( = C + C /L + C.../L2 + C,/L3 + C97/L4 L>3.60A. (522)
v i ws' 23 24 25 26 27
The details of the analytical integration of equation (520) and
expressions for the coefficients in equation (522) can be found in
Appendix B.
D. Consider now the case of a differential potential, d
ws
C **" dx
= 4Â£wg ^ws(r) > continuously distributed along the spherocylinder
from y = 0 to y = L interacting with molecular centers in y < 0 and
y > L as shown in Figure 5ld.
2 2 2
For the region y < 0, r =x +w +2xw cos 0 and
dV = 2tt w sin 8 d0 dw.
MD1= p
i w
fL
tt/2
dx
d0
J0 j
o 4
dw 8tt w sin 0
r C
e
ws
12
7
ws
(x2+w2+2x w cos 0)^
ws
(x24w2+2x w cos 0)3
hs. *
8s(w >
(523)
where w = w + a a
w ws
2 2 2
For the region y > L, r =(Lx + wf cos 0f) + (wf sin 0')
2
and dV = 27Tw' sin 0' d0' dwf. Letting xr = Lx yields
207
Table Blc (Continued)
a
s
O
= 3.80 A
Coefficient
A
n
B
n
D
n
C16
0.384936D+00
0.682190D04
0.343000D06
C17
0.154952D+00
0.124761D03
0.900000D07
C18
0.195367D01
0.237170D04
0.220000D07
C19
0.123251D+01
0.207174D01
0.352720D04
C20
0.437732D+01
0.274843D01
0.446380D04
C21
0.211111IH01
0.134720D01
0.220160D04
C22
0.296671D+00
0.193113D02
0.316500D05
C23
0.216043D+01
0.365620D02
0.724300D05
C24
0.442062D+01
0.437975D01
0.724830D04
C25
0.985060D+02
0.624044D+00
0.102510D02
C26
0.504996D+03
Q.327473D+01
0.537615D02
C27
. 0.873619D+03
0.573767D+01
0.941360D02
C28
0.108286D+02
0.509862D01
0.989470D04
C29
0.199940D+01
0.185842D01
0.259000D04
C30
0.187161D+01
0.583554D02
0.670900D05
C31
0.272927D+00
0.780400D03
0.920000D06
C32
0.279471D+02
0.890740D01
0.194720DQ3
C33
0.836984D+00
0.689649D01
0.122525D03
C34
0.247125EH02
0.768669D+00
0.141372D02
C35
0.132752D+03
0.162651EH01
0.276010D02
Table Cla (Continued)
Solute
T(K)
AG
c
RT
AG
i
RT
AG
r,v
RT
C4H10
277.15
42.31
35.57
3.05
298. 1 5
40.46
32.77
3.03
323.15
38.13
29 .72
2.97
358.15
34.80
25.94
2.83
AG 
AG
cal
exp
RT
RT
9. 79
9. 809
+ .011
10.73
10.729
+.006
1 l .38
11.39 5
+.005
11.69
11.775
+. Oil
227
83
The = 0 dividing surface is at 0. Therefore Z^ Z^ =
 (R^ + 6 + 6^) and
Y Y = (R. + 6 + 6J C' R T.
w r,w i 1 r g
(446)
Considering now the case of a solute molecule occupying the
cavity, the r = 0 dividing surface is again at 0. The location of the
T =0 dividing surface may change upon addition of the solute to
w
R^ + 6 + 6^, where S' denotes the surface of tension in the presence of
the solute. Thus,
Y Y = (R. + S' + S')C' R T. (447)
w s,w i 1 s g
Since there is only one rcule or solute molecule in the volume
4tt
3 V
C' = C' =
r s 4ir
3 '
.NR.
3 o i
(448)
Subtracting equation (44$) from (447) we obtain
[(66') + (66')]
Y Y =  R T.
rw rw 4tt 3 g
3 o i
(449)
Since (66') and (6^6^) are probably very small relative to
3
R^, the change of interfacial tension upon addition of the solute should
O O
be negligible. For example, if (66') = (6^6.p = 0*10 A and R, = 3 A,
2
Y Y 1.3 dynes/cm .
s 9 w r, w
APPENDIX A
PROGRAM FOR CORRELATION OF SPHERICAL GAS
SOLUBILITY PROPERTIES
Table 4lb
Enthalpy and Heat Capacity Contributions
Solute
Temperature
AH
c
RT
AH?
i
RT
AH .
cal
RT
AH
exp
RT
ACp
c
R
ACp
exp
R
Helium
298.15
0.15
1.67
0.82
0.29
9.78
14.18
323.15
0.56
0.61
0.05
0.82
10.34
13.84
Neon
298.15
0.08
1.65
1.73
1.54
10.73
18.13
323.15
0.70
1.52
0.82
0.37
11.27
17.70
Argon
298.15
0.20
4.35
4.15
4.83
15.13
23.48
323.15
1.31
3.99
2.68
2.64
15.65
23.11
Krypton
298.15
0.29
5.80
5.51
6.28
16.69
25.60
323.15
1.53
5.24
3.71
3.84
17.24
25.08
Xenon
298.15
0.56
8.06
7.50
7.77
20.86
30.56
323.15
2.14
7.36
5.22
4.83
21.64
29.98
Methane
298.15
0.41
5.66
5.25
5.55
18.38
25.55
323.15
1.79
5.19
3.40
3.19
19.11
24.48
274
Franks, F., "SoluteWater Interaction and the Solubility Behavior of
LongChain Paraffin Hydrocarbons," Nature, 210, 87 (1966).
Gibbons, R. M., "The Scaled Particle Theory for Particles of Arbitrary
Shape," Mol. Phys., 17, 81 (1969).
Gibbons, R. M., "The Scaled Particle Theory for Mixtures of Hard Convex
Particles," Mol. Phys., JL8, 809 (1970).
Gmehling, J., D. D. Liu and J. M. Prausnitz, "High Pressure VaporLiquid
Equilibria for Mixtures Containing One or More Polar Components.
Application of an Equation of State Which Includes Dimerization
Equilibria," submitted to Chem. Eng. Sci. (1979).
Gubbins, K. E. and J. P. O'Connell, "Isothermal Compressibility and
Partial Molal Volume for Polyatomic Liquids," J. Chem. Phys., 60,
3449 (1974).
Hall, D. G., "Exact Phenomenological Interpretation of the Micelle Point
in MultiComponent Systems," Trans. Faraday Soc., j68, 668 (1972).
Hall, D. G. and B. A. Pethica, "Thermodynamics of Micelle Formation,"
in Nonionic Surfactants, Ed., Martin J. Schick, Marcel Dekker,
New York, N. Y. (1970).
Helfand, E., H. L. Frisch and J. L. Lebowitz, "Theory of the Two and
OneDimensional Rigid Sphere Fluids," J. Chem. Phys., _34, 1036 (1961)
Helfand, E., H. Reiss, H. L. Frisch and J. L. Lebowitz, "Scaled Particle
Theory of Fluids," J. Chem. Phys., 33, 1379 (1960).
Hill, T. L., Statistical Mechanics, McGrawHill, New York, N. Y. (1956).
Hill, T. L., "Holes and Cells in Liquids," J. Chem. Phys., 2Q, 1179 (1958)
Hill, T. L., Thermodynamics of Small Systems, Vols. 1 and 2, Benjamin,
New York, N. Y. (1963).
Hobbs, M. E., "The Effect of Salts on the Critical Concentration, Size
and Stability of Soap Micelles," J. Phys. & Colloid Chem., 5J5, 675
(1951).
Israelachvili, J. N., D. J. Mitchell and B. W. Ninham, "Theory of Self
Assembly of Hydrocarbon Amphiphiles into Micelles and Bilayers,"
J. Chem. Soc. Faraday Trans. 2, _72, 1525 (1976).
Kalyanasundaram, K. and J. K. Thomas, "On the Conformational State of
Surfactants in the Solid State and in Micellar Form. A LaserExcited
Raman Scattering Study," J. Phys. Chem., 8Â£, 1462 (1976).
Kaneshina, S., M. Tanaka, T. Tomida and R. Matuura," Micelle Formation of
Sodium Alkylsulfates Under High Pressure," J. Colloid Interface Sci.,
48, 450 (1974).
FORTRAN IV G LEVEL 21
MAIN
DATE
79 loa
01/01/23
0204
0205
0206
0207
0206
0209
021 0
021 1
0212
02 13
0214
0215
0216
0217
0218
0219
0 22 0
0221
0222
0223
0224
0 22 5
0226
022 7
0228
0229
0230
GO TO 102
117 WRITE (6*118) T(J ),WMW( I, J),WMCC{ I J) *WMIC(I ,J) ,WRV( I J) ,WMMI( I J)
2.WCA{I,J),WT(I.J),EXMG(I)
118 FORMAT { / 1 CX*F7. 2 ,3X .F6.2.2X *F 72.2X* 6(F6 .2 ,2X ) )
102 CONTINUE
1C1 CONTINUE
WRITE (6.105)
105 FORMAT (1*///////////)
DC 106 1=1.3
DO 107 J=1,4
IFJ.EQ.2) GO TO 119
WRITE (6.104) T(J) .HMW{I .J) ,H MCC{I ,J) HMIC(I .J) ,HRV( I J),HMMI( I,J )
2 FCA ( I J ) HT ( I J )
GO TO 107
119 WRITE6 118) T(J) HMW( I.J) ,HMCC(I,J)HyiC(I,J).HRV(I,J).HVMl(I,J),
2HCA( I,J).HT(I J),EXMH{ I )
107 CONTINUE .
106 CONTINUE
WRITE (6,103)
108 FORMAT (1///////////)
DO 109 1=1,3
DO 110 J=l .4
IF (J.E0.2) GO TO 120
WRITE (6,111) T(J),SMW{I,J),SMCC( I,J ) SMICI J) ,SRV{ I J) ,SMMI {I J )
2. SCA ( I J) ST( I J)
111 FORMAT (/. 10X.F7.2,3X,7(F6.2,2X))
GO TO 110
120 WRI TEC 6,121 ) T( J) SMW( I J) .SMCC (I J) ,SVIC{ I, J) SRV ( I, J) ,SMMI ( I, J ),
2SCA( I ,J)ST( I ,J ) ,EX MS ( I)
121 FORMAT (/, l OX ,F7.2 ,3X.d (F6.2,2X ) )
110 CONTINUE
109 CONTINUE
11
In order for equation (230) to yield equation (225), the value of a
must be unity.
This conclusion has two important consequences for theoretical
analysis. Since a has not been assumed to be unity in the semiempirical
expression and data analysis of Phillips (1955), Molyneaux and Rhodes
(1972) and others, it is not clear what interpretation should be placed
on their results.
The second consequence involves the work of Sexsmith and White
(1959ab) which, when assuming a < 1, gives a maximum in the monomeric
amphiphile concentration. Using equations (22) and (227) with mole
fractions for activities at all concentrations plus
x = x a(x x,) (231)
cool
yields the relation
x, 
x x,
o 1
. 
1/N
K[x a(x x,)1
o o 1
a
(232)
where
K = exp [AG/NRT].
m
(233)
At small values of x x, = x but at larger values of x >> x, two
o 1 o o 1
limiting cases appear
x, l/xa K(la)a a < 1
1 o
, /.1/2N.1/2
x^ (xq/N) /K a = 1.
(234)
(235)
Equation (234) is chosen by Sexsmith and White (1959b) which indicates
a rapidly decreasing monomeric concentration with total amphiphile while
equation (235) gives a slowly increasing monomer concentration. Experi
mental evidence is mixed (Hall and Pethica, 1970).
119
Table 54
Energy Parameter Values and Length Function
Hydrocarbon
e /k (K)
ws
A E /k (K)
ws
Methane
232.5
166
Ethane
398.5
172
Propane
570.6
179
Butane
750.0
175
Pentane
925.1
173
Hexane
1098.4
183
Heptane
1281.4
175
Octane
1456.0
181
Nonane
1645.9
178
Decane
1823.9
196
Undecane
2020.1
180
Dodecane
2199.8
201
Tridecane
2401.1
181
Tetradecane
2582.3
L = 0.52778918 + 3.194678D+02/T 2.8715619W04/T2
with T in K.
202
Table Bla
Parameters for Temperature Dependence of
Interaction Correlation Coefficients
a
s
= 3.40 A
Coefficient
A
n
B
n
D
n
V
0.341711DQ1
0.119371D01
0.209290D04
C2
0.218037D+01
0.118486D01
0.186620D04
C3
0.104553EH01
0.556065D02
0.903900D05
V
0.141446D+00
0.814235D04
0.131700D05
C5
0.148324D+01
0.297438D02
0.623100D05
C6
0.532135D+Q0
0.485514D02
0.815600D05
C7
0.100441D+02
0.755970D01
0.124682D03
C8
0.543575D+02
0.493098D+00
0.813850D03
C9
0.103527D+03
0.4 53733D+00
0.688320D03
C10
0.471167D+01
0.00 D+00
0.00 D+00
C11
0.196545D01
0.00 D+00
0.00 D+00
C12
0.387170D04
0.00 D+00
0.00 D+00
C13
0.135525D+01
0.266010D02
0.572700D05
C14
0.480870D+00
0.174845D03
0.522000D06
C15
0.385621IHO0
0.115702D03
0.190000D06
FORTRAN
0 123
O 124
0125
O 126
0127
0128
0129
0 130
0 131
0132
0133
0134
0135
0 136
0137
0138
0 139
014 O
014 1
0142
0 143
0 144
IV G LZVcL 21 CALFUN DATE = 79109 22/43/30 R
GC2=XTC{ 85) +XTC {86)*T{'j )+XTC(87 )*T< J )**2
GC3=XTC [ 33)+XTC( 39 ) *~( J MXTC ( 90 > *T( j ) **2
GC4 = XTC(91 )+ XTC(92)*T{J)fXTC{93 ) *TCJ )**2
GC 5= XT C(94)+XTC(95)*T(J) + XTC 96)*T(J )**2
GC6=XTC(97>+XTC{98)*T(j )+XTC(99)*T(J )**2
GEE=DEXP( G C1 + GC2* DLQG (UL( J) )+GC3*DLCG (UL ( J ) ) **2+ GC4 DLOG(UL (J))**
23+GC5*DL0G(UL ( J ) ) **4+GC 6 *D L3G ( UL ( J ) ) ** 5)
GDE=GEE+0.50*(XTC(79)+XTC{80)*T(J)+XTC{3l)*T(J)**2)
SCi=XTCC33)+2.0*XTC(84)*T(J)
SC2=XTC (86 >+2.0 *XTC( 87) *T { J )
SC3 = XTC{89)+2 0*XTC(90)*T(J)
SC4=XTC(92J+2.0*XTC(93)*T(J)
SC5 = XTC (95 )+2.0 *XTC (96 ) *T ( J )
SC6=XTC(98)+2.O*XTC(99)*T(J)
S DE=GEE*(SCI+ S C2*0L0 G(UL(J) )+ GC2*DUL/UL(J)+ SC3*DLOG(UL(J))**2 + 2.*
2 GC3 DLOG(UL(J) )*DUL/UL(J)+SC4*DLQG(UL(J) )**3 + 3.O*GC4*DL0G(UL{J >)**
32*DUL/UL(J)+SC5*DLCG(UL(J)>**4+4.0*GC5*DL0G(UL(J))+*3*QUL/UL(J)+SC
4 6 *DLOGCUL(J ))**5 + 5.O*GC6*DL0G(UL { J ) )**4*DUL/UL(J) )+0.5 0*(XTC(80)
52 O* XTC ( 8.1 ) *T( J ) )
G I (J I)=(DF*EPSI(I)*(GDOLLI+GD E)+CF*EPSI( I )* (GCOLL I + GCGLLI ) )/(T(J)
2)
SI(J .1 )=DF*EPSI(I)*(SDOLLI + SDE)+CF+EPSI(I )*{SCOLLI+SCGLLI )0DF*EP5
1 I ( I )*{ GDOLL I + GDE)DCF*EPS I ( I )*( GCOLL I + GCGLLI >
GO TO 7
C SMALL UL VALUES
C CONTINUOUS DISTRIBUTION
C 0
5 GC1 = XTC(31)+X7C(32)*T(J)+XTC(33)* T{ J)* *2
GC2=X'r C (34 ) +XT C (35 ) *T(J )+XTC(36 ) *+( J )**2
GC3=XTC(37)+XT C(38)*T(J) + XTC(39)*T(J)**2
GC4= XT C(40)+XTC{41)*T{J)+XTC( 42)*T(J)**2
GCOL SI=(GC1*UL(J)+GC2*UL(J)**2+GC3*UL(J )**3+GC4*UL(J )**4 )
218
3
Attempts at development of a model for the thermodynamic process
of micellization are discussed in Chapter 6. Some contributions such as
electrostatic repulsions between head groups in the micelle surface and
watermicelle interactions are omitted from the model and other contribu
tions are likely to be inadequate. Comparison of the contributions of
all effects is discussed and suggestions for improvement are given.
Chapter 7 presents the results of a limited experimental investi
gation not directly related to the theoretical study. The objective of
the experiment was to determine the effect of temperature, pressure and
dissolved gas on the isotropicanisotropic phase transition in lyotropic
liquid crystals. A study of a single system containing potassium oleate
showed a twophase region between about 40 80C and no reproducible
effect of pressure or dissolved gas on the phase behavior.
Chapter 8 provides a summary of conclusions and suggestions
for future research as discussed in detail in the earlier chapters.
169
3. Extension of this model to aliphatic hydrocarbons involved
development of an expression for the total interaction energy between
a spherocylindrical solute and a spherical solvent. In the perturbation
theqry used, .the radial distribution function was considered as a function
of distance from the spherocylinder surface while the intermolecular
potential was distributed along the spherocylinder axis.
4. Correlation of solution properties for the gaseous aliphatic
hydrocarbons (C^C^) is quite good and predicted trends for the liquid
hydrocarbons,for which little data are available, are reasonable. The
results are quite sensitive to the chain segment length L which has
CH2
a stronger temperature dependence than the hard sphere diameter of the
spherical solutes. This leads to a large interaction entropy contribution
which actually dominates the cavity contribution at higher temperatures.
5. A partial model for the thermodynamic properties of ionic
micellization was developed based on the aqueous solubility model.
Contributions to the thermodynamic process which were not modeled as well
as present inadequacies in modeled effects resulted in poor quantitative
agreement with experiment. The desired result is a small difference of
large contributions and thus good agreement may not be obtainable. The
large entropy increase due to loss of monomerwater interactions upon
micelle formation suggests that the excluded volume effect is not the only
i .
I
significant entropy driving force for micellization.
6. A limited experimental investigation, on the potassium
oleate system, into the effect of temperature, pressure and dissolved
gas on the isotropicanisotropic transition for lyotropic liquid crystals
I
j
was conducted. Tentative results show a twophase region between 40 and
38
Stillinger (1973) shows compelling evidence that the agreement
is not necessarily because of the correctness of the physical assump
tions. He lists values of Ym and 5 for water using equations (324)
and (325) at several points along the saturation curve for water. He
also includes measured liquidvapor interfacial tension for comparison.
Table 31 shows they do not agree. Of particular significance is the
improper temperature dependence of the interfacial tension which may
contribute to error in the predicted entropy of solution. Stillinger
argues that the sign of 5 is incorrect, although an analysis of the work
of Koenig (1950) shows that 6 is positive so long as the radius of curva
ture of the cavity r has a positive sign convention.
Stillinger developed a revised G(r) for water using the exper
imental liquidvapor interfacial tension, YÂ£v> and the radial distribu
tion function, g(r), as input data.
The most accurate determination to date of the oxygenoxygen
(2)
pair correlation function g (r) in liquid water can be determined
from Fourier transformation of the structure function data as determined
from Xray diffraction by Narten and Levy (1969, 1971). Details of this
method will be discussed later in this chapter. Their results show that
O
essentially no pairs of oxygen nuclei occur closer than 2.40 A. There
O
fore, equation (316) will be correct for G(r) in the range 0
For larger r, at least the pair term in P(r), equation (315), should
contribute, and so the same would be true in G(r).
In ice, strong directional forces between neighbor molecules
produce characteristic isosceles triangles of oxygen nuclei. The apex
angle is the tetrahedral angle 0^ = 109; since the hydrogen bonds in
135
The first term in equation (61) is the ideal gas contribution.. Table 61
gives expressions for a^, b and for particles of several common shapes.
An equation of state for an mcomponent mixture of arbitrary
shaped hard particles can be obtained from equation (61) using
3P "
8\ i1 Pi 3k '
(62)
For a mixture of hard spheres equations (61) and (62) yield the Percus
Yevick compressibility equation of state for mixtures (Lebowitz et al.,
1965).
Gibbons (1970) pointed out that a thermodynamic inconsistency
exists in equation (61). In order to obey the GibbsDuhem equation the
following must apply
3y.
3n.
J
TV'ni
3n.
(63)
Equation (61) will not obey this relation except in the limit
of a mixture of particles of the same shape. Boublik (1975) later gave
an alternative to equation (61) which is thermodynamically consistent.
It will be developed here in a different manner. Also presented will be
expressions for the PercusYevick pressure equation of state and the
CarnahanStarling equation of state for mixtures of arbitrary shapes.
A. PercusYevick Compressibility Equation of State for
Mlxtures of ArbitraryShaped Blg.i'd Bod 1 itn
The compressibility equation is
P
(1Y)
1 r.2^,
3 B C
(1Y)'
+
AB
(1Y)2
(64)
Table 55a (Continued)
Solute
T(K)
AG
c
RT
AG?
i
RT
AG
r, v
RT
AGcal
RT
AS
c
R
AS?
i
R
AS
R
As
cal
R
C8H18
277.15
75 .54
71.47
6.56
10.63
20.67
7. 49
6. 56
34.72
298.15
71 .31
65.75
6. 52
12.07
10.42
12.02
5.32
27.76
323.15
66.19
59.53
6.38
13.03
0.16
17.29
4.14
21.59
358.15
59.15
51.63
6.09
13.56
11.31
24.27
2. 71
15. 66
C9H20
277.15
83 .04
80.21
10. 45
13. 29
21.18
8.57
7.65
37.40
298.15
78.28
73.78
10.21
14.71
9.3 7
13.66
6.20
29.73
323.15
72.53
66.78
9.93
15.63
1. 43
19. 56
4. 83
22.95
353.15
64.66
57. 96
9.57
16.27
14.01
27.36
3.16
16.51
C10H22
277.15
90 .55
38.35
8.55
10.74
21.69
9. 60
8. 55
39. 83
293.15
85.25
81.26
8.49
12.48
9. 3 1
1 5. 20
6. 93
31.44
323.15
7 8.83
73. 54
8. 31
13.65
3.03
21.69
5.39
24.05
35e.l5
70.18
63.81
7.93
14.29
16.70
30.24
3. 53
17.07
A MOLECULAR THERMODYNAMIC MODEL FOR AQUEOUS SOLUTIONS
OF NONPOLAR COMPOUNDS AND MICELLE FORMATION
By
ROBERT J. BRUGMAN
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1979
ACKNOWLEDGMENTS
I wish to express my deepest appreciation to Dr. John OConnell
for his interest and enthusiastic guidance throughout my graduate studies.
Working with him has been a truly valuable and enjoyable experience.
I also wish to thank Drs. J. C. Biery, G. Y. Onoda and F. A.
Vilallonga and. Prof. R. D. Walker for serving on my supervisory committee
It is a pleasure to thank the faculty and students of the Depart
ment of Chemical Engineering for providing an enjoyable, pressurefree
environment in which I could pursue my research.
I am extremely grateful to Mrs. Thomas Larrick for her excellent
typing despite the author's poor handwriting and the highly technical
nature of the task.
Finally, I am grateful to the Department of Energy and the members
of the Enhanced Oil Recovery Consortium at the University of Florida
who provided financial support for this work.
ii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
LIST OF TABLES vi
LIST OF FIGURES ix
KEY TO SYMBOLS x
ABSTRACT .....' xiv
CHAPTER
1 INTRODUCTION 1
2 THERMODYNAMICS OF MICELLIZATION 4
Introduction 4
Thermodynamic Formalism of Micellization 4
Effect of Added Salt on Micellization 12
Some Theories for Free Energy Changes Upon Micellization 15
Contributions to Thermodynamic Properties of Micellization
from Various Species 20
A Thermodynamic Process for Micelle Formation 25
3 SCALED PARTICLE THEORY MODIFIED FOR AQUEOUS SOLUTIONS . 29
Introduction 29
Basis of Scaled Particle Theory 29
Scaled Particle Theory and Aqueous Solutions 37
Curvature Dependence of Surface Tension ......... 43
Determination of the Radial Distribution Function and
the Direct Correlation Function for Liquid Water
from XRay Diffraction 45
4 MODELING OF SPHERICAL GAS SOLUBILITY . . 62
Introduction 62
Thermodynamic Properties of Solution from
Experimental Data 63
Application of ScaledParticle Theory to Aqueous
Solubility 66
Contributions to the Thermodynamic Properties
of Solution from Cavity Formation 72
iii
TABLE OF CONTENTS (Continued)
CHAPTER Page
4 (Continued)
Contributions to the Thermodynamic Properties
of Solution from Intermolecular Forces 78
Analysis of Spreading Pressure of a Solute Occupying
a Cavity 81
Discussion of Results and Suggestions for Future
Research 84
5 AQUEOUS SOLUBILITY OF ALIPHATIC HYDROCARBONS ...... 92
Introduction 92
Calculation of Thermodynamic Properties of Cavity
Formation forAliphatic Hydrocarbons 93
Free Energy of Interaction Between a Spherocylindrical
Solute and Spherical Solvent 94
Changes in Rotational and Vibrational Degrees of Freedom
of Aliphatic Hydrocarbons Upon Solution 104
Results of the Model for Aqueous Solubility of
Aliphatic Hydrocarbons 110
Comparison with Infinite Dilution Properties
of Surfactants 129
Suggestions for Future Work 129
6 MODELING OF THE THERMODYNAMIC PROPERTIES OF MICELLIZATION 132
Introduction and Review of Thermodynamic Process
for Micelle Formation 132
Derivation and Application of an ArbitraryShape
Hard Body Equation of State 134
Contributions to a Model for the Thermodynamics
of Micellization 139
Discussion and Suggestions for Future Research 149
7 EXPERIMENTAL INVESTIGATION OF PHASE BEHAVIOR AND
TRANSITIONS FOR CONCENTRATED SURFACTANT SOLUTIONS .... 156
Introduction 156
Experimental Objectives ..... 156
Description of Experimental Apparatus 156
Operating Procedures 160
Pertinent Calculations for Dissolved Gas Experiments . 162
Results and Suggestions for Future Work 166
8 SUMMARY AND CONCLUSIONS 168
iv
TABLE OF CONTENTS (Continued)
APPENDIX Page
A PROGRAM FOR CORRELATION OF SPHERICAL GAS
SOLUBILITY PROPERTIES 172
B HELMHOLTZ FREE ENERGY OF INTERACTION BETWEEN
A SPHEROCYLINDRICAL SOLUTE AND SPHERICAL SOLVENT .... 184
Integrations of the Components of the Helmholtz
Free Energy of Interaction 194
Correlation of the Helmholtz Free Energy of
Interaction with O ,L and Temperature 201
s
C PROGRAMS FOR GAS AND LIQUID HYDROCARBON
SOLUBILITY PROPERTIES 209
D PROGRAM FOR CALCULATION OF THERMODYNAMIC
PROPERTIES OF MICELLIZATION 258
BIBLIOGRAPHY 272
BIOGRAPHICAL SKETCH ......... . 280
v
LIST OF TABLES
Table Page
31 Surface Tension and Curvature Parameter Calculated
for Liquid Water at Its Saturated Vapor Pressure
Using the Pierotti Approximation 39
32 Radial Distribution Function for Liquid Water 47
33 Direct Correlation Function for Liquid Water ...... 50
34 Direct Correlation Function for Liquid Water 53
35 Reduced Direct Correlation Function for Liquid Water . 58
4la Solution Properties from Pierottis Model 70
4lb Enthalpy and Heat Capacity Contributions 71
42 Universal Correlation for the Reduced Hard
Sphere Diameter 76
43 Characteristic Parameters 77
44 Intermolecular Potential Energy Parameter 80
45a Contributions to Free Energy and Entropy of Solution . 85
45b Contributions to Enthalpy and Heat Capacity of Solution 88
51 c Parameter Values for Aliphatic Hydrocarbons 107
52 Properties Required to Analyze Liquid Hydrocarbon
Solubility ..... .......... 109
53a Contributions to Free Energy of Solution of
Gaseous Hydrocarbons Ill
53b Contributions to Enthalpy of Solution of
Gaseous Hydrocarbons 113
53c Contributions to Entropy of Solution of
Gaseous Hydrocarbons .... 115
53d Contributions to Heat Capacity of Solution
of Gaseous Hydrocarbons 117
vi
LIST OF TABLES (Continued)
Table Page
54 Energy Parameter Values and Length Function 119
55a Contributions to Free Energy and Entropy of
Solution of Liquid Hydrocarbons . . 121
55b Contributions to Enthalpy and Heat Capacity of
Solution of Liquid Hydrocarbons 125
56 Infinite Dilution Heat Capacity of Surfactants
in Water at 298.15K 130
61 Comparison of Properties of Hard Spheres with Those
of Some NonSpherical Particles 137
62a Contributions to Gibbs Free Energy of Micellization . 141
62b Contributions to Enthalpy of Micellization . 142
62c Contributions to Entropy of Micellization 143
63 Parameter Values for Micellization Model . 145
71 Temperature Dependence of Two Phase Region . 164
72 Pressure Dependence of Two Phase Region 165
73 Effect of Dissolved Methane 165
Ala Contributions to Free Energy and Entropy of Solution . 187
Alb Contributions to Enthalpy and Heat Capacity
of Solution 190
Bla Parameters for Temperature Dependence of 0
Interaction Correlation Coefficients (a^ = 3.40 A) . 202
Blb Parameters for Temperature Dependence of 0
Interaction Correlation Coefficients (a = 3.60 A) . 204
s
Blc Parameters for Temperature Dependence of 0
Interaction Correlation Coefficients (a = 3.80 A) . 206
Cla Contributions to Free Energy of Solution of
Gaseous Hydrocarbons 226
Clb Contributions to Enthalpy of Solution of
Gaseous Hydrocarbons .... 228
vii
LIST OF TABLES (Continued)
Table Page
Clc Contributions to Entropy of Solution of
Gaseous Hydrocarbons 230
Cld Contributions to Heat Capacity of Solution of
Gaseous Hydrocarbons 232
C2a Contributions to Free Energy and Entropy of
Solution of Liquid Hydrocarbons 248
C2b Contributions to Enthalpy and Heat Capacity of
Solution of Liquid Hydrocarbons 252
C3 Energy Parameter Values and Length Function 256
LIST OF FIGURES
Figure Page
21 Contributions of Species to Property Changes
of Micellization 22
22 A Thermodynamic Process for Micelle Formation 26
31 Contact Correlation Function; Comparison of
Different Models . 42
32 Reduced Direct Correlation Functions .... 57
5la Fixed Potential at y = 0 Interacting with Molecular
Centers in 0 < y < L and 0 < z < 00 95
5lb Fixed Potential at y = 0 Interacting with Molecular
Centers in y < 0 and y > L 95
5lc Distributed Potential Along Spherocylinder Axis
from y = 0 to y = L Interactint with Molecular
Centers in 0 < y < L and 0 < z < 00 . 96
5ld Distributed Potential Along Spherocylinder Axis
from y = 0 to y = L Interacting with Molecular
Centers in y < 0 and y > L 96
61 A Thermodynamic Process for Micelle Formation ...... 133
71 High Pressure Experimental Apparatus 158
ix
KEY TO SYMBOLS
A
A
a
a
c(r)
Cdis
Cp
d
f
g(r)
G
G(r)
H
H (s)
m
J
K
L
lch2
P
P
o
Q
= Helmholtz free energy
= Helmholtz free energy
= activity
= cavity surface area
= direct correlation function
= dispersion coefficient in intermolecular potential
= heat capacity
= hard sphere diameter
= fugacity
= radial distribution function
= Gibbs free energy
= contact correlation function
= enthalpy
= scattering structure function
= arithmetic mean curvature
= Henry's constant
= spherocylinder length
= segmental length
= average micelle aggregation number
= pressure
= probability of an empty cavity
= canonical partition function
x
gas constant
separation between molecules
entropy
temperature
characteristic temperature
interaction energy
volume
characteristic volume
work of cavity formation
mole fraction
reduced solvent density
Greek Letters
a = fraction of counterions bound to micelle
= solvent coefficient of thermal expansion
(*2 = solute polarizability
3 = 1/KT
y = surface tension
CO
Y = planar surface tension
T. = relative adsorption of i to water
i,w
S = curvature dependence parameter for surface tension
A = denotes a property change
Â£ = interaction energy parameter
y = chemical potential
p = number density
a = potential distance parameter, hard sphere diameter
xi
Subscripts
O = overall mole fraction of surfactant
1 monomer or solvent property
2 = solute property
c = cavity property
ca = counterion adsorption
ci = counterion cavity property
cal = total calculated property
exp = experimental proprty
HS = hard sphere property
hs = hard sphere property
i = interaction property
m = micelle property
mic = micelle property
mmi = monomermonomer interaction property
0 = absence of added salt
ref = refers to reference solute
s = solvent property
w = water
ws = watersolute property
Superscripts
o = standard state
+ = property value at CMC
AQ = aqueous solution
diss = dissolved state
xii
g = gas phase
el = electrostatic quantity
hs = hard sphere quantity
L = liquid phase
ref = refers to reference solute
V = vapor phase
Overline
= denotes partial molar, average, or reduced property
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor Philosophy
A MOLECULAR THERMODYNAMIC MODEL FOR AQUEOUS SOLUTIONS
OF NONPOLAR COMPOUNDS AND MICELLE FORMATION
By
Robert J. Brugman
June 1979
Chairman: John P. O'Connell
Major Department: Chemical Engineering
A thermodynamic process for micellization has been developed
which provides a basis for better understanding of molecular mechanisms
important in the formation of micelles as well as other processes of
aqueous solution. The foundation of a model of the micelle formation
process is a unified molecular thermodynamic theory of aqueous solutions.
For spherical gases, a modification of scaled particle theory
has been used for the excluded volume contribution while a mean field
theory approach has been used for intermolecular interactions. Very
good correlation of experimental thermodynamic properties was achieved
over a considerable temperature range (277K 358K).
Extension of the model to aliphatic hydrocarbon solutes required
development of an expression for the total interaction energy between
a spherocylindrical solute and a spherical solvent. The radial distri
bution function was considered to be a function of distance from the
spherocylinder surface and the intermolecular potential was distributed
along the spherocylinder axis. The thermodynamic properties of solution
for gaseous hydrocarbons (C^ C^) are well correlated and predicted
xiv
trends in solution properties for the liquid hydrocarbons were in
reasonable accord with the few experimental data available. The entropy
contribution from intermolecular interactions was unexpectedly large,
actually dominating the cavity (excluded volume) contribution at higher
temperatures.
A primitive model for the thermodynamic properties of ionic
micellization was tested using the aqueous solubility model and electro
static theories. Quantitative agreement with experiment was not found
due to contributions to the thermodynamic process which were omitted
as well as model inadequacies. Suggestions for improvement have been
made.
A limited experimental investigation was conducted into the
effect of temperature, pressure and dissolved gas on the isotropic
anisotropic transition for lyotropic liquid crystals. Tentative results
indicate a two phase region between 40 and 80C and little or no effect
of pressure or dissolved gas on the isotropicanisotropic transition,
except for transients.
xv
CHAPTER 1
INTRODUCTION
The objective of this investigation is development of a unified
molecular thermodynamic theory of aqueous solutions with particular inter
est in the aggregation of surfactant monomers to form micelles. This
development is based upon building stepwise from a treatment of aqueous,
solubility of spherical nonpolar gases to consideration of the numerous
molecular effects important for a rigorous model of the thermodynamics of
micellization.
Chapter 2 progresses from a discussion of the thermodynamic
formalism of micellization through a review of previous models to develop
ment of a thermodynamic process for micellization. Previous models
(Tanford, 1974b) considered micelle formation as a result of a balance
between favorable (AG^ < 0) "hydrophobic forces" arising from minimization
of waterhydrocarbon contact area and unfavorable (AG > 0) electrostatic
m
forces between surfactant head groups in the micelle. While these models
provide reasonable correlating expressions, they present an overly sim
plistic picture of the important molecular effects involved. The present
process analyzes the situation in considerably more detail, accounting
for several other significant effects such as changes in intermolecular
interactions upon aggregation, entropy and enthalpy changes upon restric
tion of surfactant head groups to the micelle surface and most importantly
1
2
excluded volume effects associated with creating and destroying the appro
priate sized cavities in water.
Interest in this excluded volume effect leads to a discussion
in Chapter 3 of the application of scaled particle theory to aqueous
solutions of nonpolar gases as originally done by Pierotti (1965), with
modifications suggested by Stillinger (1973) to account for the special
effect of the hydrogen bond structure of water. Fundamental difficulties
arise when considering the appropriate matching of the microscopic expres
sion of Stillinger for the work of cavity formation with the macroscopic
expression resulting from the relation of Koenig (1950) for the curvature
dependence of the surface tension. Rather than address this controversy
directly, we have chosen to use argon as a reference solute and use only
the macroscopic expressions to obtain properties for the other solutes
from those of argon.
Application of the model to aliphatic hydrocarbons is presented
in Chapter 5. To model the linear hydrocarbons as spherocylinders, an
expression is derived from perturbation theory for the total interaction
energy between a spherocylindrical solute and a spherical solvent. This
derivation considers the radial distribution function to be a function
of distance from the spherocylinder surface rather than a centertocenter
distance (spherical case) and distributed the intermolecular potential
along the spherocylinder axis.
The correlation for the gaseous hydrocarbons is quite good and
predicted trends in liquid hydrocarbon solubility properties seem reason
able. Sensitivity of the results to chain segment length is examined.
3
Attempts at development of a model for the thermodynamic process
of micellization are discussed in Chapter 6. Some contributions such as
electrostatic repulsions between head groups in the micelle surface and
watermicelle interactions are omitted from the model and other contribu
tions are likely to be inadequate. Comparison of the contributions of
all effects is discussed and suggestions for improvement are given.
Chapter 7 presents the results of a limited experimental investi
gation not directly related to the theoretical study. The objective of
the experiment was to determine the effect of temperature, pressure and
dissolved gas on the isotropicanisotropic phase transition in lyotropic
liquid crystals. A study of a single system containing potassium oleate
showed a twophase region between about 40 80C and no reproducible
effect of pressure or dissolved gas on the phase behavior.
Chapter 8 provides a summary of conclusions and suggestions
for future research as discussed in detail in the earlier chapters.
CHAPTER 2
THERMODYNAMICS OF MICELLIZATION
Introduction
The purpose of this chapter is to discuss the fundamental
thermodynamics with which a theory of micellization must be consistent.
In the first section the basic formalism is developed, along with dis
cussion of the effects of the distribution of micelle aggregation number.
Consideration is then given to the effect of added electrolyte.
Previous theories for the free energy change upon micellization are
critically reviewed, leading to a reconsideration of contributions to
thermodynamic properties of micellization from the various species
involved and development of a novel thermodynamic process for micellization
Thermodynamic Formalism of Micellization
For nonionic amphiphiles, Hall and Pethica (1970) show a relation
derived from Hill's small systems thermodynamics (1963) for the Gibbs'
free energy of formation, AG, of a system of micelles of average aggre
m
gation number, N, in a standard state of infinite dilution, from monomeric
amphiphiles, also in a standard state of infinite dilution, in terms of
the mole fraction of the monomers, x,. and of the micelles, x when
1 m
ideal solution is assumed
4
5
AG \i
m m
RT RT
P1
Â£n x
RT L
1
&n x
m
(21)
At the critical micelle concentration (CMC), relatively abrupt
changes in properties are observed because the concentration of monomeric
species begins to change very little with the mole fraction of added
amphiphile, xq. (micelles are being replicated)
x = x,. + N x .
o 1 m
(22)
Around the CMC the value of 9x.,/9x _ falls rapidly from near
1 o ' T,r
unity to near zero. The CMC definition of Phillips (1955), explored by
Hall (1972), is
lim
\ +
x * x
o o
T,P
(23)
where xq is the CMC and c is an "ideal colligative property" which depends
only on the number of solute species (monomers and micelles). As noted
by Chung and Heilweil (1970), this definition is difficult to interpret
unambiguously in terms of the experiments done with micellar solutions.
As an alternative, the expression of Hall and Pethica (1970) can
be used
lim
9(x +x )
1 m
, 9 x
^ + o
X > X
o o
= 0.5
(24)
T,P
which is essentially equivalent to equation (23) for sharp CMC points.
This expression can be used to obtain (Hall and Pethica (1970))
+ 2 +
x = (N 2N)x .
1 m
(25)
6
From the definition of x in equation (22)
o
+ + 2
x = x /(N N)
m o
+ +2 2
x = x (N 2N)/(N
1 o
N)
(26)
(27)
which then yields
AC + ,
m 0 + I
32 X'll x ~
NRT N
 Â£n x+ + Â£n + Â£n(l N/N^)
o
+ Â£n Q(12N/N2)/(1N/N2) 3
To assume that polydispersity is unimportant the righthand
(28)
side of equation (28) should be small enough to neglect, which means its
value is approximately 0.05 for 5% error in x+ (typical experimental
uncertainty). By taking
1 1/N N/N
(29)
equation (28) becomes, in first order approximation
AG Â£n x+ + 2Â£n 1
m + .. o
xn x
NRT N
(210)
The righthand side of equation (210) is always positive so
Â£n x+ represents a lower algebraic bound to AG/NRT. For the effect to
o m
be less than 5%, N should be greater than about 300 to 600. However, it
ranges upward from 10, so polydispersity is normally important and thermo
dynamic theories for AG/NRT should give an equation which can yield
m
2 _
values of N and N. Further, data analysis to obtain values of Ag from
m
CMC values must allow for these terms.
The division of the free energy change of micellization into
enthalpy and entropy components is accomplished by the relations
7
All1
m
, 9AG/RT
1 m
NR
N
31/T
AG
m 9Â£nN
P,n NRT 81/T
P.n
(211)
AH 9Â£n x+
m o
9Â£nT
1 J9Â£nN
NRT
AS AH AG
m m m
NR
NRT
_ \9Â£nT
P,n N v.
i 9Â£n x
+
Â£n x + 3 2Â£nN
o
9Â£nT
P,n.
(212)
(213)
AS
9Â£n x
+
NR
; Â£n x 
o
3Â£n T
P.n
N
"9[T Â£n x+]
o
9T
P.n
 2
9[T Â£n N]
9T
+ 1 Â£n
+ 9Â£nN
P.n
o 9Â£nT
P,n.
(214)
The last term in equation (212) is small when the righthand side
of equation (210) is, so the standard state enthalpy change is normally
close to the temperature derivative of the CMC. As determined by Desnoyers
et al. (1978) from direct calorimetry, at lower temperatures (T < 25C)
the micellization process is entropy driven (TAS<0, AH>0), whereas at
m m
higher temperatures it is enthalpy driven (TAS >0, AH<0). This con
m m
siderable variation of AH with temperature is reflected in a large nega
m
tive heat capacity change (ACp) upon micellization. These trends agree
m
qualitatively with those determined using equations (212) and (214) by
Moroi et al. (1975) for sodium alkyl sulfates. Thus theories which
attempt to correlate the data must exhibit considerable flexibility in
their temperature dependence.
The above analysis was developed by Hall and Pethica (1970) for
nonionic species. The small systems analysis has not been applied to
ionic systems where the effect of counterions on the thermodynamics must
be included. This may be due to the extreme complexity of such an effort.
8
At present, the thermodynamics of micellization for ionic systems
is divided into two formalisms based on the mass action approach. These
are reviewed by Mijnlieff (1970). In the first, such workers as Stigter
(1964, 1974ab, 1975ab), Emerson and Holtzer (1965, 1967ab), and Mukerjee
(1969) focus on the changes associated with the amphiphilic ions forming
an aggregate. Thus, for a singly charged anionic monomer, the reaction is
N
N M + M
1 m
and the thermodynamic formulation is
hi = + RT Jin a^
y = p + ]iel + RT Â£n a
m m m m
(215)
(216)
(217)
y+ Ny* = 0 = (y Ny) + (yel Ny1) + RT (n a+ N Jin at) (218)
ml m lml m 1
where y is the standard state (infinite dilution) chemical potential of
the uncharged monomer, y is the standard state chemical potential of the
m
el 0I
uncharged micellar aggregate, and (y N y ) is the chemical potential
m 1
difference associated with changing the charge on the micelle and the
monomers from zero to full value while it is in the presence of the ionic
atmosphere of the counterions. Such a change involves the response of the
counterions and is sensitive to the detailed molecular structure assumed,
as the calculations of Stigter (1975ab) show. It is particularly sensi
tive to the fraction of ions assumed bound to the micelle in the Stern
layer as related to electrophoretic and electrical conductance measure
ments (Stigter, 1964). This fraction is apparently of the order of one
half to threefourths when the micelle is fully charged but how this value
depends upon the charging process is unclear.
9
el
The value of y^ is determined from some expression such as that
of DebyeHuckel theory leading to
el
y1 = + RT Â¡L n y1
(219)
el
The relationship of the counterions to y^ is one of equilibrium between
those in bulk solution and those in the Stern (bound) layer and the Gouy
Chapman (diffuse) layer (Stigter, 1964)
y (solution) = y(s) + yf^ + RT in a (s)
c c 1 c
el
y (solution) = y (micelle) = y(m) 1^ ac^m^
c c c I
or
el
(220)
(221)
(222)
p(m) i(s) + A ET
CCS
el el
Substituting for y ^ Ny^ in equation (18) and combining the standard
state chemical potentials yield
Ja+ a+(m)
y N(y + y) + RT Un > = 0
m + +,
(a a (s) ]
1 c
(223)
Assuming that we can replace a^ and ac(ra) by unity (micelles) and a^
and a*(s) by mole fraction (solution), using the definition of Ag from
equation (21) where all the species are uncharged gives
AG
NRT
= Â£n x, x
1 c
(224)
which for no added salt (x+ = x*) is
c 1
AGm +
Z = 2 Â£n x
NRT 1
(225)
10
This relation also appears in the work of Shinoda and Hutchinson (1962).
It is important to note that all standard state chemical potentials and
activities given above are for neutral species.
The second approach to the thermodynamic relationships for ionic
amphiphiles (Sexsmith and White, 1959ab; White, 1970; Phillips, 1955;
Molyneaux and Rhodes, 1972; Kaneshima et al., 1974) writes the reaction as
M~ + NaC+ + M_N(1_a) (226)
I m
where a is the apparent fraction of amphiphiles whose charge is neutral
ized by bound counterions. The chemical potential relation is then
+ + +_ o~o o
y Ny, Nay = 0 = y Ny, Nay
m 1 c m 1 c
+ RT[Â£n a+ N Â£n at(a+)a] (227)
m 1 c
where the standard state is the charged species at unit activity. Again
f* *1*
assuming that we can replace a^ by unity and ap and a^ by mole fractions,
using the definition of AG from equation (2^1) gives
AG ,
= in x (x )a (228)
RT 1 c
In these relations, the chemical potentials are for ionic species, a
concept which is tenuous since in the definition
Hi ^/3n.)T5P)n
w
(229)
charge neutrality prevents holding all n^ constant while np is varied if
species i is charged. For the case of no added salt equation (228) yields
AG
= (1+a) Â£n x,
NRT
(230)
11
In order for equation (230) to yield equation (225), the value of a
must be unity.
This conclusion has two important consequences for theoretical
analysis. Since a has not been assumed to be unity in the semiempirical
expression and data analysis of Phillips (1955), Molyneaux and Rhodes
(1972) and others, it is not clear what interpretation should be placed
on their results.
The second consequence involves the work of Sexsmith and White
(1959ab) which, when assuming a < 1, gives a maximum in the monomeric
amphiphile concentration. Using equations (22) and (227) with mole
fractions for activities at all concentrations plus
x = x a(x x,) (231)
cool
yields the relation
x, 
x x,
o 1
. 
1/N
K[x a(x x,)1
o o 1
a
(232)
where
K = exp [AG/NRT].
m
(233)
At small values of x x, = x but at larger values of x >> x, two
o 1 o o 1
limiting cases appear
x, l/xa K(la)a a < 1
1 o
, /.1/2N.1/2
x^ (xq/N) /K a = 1.
(234)
(235)
Equation (234) is chosen by Sexsmith and White (1959b) which indicates
a rapidly decreasing monomeric concentration with total amphiphile while
equation (235) gives a slowly increasing monomer concentration. Experi
mental evidence is mixed (Hall and Pethica, 1970).
12
A possible resolution of this conflict between the two approaches
to the thermodynamics of ionic amphiphiles lies in the new specific ion
surfactant electrode technique (Evans, 1978). When combined with ion
specific electrodes for the counterion, measurements on long chain sur
factants permit the fractional charge on the micelle to be determined.
Present estimates yield values of a = 0.8, whereas inferred results gave
values closer to 0.5.
Effect of Added Salt on Micellization
An extremely important aspect of the thermodynamics of micellization
is concerned with the effect of added salt on the CMC of ionic amphiphiles.
Examination of the better data reported by Mukerjee and Mysels (1971) for
systems such as alkyl ammonium chlorides and bromines, sodium alkyl sulfates
and alkyl sulfonates with added salts such as the sodium halides, potassium
nitrate and potassium bromide up to 1M, confirm the relation first given by
Corrin and Harkins (1947), Hobbs (1951) and by Shinoda (1953) and described
in detail by Mijnlieff (1970) and Lin and Somasundaran (1971). The data can
be reproduced to within the estimated experimental error with
&n X^/x+ = Kr Ln
~(x+ + x2)'
+o
(236)
fo "f*
where x^ is the CMC without added salt, x^ is the value with added salt
of mole fraction x^ and Kr is a constant independent of the salt whose
value is 0.66 .03 for anionic amphiphiles and 0.58 .03 for
cationics. The data are insufficient and probably of inadequate accuracy
to properly test this correlation for multiply charged salts. Mijnlieff
13
writes the reaction for the neutral species (M^ = Amphiphilic Salt,
= Added Salt, = Micelle)
Mx + Q S2 J Mmq (237)
and the mass action relation for amphiphile (1) and salt (2) as
V'l + Q u2 vm
(238)
Now for an ideal solution where the added salt has a common ion with the
amphiphilic salt
^MQ ^MQ
JJ = y + RT An xx
11 1 c
y = y + RT An x x
2 2 2 c
(239)
(240)
(241)
where xc = x^ + x^ is the mole fraction of counterion in the system from
both a 11 amphiphilic salt and a 11 added salt. The relations for other
salts would be similar in form but more complex in detail. The equilib
rium relation is then
AGm 4 N ^ Q **2
RT RT
In the limit x^ = 0, Q = 0
= Â£n^xfX + K2)<1+(>/'J.
o o o
yM0 N yi n +o
= 2 An x,
o 1
N RT
(242)
(243)
where N is the micelle number in the absence of added salt and the
standard state chemical potential of the micelle without salt y^ may
differ from that with salt, y^. Again, these are neutral species, not
charged.
14
Finally, this may be rearranged to give
Â£n(x^/x^) = Jin
K1 + x2
fo
L xl ..
j \j
MQ % Q o
+ 
N
 y2
N
RT
 ^ Jin [x (x* + x )]. (244)
N
For the correlation of equation (236) to hold, the form of
the standard state chemical potential must be
U U
^MQ ^MO C) o
n ho _
N N N Q
: ; = ^ Jin x +
RT N
Q
L
+ 1 + K.' Jin (x{ + x^)
 (1 + K') Jin x
+o
(245)
Mijnlieff shows that the reciprocity relation
9n
9n,
T,P,n
T,P,nr
(246)
leads to
Q =
.
(1 + Kf)
2 + (lKf)x^/x2
< 0.
In the limit x2/x^ K< ^
Q (1+K'>*2
(1 K')
+
(247)
(248)
This equals zero when x2 = 0. In the limit 1, Q/N = 0.16 for
anionic amphiphiles and 0.21 for cationics. The fact that it is
constant, but different for the charge types must be of significance.
15
Finally, after some rearrangement
o o _
yMQ yM0 Q o
 ~ y2
N N N
RT
+
= (1+K)
2 + (1K)x^/x2
(1K')
Â£n
+ (l + (lK'Jx^/x^ Â£n (1 + x^/x2)
 Â£n x
+o
(249)
Thus, theories for the standard state Gibbs' free energy change should be
of the above form. When x2/x^ 1 or high salt concentration equation
(249) becomes
o o
^MQ _y_H0 + (1 + K,)y2
N N
RT
 (1 + Kf) Â£n x+ = constant .
(250)
Some Theories for Free Energy Changes
Upon Micellization
Before proceeding to describe the theories for calculating Ag,
consideration should be given to a significant phenomenological observa
tion previously discussed by Lin and Somasundaran (1971). From tables
of Mukerjee and Mysels (1971), the critical micelle concentration for
amphiphiles with paraffinic tails varies with the number of carbons in the
following way:
a) for all ionics such as sulfates, sulfonates and alkanoates, each
+
additional carbon changes fn x^. within experimental error by a value of
0.69( 0,02) with negligible effect of temperature and added salt concen
tration (Lin and Somasundaran (1971) cited values essentially the same);
b) for nonionics such as oxyethylene 3 alcohols, 6 alcohols, and n and
cbetaines the variation is from 1.09 to 1.28 although it is constant
for each compound;
16
c) by contrast, Tanford (1973) quotes the results of McAuliffe (1966)
for each carbon group changing the alkane solubility, i,n x^, in water at
25G by 1.49( 0.02). It is not surprising that the nonionics should
show some difference of carbon number effect with head groups and/or
perhaps mean aggregation number. However, it is quite surprising that
these effects do not appear for the ionics.
One explanation which can be advanced is that, except for small
differences in potential energy and in conformational entropy of the hydro
carbon tails in bulk alkanes compared to micelles of amphiphiles (and
even these should probably vary proportionally to the carbon member), the
carbon number dependence of micelles and alkane solubility should be the
same since it is caused solely by the "hydrophobic" effect on the monomers
The variation to be described is
A Â£n x
An
A(AG/N(1 + a)RT)
m
An
(251)
c c
where a = 0 for alkanes and nonionics and a = 1 (?) for ionics and
x is x+ for micelles and x^ for alkane solubility. As noted above, the
value for ionics (0.69) is slightly less than onehalf that for alkane
solubility (1.49). While it is unclear why the nonionics do not have
a value equal to twice that for the ionics, it is possible that the volume
excluded to water by nonionic micelles varies with the nature of the com
pound due to differences of penetration of water around the head groups
which causes differences in the "hydrophobic" effect.
The above observation has been considered by Tanford (1972,1974ab)
in which a theory of Tartar (1955) is expanded for micelle formation and
size distribution. Tanford separates AC^/NRT into a portion linear in the
17
carbon number, nc, a portion which depends upon the area of the hydro
carbon core in the micelle, A^, plus a portion dependent only on the area
per head group A Tanford's empirical expression for an ideal solution is
1 i ag
 Â£n x + Un N + Â£n x1 =
N m N 1 NRT
 [krk2nc + k3V + i/A <252>
where the constants are positive, the <5_^ are constants and there may be
as many as three different terms in the 6^. sum. The first group
of terms on the righthand side is the same as (p/N ]J) in equation (18)
ml
g1 gI
while the summation is apparently (p /N p, ). No distinction is made
ml
by Tanford between enthalpy and entropy contributions. However, the second
term is likely to be enthalpy dominated, whereas the first is mixed imply
ing certain temperature variations of the k^ and 6^. Tanford identified
^RM t^iat f an ellipsoid whose minor axis is that of the flexible
hydrocarbon chain length plus 3A. (An extensive discussion of the role of
geometric constraints in micellization can be found in Israelachvili,
Mitchell and Ninham (1976)..) Tanford's values for k^ and k^ are apparently
derived empirically for micelles since they are not the same as those for
alkane solubility.
A preferable approach is the concept of BenNaim (1971) Tenne and
BenNaim (1977) and Pratt and Chandler (1977), who indicate that the
"hydrophobic" effect arises from aggregation of the volume (cavity) occu
pied by the hydrocarbon which is excluded from occupancy by the water.
As calculated using scaled particle theory (the application of scaled
particle theory and subsequent modifications to aqueous solutions is dis
cussed in considerable detail in Chapter 3), this effect is essentially
18
entropic at low temperature (T 10C) and becomes increasingly enthalpic
with increasing temperature as reflected in a large heat capacity (Pierotti,
1965). As previously noted, the thermodynamic properties of micellization
follow this same temperature trend.
Early approaches beginning with Debye (1949) tended to view micelli
zation as an enthalpic rather than entropic process. Debye ascribed micelle
formation to the opposition between hydrocarbonchain attraction and ionic
repulsion. The more complex approach of Poland and Scheraga (1965, 1966)
attributed the solvent contribution to micelle formation to the theory of
hydrophobic bonding of Nemethy and Scheraga (1962ab). This theory attrib
utes hydrophobic bonding to an increase in:the amount of hydrogen bonding
of the water near a solute over its average value in pure water and is
similar to the concept of: "iceberg formation" around nonpolar solutes of
Frank and Evans (1945). It should be noted that there are both strong
entropy and enthalpy contributions in this approach. The calculated stan
dard free.energies, enthalpies and entropies of solution for alkanes and
benzene homologs agree well with experimental values within the temperature
range of 0 to 70C.
The principal difference between this approach and that of Ben
Naim (1971) is that scaled particle theory is applicable, to any solvent
and does not rely explicitly on microscopic details concerning change in
water structure or conformation upon inclusion of the solute. These effects
may well be implicit in the details of scaled particle theory (particularly
with the modifications discussed in Chapter 3).
The hydrophobic free energy change Ay /RT associated with a sphere
rib
of diameter O being inserted into a solvent of diameter is given by
19
Ay
HS 0 ft N 9
__ = Â£n (1y) + 
(iy)'
3z
+
9z.
(1y) 2
&
i / 12y i I8y 1
l(1y) (ly)2J
k!
(253)
where y = jp a2 and p is the solvent number density (a very small term
which varies as the pressure has been ignored). At constant temperature
this means that Ay /RT is a quadratic in the ratio of solute to solvent
Ho
diameter.
Ay /RT = a+ b(/ ) +c(a/CT ) .
HS s s
(254)
To create a micelle of diameter O from N solutes of diameter a,,
m 1
the free energy change per solute will vary as
ag*hs AAm
NRT
NRT
AA,"
[~1 ,"1
RT
= a
1
N
HS J
HS
b(a /N a.) (a2/N oh
I ^ 1 m 1 f ^ r r~ \
+ + c 2 (255)
where a is the solvent diameter,
s
For this to coincide with Tanford's concept the first bracketed
terms must vary linearly with the amphiphile carbon number n^ and there must
be a direct correspondence between the hydrocarbon water contact area
2
and cr^. The first is precisely what is appropriate for an amphiphile
monomer cylinder of constant radius r whose length I is proportional to
2
n^ modelled as a sphere of equal area. (The term in O inequation (253) is
the most important.) Thus
^ = ira. = 2Tr2,r = A ...
sphere 1 cylinder
(256)
with
so
II
n
+
c
n
i
2
c
.2 ,
. c'
+
c' n
1 1
2
c
(257)
(258)
20
Second, the ellipsoidal micelle geometry is close to spherical so
2
should be proportional to O .
m
To add further evidence to this assertation, the results of
61 Q JL
Stigter (1975ab) who has developed a very detailed theory for [y /N y ]
m 1
should be examined. When these calculated contributions are subtracted
from experimental in x+ values, a correlation is found with the amphiphile
water contact areas of monomer and micelle of the form
el
AG
m
y
m el
yx
N
NRT
RT
= k! k' A. + k' A /N
1 2 1 3 m
= [af + b'n ] + c' a2/N
c m
(259)
(260)
where the constants are all positive. The second form is again entirely
consistent with the "hydrophobic" concept. Thus the form of the hydrophbic
effect for micelle formation is accounted for by rigid body effects.
Thus, calculations of the nonelectrostatic contributions to
micellization from rigidbody volumes excluded to the solvent (water)
appear to be consistent with present data and knowledge. In other solvents,
the effect will be significantly smaller due to being larger; this may
explain why nonaqueous micellization is of considerable less importance
and the values of N are much smaller (Kitahara, 1970).
Contributions to Thermodynamic Properties
of Micellization from Various Species
Figure 21 describes a categorization of the thermodynamic contribu
tions which must be taken into account in development of theories describing
21
micellization. The breakdown is into overall energetic effects in various
regions of the system, entropic contributions due to changes in molecular
conformation and excluded volume effects. The last is chosen as a sepa
rate category (which has enthalpy and entropy contributions) because it
has been modeled in several theories as the "hydrophobic" effect. Electro
static effects are indicated which would be restricted to ionic amphi
philes. Finally, a distinction is made between those contributions which
are associated with aspects internal to a micelle, including conforma
tional constraints and those which occur across the interface between the
micelle and the aqueous environment.
Effects arising from the properties of salt species are restricted
to ionic amphiphiles. Important enthalpic effects are possible changes
in ion hydration (uncertain at present) upon binding to the micelle sur
face as well as increased electrostatic interaction of the ions with the
much greater surface charge density of the micelle relative to the amphi
phile monomer. The latter effect has been extensively modeled; the most
recent and detailed being that of Stigter (1975ab). Unfortunately, due
to the great complexity of detail Stigter's model is not readily adapt
able to a general theory of micellization.
Two significant entropy effects are attributable to the salt
species, both resulting from binding of oppositely charged ions to the
micelle surface. The volume excluded to the solvent by the ions is
aggregated at the micelle surface resulting in an entropy increase as mod
eled by the scaled particle theory. Secondly, the ion configurational
entropy is considerably decreased as modeled by Stigter.
Fig. 21. Contributions of Species to Property Changes of Micellization
hO
ro
23
Enthalpic contributions due to properties of the amphiphile
species include repulsive interaction (AH > 0) between charged or polar
amphiphile head groups crowded at the micelle surface and attractive
(AH < 0) Van der Waals interactions between hydrocarbon chains in the
micelle interior. Theories of electrostatics such as Stigter's attempt
to model the repulsive interactions which have also been handled semi
empirically by Tanford. Unfortunately, the Van der Waals attractive
interactions are frequently considered as part of the "hydrophobic force"
(Tanford 1974a).
A significant entropy effect attributable to the amphiphile is
the change in constraints on the motion of the hydrocarbon chains in the
micelle compared to monomeric form. Considerable evidence indicates that,
at least near the polar head group, the hydrocarbon chains in micelles
exhibit a considerable degree of rigidity (Kalyanasundaram and Thomas,
1976; Roberts and Chachaty, 1973). Unfortunately, knowledge of the
configuration of the hydrocarbon chains in water is inconclusive.
A partial model of this effect is possible using a hardbody equation
of state (Gibbons, 1969; Boublik, 1975) to calculate the entropy change
associated with the change in hydrocarbon density from the monomer
solution to the micelle. More specific effects associated with rota
tional freedom of the hydrocarbon chains would best be modeled from
a lattice approach (Poland and Scheraga, 1965).
Enthalpic effects associated with the solvent include changes
in binding of the solvent to the amphiphile head groups and ions upon
micellization. A lack of knowledge concerning this effect is compounded
by considerable debate concerning the location of the micellesolvent
24
interface relative to the head group position (Tanford, 1972; Stigter,
1975ab) and thus whether the head group should be considered hydrated
upon micellization.
As noted in Figure 21, significant entropy and enthalpy
(relative importance is dependent on temperature as previously noted)
contributions to micellization are due to a change in the volume excluded
to the water molecules upon micelle formation. This effect can be readily
calculated using scaled particle theory, which will be discussed in
Chapter 3, both in the original form and modified for unusual structural
properties of water.
In principle, all of these effects should be considered in
development of a theory for micellization. However, such a develop
ment would lead to models containing too many parameters with exces
sively complex expressions. Empirically, it has been observed that
the entropy change upon micellization is large and positive at low tem
peratures and decreases, eventually becoming negative at higher temper
atures. The enthalpy change is positive at low temperatures, changes
sign at approximately 25C and becomes quite negative at higher temper
atures. Thus micellization switches from an entropy to an enthalpy
driven process with increasing temperature. This temperature behavior
is closely followed by both the water structure approach of Nemethy. and
Scheraga (1962ab) and the excluded volume concept of BenNaim (1971).
As noted previously this latter approach may macroscopically utilize
some of the microscopic detail of the former.
For micellization, the exluded volume which is dispersed with
monomers is coalesced when micelles are present. While it may seem
unusual for coalescence to lead to increased entropy, it should be
25
remembered that the species whose entropy is increased is the solvent
water not the amphiphile. When the excluded volume is coalesced many
more configurations are available for the water molecules than when
it is dispersed.
A Thermodynamic Process for Micelle Formation
Development of a meaningful theory of micellization requires
integration of the various effects discussed in the previous section
into a thermodynamic process for micelle formation. Such a process is
illustrated in Figure 22. The calculation of the change in a thermody
namic property between two equilibrium states is independent of the
path followed between these states. Thus the process in Figure 22 can
be developed for conceptual expedience rather than physical reality.
Note that the process involves three parallel paths for the amphiphile,
counterions and solvent.
The initial step of the process involves removal of the
amphiphiles and an appropriate fraction of the counterions from their
cavities in solution at constant density. Since the density is main
tained constant, there is essentially no change in the entropy of the
monomers and counterions. However, enthalpy and entropy changes in the
solvent will occur due to removal of these species from their excluded
volume cavities. The enthalpy change will be positive due to elimination
of the intermolecular interactions between the amphiphile and water.
An entropy change (probably negative) will arise due to rearrangemnt of
local water structure around the cavities. Frank and Wen (1957) have
examined this effect for ions.
%..0
^ 67,
Step la
AH = ?
Qi?*
Step 2
AH < 0
Step 4
AH > 0 .
(
Step 5a
AH = ?
AJA
AS = ?
AS > 0
AS < 0 >
V
@
AS = ?
vN
 (JS
Dispersed
Counterions
Fig. 22. A Thermodynamic Process
for Micelle Formation
27
The next step in the process involves collapsing the amphiphile
and counterion cavities. Scaled particle theory calculations yield
significant entropy and enthalpy contributions, whose relative magnitude
is temperature dependent.
The third step involves compression of the dispersed monomers to
micellar density, with restriction of the monomer head group to the
micelle surface. An appreciable entropy decrease will occur due to both
the compression and the restriction placed on head group location. The
enthalpy will increase since the electrostatic repulsion between the
head groups at the micelle surface will more than counteract attractive
Van der Waals interactions between the compressed hydrocarbon chains.
Step 4 is essentially the reverse of step,2 with creation of
a micelle cavity. This has small entropy and enthalpy changes because
of the surface area and curvature dependence of the excluded volume
effect. On a monomer basis the magnitude of the changes in step 4 is
much less than those of Step 2.
The final step of this thermodynamic process for micelle
formation involves placement of the compressed monomers in the micelle
cavity and binding of the dispersed counterions to the micelle surface.
The enthalpy decrease accompanying this step results from reduction of
the head group repulsion at the micelle surface because of the counterion
binding and creation of attractive interaction between the head groups
and water.
Entropy changes are mixed with an entropy decrease due to binding
counterions to the micelle surface and a possible entropy increase due to
water structure rearrangement around the occupied cavity.
28
As previously noted, experimentally the overall process of
micelle formation is entropy driven at lower temperatures and become
enthalpy driven with increasing temperature. Since step 2 is the only
one with an appreciable entropy increase at lower temperatures (T 35C)
the driving force (at least at lower temperatures) for micellization
must be aggregation of the volume excluded to the solvent by the amphi
phile monomers. Since there are likely to be only weak temperature
variations in the other steps of the process, step 2 must also reflect
the change to an enthalpy driving force at elevated temperature. Since
there is a large heat capacity effect in it, agreement with this trend
is expected and observed. Further discussion and calculations involving
the more significant stages of this process are included in Chapter 6.
CHAPTER 3
SCALED PARTICLE THEORY MODIFIED
FOR AQUEOUS SOLUTIONS
Introduction
The initial section of this chapter provides a brief summary of
scaled particle theory with particular emphasis on aspects important in
extension of the theory to aqueous solutions. Previous efforts at such
an extension (Pierotti, 1965; Stillinger, 1973) are discussed critically,
particularly with regard to treatment of the contact correlation function
G(r) at macroscopic r values.
The expression of Koenig (1950) for the curvature dependence of
the surface tension is then utilized to derive an exact relation for G(r)
in the macroscopic region.
Finally, structural aspects of liquid water are investigated
through calculation of the radial distribution function and direct corre
lation function from Xray diffraction data of Narten and Levy (1971).
Structural features are particularly apparent when comparing the direct
correlation function with that of liquid argon at similar density.
Basis of Scaled Particle Theory
The intent of this section is to provide a brief survey of scaled
particle theory with particular emphasis on the assumptions involved and
29
30
and applicability of the theory to aqueous solutions. Several more exten
sive reviews are available (Reiss, 1965, 1977).
Scaled particle theory was originally developed and used for the
study of hard sphere fluids (Reiss et al., 1959). Application was also
made to one and twodimensional systems [rods and rigid disks] (Helfand
et al., 1961; Cotter and Martire, 1970 ab; Cotter and Stillinger, 1972)
as well as to mixtures of disks and spheres (Lebowitz et al., ,1965). An
attempt was also made to rigorously extend the formalism to real fluids
(Helfand et al., I960) along with some application to simple fluids such
as the inert gases in the liquid state (Yosim and Owens, 1963, 1964).
More recently scaled particle theory has been applied to aqueous solutions
(Pierotti, 1965; Stillinger, 1973).
The starting point of scaled particle theory is consideration of
the work of creating a cavity at some fixed position in the fluid. In a
fluid consisting of hard spheres of diameter a, a cavity of radius r cen
tered at R^ is the same as a requirement that no centers of. the hard
spheres can be found in a sphere of radius r at Rq. Thus, creation of a
cavity of radius r at Rq is equivalent to placing a hard sphere solute of
diameter b at R^ such that r = (a + b)/2. Hence the work required to create
such a cavity is also the work required to introduce a hard sphere solute
at Rq. This work is computed by using a continuous process of "building up"
the solute in the solvent. Hence the name "scaled particle theory."
It is important to note that a cavity is considered "empty" in
scaled particle theory if no centers of particles are found in it. Also it
is worth noting that a hard sphere of zero diameter produces a cavity of
radius a/2 in the system, whereas a cavity of zero radius is equivalent to
placing a hard sphere of negative diameter b = a in the system.
31
The fundamental distribution function in scaled particle theory
is PQ(r), the probability that no molecule has its center within the
spherical region of radius r centered at some fixed Rq in the system.
This function was originally introduced by Hill (1958).
Let PQ(r+dr) be the probability that the centers of all molecules
are excluded from the sphere of radius r + dr. Now the probability that
2
the spherical shell of thickness dr and volume 4irr dr contains a particle
2
center is 4irr pG(r)dr where G(r) is defined so that pG(r) measures the con
centration of molecular centers just outside the sphere. Thus G(r) mea
sures the conditional probability that the center of a molecule will be
found within the spherical shell at r when the region enclosed by the
shell is known to be empty. The probability that the spherical shell is
free of molecular centers is
1 4irr2pG(r)dr (31)
For the volume of radius r + dr to be devoid of centers, it is
necessary that the volume of radius r, and the shell of thickness dr, be
simultaneously free of centers. Thus the probability PQ(r+dr) is given by
PQ(r+dr) = Pq(r) [l47rr2pG(r)dr].
Expanding PQ(r+dr) to first order in dr yields
3P
P (r+dr) = P (r) + r^ dr + ..
o o 9r
Combining equations (32) and (33) yields
9 Â£n P 9
5 = 4irr pG(r).
dr
Upon integration
Po(r)
exp [
4'ITr^pG^r,)dr,
y o
(32)
(33)
(34)
(35)
32
where the initial condition P^iO) = 1 has been applied (a cavity of zero
radius is always empty).
An important relationship can be derived between PQ(r) or G(r)
and the work of cavity formation W(r). This relation shall be derived
in the canonical (T,V,N) ensemble. The probability density of finding
N
a specific configuration R = R^,...,R^ is given by Hill (1956) and
BenNaim (1974) as
P(RN)
exp f g u(RN)]
 exp [ 3 U(RN)]dRN
(36)
1 N
where 3 = (kT) and U(R ) is the interaction energy among the N particles
N
at.the configuration R Thus, the probability of finding an empty spher
ical region of radius r, centered at Rq may be obtained from equation (36)
by integrating over all the region Vv(r) where v(r) denotes the spher
ical region of radius r.
Po(r) =
Vv(r)
P(RN)dRN
(37)
The following relation exists between the Helmholtz free energy,
of a system and the corresponding partition function in the canonical
ensemble
exp [BU(RN)]dRN (38)
3
where A is the momentum partition function, and no internal degrees of
freedom are ascribed to the particles.
Similarly the free energy of a system with a cavity of radius r
at R is given by
o
exp [(3A(T, V,N) ] =
. 3N
exp H3A(T,V,N;r)] =
N!A
3N
Vv(r)
exp [6U(RN)]dRN. (39)
33
Thus the ratio of (38) and (39) gives
exp {Â¡3[A(T,V,N;r) A(T,V,N)]} =
Vv(r)
N N
exp [BU(R )]dR
exp [3U(RN)]dRN
(310)
V
=po(r).
Equation (310) is an important connection between the work (at
given T,V,N) of creating a cavity of radius r, and the probability of
finding such a cavity in the system. This relation can be rewritten as
W(r) = A(r) A = kT Â£n PQ(r)
= kTp
47Trf GiOdr'
(311)
Since the work required to create a cavity of radius r is the
same as that required to place a hard sphere of diameter b = 2r a at
R^, the chemical potential of this added particle is equal to the work
plus the translational free energy
u = W(r) + kT n K
b b b
W(r) = kTp
(a+b)/2
4TTr'^ G(r')dr'.
(312)
(313)
Here, p, = 1/V is the "solute" density, whereas p = N/V is the "solvent"
b
density.
An exact expression is available for P^ir) at very small r (Hill,
1958). If the diameter of the hard sphere particles is a, then in a
sphere of radius r < a/2 there can be at most, one center of a particle
at any given time. Thus for such a small r, the probability of finding
3
the sphere occupied is 47rr p/3. Since the sphere may be occupied by
34
at most, one center of a hard sphere, the probability of finding it
empty is
P(r) 1 p
o 3
for r < a/2.
(314)
For spheres with a slightly larger radius, namely for r < a//3
at most two centers of hard spheres can be accommodated. The correspond
ing expression for P (r) is
t, / \ 4iTr
Po(r) = 1 ~ ~J~P + 2
2
Â£_
gC^.R^d^d^
(315)
V(r)
where giR^jR^) is the pair correlation function, and the integration is
carried out over the region defined by the sphere of radius r.
The probability that a cavity of radius r is empty, in the terms
/ \
of pair correlation functions g for molecular centers in the pure
solvent is
P(r)
1 + E
n=l
[(p)n/n!]
V(r)
dR.
1'
.dR g^ (R . .R ) .
n 1 n
(316)
The terms in this series will all vanish for n exceeding the maximum
number of solvent molecule centers that can be packed in a sphere of
radius r. Equation (315) represents the first three terms in this series.
When 0 < r < a/2 all terms in equation (316)' beyond n = l vanish. In this
range equation (314) applies.
As r begins to exceed a/2, two solvent centers can fit into the
cavity, so the n = 2 term in series (316) begins to contribute. However,
P(r), W(r) and G(r) all remain continuous and differentiable at r = a/2.
As r + , W(r) becomes dominated by work against the external
pressure P and against the surface tension y of the cavitysolvent inter
face. Thus
35
W(r) =(4Trp/3)r3 + (47TYJr2 (liy^)r + 0(1) (317)
Here Yra is the surface tension in the planar interface limit, and 6 pro
vides the leading term in the curvature dependence of the surface tension
Y (Buff, 1951)
Y = yjl 26/rJ. (318)
The integral relation (equation.311) between W(r) and G(r) results
in the following larger behavior for G(r)
p X J
G(r) ikx + Sri ^2 + (319)
Subsequent efforts (TullySmith and Reiss, 1970) showed that the coeffi
(320)
(321)
(322)
Thus exact results exist for G(r) at very small and very large r.
Reiss et al. (1959) and Pierotti (1963) adopted the procedure of bridg
ing these two extremes with a smooth function of r. They assumed that
G(r) is a monotonic function of r in the entire range of r. They sug
gested the form of (319) with empirical parameters
Since this is a drastic linearization of the rigorous relation
of Koenig (1950), a more rigorous approach to the curvature dependence
of the surface tension is presented in a later section.
3
cient on the r term in the expansion for G(r) must be zero.
For small cavities, equations (34) and (314) yield
for r ^ a/2
and for W(r) from equation
G(r) = (1 47Tr3p/3) 1
W(r) = kT Â£n(l 4pr~p/3) for r < a/2.
For very large cavities, r , from equation 019)
P
G(r) =
pkT '
36
G(r) = A + B/r + C/r2
(323)
The coefficient A in equation (323) was determined from
equation (322). Expressions for B and C were determined by matching
values and derivatives of equations (319) and (320) when r equalled
a/2. If P is the experimental value the expressions lead to
^ A
Y 3i_kT _1_ 3
Y 2 Uy 2 2
ira ^ } (1y)
pkT
1 +
3y
(324)
(325)
2+y2(ly) (P/pkT)^
where y = Trpa3/6.
The lower solid curve in Figure 31 shows the resulting G(r)
O
function. Its most distinctive feature is the maximum at r ~ 2.0 A.
Similar maxima occur for other temperatures, but always at r = 46 in
the Pierotti approximation.
Integration of equation (311) with this expression for G(r)
yields
W(r) = K + K,r + K.r2 + K0r3 (326)
o 1 2 3
where the coefficients are
K = kT[ Â£n(ly) + 4.5z2] \ ir Pa3 (327a)
o 6
K = (kT/a) (6z + 18z2) + TT Pa2 (327b)
K2 = (kT/a2) (12z + 18z2) 2tt Pa (327c)
K3 = 4ir P/3 (327d)
where z = y/(1y).
Thus, an approximate expression is obtained for the work required
to create a cavity of radius r in a hard sphere fluid of diameter a.
37
So far all relations are derived for a hard sphere fluid. The
application of this theory to real fluids proceeded along three lines.
One was by Yosim and Owens (1963) which involved using hard sphere
diameters for real fluids determined from experiment on one property
such as surface tension, isothermal compressibility or thermal expan
sivity in expressions for the other properties. The results were
reasonably good for nonpolar species but poor for polar substances, par
ticularly water. Another procedure was to predict entropy of phase
change by a clever thermodynamic cycle. The same experience was encoun
tered here as with the other procedure. Finally, Pierotti (1963) explored
the results of the theory for gas solubility in liquids. The process he
used consisted of creation of a cavity in the liquid, with the free energy
change calculated from the above expressions. The second step was to fill
the cavity with the solute, calculating the free energy effects as the
sum of contributions of pairwise intermolecular forces.
Scaled Particle Theory and Aqueous Solutions
Pierotti (1965) applied his scaled particle theory for solubility
to aqueous solutions of nonpolar gases.. Somewhat surprisingly, he found
it possible to predict enthalpies, entropies and heat capacities of solu
tion with the physical assumption that water molecules arrange themselves
spatially in the pure liquid as would hard spheres of an appropriate size
and several mathematical approximations. Considering the complexity of
water structure and interactions, this success seems quite fortuitous, since
the only explicit information required about the molecular structure of
O
water is a, the distance of closest approach which he set at 2.75 A.
38
Stillinger (1973) shows compelling evidence that the agreement
is not necessarily because of the correctness of the physical assump
tions. He lists values of Ym and 5 for water using equations (324)
and (325) at several points along the saturation curve for water. He
also includes measured liquidvapor interfacial tension for comparison.
Table 31 shows they do not agree. Of particular significance is the
improper temperature dependence of the interfacial tension which may
contribute to error in the predicted entropy of solution. Stillinger
argues that the sign of 5 is incorrect, although an analysis of the work
of Koenig (1950) shows that 6 is positive so long as the radius of curva
ture of the cavity r has a positive sign convention.
Stillinger developed a revised G(r) for water using the exper
imental liquidvapor interfacial tension, YÂ£v> and the radial distribu
tion function, g(r), as input data.
The most accurate determination to date of the oxygenoxygen
(2)
pair correlation function g (r) in liquid water can be determined
from Fourier transformation of the structure function data as determined
from Xray diffraction by Narten and Levy (1969, 1971). Details of this
method will be discussed later in this chapter. Their results show that
O
essentially no pairs of oxygen nuclei occur closer than 2.40 A. There
O
fore, equation (316) will be correct for G(r) in the range 0
For larger r, at least the pair term in P(r), equation (315), should
contribute, and so the same would be true in G(r).
In ice, strong directional forces between neighbor molecules
produce characteristic isosceles triangles of oxygen nuclei. The apex
angle is the tetrahedral angle 0^ = 109; since the hydrogen bonds in
39
Table 31
Surface Tension and Curvature Parameter Calculated
for Liquid Water at Its Saturated Vapor Pressure
Using the Pierotti Approximation
T
^v(expt.)
(K)
(dyne/cm)
277.15
75.07
298.15
72.01
323.15
67.93
348.15
63.49
373.15
58.78
473.15
37.81
573.15
14.39
YjEq.(324)]
(dyne/cm)
6[Eq.(32
0
A
51.44
0.5026
54.97
0.5022
58.35
0.5010
60.96
0.4992
62.86
0.4970
63.82
0.4845
52.18
0.4648
40
ice have length approximately 2.76 A, the smallest sphere which could
0
enclose these triangles would have radius r = 2.25 A. Thus, for ice
nothing beyond the pair (n=2) terms in P(r) and G(r) would be required,
O
provided r does not exceed 2.25 A.
Certainly the hydrogen bond pattern present in ice is severely
distorted upon melting. However, the coordination number in the liquid
remains low and thus it seems reasonable that the tendency toward tetra
hedral bonding persists in the liquid (Narten and Levy, 1969). Stillinger
assumed that for cold liquid water triplets of oxygen nuclei are seldom
distorted into a more compact configuration than would result from reduc
ing 0 to 90. The resulting triplet will fit into a sphere of radius
r = 1.95 A.
The repressions for G(r) are thus the following from equations
(34) and (314):
G(r) = [1 (4TT/3)pr3] 1
while from equations (34) and (315)
(2x
TT p
(0 < r < 1.20 A)
(328)
1 +
dt g(2)(t)t2(t2r)
G(r) =
o
, 4tt 3 n2
1 pr + (up)
( 2r
, (2) 2,1 3 2 8 3.
dt gv '(t)t (^ t 2r t+r )
(329)
(1.20 1 r < 1.95 A).
O
In order to specify G(r) beyond r 1.95 A in terms of correla
tion functions, knowledge of g^3^,g^\... would be required. In the
absence of such knowledge Stillinger relied on the conventional series
expressions for G(r) (equation 319) truncated after the fourth term
P 2^Â£v ^2 ^4
G(r)" m+ ^+ i +f
r r
(1.95 A < r < ) .
(330)
41
3
The r term is missing in equation (330) as required by the
general theory (Stillinger and Cotter, 1971) so that W(r) does not
have contributions proportional to to r, and are adjustable
parameters. Matching the magnitude and first derivative of G(r) at
O
r = 1.95 A between the exact microscopic expression (329) and the macro
scopic series (330) can be used to fix their values. Series (310) can
be expanded and the further parameters obtained by matching higher order
derivatives.
The function G(r) as calculated using this procedure at 25C is
the upper solid curve in Figure 31. The present procedure tends to give
G(r) an appreciably larger maxima than the Pierotti hard sphere approach.
Also the Pierotti approximation is less temperature sensitive (it depends
essentially on the temperature dependence of the number density alone).
Stillinger (1973) postulated a relatively simple physical explan
ation for the larger G(r) maxima in his approach. Unlike the Pierotti
approximation, it accounts for the strong and directional hydrogenbonding
(2)
forces in water, not only through the pair correlation function g but
also in the selection of the r value at which triplets first contribute.
As the exclusion sphere expands, it is forced to stretch and tear the
hydrogenbond network in its neighborhood. While this process occurs,
the remaining hydrogen bonds probably reach around the exclusive sphere
f
in a tight net, which enhances G(4).
Although the Stillinger modification appears to be an appropriate
extension of the Pierotti concepts, it suffers a fundamental flaw which is
discussed in the next section. Therefore, it yields results for gas
solubility which are not significantly more meaningful.
42
I'ig. 31. Contact Correlation Function; Comparison of
Different Models
43
Curvature Dependence of Surface Tension
The expression for the curvature dependence of the surface
tension (equation 318) is an approximation to the rigorous relation of
Tolman (1949) and Koenig (1950). From the Gibbs theory of surface
tension Koenig developed general equations for the change of surface
tension with curvature for systems having an arbitrary number of com
ponents. For the special case of a spherical surface Koenig's expres
sion reduces by an appropriate choice of Gibbs surfaces to a form iden
tical with that of Tolman for a onecomponent system.
9 &n y
3(6q)
T
 2(l + 6q+i <$2q2
l+26q(l+<5q + j <5^q2)
(331)
where q = 1/r. For 6q 1 equation (331) reduces to equation (318).
<5 is a measure of the distance from the surface of tension to a Gibbs
auxiliary surface measured from the spherical phase outward. It is
reasonable to assume that both the surface of tension and the auxiliary
surface lie within or very near to the interface layer and consequently
<5 is of the order of magnitude of the thickness of the interface layer.
Kirkwood and Buff (1949) performed approximate calculations for liquid
argon based upon statistical mechanics which suggested that 6 is of the
O
order of 3 A.
However, Lovett (1966) in his analysis of fluctuations about the
mean position of the interface shows that 6 should be related to the
third moment of the fluctuations, whereas the interfacial thickness is
related to the second moment. Because of this uncertainty, we will view
6 as a parameter in the study of aqueous solutions.
44
Equation (331) can be rearranged to yield
9 Â£n Y = 2[(1+ 6/r) 1]
8 ln r 1 + 2[(1 + 6/r)]3
(332)
Equations (318) and (319) imply that G(r) in the macroscopic
region should be written as
G(r) =
JiX
pkT pkTr
Since the first term is negligible at atmospheric pressure
2Y
(333)
G(r)
pkTr
(334)
This leads to
9 r G(r) 2 9y
9r pkT 9r
2y 9 Â£n y
pkTr 9 Â£n r
(335)
(336)
Substituting equation (332)
9 r G(r) = r G(r)
9r r
Upon integration
rG
2[(1 + 6/r)J 1]
Ll + 2(1 + 6/r)3
(337)
r r
Â£n
(rG)
2[(1 + 6/r ') 1] dr '
[1 + 2(1 + 6/r')3] r'
(338)
where (rG)^ = 2 Y^/P^T.
Substituting X = 6/r', equation (338) can be rewritten in
dimensionless form with finite limits
/6/r _
Â£n
rG
(rG)
2 [ (1 + X)
1]
dX.
(339)
Jo X[1 + 2(1+X) ]
An exact expression is now available for the contact correlation
function G(r) rather than the series approximation of equation (319).
45
From equation (334 )
9 G(r)
3r
2.
+
3Y
pkTr
=
2 pkTr 3r
1
pkTr
(340)
(341)
1 + 2(1 + 6/r)'
From the form of equation (341) it is apparent that this exact
expression for the macroscopic G(r) cannot go through an extremum
(3G/3r = 0) and match the microscopic G(r) on the lower side of the
extremum as does the series approximation. This behavior is illustrated
in the middle curve of Figure 31 where 8 = 0.80. Means of dealing with
.this difficulty will be discussed in Chapter 4.
Determination of the Radial Distribution Function and the Direct
Correlation Function for Liquid Water from XRay Diffraction
The total scattered intensity in electron units per molecule,
I(s), obtainable from Xray diffraction measurements on liquid water
(Narten and Levy, 1971), is related to the orientationally averaged
radial distribution function g(r) through
.00
I(s) = + 2 _o 4Trr2p(g(r) l) S:U1JSr) dr (342)
from which g(r) can be obtained by Fourier transformation. The quantity
s is the scattering vector (magnitude s = (4tt/A) sin 0, with X the wave
length and 28 the scattering angle). The quantity <^F ^>is the average
scattering from one independent molecule, depending only on the intra
2
molecular distribution of scattering density, whereas <(Fdescribes
the average scattering from a molecule of random orientation with respect
to any other molecule taken as the origin. Narten and Levy obtained
2 2
2 2
approximation with the result that <(F ^> ~ <^F)> for intermediate values
of s.
46
In practice the accessible range of scattering angles is limited
to finite values of the. variable s < s
max
Fourier transformation of
the structure function,
Hm(s) H [I(s) ]/2
(343)
yields a correlation function
gm(r) = 1 + (2rr2pr) 1
r s
max
s H (s) sin (sr)ds
m
(344)
which becomes exactly equal to the function g(r) only if s^ > , Also
since the Xray scattering center of a water molecule is so close to the
oxygen atom the i?m(r) determined is essentially the oxygenoxygen atom
correlation function in liquid water.
A direct correlation function, as proposed by Ornstein and
Zernike (Reed and Gubbins, 1973), may be defined by the following equation
(r12) = (g(r19) l) 
12
c(r ) (g(r ) l)dr.
(345)
13' 23 J 3
where c(r) is the direct correlation function.
The direct correlation function can be obtained from the structure
function H (s) as follows (Fisher, 1964)
m
c (r) = (27TZpr) ^ f s H (s) l + H (s)l X sin (sr)ds.
m m v m '
(346)
Tables 32 and 33 contain g (r) and c (r) calculated from the
m
m
structure function H (s) data of Narten and Levy (1971) at several temper
m
atures. Table 34 contains an expansion of Table 33 in the region of prin
cipal structural features. Note the unusual local extremum in c (r)
m
O
at r s; 2.9 A.
Gubbins and O'Connell (1974) present a remarkable correspondence
between the reduced isothermal compresibility for several molecules
47
Table 32
Radial Distribution Function for Liquid Water
r(A) g (r)
m
Temperature (C)
4
20
25
50
75
100
0. 10
0.06
0.09
0.30
C.30
0.2 1
0. 1 5
0. 20
0.02
0.03
0.03
0.07
0.06
0.00
0. 30
0.02
0.06
0. 11
0. 16
0.11
0.04
0.40
0.03
0.00
0. 02
0.04
0.0 2
0.04
0.50
0.06
'<1.03
0. 06
0.02
0.0 5
0.09
0. 60
0.03
0.03
0. 06
0.16
0.03
0.0 6
0.70
0.01
0.09
0. 11
0. 22
0.06
0.04
0.80
0.05
0.05
0.02
0.16
0.01
0. 12
0.90
0.05
0.02
0. 01
0. 18
0.0 3
0. 1 8
1. 00
0.04
0.01
0.10
0. 30
0.07
0.12
1. 1C
C. 14
0.06
0.20
0. 36
0.17
0.03
1. 20
0. 14
0.06
0.16
0.29
0.19
0. 10
1.30
0.04
0.03
0.07
0.23
0 17
0.08
1. 40
 0.02
0.10
0. 07
0. 23
0.16
0.05
1.50
0.02
0.07
0. 12
0. 20
0. 17
0. 1 1
1.60
0.09
0.01
0.12
0.14
0.16
0. 19
1.70
0.14
0.04
0. 10
0. 12
0. 15
0.21
1. 80
0. 1 5
0.07
C. 12
0. 10
0. 1 1
0.15
1.90
0.15
0.09
0. 14
0.02
0.0 3
0.0 8
2. 00
0.14
0.08
0.12
0. 16
0.0 1
0.08
2. 10
0.15
0.09
0. 12
0. 15
0.06
0.14
2. 20
0. 17
0.14
0.14
0. 07
0.11
0.15
2. 30
0.13
0. 13
0. 05
0. 14
0.01
0.09
2. 40
0.04
0.03
0. 1 1
0. 29
0.12
0.06
2. 50
0.15
C. 08
0.03
0.09
0.05
0.26
2. 60
0.72
0.63
0.57
0.69
0.66
0.76
2.70
1.62
1.59
1. 51
1.66
1.4 4
1.3 7
2. 80
2.29
2.34
2.22
2.22
1.96
1.78
2.90
2.29
2.41
2. 29
2. 22
2. 07
1 .86
3. 00
1.76
1 .92
1.88
1.90
1.85
1.70
3. 10
1.22
1. 40
1.4 1
1.54
1. 52
1.49
3.20
0.97
1.10
1.13
1.22
1. 23
1. 35
3. 30
0.90
0.93
0.99
1.02
1.09
1.2 3
3.40
n.83
0.80
0. 89
0.97
1.07
1.11
3. 50
0.80
0.78
0.85
1.02
1.07
1.02
48
Table 32 (Continued)
r (A) 8m(r)
Temperature (C)
4
20
25
50
75
100
3.60
0.84
0.87
0.89
1.04
1.03
0.97
3.70
0.92
0.94
0.95
0.99
0.98
0.93
3.80
0.94
0.95
0.96
0.95
0.97
0.91
3.90
0.92
0.94
0.94
0.95
0.97
0.91
4.00
0.96
0.98
0.94
0.96
0.97
0.95
4.10
1.03
1.03
0.99
0.97
0.97
0.99
4.20
1.10
1.07
1.05
0.99
0.99
1.02
4.30
1.12
. 1.09
1.08
1.02
1.00
1.03
4.40
1.13
1.10
1.10
1.03
0.99
1.02
4.50
1.15
1.11
1.12
1.03
0.98
1.02
4.60
1.17
1.13
1.13
1.06
1.02
1.01
4.70
1.17
1.15
1.13
1.11
1.05
1.02
4.80
1.14
1.14
1.10
1.11
1.04
1.02
4.90
1.09
1.07
1.07
1.06
1.01
1.01
5.00
1.05
1.00
1.04
1.00
1.00
1.00
5.10
1.01
0.97
1.00
0.98
1.00
0.99
5.20
0.96
0.96
0.96
0.96
0.98
0.99
5.30
0.91
0.92
0.93
0.93
0.95
0.98
5.40
0.88
0.88
0.91
0.91
0.95
0.96
5.50
0.86
0.87
0.89
0.91
0.96
0.95
5.60
0.86
0.89
0.87
0.92
0.95
0.94
5.70
0.86
0.89
0.86
0.92
0.94
0.92
5.80
0.86
0.87
0.87
0.92
0.94
0.92
5.90
0.89
0.89
0.93
0.95
0.95
0.94
6.00
0.92
0.93
0.93
0.95
0.96
0.98
6.10
0.95
0.96
0.96
0.95
0.96
1.01
6.20
0.97
0.98
0.98
0.96
0.97
1.02
6.30
0.99
1.01
0.99
0.98
1.00
1.03
6.40
1.02
1.04
1.01
1.00
1.01
1.03
6.50
1.04
1.05
1.04
1.02
1.02
1.03
6.60
1.06
1.05
1.06
1.04
1.03
1.02
6.70
1.07
1.06
1.06
1.06
1.04
1.01
6.80
1.07
1.07
1.05
1.07
1.04
1.02
6.90
1.06
1.07
1.05
1.06
1.04
1.03
7.00
1.06
1.05
1.06
1.05
1.04
1.03
49
Table 32 (Continued)
r(A) g (r)
m
Temperature (C)
4
20
25
50
75
100
7. 10
1.0 5
1. 04
1.06
1.05
1.04
1.03
7.20
1.04
1.05
1.05
1.04
1.04
1.04
7. 30
1.03
1 .04
1.03
1.03
1.03
1.03
7.40
1.02
1.01
1. 02
1.02
1.02
1. 02
7. 50
1.01
0.99
1.01
1.01
1.02
1.00
7.60
1.00
0.98
1.00
1.no
1.00
0.99
7. 70
0.98
0.98
0. 98
0.99
0.98
0.98
7. 80
0.98
0.98
0. 98
0. 99
0.98
0. 98
7.90
0.98
0.98
0. 98
0. 99
0.98
0.93
8.00
0.98
C. 99
0. 98
0.98
0. 99
0.98
3. 10
0. 98
0.98
0.99
0. 97
0.98
0.99
8.20
0.97
0.97
0. 99
0. 97
0.98
0.99
8. 30
0.99
0.98
0.98
0. 98
0.98
0.99
8.40
0.99
1.00
0. 99
0. 99
0. 99
0.99
8. 50
0.99
0.99
0. 99
0. 99
0.99
0.99
8.60
0.99
0. 98
0. 99
0. 99
1.00
0. 99
8. 70
1.00
0.99
0. 99
1.00
1.00
0. 99
8. 80
1.0 1
1.01
1.00
1.00
0. 99
0.99
8. 90
1.01
1.01
1.00
1.00
1.00
1.00
9.00
1.00
1.00
1. 00
1.00
1.00
1.01
9. 10
1.00
1 .00
1.00
1.00
1.01
1.01
9.20
1.01
1.01
1.00
1. 0 1
1. 00
1.02
9. 30
1.00
1.01
1.01
1.01
1.00
1.02
9.40
0.99
1.00
1.00
1.00
1.00
1.00
9.50
0.99
0.99
1.00
1,00
1.01
0.99
9.60
1.00
1.00
1.00
1.00
1.00
0.99
9.70
0.99
1 .00
1.00
1.00
1.00
1.00
9.80
0.99
1.00
1.00
1.00
1.00
1.00
9.90
0. 99
1 .00
1.00
1.01
1.00
1.00
10.00
1.00
1.00
1.00
1.0 1
1.00
1.00
50
Table 33
Direct Correlation Function for Liquid Water
r(A) cm(r)
Temperature (C)
4
20
25
50
75
100
0.10
12.53
15.61
13.18
13.21
11.17
9.74
0.20
12.07
15.05
12.45
12.45
10.64
9.41
0.30
11.54
14.42
11.84
11.83
10.23
9.12
0.40
11.05
13.84
11.41
11.37
9.94
8.90
0.50
10.56
13.26
10.90
10.80
9.57
8.64
0.60
10.06
12.63
10.29
10.13
9.10
8.30
0.70
9.81
12.05
9.81
9.58
8.70
7.97
0.00
9.29
11.61
9.50
9.20
8.43
7.74
0.90
8.95
11.24
9.17
8.77
8.12
7.51
1.00
8.53
10.79
8.70
8.25
7.71
7.17
1.10
8.10
10.29
8.26
7.82
7.29
6.76
1.20
7.79
9.85
7.98
7.54
6.97
6.42
1.30
7.57
9.51
7.74
7.27
6.70
6.19
1.40
7.33
9.15
7.42
6.94
6.41
5.96
1.50
6.99
8.69 .
7.06
6.65
6.12
5.65
1.60
6.62
8.20
6.74
6.40
5.84
5.31
1.70
6.28
7.72
6.45
6.11
5.56
5.04
1.80
5.97
7.16
6.12
5.82
5.31
4.84
1.90
5.66
6.81
5.78
5.63
5.10
4.65
2.00
5.36
6.36
5.46
5.46
4.85
4.38
2.10
5.04
5.93
5.13
5.13
4.50
4.07
2.20
4.69
5.44
4.79
4.72
4.14
3.79
2.30
4.41
5.00
4.53
4.45
3.97
3.60
2.40
4.17
4.65
4.26
4.28
3.82
3.38
2.50
3.74
4.19
3.95
3.75
3.37
3.94
2.60
2.86
3.22
3.02
2.66
2.50
2.21
2.70
1.56
1.87
1.77
1.39
1.47
1.37
2.80
0.70
0.74
0.77
0.54
0.71
0.74
2.90
0.44
0.34
0.43
0.30
0.38
0.47
3.00
0.72
0.51
0.55
0.38
0.39
0.43
3.10
1.04
0.75
0.83
0.54
0.54
0.49
3.20
1.09
0.79
0.91
0.67
0.65
0.49
3.30
0.99
0.74
0.88
0.71
0.64
0.4 7
3.40
0.89
0.66
0.81
0.62
0.52
0.46
3.50
0.78
0.50
1
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0.44
0.40
0.44
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Table 33 (Continued)
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tn
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
ro
U)
N>
ro
ro
to
>
tn
tn
Ov
o
'O
o>
cv
tn
4>
4^
to
ro
h1
o
o
o
o
o
O
a
i
o
1
o
1
o
1
o
1
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
O
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
O
o
o
o
o
O
o
o
o
o
o
tn
M
o
{*
h
o
o
o
o
ro
ro
ro
to
to
to
to
ro
ro
ro
ro
K*
to
to
4>
4>*
4>
tn
o
1
o
o
1
o
1
o
1
o
o
o
o
1
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
O
O
o
H*
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
O
o
o
o
o
O
o
O
ro
ro
ro
to
to
M
o
o
o
o
(1
o
o
!1
ro
ro
o
o
ro
to
4>
CTv
Crv
o>
Table 33 (Continued)
53
Table 34
Direct Correlation Function for Liquid Water
O
r(A) c (r)
m
Temperature (C)
4
20
25
50
75
100
2.50
3.74
4.19
3. 95
3.75
3. 3 7
2. 94
2. 53
3.53
3.96
3.73
3.48
3.15
2.74
2.56
3.27
3.67
3. 46
3. 15
2.89
2.53
2.59
2.97
3.34
3. 14
2.79
2.60
2. 29
2. 62
2.63
2.97
2.78
2.40
2.29
2.04
2.65
2.26
2.56
2.4 1
2. 0 1
1.98
1.78
2.66
1.89
2.15
2.02
l .63
1 .67
1.53
2.71
 1.53
1.74
 1. 65
1.28
1.38
1.30
2.74
 1.20
1.35
1.3 1
0.98
1.12
1.08
2. 77
0.92
1.02
1.01
0.73
0.89
0.90
2.80
0.70
0.74
0. 77
0. 54
0. 7 1
0. 74
2. 83
0.54
0 .54
0.60
0.4 1
0.56
0.63
2. 86
 0.46
0.41
0. 49
0.34
0.46
0.54
2.89
0.43
0.35
0.44
0. 30
0. 40
0.49
2.92
0.47
0.34
0.44
0.30
0 .36
0.46
2.95
0.54
0.38
0. 47
0.32
0.36
0.44
2.98
0.65
0.4 5
0.54
0.36
0.37
0. 45
3. 01
0.76
0.54
0.62
0.40
0 .40
0.46
3.04
0.87
0.62
0. 70
0. 45
0. 4 4
0.47
3.07
0.97
0 .69
0.77
0.49
0.49
0.48
3.10
1.04
0. 75
0. 83
0. 54
0.5 4
0 .49
3.13
1 .08
0.78
0.88
0.58
0.58
0. 49
3. 16
1.10
0. 80
0.90
0 .62
0.62
0.49
3. 19
l l 0
0. 80
0.91
0. 66
0. 65
0.49
3.22
1 .08
0 .79
0.91
0.69
0.66
0.48
3. 25
 1.04
0.77
0. 90
0.7 1
*0.66
0.48
3.28
1.0 1
0.75
0. 89
0.71
0.65
0.47
3.31
0.97
0.73
0.87
0.71
0.63
0.47
3.34
0.94
0. 71
0. 85
0. 69
0.60
0.47
3.37
0.91
0.69
0.83
 0. 66
0.57
0.47
3. 40
0. 89
0.66
0.81
0.62
0.52
0.46
3.43
0.86
0.62
0. 79
0. 56
0.48
0.46
3.46
0.83
0.58
0.76
0.51
0.4 5
0.46
3. 49
0.79
0.52
0.72
0.45
0.4 1
0.45
3.52
0.75
0.46
0. 68
0. 40
0. 39
0.44
54
Table 34 (Continued)
r(A) c (r)
m
Temperature (C)
4
20
25
50
75
100
3. 55
0.70
0.39
0. 63
0.36
0. 37
0.43
3.58
0.64
0 .3 1
0.58
0.32
0.35
0.4 1
3.61
 0.58
0. 24
0. 52
0.29
0.3 4
0.40
3.64
0.52
0.17
0.47
0.28
0.32
0. 38
3.67
0.47
0.10
0.42
0.26
0.3 1
0.37
3.70
0.42
0. 05
0. 37
0.25
0.30
0.35
3.73
0.37
0.00
0.33
0.24
0.28
0. 33
3. 76
0.33
0 03
0.29
0.23
0.26
0.32
3.79
0.30
0.06
0. 27
0.22
0. 24
0.30
3.82
0.28
0.09
0.24
0.20
0 .2 1
0.29
3.85
0.26
0. 12
0. 22
0. 1 8
0.19
0 .27
3.88
0.23
0.15
0.21
0.15
0. 1 7
0. 24
3. 91
0.21
0.18
0. 19
0.13
0.15
0.22
3.94
0.18
0.2 1
0. 1 7
0. 1 0
0. 1 3
0. 1 9
3.97
0.15
0 .25
0.14
0.08
0.11
0. 16
4. 00
0.11
0.29
0. 1 1
0.06
0.10
0.13
4.03
0.07
0 .33
0. 08
0. 04
0. 08
0. 1 0
4.06
0.02
0.36
0.04
0.02
0.0 7
0.07
4.09
0.02
0.40
0. 01
0. 00
0.05
0.04
4.12
0.06
0.43
0. 03
0.02
0.0 3
0. 0 2
4. 15
0. 1 0
0.46
0.07
0.04
0 .0 1
0.0 1
4. 18
0.14
0.48
0. 10
0. 06
0. 0 l
0. 03
4.21
0.16
0 .50
0.13
0.08
0.03
0.0 4
4.24
0.19
0 52
0. 16
0. 1 1
0.05
0.06
4.27
0.2 1
0.53
0.18
0. 1 3
0. 06
0.07
4.30
0.22
0.55
0.20
0.15
0.07
0.08
4.33
0.24
0. 56
0. 22
0. 1 6
0.08
0.09
4.36
0.25
0.57
0.23
0. 17
0.09
0. 1 0
4. 39
0.26
0.58
0.25
0.18
0.09
0.10
4. 42
0.28
0.59
0. 26
0. 19
0. 09
0. 1 1
4.45
0.29
0 .59
0.28
0.20
0.10
0.11
4.46
0.31
0.60
0. 29
0.2 0
0.1 L
0.12
4.51
0.32
0.60
0.30
0.21
0. 1 2
0. 1 2
4. 54
0. 33
0.61
0.32
0 .22
0.13
0.12
4.57
0.35
0.62
0. 33
0.24
0. 1 5
0. 13
55
Table 34 (Continued)
r(A) c (r)
m
Temperature (C)
4
20
25
50
75
100
4. 60
0.36
0.62
0.33
0 .25
0.16
0.13
4,63
0.36
0.63
0. 34
0.2 7
o. i a
0.14
4.66
0.37
0 .64
0.34
0.29
0.19
0.14
4. 69
0.37
0.64
0. 34
0.30
0.20
0.15
4.72
0.37
0.64
0.34
0.3!
0. 2 1
0. 1 5
4.75
0.36
0.63
0.33
0.31
0 .21
0.15
4.78
0.35
0.62
0. 33
0. 30
0. 20
0.15
4.81
0.34
0.60
0.32
0.29
0.19
0. 15
4. 84
0.33
0.58
0.31
0.28
0.18
0.15
4.87
0.3 1
0.54
0.30
0. 25
0. 1 7
0.14
4.90
0.3 0
0.5 1
0.29
0.23
0.16
0.14
4.93
0.28
0. 47
0. 27
0.21
0.15
0.13
4.96
0 .27
0.44
0. 26
0. 1 8
0. 1 4
0. 1 3
4. 99
0.25
0.4 1
0.24
0.16
0.14
0.13
5.02
0.23
0.38
0. 23
0.14
0.14
0. 1 2
5. 05
0.22
0.36
0.21
0.12
0.13
0.12
5. 08
0.20
0.34
0. 19
0.10
0.13
0.11
5.11
0.18
0.32
0. 17
CO
o
o
0. 1 2
0. 1 1
5. 14
0.16
0.30
0.15
0.07
0.1 l
0.11
5. 17
0. 1 4
0. 28
0. 13
0. 05
0.09
0.10
5.20
0.12
0 .26
0.11
0.03
0.08
0. 1 0
5. 23
0.10
0.24
0.09
0.0 l
0.06
0.09
5. 26
0.07
0.21
0. 07
0. 0 1
0. 04
0.08
5.29
0.05
0.18
0.05
0.04
0.03
0.07
5.32
0. 03
0. 15
0. 04
0. 06
0.02
0.06
5.35
0.0 1
0.12
0. 02
0.07
0.0 1
0.05
5.38
0.01
0.09
0.00
0.09
0.00
0 .04
5.41
0.02
0. 07
0. 01
0. 1 0
0. 01
0.03
5.44
0.03
0.05
0.03
0.11
0.0 1
0. 02
5. 47
0.04
0.04
0.04
0.12
0.0 1
0.01
56
including water with diverse intermolecular forces and orientational
effects. The reduced isothermal compressibility can be related to the
direct correlation function by
1 = 3P/RT
PRTKT 3p
1 4irp
2
r
c(r)dr
(347)
Figure 32 and Table 35 represent an attempt to extend this
investigation to a microscopic level using a reduced direct correlation
function. The reducing parameters obtained by Mathias (1978) by fitting
pure component compression data are utilized
&
Argon : V =75.4 cc/gmole T = 150.7K
Water : V =46.4 cc/gmole T = 438.7K.
The ratios of these values are similar to those found by Gubbins
and O'Connell (1974) although the temperature ratio is lower.
The reduced direct correlation function (integrand in equation
*1/3 2 *1/3
(347) 4irpV r c(r) as a function of reduced distance r = r/V at
several temperatures is presented in Table 35. Figure 32 shows the
reduced direct correlation function for water compared with two states of
liquid argon whose reduced density bound that of water. (Argon data of
Yarnell et al., 1973, and Smelser, 1969).
Of particular interest in Figure 32 are the unusual extremum in
the reduced correlation function for water at r*x0.8 and the shift of
the maximum in the water results to considerably larger distances (r* = 1.3)
than that of argon (r = 0.9). While any interpretation of these results
is highly speculative, the dual nature of liquid water as proposed by
Narten and Levy (1969) may offer some assistance since it suggests the use
*1/3
Fig. 32. Reduced Direct Correlation Functions
Ui
^4
58
Table 35
Reduced Direct Correlation Function for Liquid Water
r*
/ .l/3 2 ,
4tt p V* r c (r)
m
Temperature (C)
4
20
25
50
75
100
0.05
0.59
0.74
0. 61
0.61
0.51
0.44
0. 10
2. 19
2.74
2. 25
2.22
1.9 1
1. 68
0. 15
4.54
5.70
4.66
4.57
4.01
3.58
0.20
7.45
9.30
7. 57
7. 32
6. 57
5. 92
0. 25
 10.90
13.67
11.14
10.56
9.65
8.78
0.30
14.36
18.20
14. 60
13.69
 12. 61
11.51
0. 35
 18.28
23.00
18.67
17.42
15.86
14.37
0.40
22.49
27.97
22.63
21.02
 19. 14
17.50
0. 45
25.89
31.96
26.34
24.76
22.27
19.88
0. 50
29.11
35 3 6
29.77
28.07
25. 28
22.63
0. 55
32.04
38.11
32.55
32.10
28.2 8
25. 15
0.60
33.98
39.60
34.52
33.92
29. 39
26.24
0.65
35.60
40.04
36.71
35.79
31.56
27.88
0.70
34.75
38.85
36.59
34.17
30.3 8
26.01
0.75
18.80
21.25
20. 08
15.72
16.27
14.87
0.80
5.45
4.6 2
5. 66
3.87
5. 17
6. 10
0.85
 12.93
9.21
10.34
6.53
6.3 8
6.41
0.90
16.76
12.27
14. 31
10.85
 10. 22
7.27
0.95
 15.37
11.30
14.07
10.32
8.69
7.79
1.00
12.05
5. 41
10.75
5.95
6.5 3
7. 56
1.05
6.90
1.02
6.03
4. 82
5.2 5
6.43
1.10
3.96
5.32
3. 70
2. 18
2.82
4.05
1. 15
2.06
11.34
1.15
0.70
0.5 3
0. 18
1.20
6.40
15. 45
5. 77
4.21
2. 12
2. 28
1. 25
9.47
18.25
8. 97
6.22
3.23
3.41
1.30
12.16
20.96
11.24
9. 46
6. 33
4. 56
1.35
11. 43
20.05
10.81
9.45
6.11
5.02
1.40
8.74
14. 14
8. 50
4.94
4. 99
4.37
1. 45
4.53
10.43
4.37
0.92
2.89
3. 64
1.50
0.56
3.77
0. 06
4.08
0. 17
1. 37
1.55
2.46
1.16
3.57
5.96
1.25
0.45
1.60
2.74
1. 20
4. 89
7.11
2. 98
1.84
1.65
0.83
1.53
2.80
6.24
1.91
0.29
1.70
1.53
0. 11
0. 28
5.74
1. 44
2. 97
1.75
2.33
0.32
0.10
4.86
0.19
4.22
Table 35 (Continued)
. *1/3 2 .
4tt p V r c (r)
m
Temperature (C)
4 
20
25
50
75
100
1.80
4.02
0.27
1.85
2.72
1.82
4.88
1.85
4.64
2.18
3.13
0.56
2.76
3.26
1.90
4.16
1.79
1.68
0.69
2.94
3.65
1.95
3.31
4.33
1.73
0.20
3.40
4.37
2.00
2.56
4.65
1.18
0.24
3.42
4.60
2.05
1.50
5.81
0.25
0.20
2.80
3.80
2.10
1.25
' 7.11
0.60
0.35
2.54
1.69
2.15
0.77
6.25
0.96
1.66
0.60
0.59
2.20
2.10
3.63
0.53
3.29
1.65
0.23
2.25
1.88
2.84
2.78
3.78
2.10
1.67
2.30
1.99
2.65
3.06
5.48
2.11
1.12
2.35
2.53
0.18
3.40
6.91
2.98
0.69
2.40
1.41
2.29
3.04
7.77
2.98
0.54
2.45
1.94
0.31
2.72
8.07
1.89
1.25
2.50
0.15
0.00
2.30
6.74
1.87
0.57
2.55
0.10
0.56
2.03
6.51
1.26
0.34
2.60
1.37
0.76
2.05
5.29
0.22
0.53
2.65
1.66
0.13
1.10
3.84
0.33
3.62
2.70
1.77
1.64
1.66
2.94
0.91
2.45
2.75
2.93
1.86
1.32
3.75
0.63
2.62
2.80
1.00
2.63
1.54
2.94
1.14
. 1.88
2.85
1.87
2.49
1.55
1.17
0.95
1.53
2.90
1.96
2.41
2.36
1.10
0.95
2.22
2.95
1.73
1.32
1.38
0.55
1.28
CO
iH
1
3.00
2.43
2.74
1.62
0.08
1.27
2.73
3.05
2.25
1.52
0.55
1.40
0.88
0.74
3.10
1.11
1.08
0.85
2.37
0.53
0.01
3.15
0.85
1.29
1.56
2.83
1.35
0.65
3.20
0.82
1.26
1.40
2.99
1.70
0.07
3.25
1.68
1.12
1.20
2.91
2.09
0.10
3.30
2.05
0.02
0.26
3.68
1.04
0.06
3.35
0.63
0.12
0.28
1.63
0.77
0.49
3.40
] .12
0.62
1.77
1.36
0.48
2.64
3.45
0.18
0.65
1.75
3.16
0.24
1.48
3.50
1.48
0.58
2.83
1.61
0.45
1.33
*
55
60
65
70
75
80
85
90
95
00
05
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
00
60
Table 35 (Continued)
, *1/3 2 ,
4tt p V r c (r)
m
Temperature (C)
4
20
25
50
75
100
1.94
0.18
2.87
2.06
0.54
0.57
0.95
1.96
1.33
2. 30
0.70
0.09
1.77
0.43
2.53
1.80
0.58
2. 15
1.62
1.10
1.68
1.62
0.55
1.07
1.51
0.37
1.61
1.42
0.49
1.63
0.54
0.75
1.06
1.70
0. 1 1
0.55
1.79
1.71
0.67
1.05
1.13
0.56
2.01
1.59
0. 50
1.51
0. 27
1. 12
0.45
1.57
0.14
0.60
0.47
0.20
0.60
0.85
0. 14
0.98
0.44
0. 11
0.55
1.47
0.55
1.39
0.07
0.64
0.04
0.75
0. 37
0.75
1.30
0.01
0.58
1.64
1.09
1.63
0.6 3
0.66
0.35
0.8 3
1. 15
1.45
0.42
0.25
1.19
1.43
1.53
0.91
1.24
1. 12
1.82
0.99
1. 28
1.34
1.37
2. 92
1.30
1.09
1.42
0.53
1.11
0.94
1.23
0.87
1. 42
0. 87
1. 33
0.45
1.78
0.78
1.04
0. 07
0.9 5
0.62
2.18
0.90
0. 91
0. 18
2. 1 1
2.00
 1.28
0.30
0,72
0.26
0.6 9
2. 25
1.32
0.61
0. 48
0.71
1.31
2. 57
0.54
0.19
0.06
2.23
2.2 2
1.85
1.06
0.47
0.11
2. 22
0.70
1.72
1.25
0. 32
0.57
0.97
0.04
1.85
1.05
0.03
0. 4 1
1. 32
0.3 2
0.27
 1.13
0.72
0.79
1.95
0.56
0.12
0.51
0.67
0. 84
2.27
0.0 3
0. 55
0.38
1.05
1.08
2.75
0.38
0.44
1.11
1.24
1. 14
0.49
0.85
0.52
61
of a combination of two distinct correlation functions to model the
results of Figure 32.
The most remarkable fact about these results is that even though
the integrands do not scale, the integrals do over wide ranges of
conditions.
CHAPTER 4
MODELING OF SPHERICAL GAS SOLUBILITY
Introduction
The initial section of this chapter outlines the derivation of
the thermodynamic properties of solution from the experimentally deter
mined Henry's constant. Since several correlating equations yield equally
accurate fits of the Henry's constant but considerable variation in the
enthalpy, entropy and heat capacity changes upon solution, they provide
reasonable bounds for the experimental properties of solution.
The second section is concerned with a theory for the thermo
dynamic properties of aqueous solutions. Pierotti (1965) considered the
solution process to consist of two steps: creation of a solutesized
cavity in the solvent and introduction of an interacting solute molecule
into the cavity. Using scaledparticle theory for the first step and
a meanfield theory using the LennardJones potential for the second,
Pierotti obtained reasonable values for the properties.
Stillinger (1973) proposed an extended analysis for the first step.
However, reexamination of this method has led to the present use of a
reference solute to encompass the smallest scale details in the cavity
formation step. A straightforward model based on macroscopic properties
then correlates the work reqired to form a different solute cavity from
the reference cavity.
62
63
The second step is modelled similarly to Pierotti but using
a full reference radial distribution function and a more appropriate
pair potential for water. The energy parameters for unlike interactions
were based on the approach of Rigby et. al. (1969) but allowed to vary to
obtain a highly accurate fit of the Henry's constant at 298.15K. Care
has been taken to provide accurate models for the temperature dependent
hard sphere diameters and mixture radial distribution function..
The possibility of contributions to the properties of solution
arising from a change in the interfacial tension upon introduction of
a solute into a cavity is discussed. Such a possible effect is shown to
be insignificant.
The last section of this chapter discusses the results of the
modeling from the viewpoint of possible inadequacies and thus provides
a basis for suggestions of future research.
Thermodynamic Properties of Solution from Experimental Data
Appropriate derivatives of the Henry's constant with respect to
temperature yield the enthalpy and entropy of solution and the difference
in heat capacity of the gas between the liquid and gaseous phases. For
states at the same temperature, the activity of the gas in each phase
is related to the partial molal free energy, or chemical potential by
C.2 = C? + RT An a2 (41)
If the standard state of the gas in the gaseous phase at any fixed
temperature is defined as that in which the gas has a fugacity of unit
pressure, the chemical potential for that phase is
Gg = G^g + RT In fg (42)
64
For the dissolved gas, the standard state at any chosen temperature is
taken to be the hypothetical state found by extrapolation of the line
representing Henrys law on an f^ vs graph to unit mole fraction for
the dissolved gas solute. The fugacity in this standard state is K, the
Henry's constant. This choice makes the activity, ^2^2 = ^2^ t^e
dissolved gas approach the mole fraction of the dissolved gas in very
dilute solutions where Henry's law is valid. Thus, for dilute solutions,
equation (41) becomes
diss o,diss
(j (j0
+ RT n x.
For equilibrium
and
or
diss g
G2 G2
Tro 770,diss 770,g _
AG
AG = RT in K
 G^,b = RT(Â£n x2 in ip
(43)
(44)
(45)
(46)
where Henry's law has been used.
The other standard state thermodynamic properties of solution
can be obtained from the appropriate temperature derivatives.
Ao
Ah = T
2 3(AG/t)
3t
AS = 
3t
and
*770 __ 3(AH)
ACP
(47)
The sources of the Henry's constant data used in this work are listed
below.
Gas
Helium, Neon, Argon, Krypton, Xenon
Carbon Tetrafluoride, Sulfur Hexa
fluoride
Methane
Neope ntane
Henry's Constant Data Source
Benson and Krause (1976)
Ashton et al. (1968)
Wilhelm et al. (1977)
Wetlaufer et al. (1964),
Shoor et al. (1969)
65
Several equations for correlating An K as a function of temper
ature are available in the literature (Benson and Krause, 1976). Two
expressions are considered here.
An (1/K) = aQ + a1 An T + a2(Â£n T)2 (48)
An (1/K) = bQ + b /T + b2/T2. (49)
A standard leastsquares routine was utilized to determine the
parameter values and their standard deviations in equations (48) and
(49). Extreme values of,the thermodynamic properties calculated at
one standard deviation of the parameters were used to provide an estimate
of the bounds on the true value. For purposes of subsequent modeling the
average value of the two extremes was used as the "true" experimental
property value. The "true" experimental values and error limits are
listed in Table 45.
66
Application of ScaledParticle Theory
to Aqueous Solubility
Pierotti (1963) considered the process of introducing a solute
molecule into a solvent as consisting of two steps. First a cavity is
created in the solvent of a suitable size to accommodate the solute
molecule. The reversible work or partial molar Gibbs free energy
required to do this is identical with that required to introduce a hard
sphere of the same radius as the cavity into solution. The second step
is the introduction into the cavity of a solute molecule which inter
acts with the solvent according to a chosen intermolecular pair potential.
Associated with each step is a set of thermodynamic functions
with which the solution process can be described. Pierotti showed that
for extremely dilute solutions
Â£n K = G /RT + G./RT + Â£n (RT/Vj (410)
c i 1
where K is the Henry's constant, G^ and G^ are the partial molar Gibbs
free energy for cavity formation and interaction, respectively, and Vp
is the solvent molar volume.
The molar enthalpy of solution is given by
Al (Hts), \ + . RT + ap RT2 (411)
where is the coefficient of thermal expansion of the solvent.
The molar heat capacity change for the solution process is
given by
/3 AW0^ o / 9a A
ACP = i9T~)p= Cpc + Cpi R + % RT + RT' (4"12)
The partial molar volume of the solute is given by
V = V + V. + 3 RT
2 c i
(413)
67
where 6 is the isothermal compressibility of the solvent.
Pierotti calculated the partial molar Gibbs free energy of
creating a cavity in a fluid using the scaled particle theory approach
of Reiss et al. (1959) (Equations 326 and 327). Appropriate temper
ature derivatives yield H^, Cp and S^, the molar entropy of cavity
formation.
The interaction energy of a nonpolar solute with a polar solvent
can be described in terms of dispersion, induction and repulsive inter
actions. Pierotti approximated the dispersion and repulsive interactions
by a LennardJonespairwise additive potential while the inductive inter
action was described by an inverse sixth power law. The total interaction
energy per solute molecule was given by
G. = C,. (Z (r "6 cr r~12) } C. L r"6 (414)
x dis V p p 12 p v md p p v '
where r is the distance from the center of the solute to the center of
P
the Pj.^ solvent molecule and is the distance at which the dispersion
and repulsive energies are equal.
Cdis 4 e12 12= 4
(415)
where and are the energy parameters for the solvent and solute,
respectively, and a and o^ are the corresponding distance parameters in
the LennardrJones potential.
Cind l 2
(416)
where is the solvent dipole moment and is the solute polarizability.
68
An alternative method of calculating discussed by Pierotti
is the KirkwoodMuller formula.
r 2 aia2
dis m c (a1/x1) + (a2/X2>
(417)
where m is the mass of an electron, c is the velocity of light and x^ and
X2 are the molecular susceptibilities of the solvent and solute.
In order to calculate G^,, Pierotti assumed the mean field
approach: the solvent is infinite in extent and uniformly distributed
according to its number density p around the solute molecule. The
number of molecules contained in a spherical shell a distance r from the
2
center of the solute molecule is then equal to 4lTp r dr where dr is the
shell thickness. Combining this with equation (414) and replacing the
summation by an integration gives
r 00
1
kT
4ttp
kT
dis + ^ind
, f4
C,. ex. 0
dxs 12
.,10
dr1
(418)
r v.v r j r
where R is the distance from the center of the solute molecule to the
center of the nearest solvent molecule.
Pierotti obtained an estimate of the distance parameter for
water, afrom a graphical extrapolation of K vs and vs to
obtain values of K and at = 0. Since, from equations (416) (418),
is proportional to
Un K = G^/RT + nRT/V^ at 02 = 0.
(419)
Thus, a can be calculated given values of K and G. Pierotti's value of
w Z
O= 2.75 A was essentially independent of temperature.
69
Pierotti's final expression for Â£n K from equations (410),
(414) (418) is
8 C. TTp _
Â£n K + G /RT Â£n (RT/V ) =
. J o
6 kT a
12
%
 (11.17p/T)(e /k)'2 (e /k)"2 a?. .
W Z .Z
(420)
Pierotti determined E /k from the best linear fit of the left
w
3
hand side of equation (420) as a function of (c^/k)z A reasonably
straight line is obtained, insuring a good fit of the experimental K val
ues. However, the value of e^/k (= 85.3) obtained seems unreasonably low
when compared to that resulting from fitting other thermodynamic property
data (e.g., Rigby et al. (1969)).
Table 41 presents the results of Pierotti's model at 298.15K
and 323.15K. Under the assumptions of his model AS_^ = 0 and ACp^ = 0.
The experimental values as previously discussed are included for compar
ison. Note that terms arising from the term Â£n (RT/V^) in equation (410)
have been lumped with the cavity terms in Table 41. Considering that
no fitting of solute parameters was done, the results are quite good
except for the heat capacities.
Table 4la
Solution Properties from Pierotti's Model
Solute
Temperature
AG
c
AG?
X
AG
exp
AS
c
AS0
exp
RT
RT
RT
RT
_R
R:_
Helium
298.15
12.20
0.67
11.53
11.86
12.35
12.15
323.15
12.18
0.61
11.57
11.84
11.62
11.02
Neon
298.15
12.63
1.65
10.98
11.71
12.72
13.25
323.15
12.60
1.52
11.08
11.77
11.90
11.81
argon
298.15
14.66
4.35
10.31
10.59
14.66
15.42
323.15
14.59
3.99
10.60
10.88
13.25
13.53
Krypton
298.15
15.33
5.80
9.53
10.00
14.97
16.29
323.15
15.29
5.24
10.05
10.41
13.25
14.25
Xenon
298.15
17.26
8.06
9.20
9.45
16.58
17.22
323.15
17.18
7.36
9.82
9.95
15.00
14.78
Methane
298.15
16.15
5.66
10.49
10.60
15.66
16.14
323.15
16.10
5.19
10.91
10.95
14.27
14.14
Table 4lb
Enthalpy and Heat Capacity Contributions
Solute
Temperature
AH
c
RT
AH?
i
RT
AH .
cal
RT
AH
exp
RT
ACp
c
R
ACp
exp
R
Helium
298.15
0.15
1.67
0.82
0.29
9.78
14.18
323.15
0.56
0.61
0.05
0.82
10.34
13.84
Neon
298.15
0.08
1.65
1.73
1.54
10.73
18.13
323.15
0.70
1.52
0.82
0.37
11.27
17.70
Argon
298.15
0.20
4.35
4.15
4.83
15.13
23.48
323.15
1.31
3.99
2.68
2.64
15.65
23.11
Krypton
298.15
0.29
5.80
5.51
6.28
16.69
25.60
323.15
1.53
5.24
3.71
3.84
17.24
25.08
Xenon
298.15
0.56
8.06
7.50
7.77
20.86
30.56
323.15
2.14
7.36
5.22
4.83
21.64
29.98
Methane
298.15
0.41
5.66
5.25
5.55
18.38
25.55
323.15
1.79
5.19
3.40
3.19
19.11
24.48
72
Contributions to the Thermodynamic Properties
of Solution from Cavity Formation
Consider the origin of the series approximation (equation 330)
to the contact correlation function G(r) when the cavity radius r is
large. As noted in Chapter 3, the exact expression for G(r) derived
from Koenig's (1950) expression for the curvature dependence of the
surface tension (339) should allow a more rigorous calculation of G(r)
than the series result of Stillinger. However, equation (341) shows
that G(r) cannot pass through an extremum (dG/dr = 0) for finite values
of 6. Thus, the extremum value of G(r) seems a natural dividing point
between microscopic and macroscopic regimes.
Now, previous expressions (329) for G(r) in the region where
r contains two water molecular centers can pass through an extremum and
be used to match equation (339). However, we have found that the assump
tion of constant 6 severely restricts the range of r values in which the
two functions can be matched. Coupled with a lack of knowledge concern
ing the value of r at which the triplet correlation function becomes sig
nificant, this casts considerable doubt on the rigor of the expressions.
While the contact correlation function must be continuous with r, the
form is truly unknown in the region of the maximum.
Considering these difficulties, it was decided to abandon
efforts at linking microscopic and macroscopic approaches and simply
use a reference solute. The experimental solution properties of the
reference solute encompass the microscopic detail and allow use of a
macroscopic expression for differences between the reference and other
solutes. Argon was chosen as the reference solute for this work.
73
For macroscopic properties, Melrose (1970) showed that for two
phases, a and 3, in contact
dU = TdS + E y. dN. PadVa P^dV^ + vda +
ill
where J is the arithmetic mean curvature.
lx
dJ
add
(421)
J.L + L
R1 R2
where and are the principal radii of curvature.
For an isothermal constant composition process, the Helmholtz
Free Energy (work) is
c a 5v'
(422)
dA = PadVa P^dV^ + yda +
IX
9J
adJ.
Since our cavity creation process is constant pressure with
dVa = dV6
d4 = yda +
lx
l9JJ
adJ .
(423)
Using the one term macroscopic approximation
y = y (16 J)
IX _
dJ
= y 6.
(424)
The work of changing the cavity from that of the reference
solute (argon) is then
Ab 'ref Y V^ref1 Y 6[
OQ
= y (a a r)
1 s ref
1 
6(a J a J ,.)
s s ref ref
(a a r)
s ref
Since for a sphere
, 2
a 4ttR and J = 2/R
26
A A c = 4tt y
s ref
2 2
R R r
s ref
(R +R J ,
s ref J
(425)
(426)
74
Since dG = cL4 + d(PV), and our cavity formation process is at
ct 8
constant pressure and constant overall volume (dV = dV ), G G
s ref
= A A
s ref
The free energy of creating the argon reference cavity is
obtained by difference between the experimental free energy of solution
and the interaction contribution discussed in the next section. The
other cavity contributions to the thermodynamic properties of solution
are obtained through the appropriate temperature derivative of equa
tion (426).
Calculation of the thermodynamic properties of cavity formation
requires a model for the hard sphere diameters of the solute and solvent
as a function of temperature. We chose the model of Mathias (1978).
From considerations involving the direct correlation function he
postulated that the reduced hard sphere diameter should be some universal
function of reduced temperature and reduced density. The functional
form was obtained by fitting the experimental reduced isothermal compres
sibility to that obtained for a hard sphere fluid from the Carnahan
Starling equation (Carnahan and Starling, 1969). The rather complicated
function is shown in Table 42. Table 43 shows the values of the char
acteristic parameters for water and solutes of interest in this chapter.
Slight adjustments in these parameters compared to those of Mathias were
sufficient (with helium and neon as notable exceptions) to provide a con
sistent fit of the solubility data. Mathias chose the particular form
because it has the following features:
a) At high reduced temperatures the hard sphere diameter is
a function of reduced temperature only.
75
b) At high densities the hard sphere diameter is a function of
reduced temperature only.
c) The first two exponential terms were chosen to represent the
minimum in the isothermal change of the hard sphere diameter with density.
The particular form was chosen since the position of the minimum seems
to shift linearly with temperature.
d) The last exponential term is used for changes in the hard
sphere diameter required to obtain an accurate representation in the
critical region.
76
Table 42
Universal Correlation for the Reduced Hard Sphere Diameter
Reduced Temperature: T = T/T*
Reduced Density:
a
p = p/V*
T 2 0.73: f = a_/T
s 7
8
T Â£ 0.73: f = a,, exp [aic T]
s 14 1j
2 3
ttN dJ ?
= fs + a2/exp [a4(p+ alT) ]
2 2
a^/exp [a^(p+a^T ag) ] + ag/exp UiotCT al3) +
an(p a12) }]
(427)
(428)
(429)
a = 0.54008832
a2 = 1.2669802
a3 = 0.05132355
a. = 2.9107424
4
ac = 2.5167259
5
a = 2.1595955
6
a? = 0.64269552
aD = 0.17565885
8
ag = 0.18874824
a1Q = 17.952388
a n = 0.48197123
a12 = 0.76696099
a13 = 0.76631363
a., = 0.809657804
a15 = 0.24062863
77
Table 43
Characteristic Parameters
T*(K)
V*(cc/g mol)
Helium
39.0
(10.3)
50.0
(37.5)
Neon
45.2
60.5
(40.3)
Argon
150.8
74.9
Krypton
209.4
88.5
(91.2)
Xenon
289.7
114.5
(118.0)
Methane
190.6
96.0
(99,0)
Carbon Tetrafluoride
227.6
147.0
Sulfur Hexafluoride
318.7
203.9
(198.0)
Neopentane
433.8
312.1
(303.0)
Water
438.7
46.4
Values in parentheses are
those of
Mathias (1978)
if different
from those utilized here.
78
Contributions to the Thermodynamic Properties
of Solution from Intermolecular Forces
From perturbation theory (Reed and Gubbins, 1973) the configura
tion integral L for a mixture can be related to the intermolecular
potential by
In L = In Lref Z I p pD V
kT a=a g=a 3
P ref 2
TaB aB
(430)
ref P
where L is the reference configuration integral, (J)^ is the differ
ence between the real pair, potential and the reference state pair poten
ref
tial, and g^g is the reference mixture radial distribution function.
We have restricted the model to a firstorder perturbation theory.
A basic relation of statistical mechanics is
3 Â£n L
p = kT
Y
3N.
Y J
(431)
T> V, N,
B^Y .
With the hard sphere as the reference state, equations (430)
and (431) yield
p = p + 4ttp
Y Yhs Kw
*vs(r) Cs
(432)
where R is the distance from the center of the solute molecule to the
center of the nearest water molecule, d> (r) is the watersolute inter
hs
molecular potential and (r) is the watersolute hard sphere radial
distribution function. Since = p^
G. = 4ttp
1 w
(r) ghs(r) r2dr.
ws ws
(433)
79
We have chosen to approximate 4> (r) by a LennardJones form
ws
with a.. = R.
1J
(r) = 4e '
ws ws
R
lrJ
12
V6
(434)
where Â£^g includes both dispersion and induction interaction. An approx
imate expression for Â£^g from Ribgy et al. (1969)
Â£' = e
ws ws
1 +
2
a M
s w
4e a3 (a + 2a )3I
WS ws ws
(435)
utilizes StockmayerKihara potential parameters E and 0 along with
ws ws
solute polarizabilities and the dipole moment of water u .
w
Table 44 lists the values of E^g calculated from equation (435)
compared to those required to obtain an exact fit of the standard free
energy of solution AG at 298.15K. The two values are in close agree
ment in most cases.
In order to evaluate the integrals in equation (433), an
approximation must be formulated for the radial distribution function of
a hard sphere mixture. We have chosen the formulation of Mathias (1978).
hs
He chose to approximate g_^.(r) in terms of the value at contact
and an equivalent pure hard sphere distribution function. An average hard
sphere diameter can be defined as
n ., n
. ,a+l. r .a
d = h x, d, / E x. d. .
av i=l 1 1 i=l 1 1
(436)
We used a = 3; equivalent to a volume fraction average.
hs.
If g..(d,.) is the contact value of the radial distribution function of
ij ij
hs P
the mixture and g (d ) is the radial distribution function of a pure
av
80
Table 44
Intermolecular Potential Energy Parameter
Helium
46.8
Neon
156.6
Argon
375.6
(375.6)
Krypton
467.2
(462.2)
Xenon
562.3
(547.5)
Methane
462.0
(495.4)
Carbon Tetrafluoride
509.7
(514.2)
Sulfur Hexafluoride
586.3
Neopentane
663.2
(707.8)
The water energy parameter e^/k ~ 170 as obtained from Rigby et al.
(1969). Values in parentheses are from equation (435) whereas other
values result from solubility data fit at 298.15K.
Pure component parameter values for equation (435) are from
O'Connell (1967).
81
hard sphere fluid whose hard sphere diameter is d and is at a reduced
aw
n 3
density of r) = E p. d., then our approximation is
i=l 1 1
hs ..
8s(r) 1
hs ,, ,
g (d ) 1
ws ws
hs P ,
8 (dav> 1
'8hS P '
(437)
This form was deduced from the results of Throop and Bearman
(1965) who numerically evaluated the radial distribution function for
various hard sphere binaries using the PercusYevick equation.
We evaluated equation (433) for several R values over the
temperature range considered in this work and obtained the following
accurate correlation.
G = (6. 7202 4.954 xl03/T + 6.548 xl05/T2 1.52R 3.17R2) (438)
The other interaction contributions to the thermodynamic
properties H_^, S_^, Cp^ can be obtained from the appropriate temperature
derivatives of equation (438).
Analysis of Spreading Pressure of
a Solute Occupying a Cavity
There is the possibility of a contribution to the thermodynamic
properties of solution due to a change in the interfacial tension upon
introduction of a solute into a cavity. This section will however illus
trate that the magnitude of such an effect is insignificant.
For an ideal dilute solution, the thermodynamic and monolayer
analysis of Chapters 7 and 12 of Defay et al. (1966), respectively, lead
to the same result
82
7T = Y Y. = r. RT
W 1,W 1,W g
(439)
where tt is the spreading pressure and I\ ^ is the relative adsorption
of i to water, independent of the choice of dividing surface.
From Defay et al. the relative adsorption can be written in terms
of the adsorption of each component
i.w
= r.
 r
w
Cr C"
i i
z C' C"
w w
(440)
where denotes the bulk i phase and the bulk water phase. For the
present case C'/ = 0 = C'
i w
r. = r
i.w i
+ r
w
c.'/c .
i w
Also for any change in the dividing surface from z^ to z^,
r.
i
and
 r.
Z
z2
= (z0z)(C'C!)
Z i z 1 i 1
r. = r.
I.W 1
z2
 (Z2Z1>C +
c:
i
Z C"
Z1 w
If z is chosen so that T. = 0 and z, such that F = 0,
2 i 1 w
and
r. = (z^z,) c:
i.w 2 1 i
(Y Y ) = (ZoZt ) C.f R T .
'w 'i,w 2 1 i g
(441)
(442)
(443)
(444)
(445)
As noted by Reiss (1974) we must fix our frame of reference at
the center of mass of the i phase. Then for an unoccupied cavity (termed
an rcule by Reiss and TullySmith, 1971) of radius R_^ bounded by water,
the cavity boundary is R^., the surface of tension is at R, + and the
equimolecular water dividing surface is at R^ + 6 + 6^.
83
The = 0 dividing surface is at 0. Therefore Z^ Z^ =
 (R^ + 6 + 6^) and
Y Y = (R. + 6 + 6J C' R T.
w r,w i 1 r g
(446)
Considering now the case of a solute molecule occupying the
cavity, the r = 0 dividing surface is again at 0. The location of the
T =0 dividing surface may change upon addition of the solute to
w
R^ + 6 + 6^, where S' denotes the surface of tension in the presence of
the solute. Thus,
Y Y = (R. + S' + S')C' R T. (447)
w s,w i 1 s g
Since there is only one rcule or solute molecule in the volume
4tt
3 V
C' = C' =
r s 4ir
3 '
.NR.
3 o i
(448)
Subtracting equation (44$) from (447) we obtain
[(66') + (66')]
Y Y =  R T.
rw rw 4tt 3 g
3 o i
(449)
Since (66') and (6^6^) are probably very small relative to
3
R^, the change of interfacial tension upon addition of the solute should
O O
be negligible. For example, if (66') = (6^6.p = 0*10 A and R, = 3 A,
2
Y Y 1.3 dynes/cm .
s 9 w r, w
84
Discussion of Results and Suggestions
for Future Research
With accurate values for the characteristic volume and temper
ature and the interaction energy parameter as previously discussed, the
final fit to the experimental data involves fitting 6 to a temperature
dependence of the form
6 = A + B/T + C/T2 (T in K) (450)
A minimum sum of squares fit resulted in A 8.3194896
B = 2,605.2103
C = 189,930.69
O
Temperature (K)
6(A)
277.1.5
1.39
298.15 
1.72
323.15
2.08
358.15
2.53
Experience has shown that a fit of similar accuracy to Table 45
can be obtained with a different set of interaction energy parameters
Â£ and consequently 6 values. An example considerably different from that
of Table 45 can be found in Appendix A along with the computer program
which determines the coefficients in the 6 function (equation (450).
The surprisingly large magnitude of 6 relative to the radius of
the cavities involved may arise from several sources: (1) inadequacy in
using the surface tension of pure water for the calculations involving an
evacuated cavity, (2) indicative of strong structural changes in water
caused by cavity formation, and (3) covers other inadequacies in the model.
Table 45a
Contributions to Free Energy and Entropy of Solution
Solute
T(K)
AG
c
RT
AG?
i
RT
AGcal
RT
AG
exp
RT
AS
c
R
AS?
i
R
AS ,
cal
R
AS
exp
R
He
277. 15
12.65
0.36
11.79
1 1.802
+.001
13.17
0. 13
13.30
 13.20
4*.
298.15
12.64
0 .78
11.86
11 .863
+.001
11.37
0. 2 1
12.08
12.15
+ .
323.15
12. 52
0.71
11.82
11.841
+.002
10.51
0.26
10.7/
11 .02
+ .
358.15
12.24
0.61
1 1 .63
1 1 .691
+.005
3.70
0.29
8. 99
9.62
+ .
Ne
277.15
14.70
3.15
11.55
11.543
+.001
 14.02
0. 48
14.50
14.59
298.15
14. 59
2. 88
11.71
11.710
+.001
12.42
0.72
13.15
13 25
+ .
323.15
14.36
2 .60
11 .76
l 1 .773
+.002
10.73
0. 89
11.61
11.81
+ .
358.15
13.89
2.25
11.64
11.689
+.035
3.51
^0. 98
9.49
10.01
+.
Ar
277.15
19.34
9. 13
10.16
l0.159
+ .003
15.78
1.3 1
 17.05
17.13
4,
298.15
19.0 1
8.42
10.59
10.588
+.002
 13.52
1.89
15.41
 15.42
4,
323.15
18.49
7.60
10. 88
10.883
+.009
1 1.09
2.27
13.37
13.53
4*.
358. 15
17.61
6 .63
10.99
1 1 .031
+.068
8.0 0
2.49
10.49
11.17
4.
08
02
09
41
11
03
13
57
08
30
.04
.96
Table 45a (Continued)
Solute T(K)
AG
c
RT
AG?
i
RT
AGcal
RT
AG
exp
RT
Kr 277.15
22. 13
12.71
9. 42
9.457
+
298.15
21.66
l l .66
10.00
10 .004
+
323.15
20.97
10.55
10.42
10.410
+
358.15
19. 85
9.22
10.63
10.657
+
Xe 277.15
26.54
17.85
8.69
3.775
+
298.15
25.85
16.40
9.45
9.448
+
323. 1 5
24. 87
l 4.86
10.01
9.953
+
358.15
23.3 4
13.03
10.31
10 .272
+
CH. 277.15
4
23. 03
12.99
10. C4
1 0.1 04
+
298. 15
22.51
11.92
10.59
10 .596
+
323.15
21.76
10.79
10.98
10.946
+ .
358.15
20. 56
9.43
1 1 13
11.133
+ ,
AS
c
R
AS?
i
R
i5Li
R
AS
Â§xÂ£
R
16.90
1.71
13.61
18.18
+.10
14.19
2.45
16.64
 16* 29
+.04
11.31
2. 93
14.23
14.25
+.18
7 .67
3. 19
10.86
11.69
.79
18.39
2. 26
20.65
19.47
+ .12
15.03
3.16
18.19
17.22
+.04
1 1 .47
3. 74
15.21
 14.76
+.24
7. 0 1
4. 06
ll.07
 11.72
+.99
17.16
1.75
18.91
18.06
+.30
 14.34
2. 47
 16.8 l
16.14
+ .04
11.33
2. 94
14.27
14.14
+ .07
7 .54
3. 19
10.73
l1 .66
+ .29
001
001
004
048
001
001
006
062
005
004
003
008
Table 45a (Continued)
Solute
T(K)
AG
c
RT
AG
X
RT
cal
RT
AG
exp
RT
AS
c
R
AS?
i
R
cal
R
AS
exp
R
CF4
277.15
29. 96
18.06
11.90
l1.891
+.003
19 .27
2.28
21.55
21.99
+ .
298.15
29.07
16.59
12.48
12.478
+.002
15.51
3. 06
18.58
18.53
+ .
323.15
27.87
15.05
1281
12.801
+.005
11.50
3.56
15.06
14.84
+ .
358.15
26. 02
13.22
12.80
12.769
.083
6.5 0
3.82
10.32
10.2 1
1.
SF6
277.15
37.02
25 .44
11.58
11.548
+.008
21. 09
3. 02
24.11
25.08
+
298.15
35.73
23.40
12.33
12.330
+.010
1 6.39
3.95
2 0 .34
20 .38
+ .
323. 1 5
34.03
21 .26
12.78
12.764
+.005
1 l .38
4.53
15.91
 15.38
+ .
358.15
31.5 1
18.72
12.79
12.708
+.101
5.18
4. 82
10.oc
9.13
+1.
nC5
277.15
47. 83
3 7. 41
1 0.42
10.400
+.087
23.1 1
4.28
2 7 .39
27.66
+1.
298.15
45.37
34 .43
1 1 .44
1 1 .441
+.010
17.07
5.37
22.44
22.75
+ .
323.15
43.3 9
31.32
12. 07
12.114
+.015
10.64
6. 04
16.63
17.61
+
358.15
39. 79
2 7.65
12. 14
12 .336
+.045
2 .73
6.38
9.10
11.19
+1.
29
06
30
38
30
08
36
74
94
58
22
30
00
v4
Table 45b
Contributions to Enthalpy and
Solute
T(K)
AH
c
ah
X
AH
cal
AH
exp
RT
RT
RT
RT
He
277.15
0. 52
0.59
1 .51
1 .3 99
.
298.15
0.77
0.99
0.23
0 .290
323.15
2.02
0.96
1 .05
0.8 18
.
358.15
3. 54
0.90
2.65
2.074
.
Ne
277.15
0.68
3.63
2.95
3.050
m
298.15
2.17
3.6 1
 1.44
 1.535
.
323.15
3.63
3.48
0.15
0 .0 37
.
358.15
5.38
3.23
2.15
1.678
+ .
Ar
277.15
3.56
10.49
6.93
6.965
298. 15
5.49
10.31
4 .82
4.825
.
323.15
7.39
9.88
2 .48
2.644
+ #
358. 15
9. 6 1
.9.11
0.50
0.138
+
Capacity of Solution
o
ACPC
AcP
A^cal
ACp
0
exp
R
R
R
R
i a. 56
1.24
17.33
14 .69
2.1
17.10
o.ao
16.80
14.18
0.6
16. 59
0.45
16.55
13.85
2.2
1 8.49
0.14
18.35
13.32
3.5
22.71
4.11
18.60
18.82
2.9
21.28
2.65
18.63
18.13
0.9
21.05
 1 .47
19.58
17.70
3.0
22. 26
 0. 45
21.81
17.13
4.6
31.75
9.82
21 .92
23.9 1
3.Q
30.38
6.25
24.14
23.4 8
2.0
29. 51
3. 39
26.52
23.11
4.8
30.47
0 .94
29 .52
22.53
7.0
. Heat
152
020
091
387
111
Q26
133
533
077
034
288
893
AH
ah
1
RT
AH
cal
RT
Solute
T(K)
c
RT
Kr
277.15
5.23
14.42
9. 19
296. 15
7.48
14.11
6.64
323.15
9.67
13.48
3.8 1
358.IS
12. 16
12.41
 C.23
Xe
277.15
3. 1 5
20.11
11 .96
298.15
10.82
19.56
8.75
323. 15
13. 40
18.60
5.20
358.15
16.32
17.08
0 .76
CH.
4
277.15
5.87
14.74
8.87
298.15
8.18
14.39
6.21
323.15
10.43
13.72
3 .29
358.15
13.0 1
 12.62
0.39
Table 45b
(Continued)
AH
exp
ACp
c
ACp
A^cal
ACp
rexp
RT
R
R
R
R
8.727
.099
38.19
12.51
25 .69
26.4 4
3.4
6.283
+ .035
36.30
7.86
28.45
25.60
0.9
3.840
.171
35. 43
4.23
31.21
25.03
4.0
1.031
.734
35.47
1 .13
34 .33
24 .66
7.1
10 .696
.125
4 7.20
15.49
31.72
31.4 1
4.2
7 .769
.040
44.95
9.59
35.36
30.56
1.3
4.829
.231
43.69
5.11
38.58
29.98
5.1
l .444
.926
43.09
 1.37
41.72
29.42
8.6
7.954
.30
39.64
12.21
27 .42
26.7 1
5.7
5 .549
.05
37.84
7.67
30.17
25.55
2.5
3.188
.07
36.96
4.10
32.86
24.48
1.5
0.526
.28
36. 89
1 .06
35 .34
23 .35
5.1
CO
Table 45b (Continued)
Solute
T(K)
AH
c
RT
AH?
i
RT
AH
cal
RT
AH
exp
RT
CF4
277.15
10 .69
20.33
9.65
 10.103
298.15
13. 5c
19.66
6.1 0
6.052
323.15
16.37
18.61
2.25
2.032
358.15
19.52
17.04
2.48
2. 563
SF6
277. 1 5
15. 92
28.45
12.53
13.531
298.15
19.34
27.35
8.01
3.040
323.15
22.66
25.79
3.13
2.616
358. 15
26.33
23.54
2 .79
3.572
nC^
277.15
24.7 2
41.69
16.97
 17.276
298.15
28. 80
39.80
11.00
11 .306
323.15
32.75
37.36
4.61
5.499
358.15
37.07
34.03
3.04
1. 152
R
ACp?
i
R
R
ACp
e*P
R
.271
52. 63
13.57
39.06
48.44
+ 7.3
.059
50 .57
8.33
42 .25
46.55
1.7
.300
49.20
 4.31
44.89
45. 44
7.2
1.29
48.1 9
0 .94
47 .25
44 .68
13.5
.298
65.81
16.25
49.56
65.8 1
8.9
.074
63. 25
9. 75
53.50
63.01
1.5
.352
61.32
4.98
56 .35
61 .37
9.0
1.64
5 9.39
1.10
58.29
60.24
17.6
2.0
84. 28
19.19
65.09
69.73
22.6
0.6
8 1.20
l l .23
69.97
65.58
14.2
0.23
78.57
5.67
72.90
63.47
8.4
1.26
75.37
1.40
73 .96
62.33
17.5
o
91
From a comparison of Tables 41 and 45 the results of the model
are considerably better than that of Pierotti. A portion of this improve
ment can be attributed to inclusion of temperature dependent hard sphere
diameters, an accurate model for the mixture radial distribution function
and more reasonable values for solutewater interaction energy parameters.
However, use of argon as a reference solute and forcing the inter
action energy parameters to provide an accurate fit of the Henry's con
stant at 298.15K gives the model a strong correlative rather than predic
tive nature. Though somewhat more predictive, Pierotti's model also has
a strong correlative element in the graphical determination of and
Several aspects of this model provide potentially fruitful future
research topics. (1) A valid microscopicmacroscopic match may be pos
sible through the use of a connecting function (possibly a spline func
tion) between Stillinger G(r) in the microscopic region and equation (339).
(2) Extension of the perturbation theory beyond the first term may lead
to improvement of the interaction calculations. Since the series tends
to alternate in sign several terms may be necessary to obtain improvement.
(3) Exploration of other functional forms for 6 (for example, density
rather than temperature dependent).may lead to improved overall temper
ature dependence and in particular better accuracy for heat capacity model
ing. (4) A smaller project would be determination of the requirements to
improve the fit of temperature derivative properties (enthalpy, entropy,
heat capacity) for methane and xenon.
CHAPTER 5
AQUEOUS SOLUBILITY OF ALIPHATIC HYDROCARBONS
Introduction
The subject of this chapter is development of a thermodynamic
model for the aqueous solubility of aliphatic hydrocarbons. The initial
section presents modifications to the expressions of Chapter 4 for the
thermodynamic properties of cavity formation to allow calculations for
a spherocylindrical solute using a spherical reference.
Modeling of the thermodynamic properties of interaction between
a spherocylindrical solute and a spherical solvent is discussed in the
second section. Derivations of the four contributions to the total
Helmholtz free energy of interaction are presented along with correlations
of the results to facilitate use in a computer program to model hydro
carbon solubility.
The third section presents a model for the thermodynamic property
changes associated with changes in rotational and vibrational motions of
the hydrocarbons upon solution. This model is based on the perturbed
hard chain theory of Beret and Prausnitz (1975).
Analysis of the experimental solubility data is discussed in
section four with emphasis on a unified approach to both gas and liquid
hydrocarbons. The hydrocarbon vapor pressure is used to convert the
92
93
mole fraction solubilized data for the liquid hydrocarbons into Henry's
constants.
The remainder of this chapter discusses results of the model of
hydrocarbon solubility and makes comparison with trends in infinite
dilution thermodynamic properties of surfactants as determined from calor
imetric data. The discussion emphasizes sensitivity to model parameters
and suggestions for future research.
Calculation of Thermodynamic Properties of Cavity
Formation for Aliphatic Hydrocarbons
The general relation (equation 425) for the Helmholtz free energy
change upon creation of a desired solute cavity from a reference cavity is
rewritten here for convenience.
A A c = y (a a r)
s ref s ref
6 (a J a ,.J ,)
s s ref ref
(a a c)
s ref
(51)
where J = 1/R^ + l/R^ with R^ being a principal radius of curvature.
For modeling purposes we chose to represent aliphatic hydrocarbons
as spherocylinders. For a spherical reference and spherocylindrical
solute
a r = 4tt R .
ref ref
J p = 2/R
ref ref
and
a = 2 7T R L
s s
= 4tt R
s
J = 1/R
s s
= 2/R
s
Combining these expressions yields
a J = 8 tt R +2ttL
s s s
a c J c
ref ref
8ir R
ref
(cylindrical portion)
(spherical caps)
(52)
and
(53)
94
where R ^ is the reference sphere radius, Rg is the solute spherocylinder
radius and L is the length of the cylindrical portion of the spherocylinder
Combining equations (51) and (53) yields
00 2 2
y [47r(RgRref) + 2it Rg L]
6(8ttR + 2ttL 8ttR c)
s ref
4tt(R2R2 J + 2ttR L
s ref s
(54)
Since the process being considered here is performed at constant
pressure and constant overall volume, GG c = A A where G G 
s ref s ref s ref
is the change in Gibbs free energy upon creation of the solute cavity from
the reference cavity. Other thermodynamic property changes due to forma
tion of the spherocylindrical cavity are calculated from the appropriate
temperature derivatives of equation (54).
Free Energy of Interaction Between a Spherocylindrical
Solute and Spherical Solvent
Expressions for the Helmholtz free energy of interaction between
a spherocylindrical solute and a spherical solvent are necessary to model
aqueous solubility of aliphatic hydrocarbons. We chose to divide the
intermolecular potential into discrete contributions fixed at Y = 0 and
Y = L (Figure 5la) and a continuous distribution of potential along the
axis of the spherocylinder. Four distinct contributions to the total
potential arise from this model and will be discussed individually.
A. Consider the case of a potential fixed at y = 0 interacting
with molecular centers in the region 0y'L and 0
obtain the Helmholtz free energy of interaction M we must integrate the
product of the intermolecular potential g(r), the solvent density
95
Fig. 5la. Fixed Potential at y=0 Interacting with Molecular Centers
in 0
Fig. 5lb. Fixed Potential at y=0 Interacting with Molecular Centers
in y < 0 and y > L
96
y = O y = L
I
Fig. 5lc. Distributed Potential Along Spherocylinder Axis from y = 0
to y = L Interacting with Molecular Centers in 0 < y < L
and 0 < z < 00
Fig. 5ld. Distributed Potential Along Spherocylinder Axis from y = 0
to y = L Interacting with Molecular Centers in y < 0 and
y > L.
97
hs
p and the hard sphere solventsolute radial distribution function g
w 6ws
over the appropriate volume. In our approach the unlike radial distri
tis
bution function is modelled as a function of z, the perpendicular
distance to the axes of the spherocylinder. The intermolecular poten
tial (r) is modelled as a LennardJones form,
ws
From Figure 5la the volume element dV = 2ir z dy dz and
r = /y2 + z2.
= p dV
i wj ws ws
(55)
Substituting a LennardJones potential form
= p
i w
dV 4e
D
ws
12
7
WS
ws
hs t f *
8a(z >
(56)
where the LennardJones size parameter awg is effectively the sphero
hs
cylinder cavity radius and is the point at which g becomes nonzero.
The discrete portion of the distributed potential is denoted as
4e (r) where
ws ws
12 6
a o
T / \ WS ws
*vs(r) 12
Substituting for dV and r yields
L
. .A
M. = p
i Kw
dy
dz 2tt z 4e
D
ws
r12
a
ws
ws
(y2+z2)6 (y2+^2)3
hs / *
(57)
Since equation (437) provides an expression for g as a
function of z' = z + a a equation (57) would be better rearranged
av ws
as a function of zr. (av a volume fraction average essentially equal
to the pure water a for dilute solutions.)
98
Upon substitution of z' equation (57) becomes
..A
M.
x
8tt Â£ p
ws w
r L r
dy
0
d2' 2 8^<2')
w
12
a
ws
ws
(y2+z2)6
(y2+z2)3
(58)
Analytically integrating over the variable y, equation (58)
yields
M^ = 8tt e p L
i ws w
,00
126a12
*
dz' Sws(z,)
ws
1
9
z
512(L2+z2)
+
2, 2n2
+
+
960(L2+z2)3 1120(L2+z2)4 1260(L+z)' 512zL
z8 ,1 1L
rT5+^77tan 7
3a
ws
1 + z
8(L2+z2) 12(l/"+z~)
. 1 1 L
~+ tan
2. z z
8zL
(59)
Upon numerical integration of equation (59) using a 10 point
A
Gauss Quadrature formula, was fit to a function of L with temperature
dependent coefficients.
Ma .cd
X ws
2 3 4
C L + C_L + C0L + C,L
1 / 3 4
L < 3.6 A
Anf / e 1 = C_+C,/L4T? /L^+C /L^+C./L** L > 3.6 A. (510)
v i ws' 5 6 7 8 9
The details of the analytical integration of equation (58) and
expressions for the coefficients in equation (510) can be found in
Appendix B.
99
B. Consider now the case of a potential fixed at y =0 inter
acting with molecular centers in the regions y < 0 and y > L as shown
in Figure 5lb.
2
The volume element dV = 2ttw sin 0 dw d0. For the region Y < 0,
r = w and
. .B1
M. = p
w
tt/2
d0
jo 2 D hs.
dr 2it r sm 0 4e g (r')
ws ws
12 6
j o
ws ws
.12.
(511)
MB1 = 87T p eD
1 w ws
, 2 hs, ,.
dr r 8s(r,)
w
12 'rr6
a a
ws ws
r12 r6
(512)
where rf = r + a a
w ws
,B1
Upon integration of equation (512), A/L was fit to the follow
ing function of temperature.
MB1= +eD (Cin + C., T + C19 T2).
i ws 10 11 12
(513)
Expressions for the coefficients in equation (513) can be found
in Appendix B.
Consider now the region Y > L. From the geometry of Figure 5lb,
2 2 2
r = (L + w cos 0) + (w sin 0)
= L2 + w2 + 2Lw cos 0.
M52 = p
x w
,.B2
M. = p
x w
/tt/2
00
f 12 6
d0
dw 2ttw2 sin 0 4Â£B g^S(wf)
a a
ws ws
'o J
ws ws
0
r12 r6
Tf/2
d0
jo 2 . D hs, ..
dw 8tt w sin 0 Â£ g (w')
ws ws
,12
WS
(514)
(L2+w2+2Lw cos 0)^
ws
(1? +w^+2Lw cos 0)B
(515)
100
Since d cos 9 = sin 9 d8, equation (515) can be written
MB2
i
= P.
w
d cos 9
'o
a
w
, o 2 D hs,
dw' 8tt w e g (w )
ws ws
12
7.
ws
(L2+w2+ 2Lw cos 9)^
ws
(L2+w2+2Lw cos 9)3
(516)
Analytically integrating over the variable cos 9, equation (516)
becomes
D
m 8n p Â£
mB2 it_ws
i L
ws
J0
w dw' gBs(w')
ws
w
,12
ws
12
7
WS
10(L2+w2)5
+
ws
lOL+w)10 4(Lfv)4
2 2 2
4(L +w )
(517)
where w = w + a a
w ws
B2
Upon numerical integration of equation (517), was fit to
a function of L with temperature dependent coefficients.
,B2 ,._D ^ 3
Â£n( M. /Â£ 1 = (T.+C, Â£n L + C1c(Â£n L) + (V (Â£n L)'
v i ws' 13 14 15 16
+ C1?(2n L)4 + Clg(Â£n L)5.
(518)
The details of the analytical integration of equation (516)
and expressions for the coefficients in equation (518) can be found in
Appendix B.
101
C. Consider the case of a differential potential, d (r) =
ws
C dx
ws(r) T* continuously distributed along the spherocylinder axis
from y = 0 to y = L interacting with molecular centers in 0 < y < L and
0 < z < co as shown Figure 5lc.
The volume element dV = 2tt z dy dz and from the geometry of
2 2 2
Figure 5lc, r = z + (yx) .
4C
i
L
o
 v
dy
dz
o J
0
o
dx C
2it z 4e
L ws
12
a
ws
(z't'+(y,x) )
2t 6
ws
hs, \
Ss(z >
(z2+(yx)2)3
Substituting z' z + O O yields
w ws
(519)
m: =
8tt e p
ws w
L
C L
dx
dy
'0 J
0
w
dz' Z ghS(z')
(z2f(yx)2)6
12
J
ws
ws
2t 3
(z +(yx)")
(520)
Upon analytical integration over x and y, equation (520) becomes
4
8tt EC p
MC = p_w
x L
a.
dz' ghs(z')
ws
126a12
ws
w
n 2 2. 2 ,_2. 2 2
(L +z ) z + (L +Z )
+
(L2+z2)3 Z6 (L2+zÂ¥ z8 L
3360(L2+z2)3 5040(L2+z2)4 512z
9 a 768(L2tz2) 1920(L2+z2)2
1 L 1 U
xt, i M
tan tan
z z 1
3a. .
ij
f 2 2 2
(L +z )z
3
z
2 2
[l2(l +z )
, L t. 1 L tan L*
8z ^ z z J
(521)
102
C
Upon numerical integration of equation (521), was fit to
a function of L with temperature dependent coefficients.
M=Â£ws(C19L + C20l2 + C21L3 + C22L4) h<3'6A
n( = C + C /L + C.../L2 + C,/L3 + C97/L4 L>3.60A. (522)
v i ws' 23 24 25 26 27
The details of the analytical integration of equation (520) and
expressions for the coefficients in equation (522) can be found in
Appendix B.
D. Consider now the case of a differential potential, d
ws
C **" dx
= 4Â£wg ^ws(r) > continuously distributed along the spherocylinder
from y = 0 to y = L interacting with molecular centers in y < 0 and
y > L as shown in Figure 5ld.
2 2 2
For the region y < 0, r =x +w +2xw cos 0 and
dV = 2tt w sin 8 d0 dw.
MD1= p
i w
fL
tt/2
dx
d0
J0 j
o 4
dw 8tt w sin 0
r C
e
ws
12
7
ws
(x2+w2+2x w cos 0)^
ws
(x24w2+2x w cos 0)3
hs. *
8s(w >
(523)
where w = w + a a
w ws
2 2 2
For the region y > L, r =(Lx + wf cos 0f) + (wf sin 0')
2
and dV = 27Tw' sin 0' d0' dwf. Letting xr = Lx yields
103
rL
rTT/2
rco
f c 1
. .D2
M = p
i w
dxf
o J
\
o
'O
o
dwf 8tt w'2
0
sin 0 r
Â£
WS
L
,
x Â£<*,
12
O
ws
ws
(x,2+w'^+2xrw' eos 0f)8 (x,2+w,2+2x,w' eos 0')8
(524)
where w = wf + O O
w ws
Clearly = M^2 and since sin 0 d0 = d eos 0
M? = MD1 + M02
i i
C
L
C1
e
ws
dx
d cos 0
J
0 J
0 +
"k 2. lis ^
dw w Â§Wg(w )
,12
O
ws
ws
oo 6 2 2 3
(x +w +2x w eos 0) (x +w +2x w eos 0)
(525)
Upon analytical integration over x and cos 0, equation (525)
becomes
M =
i
C
r12
f Q Q
P e
w ws
* hs *
d gus(w )
w
ws
(L+w) w
L
9
lOw
9 (L+w)9
8 8
,7 7
+
+
8(L+w)8 7(L+w)7
+ (L+w) + (L+w)8w~* + (L+w^w^ t (L4w) 1
6(L+v)8 5(L+w)^ 4(L+w)^ 3(L+w)8 2
2w+L
w+L
+ 2
2 /t 22\ 4 2 2.2 6 2 2.3 8 ,2 2,4
+ w ~(L +w ) w (L +w ) w (L +w ) w (L +w )
9,T2.2, ,,.,2,2,2 .^2^ 2x3 q 2 2.4
2 (L +w ) 4(L +w ) 6 (L +w ) 8 (L +w )
L+W
/_ 22.%
i(L +w ) 2J
a
ws
8w~
w
(L2+w2) (
2w+L
w+L
+ 3+ + in
J (L+u)
((L+w)2
2 2
UL +w )J
. (526)
104
Upon numerical integration of euqation (526), was fit to
a function of L with temperature dependent coefficients.
M1 el(C28 + C29 L + C30 + Si L3)
L < 3.60 A
= eÂ£g(C /L + C33/L2 + C34/L3 + C35/L4) L > 3.60 A. (527)
The details of the analytical integration of equation (525) and
expressions for the coefficients in equation (527) can be found in
Appendix B.
Changes in Rotational and Vibrational Degrees of Freedom
of Aliphatic Hydrocarbons Upon Solution
The canonical partition function Q is related to the Helmholtz
free energy A by the simple relation
A = k T in Q. (528)
For a pure fluid, the generalized van der Waals partition func
tion is
Q =
r iN
V '
N!
A'
( )N ( )N ( )N
4repulsiony '^attraction ',"r,v'/
(529)
where
A =
h
(2tt m kT)
and
Van der Waals suggested that
= Zf
^repulsion V
q = exp
attraction
zi.
2kT
(530)
where the free volume is the volume available to the center of mass of
a single molecule as it moves about the system holding the position of all
105
other molecules fixed and 0/2 is the intermolecular potential energy of
one molecule due to the presence of all other molecules.
For a polyatomic molecule with n atoms, there are a total of
3n degrees of freedom. However, many vibrations are of such small ampli
tude and high frequency that at normal densities they do not affect inter
molecular interactions. Prigogine (1957) postulated that for each mole
cule there are 3c external degrees of freedom which affect intermolecular
interactions. For argon (or methane), c = 1; for more complex molecules,
c > 1. For all molecules, c/n <1.
The rotational and vibrational partition function can be factored
into external (densitydependent) and internal (densityindependent)
terms.
(q ) = (q ) (q ) .
Hr,v Hr,v ext r,v int
(531)
Beret and Prausnitz (1975), following Prigogine, assumed that
contributions to the partition function from rotational and vibrational
motions may be calculated as contributions from equivalent translational
motions. Each translational degree of freedom contributes
r 1/3
V,
T 6XP 2kT
to the nonideal part of the equation of state. Treating the external
rotational and vibrational motions similarly, Beret and Prausnitz obtained
3(cl)
(q )
t,v ext
0
3
2kT
(532)
106
Therefore, the partition function for a polyatomic pure fluid is
rNc
Q =
N!
N
V
A3
\ J
~4>
T exp 2kT
f(T) .
(533)
Consider the change in Helmholtz free energy due to changes in
rotational and vibrational motions when a polyatomic hydrocarbon is taken
from gas phase density to that of pure water. We set : = 0, assuming
that this contribution has been accounted for in the interaction Helmholtz
free energy.
From equation (528) it is apparent that
AA
= H KTIn ^rV'0lutl"
r> V ^r,v ideal gas
(534)
AA = RT Â£n
r, v
If
V
i c1
AG = AA = RT(cl) Â£n (V^/V).
r, v r, v f
(535)
The other thermodynamic property changes are determined from the
appropriate temperature derivatives of equation (535).
Beret and Prausnitz chose the free volume expression of Carnahan
and Starling (1972)
r \ 2
T
*4=
 4
i4
v J
here x = = 0.7405 and v =
(536)
In this case v is the water molar
v
* 2
volume and v = 1.2227 x 10 liter/mole (Gmehling et al., 1979).
Table 51 lists values of c for aliphatic hydrocarbons from Gmehling et al.,
1979) obtained by fitting PVT data.
Table 51
c Parameter Values for Aliphatic Hydrocarbons
Hydrocarbon
c
Methane
1.000
Ethane
1.2527
Propane
1.4362
Butane
1.6058
Pentane
1.8018
Hexane
1.9521
Heptane
2.1379
Octane
2.3022
Nonane
2.5190
Decane
2.6969
Dodecane
2.9004
Hexadecane
3.5996
Undecane*
2.7987
Tridecane*
3.0752
Tetradecane*
3.2500
:k
Interpolated from other values.
108
Analysis of Hydrocarbon. Solubility Data
Henry's constants for the gaseous hydrocarbons (methane, ethane,
propane and butane) wee obtained from the correlation of Wilhelm et al.
(1977) which was based on the experimental data of several workers.
These data were correlated with two temperature functions.
Â£n (1/K) = AQ + a^/T + AÂ£/T2 (537)
Â£n (1/K) = A + A Â£n T + A2 (Â£n T)2 (538),
which gave comparable accuracy.
As shown in Chapter 4,
A G = RT Â£n K (K in atm)
exp
or A G = RT Â£n (K/P ) (539)
exp atm
if Kis in units other than atmospheric. P m is atmosphereic pressure
in corresponding units. The other thermodynamic properties of solutions
can be obtained from appropriate temperature derivatives of equation (539)
Extreme values of the properties calculated with equations (537)
and (538) supplied a bound on the "true" experimental value. These
extreme values included using one standard deviation of the least squares
fit of equations (537) and (538). The "true" experimental values are
shown along with error limits in Table 53.
Determination of the thermodynamic properties of solution for
liquid hydrocarbons requires conversion of mole fraction solubilized
data into Henry's constant data. This can be accomplished by noting
that the liquid hydrocarbon is in equilibrium with its vapor at its
vapor pressure. At equilibrium
109
CL V rAQ ...
= 2 = f2 which can be rewritten as
L s
~ K *2 ^or negligible water solubility
and an ideal aqueous solution.
K = ?S2/*2 (540)
Thus the Henry's constant can be calculated from the mole frac
tion of hydrocarbon in solution (McAuliffe, 1966) and the hydrocarbon
vapor pressure at the desired temperature. Table 52 lists the mole
s
fraction x2, the vapor pressure and Â£n ^ for pentane through decarte
at 298.15K. Unfortunately, this is the only temperature at which data
are available, preventing determination of thermodynamic properties other
than A Gxp. The vapor pressure was calculated using the Antoine equa
tion as described in Reid et al. (1977). Beyond decane the accuracy of
existing data appears insufficient to warrant analysis.
Table 52
Properties
Required to Analyze
Liquid Hydrocarbon Solubility
Hydrocarbon
Aqueous
Mole Fraction
Vapor
Pressure
(k Pa)
Â£n (K/P
atm'
Pentane
0.961D05
0.6840D+02
0.1116D+02
Hexane
0.199D05
0.2016D+02
0.1151D+Q2
Heptane
0.527D06
0.6080D+01
0.1164D+02
Octane
0.104D06
0.1864D+01
0.1208D+02
Nonane
0.309D07
0.5715D+00
o.mim02
Decane
0.659D08
0.1733D01
0.1247D+02
110
Results of the Model for Aqueous Solubility
of Aliphatic Hydrocarbons
With the curvature dependence parameter <$ of equation (54)
previously determined from modeling spherical gas solubility as dis
cussed in Chapter 4, parameters remaining in the hydrocarbon solubility
model were the interaction energy parameter e^s for each solute and the
spherocylinder length L which can be broken down into a CH^ group length.
Given a value of L, the energy parameter e was determined by
fitting the experimental standard Gibbs free energy of solution A G
for the desired solute at 298.15K. The remaining gas hydrocarbon solu
bility data were fitted to a segment Lnu as a quadratic function of
Cn.2
inverse temperature
L = A + B/T + C/T2. (541)
t*ri2
Table 53 presents the model results for the gaseous hydrocarbons
using the value of the curvature parameter 6 given in Chapter 4. To illus
trate the lack of sensitivity to parameter values, Appendix C presents a
parallel set of results using the <$ values of Appendix A.
In both cases the model accuracy is very good with the exceptions
of the results at 358.15K. It should however be remembered that the
least accurate experimental data occur at 358.15K and that the correla
tions for the interaction terms in the model are least accurate at this
temperature.
The energy parameter c was for the liquid hydrocarbons shorter
then undecane was determined by fitting A G at 298.15K. With the use
exp
of equation (541) (determined from the smaller hydrocarbons) for L,
Table 53a
Contributions to Free Energy
AG
AG
Solute
T(K)
c
.1
RT
RT
CH.
4
277. 15
23.03
12
2 9 8. 1 5
22 .52
11
323.15
21.76
 10
3 5 8. 15
20. 55
9
C2H6
277.15
30.53
22
298.15
29.49
20
322.15
28.11
18
358.15
26 .07
16
C3H8
277.15
38.04
30
298.15
36.46
28
323.15
34.46
25
258. 15
31.58
"22
Solution of Gaseous Hydrocarbons
AG AG AG
r,v cal exp
RT
RT
RT
o
o
1 0. 05
10.104
+.005
0 .0
10.60
10.596
+.004
O
o
t o. ga
10.946
+ .003
o
o
11.12
11.134
+.008
l .27
9.62
9.60 1
+.008
1.26
1 0. 31
10.307
+.006
1 .24
10.79
10.807
+ .003
1.18
11.09
1 1.072
+ .010
2. 20
9. 73
9 .708
+ .009
2.18
10.52
10.521
+ .007
2. 14
11.09
11.081
+.006
2.04
11.45
11 .342
+ .010
of
99
92
79
43
18
44
55
16
5 1
12
51
1 7
111
Table 53b (Continued)
Solute
T(K)
AH
c
RT
AH
X
RT
AH
r,v
RT
C4H10
277.15
26.89
42.36
0.00
298.15
30.73
41.71
0.56
323.15
34.26
41 .27
1 .04
358.15
37.64
43.41
1.57
cal
RT
AH
exp
RT
15.47
14.324
+.54
10.33
10.452
+.08
5 .97
6.16 3
+.13
4.20
1.325
+ .52
114
Table 53c
Contributions to the Entropy of Solution of Gaseous Hydrocarbons
AS
AS?
Solute
T(K)
c
R
X
R
CH,
4
277. 1 5
17. 13
1.75
298.15
14.32
2.4 7
323.15
11.32
2. 94
358.15
7.54
3.19
C2H6
277. 15
17 .63
1 .85
293. 15
13.76
3.20
323.15
9.73
4 .8 1
353.15
4.84
7.0 1
C3H8
277. 15
18.14
2.3 0
298.15
13 .21
4 .43
323.15
8.13
6.90
353. 15
2.15
10.24
o
tV)
<
AS
AS
r,v
cal
exp
R
R
R
0.0
13.38
 18 .057
+ .30
0.0
 16.79
16.145
+.04
0.0
14.26
14.137
+ .07
0.0
10.73
11 .66 1
+.29
1.27
20. 76
21.042
+.42
1.03
13.00
1 8 .259
+.06
0 .30
15.34
 15.340
+.11
0.53
12.30
 1 l 751
+ .4 2
2.20
22.64
22.995
+.52
 1 .78
19.42
 19.577
+ .08
1.39
16.42
15.939
+.13
0.91
13 .30
11 .59 0
+.51
115
Table 53a (Continued)
Solute
T(K)
AG
c
RT
AG?
X
RT
AG
r>v
RT
C4H10
277,15
45.54
38.80
3.05
29Â£. 15
43. 43
35.73
3.03
323.15
40 .80
32.39
2.97
355.15
37. 1 0
28.20
2. 83
AG
cal
 RT
AG
exp
RT
9.79
9.809
+ .011
10.73
10.729
+ .006
1 1.38
11.395
+.005
11.73
11.775
+. Oil
112
Table 53b
Contributions to Enthalpy of Solution of Gaseous Hydrocarbons
Solute
T(K)
AH
c
RT
AH
. X
RT
AH
rv
RT
AH
cal
RT
AH
exp
RT
CH4
277.15
5 .91
14.74
0.0
8.84
7.954
+.30
298.15
8.20
 14.39
0. 0
6. 19
5.549
+ .05
323.15
1 0 .44
13.72
0 .0
3.28
3.188
+ .07
358.15
13.02
 12.62
0.0
0. 40
0.526
+.28
C2H6
277. 15
12.90
24.03
0.00
11.13
11.441
+ .43
298.15
15.72
23.65
0 .23
7.69
7.952
+.07
323.15
13.38
23.36
0.43
4. 55
4.53 5
+.10
3 5 8. 1 5
21 .22
23.17
0.66
1 .29
0 .683
+ .41
C3H8
277.15
19.90
32.81
0.00
12.91
 13.234
+ .53
298.15
23.25
32.55
0. 40
8.90
9.052
+ .08
323.15
26.32
32.41
0.75
5.33
4 .909
+ .13
358.15
29.43
32.42
1.13
 1. e6
0.248
+.50
113
Table 53c (Continued)
Solute
T(K)
As
c
R
AS?
i
R
AS
r,v
R
AS
cal
R
AS
exp
R
C4H10
277. 15
18.65
3. 56
3.05
25.26
24.635
+.53
298.15
12.65
5 .98
2.47
21.10
21.183
+.08
32315
6.54
8.88
 1.92
17.34
 17.559
+.13
358. 15
0.54
15.21
1 .26
15.93
13.100
+ .52
116
Table 53d
Contributions
to Heat
Capacity of
Solution of
Gaseous
Hydrocarbons
Solute
T(K)
ACP
ACp
ACp
r,v
a5Ci
ACp
exp
R
R
R
R
R
CH,
4
277.15
39.33
12.21
0.0
27. 17
26.709
+5.7
296.15
37.71
7.6 7
0.0
30 .04
25 .551
+2.5
323.15
36.88
4.10
0.0
32.78
24.484
+1.5
356.15
3 6.35
1.06
0. 0
35.80
23 .845
+5.1
C2H6
2 7 7. 1 5
54 .64
17 .86
3.77
40.54
38.777
+8.3
293. 15
51 .52
 19.24
2.99
35. 28
37.091
+3.6
323. 1 5
43. 32
20.63
2.74
30.93
35.496
+2.1
356. 1 5
46 .29
22.11
2.69
26.87
34.580
+7.4
C3H8
277.15
69.89
28.36
6.50
48. 03
47.700 +10.2
298. 15
65.33
29.87
5.17
40 .63
45.591
+4.4
323.15
60 .76
31.48
4.72
34.00
43.598
+2.6
356.15
55.72
33. 57
4. 65
26.80
42.416
+9.0
117
Table 53d (Continued)
Solute
T(K)
ACPc
R
ACp
i
R
ACp
r,
R
C4H10
277.15
85.14
31.78
9.03
298. 15
79.13
34.46
7.18
323.15
72.70
37.54
6.56
358.15
65.15
53.11
6.46
A^cal
R
ACp
R
62. 39
48.133
+10.4
51.85
46.059
4.5
41.72
44.117
+ 2.6
18. 50
42.989
+ 9.2
118
119
Table 54
Energy Parameter Values and Length Function
Hydrocarbon
e /k (K)
ws
A E /k (K)
ws
Methane
232.5
166
Ethane
398.5
172
Propane
570.6
179
Butane
750.0
175
Pentane
925.1
173
Hexane
1098.4
183
Heptane
1281.4
175
Octane
1456.0
181
Nonane
1645.9
178
Decane
1823.9
196
Undecane
2020.1
180
Dodecane
2199.8
201
Tridecane
2401.1
181
Tetradecane
2582.3
L = 0.52778918 + 3.194678D+02/T 2.8715619W04/T2
with T in K.
120
the other thermodynamic properties were predicted for the liquid hydro
carbons. Due to severe uncertainty in the experimental solubility
(Baker,. 1959; Franks, 1966; Sutton and Calder, 1974) for hydrocarbons
larger than decane, Â£ was determined by extrapolation of values for
the shorter hydrocarbons. Unfortunately, such extrapolation is also
quite uncertain due to fluctuations in e as a function of carbon number.
ws
Table 55 presents results of predictions for larger hydrocarbons
for which an extremum occurs in AGa^ at decane. Appendix C again con
tains a parallel calculation using the S values of Appendix A.
As demonstrated by the results of this chapter and Appendix C,
a lack of uniqueness problem exists in the model parameters since both
E and Lare unknown, or at least uncertain, and model results are not
ws
highly sensitive to either. A modest variation in one parameter can
easily be compensated for by a modest variation in the other.
Thomas and Meath (1979) present an excellent discussion concern
ing the variation of dispersion energy coefficients (= ea^) when dif
6
ferent thermodynamic properties are fitted. Thus we would not expect to
be able to find e values obtained from fitting a thermodynamic property
to model gas solubility. We have done this because no alternative approach
is readily available.
The temperature dependence of L obtained in the modeling is much
larger than that of the hard sphere diameter for the spherical solutes.
While the complexity of the model for the interaction terms prevented using
a temperature dependent spherocylinder radius and thus, the large temper
ature dependence of L may be partially compensating for this, we expect
that the major reason is coiling of the chain, with resultant shortening,
as the temperature increases.
Table 55a
Contributions to Free Energy and Entropy of Solution of Liquid Hydrocarbons
Solute
T(K)
AG
c
RT
AG
i
RT
AG
LiZ
RT
cal
RT
AS
c
R
AS?
i
R
AS
LiZ
R
cal
R
C5H12
277.15
53 .04
46.99
4.04
10.08
19.15
4. 57
4. 04
27.76
298.15
50. 40
43. 26
4. 01
1 1 15
12.09
7.49
3 .27
22.36
323. 1 5
47.15
33.20
3.93
l 1 .87
4.94
10.91
2.55
18.40
353.15
42 .61
34.08
3.75
12.23
3. 23
 15. 52
1.67
13.95
C6H14
277.15
60.54
55.01
4.80
10.33
1 9.66
5.49
4.80
29.95
298.15
57 .37
50.63
4.76
11.50
11.54
8.96
3. 89
24.38
323.15
53 .49
45.86
4. 66
12.30
3.35
 12.99
3.03
19.36
353.15
4a. 12
39.84
4.45
12.73
5 .93
18.38
1.98
14.43
C7H16
277.15
68 .04
63.46
5.73
10.32
20.17
 6. 49
5. 73
32.39
298.15
64.34
53.39
5. 69
1 1.64
10.98
10.52
4.64
26.14
323.15
5 9.84
52.88
5.57
12.54
1 .76
15.19
3.62
20.56
353.15
53 .64
45.92
5.32
13.04
8.62
2 1. 40
2.37
15.15
121
Table 55a (Continued)
Solute
T(K)
AG
c
RT
AG?
i
RT
AG
r, v
RT
AGcal
RT
AS
c
R
AS?
i
R
AS
R
As
cal
R
C8H18
277.15
75 .54
71.47
6.56
10.63
20.67
7. 49
6. 56
34.72
298.15
71 .31
65.75
6. 52
12.07
10.42
12.02
5.32
27.76
323.15
66.19
59.53
6.38
13.03
0.16
17.29
4.14
21.59
358.15
59.15
51.63
6.09
13.56
11.31
24.27
2. 71
15. 66
C9H20
277.15
83 .04
80.21
10. 45
13. 29
21.18
8.57
7.65
37.40
298.15
78.28
73.78
10.21
14.71
9.3 7
13.66
6.20
29.73
323.15
72.53
66.78
9.93
15.63
1. 43
19. 56
4. 83
22.95
353.15
64.66
57. 96
9.57
16.27
14.01
27.36
3.16
16.51
C10H22
277.15
90 .55
38.35
8.55
10.74
21.69
9. 60
8. 55
39. 83
293.15
85.25
81.26
8.49
12.48
9. 3 1
1 5. 20
6. 93
31.44
323.15
7 8.83
73. 54
8. 31
13.65
3.03
21.69
5.39
24.05
35e.l5
70.18
63.81
7.93
14.29
16.70
30.24
3. 53
17.07
Table 55a (Continued)
Solute
T(K)
AG
c
RT
AG
i
RT
AG
r v
RT
C11H24
277.15
93.05
97.36
9.06
298.15
92 .22
89.53
9.00
323.15
85.23
81.02
8. 81
358.15
75 .69
70.29
8.41
C12H26
277.15
105 .55
 105.57
9. 58
293.15
99.1 9
97.07
9.51
323.15
91 .57
37.33
9.30
358.15
81.20
76. 19
8. 88
C13H28
277.15
1 1 3 .05
114.79
10.46
298.15
106.16
105.54
10 .38
323.15
97.92
95. 49
10.16
358.15
36 .72
82.82
9.70
AG
AS
AS?
o
AS
AS
cal
c
i
r,v
cal
RT
R
R
R
R
9.75
22.19
10.72
9.06
41.97
11.69
3. 75
 16. 90
7.34
32.99
13.01
4 .62
24.03
5.72
25.12
13.31
19.39
33. 40
3. 74
17. 75
9. 55
22.70
11.75
9.57
44.03
1 1 .63
8.2 0
18.45
7.76
34.40
13.05
6. 22
26. 17
6. 04
25.99
1 3.90
22.08
36.30
3.95
18.16
8.7 1
23.20
 12.90
 10.45
46. 56
11.00
7. 64
20. 18
8.47
36.29
12.59
7.8 1
23.56
6.59
27.35
13.60
24.78
39.54
4. 32
 19. 08
123
Table 55a (Continued)
Solute
T(K)
AG
c
RT
AG
i
RT
AG
r>v
RT
AG 1
cal
RT
AS
c
R
AS?
X
R
AS
r,v
R
A5oal
R
C14H30
277.15
120.55
123.06
1 1 .34
0.83
23.71
13.94
11.33
48.98
298.15
1 13.13
113.13
11.26
11.25
7. 08
21. 75
9. 1 8
38.01
323.15
104.27
 102.35
11.02
12.93
9.40
30.72
7.15
28.47
358.15
92.23
88.76
10.5 1
13.98
27.47
42.45
4. 68
 19.6
124
Table 55b
Contributions to Enthalpy and Heat Capacity of Solution of Liquid Hydrocarbons
Solute
T(K)
AH
c
ah
1
AH
r,v
45al
ACp
c
ACp?
ACp
r ,v
^cal
RT
RT
RT
RT
R
R
R
R
C5H12
27715
33. 89
51.56
0. 00
1 7.67
100.38
38.8 l
1 1.95
73.52
298.15
38 .30
50.75
0.74
l1.71
92.93
41.25
9. 50
61.18
323.15
42 .20
50. 1 1
i .38
6.53
84. 64
43. 59
8.68
4 9.73
358.15
45.84
49. 59
2.08
1.67
74.58
45.89
8.55
37.23
C6H14
277.15
40 .88
60.50
0.00
19.62
1 15.63
46.05
1 4. 19
83. 77
298.15
45 .83
59.58
0.83
12.87
1 06. 74
48.80
1 1 .28
69.22
323.15
50.15
58.85
1.64
7 .07
96.57
51.26
10.31
55.62
358.15
54 .05
58.22
2.47
 1.70
84. 0 1
53. 24
10.15
40.91
C7H16
2 77.15
47. 88
69. 95
0. 00
22.08
130.88
53.45
16.96
94.39
298. 1 5
53 .3 6
68.91
l .05
14.50
120.54
56.59
13.48
77.44
323.15
58 .09
68.07
1 .96
 8.02
108.5 1
59. 3 1
12.32
61.52
358.15
62.26
 6 7. 32
2.95
2.11
93.44
61.23
12.13
44 .33
125
Table 55b (Continued)
Solvent
T(K)
AH
c
AH?
l
AH
r,v
cal
ACp
c
ACp?
i
ACp
r ,v
A^cal
RT
RT
RT
RT
R
R
R
R
C8H18
277.15
54. 87
78. 96
0. 00
24.09
146.13
60.33
19.40
105.21
298.15
60.88
77.77
1 .20
15.69
134.35
63.31
15.43
85.97
323.15
66.03
76 .82
2.24
8.55
120.45
66. 75
1 4. 1 0
67.80
358.15
70.46
75. 54
3.38
2. 1 0
102.87
68.61
13 .88
48.14
C9H20
277.15
61 .87
38.73
0.00
26 .9 1
161.38
67.75
22. 63
116.26
298.15
68.41
87.44
1.40
 17.63
148.15
71.56
18.00
94.59
323.15
73.97
86.34
2.61
9.76
132.39
74.72
16.45
74.12
358.15
78 .67
85.32
3.94
2.71
l 12.30
76.49
16. 19
52. 0 0
C10H22
277.15
68 .86
97. 95
0. 00
29. 09
1 76.63
74.65
25.29
127.27
298.15
75.94
96.46
1.56
18.96
161.96
78.74
20.11
103.33
323.15
8 1 .9 1
95.23
2.92
 10.40
144. 33
32. 05
1 8.37
80.65
358.15
86 .83
94. 05
4.40
2.77
121.73
83.68
13.09
56.14
126
Table 55b (Continued)
Solute
T(K)
AH
c
ah?
1
AH
r,v
AH
cal
>
Ol
O O
ACp?
i
ACp
r,v
A^cal
RT
RT
RT
RT
R
R
R
R
C11H24
277.15
75 .86
 108.08
0. 00
32.22
1 91.88
82.28
26.80
136.40
298.15
83 .46
 106.43
1 .66
21.30
175.75
36.67
2 1.31
110.41
323.15
39 .85
105.04
3.09
12.11
156.26
90. 14
19. 48
85. 60
358.15
95.08
 103.69
4.67
3.95
131.1 6
91.61
19.17
58.72
C12H26
277. 15
82.85
ll7.32
0.00
34.47
207.13
89.26
28.32
146.19
298.15
90 .99
115.52
1.75
22.78
189.57
93. 90
22.52
118.19
323.1.5
97.79
 1 1 4. 00
3.27
12.95
168.20
97.49
20.58
91.29
358.15
103.29
112.48
4 .93
4.27
140.59
98.75
20.25
62. 10
C13H28
277.15
89.85
 127.69
15. 68
22.17
222.3 8
97. 12
10.82
136.09
298.15
98.52
 125.72
15.32
1l.89
203.38
102.05
10.21
111.54
323.15
1 05 .73
124.05
14.90
3.43
180.14
 105.77
9.63
84. 0 0
358.15
111 .49
122.36
14. 35
3. 49
150.02
1 06.82
9.02
52.23
127
AH
AH?
X
Solute
T(K)
c
RT
RT
C14H30
277.15
96.84
137.00
298.15
106.05
134.8
323.15
1 13.67
133.07
358.15
11970
 131.21
Table 55b (Continued)
AH
r,v
ABcal
ACp
c
ACp
i
ACp
r,v
aS4i
RT
RT
R
R
R
R
0.00
40.16
237.63
 104. 19
33. 53
166.97
2.07
26.76
217.18
109.37
26.66
134.47
3.37
15.54
192.08
 113. 19
24. 36
103.25
5. 84
5.67
159. 46
l14.00
23.93
69.44
128
129
Comparison with Infinite Dilution
Properties of Surfactants
Since the ultimate objective of this work was to lay the founda
tions for a molecular theory for thermodynamic properties of micelliza
tion, a comparison between properties of surfactant solutions at infinite
dilution and the present model as a function of carbon number would be
instructive. Unfortunately only infinite dilution heat capacity data
were found in the literature.
Table 56 presents a comparison hetween infinite dilution heat
capacities for nalkylamine hydrobromides and the present model at
298.15K. The surfactant data were obtained, using calorimetry, by
Leduc et al. (1974). The absolute values should differ in the two cases
due to the amine hydrobromide group on the surfactant. However the
incremental change with carbon number should agree. For the hydrobromides
AC p/R is over 10 for each additional carbon, whereas the model predicts
a value of less than 9 for the longer chains with considerable differ
ence for short chains. Since Cp is a second temperature derivative, it
will strongly reflect model inadequacies. These seem to be present to a
certain extent.
Suggestions for Future Work
Beyond suggestions made in Chapter 4 concerning aspects common
to the spherical gas model, several possibilities exist for interesting
future work. Of a short term nature would be combination of the several
correlations used for the contributions to AA. into one each for discrete
i
130
Table 56
Infinite Dilution Heat Capacity of Surfactants
in Water at 298.15K
Chain
Carbon Number
1
2
3
4
5
6
7
Alkylamine Hydrobromides
Hydrocarbon Model
(Leduc et al.,
1974)
Cp/R
ACp/R
Cp/R
ACp/R
0.93
11.3
30.36
4.2
12.22
10.8
34.52
6.6
23.04
9.9
41.15
10.1
32.92
10.3
51.19
9.5
43.23
10.6
60.69
8.5
53.85
10.5
69.15
8.2
64.32
10.7
77.35
8.8
75.09
86.11
8
131
and continuous potential distributions. This would reduce correlating
inaccuracy and computation time. Also the possibility of using a tem
peraturedependent spherocylinder radius should be studied with the goal
of determining the effect of this change on the temperature dependence of
the length parameter L. Also a deeper literature search, particularly in
foreign sources, for both long chain hydrocarbon solubility data at sev
eral temperatures and infinite dilution surfactant properties may be very
helpful in determining the validity of model predictions for long chain
hydrocarbons.
A more fundamental effect would be correlation of the sperocyl
inder length L with other thermodynamic property data. The work of
Bienkowski and Chao (1975) concerning the hard cores of normal fluids
may provide an initial guide. An appropriate property to model would be
the solute partial molar volume.
CHAPTER 6
MODELING OF THE THERMODYNAMIC PROPERTIES
OF MICELLIZATION
Introduction and Review of Thermodynamic
Process for Micelle Formation
A brief review of a thermodynamic process for micelle formation
(Figure 61) would be appropriate at this point. This process was devel
oped along three parallel branches for surfactant monomers, counterions
and water. The first step involves removal of counterions and surfac
tant monomers from cavities in solution to a gas state of the same density.
Steps 2 and 4 involve aggregation of these dispersed cavities into a
micelle size cavity, while step 3 involves compression of the gas phase
monomers to the density of the micelle interior with restriction of the
monomer head groups to the micelle surface. Step 5 places the compressed
monomer aggregate and counterions into the micelle cavity. A detailed
discussion of the thermodynamic property changes for each step of this
process was presented in Chapter 2.
The objective of this chapter is to present and discuss results
of modeling several steps of this process using the previously discussed
solubility models and models available in the literature. The ultimate
goal is a unified theory for aqueous solutions including micelle formation.
The initial section of this chapter discusses rigid body equations
of state and presents thermodynamically consistent expressions for the
132
Step lb
AS > 0,
'AH > 0
Step 3
AH < 0?
AS < 0
Dispersed Monomers
Compressed Monomers
Ah < o
Mo
@
Step la
AH = ?
Step 2
AH < 0
Step 4
AH > 0 .
r
Step 5a
Ah = ?
0
AJA
AS = *
AS > 0
AS < 0
V
J
AS = ?^
Yiy
Dispersed Monomers'
and Counterions in
Water
Water With
lispersed Cavities
Step lc
AH > 0
AS ~ 0 ?
Water
Micelle in Water
With Bound Counterions
Dispersed
Counterions
Fig. 61. A Thermodynamic Process for Micelle Formation
133
134
chemical potential of a component of a rigid body mixture. Chemical
potential expressions corresponding to the PercusYevick pressure, Percus
Yevick compressibility, and CarnahanStarling equations of state are
presented. The CarnahanrStarling equation is then used to obtain an ex
pression for the thermodynamic property changes upon compression of the
surfactant monomer to the micelle density.
The second section discusses models and specific input data for
other contributions to the overall thermodynamic properties of micelliza
tion. The final section discusses results with a particular emphasis on
the trend in calculated properties with monomer chain length and discus
sion of model inadequacies.
Derivation and Application of an ArbitraryShape
Hard Body Equation of State
Gibbons (1969, 1970) and Boublik (1974) derived an expression
for the chemical potential of a component of a mixture of hard particles
of arbitrary shapes of the form
3
= Â£n
P h
(27r m kT)
3/2
V. p V. AB i V. B2C
+ + x + ~ Â£n (1Y)
(1Y) (1Y)'
(1Y)'
? 12 2 2
a. R.B b. RIA a7 RTB
+ _i_i_ + _k_L_ + 1 i x. ,
(1Y)
(1Y)
(1Y)'
(61)
where p^ = number density of i in the mixture and
P = ? P,
i i
A = E a. R.p.
i 1 IX
B = E b. R?p.
Â¡ill
2 2
C = I a; RTp.
i i iKi
Y = l V^p. = E c. R?p.
i 1 1 11
3 = 1/kT
135
The first term in equation (61) is the ideal gas contribution.. Table 61
gives expressions for a^, b and for particles of several common shapes.
An equation of state for an mcomponent mixture of arbitrary
shaped hard particles can be obtained from equation (61) using
3P "
8\ i1 Pi 3k '
(62)
For a mixture of hard spheres equations (61) and (62) yield the Percus
Yevick compressibility equation of state for mixtures (Lebowitz et al.,
1965).
Gibbons (1970) pointed out that a thermodynamic inconsistency
exists in equation (61). In order to obey the GibbsDuhem equation the
following must apply
3y.
3n.
J
TV'ni
3n.
(63)
Equation (61) will not obey this relation except in the limit
of a mixture of particles of the same shape. Boublik (1975) later gave
an alternative to equation (61) which is thermodynamically consistent.
It will be developed here in a different manner. Also presented will be
expressions for the PercusYevick pressure equation of state and the
CarnahanStarling equation of state for mixtures of arbitrary shapes.
A. PercusYevick Compressibility Equation of State for
Mlxtures of ArbitraryShaped Blg.i'd Bod 1 itn
The compressibility equation is
P
(1Y)
1 r.2^,
3 B C
(1Y)'
+
AB
(1Y)2
(64)
136
Using equations (62) and (63) the corresponding thermodynamically
consistent relation for the chemical potential of a component of the
mixture is
eu!r = An
P h~
(2ir m kT)
3/2
V. p V. AB B2C V.
+ + ^ + 3 _L
(1Y)
b.
(1Y)'
(1Y)'
2 1 2 12 2 2
a. R.B b. R;A b. R.BC j; a? RTB
, ii i i 3 i i 6 r i
+ Jen (1Y) H ^ + z H .
(1Y)
(65)
(1Y)
(1Y)'
(1Y)'
Note the only difference between equations (65) and (61) lies in
splitting the last term into two similar terms.
B. PercusYevick Pressure Equations of State for
Mixtures of ArbitraryShaped Rigid Bodies
The pressure equation is
BP
P P_
1 2
, 3 B AB
(1Y) (1Y)2
(1Y)
2
(66)
Using equations (62) and (63) the corresponding thermodynam
ically consistent relation for the chemical potential of a component of
the mixture is
By.P = An
P h
(27rmi kT)
3/2
V. p V. AB B2C V. a. R.B
+ __ + A + __l + Â£n (1Y)
(1Y) (1Y)
(1Y)
1 2 12221222 3
+ + 3 bi R1 BC + 6*1 *1 + I *f RI B 2 Y)
(1Y> (1Y)2
(1Y)'
(1Y)'
+
in qYn I fhill + in qY)
(1Y)'
i vi 2c r i
(12Y) 2 An (1Y)
1Y)2 (1Y)3 Y
+
(67)
Table 61
Comparison of Properties of Hard Spheres with Those
ofSome NonSpherical Particles
Particle
Sphere
Spherocylinder
Oblate
Ellipsoid
Prolate
Characteristic
Dimension
a.
i
R
(radius)
R, L = aR
(radius)(length)
1 + a/4
a = major axis
b = minor axis
, 2 .2. 2
Â£ = (a b )/a
1
2
(lÂ£2)
1/2
sin ^ Â£
a = major axis
b = minor axis
, 2 V2W 2
E = (a b )/a
1
2
1Â£
1/2
+
IkÂ£_I
2e
1/2
Jin
l+Â£ I
1Â£ I
b.
i
4tt
(4+2a)TT
1Â£ l+Â£
2rr < 1 + Jin
2Â£ 1Â£
2ir < 1 +
sin Â£
2.1/2
Â£(!Â£ )
c.
i
4tt/3
(4/3 +a)n
4u(lÂ£2)1/2/3
4it
3(1e2)1/2
137
138
C. CarnahanStarling Equation of State for Mixtures
of ArbitraryShaped Rigid Bodies
The CarnahanStarling equation is
cs p AB
BP
(1"Y) + (1Y)2
2 2 12
 B C g B C
+ r + 1 ;
(68)
(1Y) (1Y)
Using equations (62) and (63) the corresponding thermodynam
ically consistent relation for the chemical potential of a component of
the mixture is
By^s = In
p h
(2Trmi kT)
3/2
V. p V. AB B2C V. ~ B2C V. a. R.B
+ .24 + _i^+9 + i1 1
UV (1Y)2
(1Y)'
(1Y)
(1Y)
2 1 2 122212 2 2 3
b. RTA 4 b. R7BC 4 af R7B 4 a7 RTB 4 Y)
 in (1Y) + 4=4r + 1 + 9 1 1
(1Y)
(1Y)
(1Y)
(1Y)'
4. (1YA 9 bj RiBC r(1 2 Y) Â£n (1Y)
Y J Y 1 (1Y)2 Y
i V. b2c
9 i
_ + (12Y) 2 Jn (1Y)
2 %3 Y
(69)
'(1Y) (1Y)'
Equations (65), (67) and (69) were obtained by trial and error
on several expressions given the corresponding expressions for hard sphere
mixtures (Reed and Gubbins, 1973), the equation of state and the constraints
of equations (62) and (63). Equation (65) is the same as that derived
by Boublik (1975).
A rigid body equation of state such as those discussed above can
be used to calculate the entropy change upon compression of the dispersed
monomers to the micelle density. This accounts only for the effect of
increased density and does not include intermolecular forces between the
chains in the micelle interior nor chain conformation change. With the
assumption that no enthalpy change is involved in this process
139
TAS = AA = 
V
pdV =
p, 2
M1 p
dp
(610)
where 1 denotes dispersed surfactants and 2 denotes surfactants in the
micellar state. To facilitate integration we let A' = A/p, B' = B/p,
C' = C/p and Y' = Y/p.
Using equation (68) we obtain
TAS
1
(1Y'p)
A 'B'
(1Y'p)2
 b'2cp
(1Y'p)2
2 2 
4 B' C'p
+ ^ T
(1Y'p>
(611)
TAS = RT
Jin
(1Y'p)
, A'B'
Pl Y'(1Y'p)
h2 2
P1 9Y2
iaY'p) +7iW
+
2B'2C'
9 Y
,2
1
) iP.
(1Y'p) 2(1_yf
P)
K
Upon rearrangement
equation (612) yields
AS P n
(1Y2)/p2
A2B2 Vt B2C2
Y2
Jin
(lY1)/p1
Y2P2(1_Y2) Y1P1(1_Y1) 9Y2p2
_dY2)2
+ in (1Y )
B1C1
9Y2P,
!k1 L(lYi)
j + in (1Y )
(612)
(613)
Contributions to a Model for the
Thermodynamics of Micellization
This section considers the contributions to a model for the
thermodynamics of micellization for a series of sodium surfactants, those
of octyl, decyl and dodecyl sulfate.
A. Cavity Aggregation
Steps 2 and 4 of Figure 61 involve monomer, counterion and
micelle cavities. First, the thermodynamic properties associated with
140
removal of the monomer cavities from solution were modeled identically
to those for the equal chain length hydrocarbon of Chapter 5. For
example, the octyl sulfate has the same chain length as octane. This
represents the assumption that the surfactant headgroup is not included
in the micelle. The Gibbs free energy expression for formation of a
monomer cavity in aqueous solution is
AG AG
Rf ET 2,r Y (V5) L + Y Wd 86Y< Wf > (614>
The sodium counterion cavity properties were modeled as equiva
lent to those for a spherical nonpolar molecule with a temperature inde
O
pendent diameter of 2.02 A. This is the bare sodium ion diameter. The
Gibbs free energy expression for formation of a counterion cavity in
solution is
AG. AG ,
Cl rpf oo oo
er = y (a a ,) 8y 6(r .r ,)
1 ci ref 1 Cl ref
(615)
aRT RT v~ci
where a is the fraction of counterions bound in the Stern layer of the
micelle. In this model a = 0.75 as suggested by the experiments of
Evans et al. (1978).
In order to alleviate overcrowding in Table 62, the monomer and
counterion cavity results were combined to yield
AG
me
RT
I_AG
m
ag;
ci
RT
RT
(616)
the negative sign indicating that the cavities have been removed from
solution.
The micelle cavity properties were modeled as equivalent to
those of a spherical solute with diameter
Table 62a
Contributions to Gibbs Free Energy of Micellization
Surf.
T(K)
A G
wm
AG
mcc
AG.
mic
AG
rv
AG .
mmi
AG
ca
cal
AG
exp
RT
RT
RT
RT
RT
RT
RT
RT
SOS
277.15
71 .47
80.55
15.11
0.86
13.83
5. 18
3.49
298. 15
65.75
76. 56
14.24
1 .1 l
12.66
5.18
5.16
 10.65
323.15
59 .53
71.67
13.24
1.28
11.44
5.18
6.43
353.15
51 .63
64. Â£6
11.91
1.36
9. 99
5.18
7 .45
SDS
277. 15
88.35
95.56
13.90
1.75
16.97
6 .64
5.39
293.15
8 1 .26
90.50
13.04
2.04
15. 55
6. 64
7.15
13.00
323.15
73.54
84.36
12. 07
2.23
14.06
6.6 4
3 .40
358.15
63.81
75.89
10.80
2.31
12.29
6.64
9 .23
SDDS
277.15
105.57
110.56
12.52
1.64
19.6 l
8. 23
5.50
298.15
97.07
 104.45
11.70
1 .99
18.05
8 .23
7 .50
15 .37
323.15
87 .83
97.06
10.78
2.24
16.40
8.23
8.85
358.15
76.19
86.92
9. 60
2.36
14.4 1
8.23
9 .67
141
Table 62b
Contributions to Enthalpy of Micellization
AH
AH
AH.
AH
AH .
AH
Ah
ah
wm
mcc
mic
rv
mint
ca
cal
exp
RT
RT
RT
RT
RT
RT
RT
RT
SOS
277.15
78.96
51.27
1 1 .70
0.00
19.48
0. 0
15.91
298.15
77.77
57. 80
12.13
0.01
18.05
0 .0
14.07
0.05
323. 1 5
76.82
63.46
12.63
0.05
16.54
0.0
5.50
358.15
75.94
68.65
13.25
0.24
14.70
0.0
e.cs
SDS
2 77. 15
97.95
65.26
11.62
0.00
23.90
0.0
20 .42
298.15
96 .46
72.85
11.87
0.0 1
22.17
0.0
13.32
C. 0
323.15
95 .23
79.34
12. 17
0.04
20.32
0.0
7 .77
358.15
94.05
85.06
12.55
0.20
18.08
0 .0
3 .66
SDDS
277.15
1 17 .32
79.24
11.13
0.00
27. 62
0. 0
21.64
298.15
115.52
87. 50
1 1.27
0.00
25.74
0.0
13.15
0.13
323.15
114.00
95.22
11.41
0 .00
23.70
0.0
6 .48
358.15
112 .48
101.48
1 l .59
0.04
21.20
0. 0
1.44
m
Table 62c
Contributions to Entropy of Micellization
Surf.
T(K)
AS
wm
R
AS
mcc
R
AS.
mic
R
SOS
277.15
7 .49
29.28
3.4 2
298.15 .
12.02
18.76
2. 1 1
223.15
1 7.29
8.21
0.6 1
358.15
24 .27
3.79
1 .34
SDS
277.15
9.60
30. 3C
2.2 8
298.15
15 .20
17.66
1.17
323.15
21 .69
5. 02
0. 09
358.15
30.24
9.17
1.75
SDDS
277.15
1 1 .75
31 .32
1 .34
298.15
18 .45
16.55
0. 43
323.15
26.17
1.84
0.62
358.15
36 .30
14.56
1 .99
AS
rv
R
AS .
mmi
R
AS
ca
R
AS
cal
R
AS
exp
R
0.87
5.65
5.18
23.39
1.12
5.39
5.18
19.23
1 0 .60
l .33
5.10
5.13
15.94
1.60
4. 71
5. 18
13.54
l .76
6.93
6.64
25.80
2.05
6.62
6.64
20. 47
1 3. 00
2.27
6.2 7
6.64
16.17
2.51
5.79
6.64
12.89
1.64
8.0 1
8.23
27. 15
1.99
7. 69
8.23
20.65
15.50
2.24
7.31
8.23
15.33
2.40
6. 79
3. 23
11.11
143
144
d3. = 6n V /it (617)
mic mxc
where n is the mean micelle aggregation number and V ^ is the partial
molar volume of surfactant monomers in the micelle. Table 63 lists
values and literature sources for V and n at 298.15K. The aggrega
mic
tion number was assumed to be temperature independent. The partial molar
volume was given a temperature dependence modeled after the results of
Shinoda and Soda (1963) for sodium tetradecyl sulfate.
V (T) = V (298.15)
mic mic
1 +
T 298.15
(
700
(618)
The Gibbs free energy of formation of the micelle cavity on a per monomer
basis is designated AG. in Table 62a.
mic
B. WaterMonomer Interactions
For step 1 of the thermodynamic process we calculated the water
monomer interacions which are eliminated upon removal of the monomer in
the same manner as the hydrocarbon solubility model. (The waterhead
group interactions are assumed unchanged upon micellization.) The inter
action free energy is designated as AG^/RT in Table 62a.
C. Counterion Adsorption
Interaction of bound sodium counterions with the water was
assumed to be unchanged upon micellization. This is equivalent to
assuming that they remain hydrated upon micellization as assumed by
Stigter (1964).
Adsorption of sodium counterions to the micelle (step 5) results
in a considerable decrease of entropy. Since the translation entropy is
proportional to Zn V where V is the specific molar volume, the entropy
decrease can be modeled as
Table 63
Parameter Values for Micellization Model
Surfactant
n
V .
mic
X
cmc
*
V
x*
(cc/mole)
(liter/mole)
(K)
Octyl Sylfate
27a
184.7b
2.36X103 C
2.3022
8.5426X102
350.56
Decyl Sulfate
41
219.2
5.59X104
2.6969
10.231 X102
365.54
Dodecyl Sulfate
64
253.1
1.46X104
2.9004
12.373 xio2
385.87
A nm
m
n
1
2
3
4
5
i
7.04677
7.22636
3.16538
14.34352
1.26227
2
3.56999
11.35209
10.85375.
3.6131
7.34334
Aniansson et al. (1976).
^Tanaka et al. (1974) with V += 6.7 cc/mole added.
Na
Q
Mukerjee and Mysels (1971),
146
AS = RT in V /V = RT in p /p
ca cm cs cs cm
(619)
where p is the number density of counterions in solution and p is
cs cm
the number density in the micelle Stern layer. The. density in solution
can be calculated from
p = (1a) p (620)
cs ss
where p is the number density of surfactant monomers in solution which
ss
can be calculated from the mole fraction at the critical micelle concen
tration X The density of counterions in the Stern layer of the
cmc
micelle can be calculated from the expression of Stigter (1964)
p = 2(1a)/v e' (621)
cm
where v is the volume per adsorption cell in the Stern layer and can be
estimated from Stigter as
v = (185 n) A3
0
for a hydrated sodium ion radius of 2.5 A. An approximate correction
for communal entropy is given by e' = 2.718. Enthalpy effects were not
modeled in the case due to the various complex conflicting models avail
able (Stigter, 1965b).
D. Monomer Compression and Interaction
This section of the model encompasses a portion of step 1 and
step 3 of Figure 61. The first process involves relaxation of the rota
tional and vibrational constraints placed on the surfactants monomers by
aqueous environment, compression of the monomers to their micellar
density and activation of rotational and vibrational constraints placed
on them by neighboring monomer chains in the micelle. Then intermolec
ular forces between the monomers are activated.
147
Both processes were modeled using the perturbed hardchain theory
of Beret and Prausnitz (1975). Although the compression step could be
modeled using a hardbody equation of state as discussed in the second
section of this chapter, we used throughout the perturbed hardchain
theory to maintain consistency.
For a pure fluid, the generalized van der Waals partition
function is
( )N( )N( )N
^ ^ideal gas^repulsion' qattraction; qr,v'
(622)
here ^repulsion VV> qr,v KV'"ezp = 1 and
^attraction exp (*/2kT)
For the initial state of the first process
N
Q. = Q., n (q )
i ideal gas r,v aq
(623)
whereas for the final state
Q, = Q., (q n )N. (q )N.
f ideal gas repulsion mic r,v mic
(624)
Since the Helmholtz free energy change for the process can he evaluated
by AA = kT Â£n Q../Q. we obtain
f i
V V V
AG = RT in (c1) RT in + (c1) RT Â£n
mc V V V
mic
mic
aq
= c RT Â£n ~~ + (c1) RT Â£n ~~ (625)
mic aq
where we have assumed, as in Chapter 5, that (j) = 0 for purposes of
vibrational and rotational calculations.
For the process of activating intermolecular interaction between
surfactant monomers in the micelle the initial state partition function is
148
( )N ( )N
^ideal gas',^repulsionmicV4r,v mic
and the final state partition function is
( )N ( )N ( )N
^f ^ideal gas^repulsion micr,v'mic4attractionmic
Thus the monomermonomer interaction Gibbs free energy is
.
AG
mmi
= RT
2kT
(626)
From the perturbed hardchain model
Vf 3
Â£n =
f 'l2
T
ll
V
(627)
where T = 0.7405, v = v/v and v is the molar volume. Also from the
perturbed hardchain model (Gmehling, 1979)
4>
2kT
5 m A
nm
2
E
n=l m=l T v
(628)
where T = T/T Values of v T and A are given in Table 62.
nm
These are values for the equivalent chain length hydrocarbons.
E. Experimental Values
The experimental standard state Gibbs free energy of micellization
can be evaluated from the mole fraction at the CMC as
;o
AG
= (1+ct) RT In X
n cmc
(629)
For a discussion of the limitations of this expression consult Chapter 2.
The other experimental properties were obtained from the appropriate
temperature derivatives of equation (629) with the assumption that a and
n were temperature independent. X data were taken from the critical
cmc
review of Mukerjee and Mysels (1971). .
149
Discussion and Suggestion for Future Research
Table 62 presents the model results for the various contributions
to the thermodynamic process for micelle formation. Probably the best
approach to a discussion of these results is to compare the entropy and
enthalpy change for each step as hypothesized in Chapter 2 with the calcu
lations of this chapter.
Step 1 involves removal of surfactant monomers and counterions
from their cavities in aqueous solution. The present model makes the
uncertain assumption that the counterions undergo no change in interac
tion with water upon micellization mainly because no method is readily
available to model such a change. Thus we assume the ion hydration
effects of steps lc and 5c cancel.
Thus, step 1 was modeled by calculating properties associated
with elimination of monomerwater interaction. No model was hypothesized
for changes in waterhead group interactions upon micellization because,
again little knowledge seems to exist concerning such changes. (Host
investigators including the author tend to exclude the head group from
the micelle proper, but some local water structure rearrangement around
the headgroup may occur.) The sign of the calculated enthalpy change is
negative, while the calculated entropy change is positive. Originally
O'/Connell and Brugman (1977) had predicted that the entropy effect of the
monomer cavity was the driving force for micellization. However, the
calculations show that, as with paraffin solubility, the major entropy
effect arises from the interactions between monomers and water. Here,
as with hydrocarbons, the entropy contribution from cavity formation
became less negative as chain length increased. This is partly from the
150
negative temperature derivative of the surface tension and partly from
the variation of the curvature parameter 6 with temperature. The monomer
cavity entropy contribution is highly temperature sensitive; results at
277K show an increasingly negative entropy with increasing chain length,
whereas those at higher temperatures show an opposite trend.
A portion of step 1 of the thermodynamic process, the relaxation
of constraints on rotational and vibrational degrees of freedom of the
monomers by their aqueous environment, has been included in the calcula
tions for step 3 and will be discussed later.
Step 2, involving removal of monomer and counterion cavities from
the water, was viewed as the dominant driving step for micellization in
Chapter 2 with a large positive entropy change and a modest negative
enthalpy change being predicted. Indeed, the calculations agree. However,
the cavity enthalpy change is large, counterbalancing the large positive
enthalpy change from watermonomer interactions. Also, the entropy change
is unusual, the magnitude for the large monomers is less than that for the
much smaller counterions. This is a reflection of the decrease in entropy
with increasing length for the spherocylindrical cavities discussed above.
Micellization is an entropy driven process with large enthalpy effects
cancelling in the range of ambient temperature. Note that in Table 62,
the properties for the monomer and counterion cavities have been lumped
to alleviate crowding and are denoted with subscript mcc.
The compression of monomers to micelle density (step 3) was
hypothesized to have negative entropy and positive enthalpy changes.
However, the present model does not consider the electrostatic repulsions
at the micelle surface and thus the attractive forces of monomermonomer
151
interaction result in a large negative enthalpy change. A significant
negative entropy change arises from monomermonomer interactions involv
ing both density and energy effects. Monomermonomer interaction contri
butions are denoted by subscript mmi in Table 62.
The compression process also involves changes in rotational and
vibrational freedom when the surfactant is transferred from the reduced
water density to the reduced micelle density. Step lb in the thermo
dynamic process involves an increase in rotational and vibrational
freedom since the reduced density decreases sharply from the aqueous
solution state to the ideal gas state. The reverse is true of step 3
where the monomer is compressed to the micelle state. The results in
Table 62 denoted by subscript rv are the combination of these two
steps. The net result is that expected since the reduced density of the
monomers is less in the micelle than in aqueous solution. However, the
extremum as a function of chain length is unexpected.
Step 4, the creation of a micelle cavity, was originally
hypothesized to involve a negative entropy change and a positive enthalpy
change. The signs of the changes are found, but the enthalpy change
dominates the small entropy change. This arises from the spherical gas
solubility model where the curvature dependence of the surface tension,
6, dominates. The temperature dependence of the micelle partial molar
volume also results in a significant temperature dependence for the
micelle diameter. Note also that the. temperature dependence of the
aggregation number has not been included, although it does vary with
chain length. The micelle cavity contribution is denoted by subscript
mic in Table 62.
152
Step 5 was modeled considering only the entropy change upon
counterion adsorption onto the micelle, yielding the expected negative
entropy change. This effect is denoted by subscript ca in Table 62.
A comparison of results from the total model (denoted by
subscript cal in Table 62) with experiment shows that the model predicts
a positive entropy change partially balanced by a positive enthalpy
change leading to a negative free energy of micellization (however not
as negative as experiment). Which significant effects have been omitted
from the model?
Changes in interaction between the counterions and water upon
micellization are likely to result in a positive enthalpy change and
positive entropy change assuming loss of counterion hydration upon
micellization. Restriction of the monomer headgroups to the micelle
surface is likely to lead to an entropy decrease which is dependent
significantly on chain length since the shorter chain is likely to have
a greater fraction of its rotational and vibrational freedom eliminated.
This should help improve the chain length variation of the total entropy
which is not strong enough in the present model. In addition, an excess
positive enthalpy change will occur due to the electrostatic repulsions
between head groups constrained to the micelle surface. The adsorbed
counterions will moderate this repulsion to some significant degree;
AH should not be zero,
ca
Appreciable negative enthalpy and entropy changes would arise
from inclusion of watermicelle interactions in the model; however, since
only a modest fraction of each monomer chain is close enough to the
micelle surface to interact with water, the enthalpy change may be
153
insufficient, when combined with the above considerations, to counter
balance the positive total enthalpy of the present model. The consequent
negative entropy change would improve agreement with experiment but
would not be likely to vary with chain length.
The overall effect of missing contributions is likely to lead
to a small (possibly negative) enthalpy contribution with increased
monomermonomer interaction in the micelle and watermicelle interac
tions cancelling to a large extent the electrostatic repulsion between
head groups and loss of hydration of counterions. The overall entropy
change should be negative due to restriction of monomer headgroups to
the micelle surface and watermicelle interaction.
If these missing contributions are insufficient to lead to an
accurate model of the properties of micellization, what aspects of the
present model are questionable and what alterations could be recommended?
As noted from Table 62, the total enthalpy is a small difference
of large numbers, the two largest contributions being the enthalpy of
monomer and counterion cavity removal AH and watermonomer interaction
mcc
AH^. However, the relative difference between the monomercavity enthalpy
and watermonomer interaction enthalpy is severely restricted since they
are the overwhelming contributions to the total hydrocarbon solubility
enthalpy. The only conceivable change would be shortening of the portion
of the hydrocarbon chain considered to be included in the micelle. From
results of the hydrocarbon solubility model this would yield improved
results and is justifiable since water may penetrate the micelle to some
extent.
Also easily noted from Table 62 is the large magnitude of the
watermonomer interaction enthalpy relative to the monomermonomer
154
interaction enthalpy AH .. The difference between these two contribu
tions is uncomfortably large. A portion of this is due to using dif
ferent methods of calculating the two contributions. Perturbed hard
chain theory (used for monomermonomer interactions) should be used to
calculate the watermonomer interaction properties and compared with the
present watermonomer calculations using the approach of Chapter 5.
Also, since the parameter c of perturbed hardchain theory is a measure
of restriction on the degrees of freedom of the molecule, c should cer
tainly be larger than that for the equal chain length hydrocarbon (value
presently used) due to restriction of the monomer headgroups to the
micelle surface. A larger c will increase the monomermonomer interaction
properties, improving the final result. Also, some specific effects may
be important since the orientation of the monomers relative to each other
is more favorable to interaction along the entire length of the chain
than in a pure hydrocarbon liquid. Also an effort should be made within
the perturbed hardchain model to ascertain the validity of the maximum
of the rotationalvibrational contributions as a function of chain length.
Ultimately a lattice model may be necessary for the micelle inter
ior if perturbed hardchain theory proves incapable of dealing with the
special orientational features present.
Dealing with those contributions completely missing from the
present model should be the most challenging and potentially rewarding.
Clearly the area of the electrostatic interactions involved in micelle
formation for ionic surfactants is in need of clarification judging from
the multitude of possible models suggested by Stigter (1975b). Probably
one should deal first with a nonionic surfactant, data permitting.
155
A model for watermicelle interactions (excluding headgroup water
interactions)should be approached initially in an analogous manner to
the watermonomer interaction model of Chapter 5. A potential can be
distributed uniformly along the monomer chain with the large micelle
radius limiting the effective interaction to the portion of the poten
O
tial within a very few A of the micelle surface.
Considering the many changes necessary for a rigorous model it
is difficult to argue that the present models is on firm ground. However,
since these changes are both unfavorable and favorable toward quantita
tive improvement of the present model, considerable possibility exists
for the model to be successful. The ultimate problem will undoubtedly
be overcoming the small difference of large numbers associated with these
numerous contributions and thus the search for high quantitative accuracy
may be fruitless. However, the knowledge to be gained, potentially
useful in other areas is likely to be worth the effort.
CHAPTER 7
EXPERIMENTAL INVESTIGATION OF PHASE BEHAVIOR AND TRANSITIONS
FOR CONCENTRATED SURFACTANT SOLUTIONS
Introduction
The initial section of this chapter discusses the objectives of,
and justification for, this experimental investigation into phase behav
ior and transitions for concentrated surfactant solutions. This is fol
lowed by a description of the experimental apparatus as well as general
operating procedures. Pertinent calculations for dissolved gas experi
ments are also included.
The final section of this chapter discusses experimental results
with a view towards suggestions for future research.
Experimental Objectives
The objective of this experimental study was to ascertain the
effect of temperature, pressure and dissolved gas (CH^ CO^) on the phase
behavior of surfactant formulations of interest in tertiary oil recovery,
particularly with regard to the isotropicanisotropic transition common
to microemulsion and lyotropic liquid crystal systems. Such a transition
is particularly important since the anisotropic phase may exhibit suffi
cient viscosity to seriously damage an oil well undergoing tertiary
recovery. It would be highly desirable to ascertain the conditions under
156
157
which an isotropic low viscosity system may be transformed into a high
viscosity anisotropic system.
The gases were investigated since considerable methane may be
present in wells which are not depleted of their natural pressure and
carbon dioxide is of considerable interest as a potential alternative to
surfactant systems for tertiary oil recovery.
Description of Experimental Apparatus
Figure 71 illustrates schematically the experimental apparatus
used. Pressure limitation is 10,000 psi; the limit for the High Pres
sure Equipment 30 cc capacity pressure intensifier. The pressurizing
medium was mercury except above HP2 where hydraulic oil was used to
protect the 15,000 psi temperature compensated Heise gauge. The gauge
has 20 psi graduations with an estimated reading accuracy of 10 psi.
A mercury reservoir and mercury level indicator were used to
control the mercury level and refill the pressure intensifier with the
aid of compressed air.
The sample under investigation was contained in an Aminco high
pressure optical absorption cell equipped with quartz windows. The cell
pressure rating is 50,000 psi at 400K, 75,000 psi at room temperature.
Total sample volume is about 9.5 cc with about 35% of this total visible
through the windows. The optical cell was housed in an insulated box
and heated with a high temperature heating tape.
Due to pressure requirements a thermocouple well was drilled at
one end of the cell. Thus considerable time (~ 57 hrs) was required
Fig. 71. High Pressure Experimental Apparatus
158
159
between temperature changes to help insure that the entire cell had
reached a uniform temperature.
Temperature measurements utilized a copperconstantan thermocouple
calibrated with boiling points of lower molecular weight alcohols, water
and anisolewater mixtures. As expected a fairly linear relationship was
obtained between the temperature T and thermocouple mf, EMF:
T(C) = 0.680 + 25.0855 EMF 0.4328 EMF2 (71)
where the EMF is in mV. The thermocouple emf was measured within
3
2 x 10 mV with a Leeds and Northrup 7556 guarded potentiometer and
9834 1 null detector. Measurement of a quantity of gas introduced into
the cell was accomplished using a section of larger inside diameter high
pressure tubing of known volume (7.86 0.07 cc) between HP9 and HP10.
Pressure was measured with a 300 psi Heise temperature compensated gauge.
The carbon dioxide was Matheson commercial grade (99.5% min. purity) and
the methane was Matheson C.P. grade (99.0% min. purity).
Auxiliary equipment not shown in Figure 71 included a Gaertner
cathetometer used to magnify the view of the optical cell and determine
relative volumes for multiphase systems. A simple photoelectric cell
detection system was constructed to investigate changes in birefringence
of the sample. Polarized plastic was mounted to fit over the cell win
dows while being protected from the heating tape. A photoelectric cell
was mounted behind one of the polarizers. A Hewlett Packard 3439 A
digital voltmeter was used to measure the amplified photoelectric cell
output voltage. However, since the experimental results are mainly
qualitative this apparatus provided little improvement over direct human
observation.
160
Operating Procedures
Introduction of mercury into the pressure intensifier was
accomplished by forcing mercury from the reservoir with compressed air
until the level in the reservoir sightglass remained constant. The inten
sifier was then used to force some mercury back toward the reservoir,
sweeping any trapped air from the intensifier region. The portion of the
system bounded by HP4, HP6, HP8 and HP10 was evacuated to about 0.4 mm Hg
pressure through HP5 to remove any air which might be inadvertently intro
duced into the sample. Mercury was then displaced from the intensifier
into this evacuated region.
The optical cell was disconnected from the system at HP10.
A small quantity of mercury (~ 12 cc) was injected into the cell prior
to the sample to provide both a dense material to facilitate mixing as
well as a material, other than the sample, which could be displaced from
the cell. The sample was introduced with a syringe in sufficient quan
tity to insure that very little air space could exist in the cell.
The cell was reattached at HP10 and pressurized with the
intensifier. Pressure relaxation was minor after a pressure change was
accomplished, indicating that very little air was trapped in the cell.
After desired temperature and pressure studies were made on the sample,
the cell was sealed from the system by closing HP10.
To begin the procedure for introducing a gas sample into the
cell, the pressure was released with the intensifier and mercury to the
right of HP4 drained into either the mercury reservoir or into a container
which could be attached at LP2 in place of the vacuum line. Considerable
161
care was taken to thoroughly remove all mercury to the left of HP10 to
prevent blockage of the gas sample from the cell. Once this was done
HP6 was closed and the system evacuated through HP5 and LP2.
With HP7 closed a sample of gas was introduced into the line
between the tank and HP10. Valve HP9 was closed and the system evacu
ated to remove all gas except in the measuring line between HP9 and
HP10. Valve HP8 was closed, HP7 opened and mercury forced from the
intensifier into the evacuated region. HP9 was then opened and the pres
sure increased to compress the gas into a very small volume near HP10.
With HP10 closed, HP11 was opened to allow expansion of trapped
air in the cell to force a small quantity of mercury into the line from
HP11 to HP12. HP10 was opened and the system pressurized to force
mercury in the connection from HP10 into the cell along with perhaps a
portion of the gas sample. HP10 was again closed and the material
trapped between HP11 and HP12 drained through HP12. A small sample of
the cell contents was again drained into the HP11HP12 line and the
cell repressurized. This process was repeated until one could be reason
ably certain that the entire gas sample was in contact with the liquid
sample in the cell. Adequate pressure was then used to insure that the
gas was dissolved in solution (note calculations in next section) and
desired conditions were investigated. Introduction of more gas into the
sample was accomplished by repeating this procedure.
162
Pertinent Calculations for Dissolved Gas Experiments
The potassium oleate microemulsion system studied had the follow
ing formulation.
1.75 g
8.8 cc
40.0 cc
16.0 cc
40.0 cc
potassium hydroxide
oleic acid
hexadecane
hexanol
water
A reaction between potassium hydroxide and oleic acid in water produced
a potassium oleate solution to which the other components were added.
On a mole fraction basis the sample composition was
0.011 potassium oleate
0.054 hexadecane
0.051 hexanol
0.884 Water
Since the pressure at which the gas sample was introduced into
the measuring volume (7.86 cc) was quite low (74.7 psia) the ideal gas
law was sufficient to determine the amount of gas added to the sample
PV 3
n = ^ = 1.63 x 10 moles.
Ri.
This corresponded to a mole fraction of added gas of about 0.0072.
The ratios of added gas to other key components was
gas/potassium oleate = 0.65
gas/hexanol = 0.14
gas/hexadecane = 0.13 .
163
A calculation of the approximate bubble point pressure was
required to insure that sufficient system pressure was maintained to
keep all the added gas in solution.
Empirically it is well established that for a sparingly soluble
gas the solubility is proportional to its vaporphase fugacity (partial
pressure)
f2 = p2 = K2,l x2 (7~ }
provided the gas pressure is not too high. Since we could, at best,
estimate the Henry's constant K for carbon dioxide or methane in this
^ 9 L
microemulsion system using the value for water, this expression was
probably sufficient for a firstorder estimate. Furthermore, since the
Henry's constant for carbon dioxide or methane in the larger alkanes is
less than that for water (de Ligny and van der Veen, 1972), this calcula
tion should provide an upper bound on the pressure requirement. The
results are shown below.
K 1 (Wilhelm et al., 1976) P
(psi) (psi)
C02 24,055 174
CH4 586,354 4230
Both these pressures are easily within the capability of the experi
mental apparatus.
164
Table 71
Temperature Dependence of Two Phase Region
Temperature
> ~ 65C
62 65C
59 62C
56 59C
53 56C
A) High Pressure Apparatus
P = 600 psia
Anisotropic Fraction Estimated Fraction
of Visible Sample of Whole Sample
0.00 0.350
0.400.450
0.43 0.50
0.45 0.55
0.55 0.65
0.00
0.15 0.35
.0.25 0.50
0.30 0.60
0.60 1.00
Temperature
> 65C
60 65C
56 60C
50 56C
45 50C
40 45C
< 38C
B) Constant Temperature Oven
P = 14.7 psia
Anisotropic Fraction
of Sample
0.00
0.00 0.20
0.15 0.25
0.25 0.40
0.40 0.67
0.67 0.75
1.00
165
Table 72
Pressure Dependence of Two Phase Region
Temperature = 67.5C
Pressure (psia)
440
1440
2000, 3500, 7000
Anisotropic Fraction
of Visible Sample
0.33
0.30
0.30
Table 73
k
Effect of Dissolved Methane
P = 7000 psi
Temperature Anisotropic Fraction
of Visible Sample
75C
68C
64C
61C
Q.25
0.30
0.50
0.90
Mole fraction of added methane was 7.2 x 10 as noted
in previous section.
166
Results and Suggestions for Future Work
Considerable difficulty maintaining a pressure seal and
interpretation of early results limited the scope and depth of this
investigation.
As apparent from Table 71, the fraction of the visible sample
which is anisotropic decreases with increasing temperature reasonably
smoothly, indicating a firstorder phase transition. The results are
presented as a range of values since considerable variation may occur,
probably due mainly to temperature control problems with sample aging as
a possibility. An independent experiment at atmospheric pressure in
a constant temperature oven confirmed these results. Since only about
35% of.the sample volume was visible between the windows of the pressure
cell, the temperature range for the twophase region is only about one
third that of the atmospheric pressure experiment in which the entire
sample was visible.
Table 72 presents results concerning the effect of pressure on
the relative phase volumes. The temperatureanisotropic fraction rela
tion differs from that of Table 71 since this is a different sample of
the potassium oleate system. The results of Table 72 clearly indicate
no effect of pressure on the relative volumes. Some earlier results
appeared to suggest enhancement of the anisotropic phase as a function
of increasing pressure. However, given early experimental difficulties
and improved technique, the latter results are probably more accurate.
Table 73 concerns the influence of dissolved methane on the
phase behavior. No significant enhancement of the isotropic or anisotropic
167
phase could be noted. No significant change in the visual appearance
of the sample resulted from methane introduction.
Table 73 concerns the influence of dissolved methane on the
phase behavior. No significant enhancement of the isotropic or aniso
tropic phase could be noted. No significant change in the visual appear
ance of the sample resulted from methane introduction.
Future efforts should attempt to reinforce or refute these
results with more detailed investigations and better temperature control.
Since interest is high in its use for tertiary recovery, investigations
into the effects of carbon dioxide should be undertaken. If future
results warrant, the effect of surfactant sample aging may be worth
investigating.
Surfactant formulations of more immediate importance to enhanced
oil recovery should be screened fairly rapidly to ascertain whether they
exhibit more sensitivity to pressure, temperature or dissolved gas. This
may provide more interesting candidates for detailed study.
CHAPTER 8
SUMMARY AND CONCLUSIONS
A thermodynamic process for micellization has been developed
which should provide a basis for a better understanding of molecular
mechanisms important in the formation of micelles as well as other pro
cesses of aqueous solution. Analysis of this process supports the
hypothesis that micellization at normal temperatures is primarily driven
by large positive entropy changes of the water when the monomers are
aggregated.
2. A model for the aqueous solution properties of spherical
gases has been developed using a modified scaled particle theory (cavity
j
creation) for the excluded volume effect and a mean field theory for the
intermolecular interaction contribution. Correlation of the experimental
data is quite good. The results are highly sensitive to the temperature
j
dependence of the curvature effect on the water surface tension. The
I
j
results show that the enthalpy, entropy and heat capacity are more sensi
I
I
tive to the temperature dependence of the curvature effect than to that
I 00
of the planar surface tension y For example, a nonzero second deriva
!
I
five of the curvature dependence is essential to obtain a reasonable
I
I
correlation of the heat capacity. Also, significant changes in the inter
!.
action energy parameter can easily be compensated by a reasonable
change in 6.
168
169
3. Extension of this model to aliphatic hydrocarbons involved
development of an expression for the total interaction energy between
a spherocylindrical solute and a spherical solvent. In the perturbation
theqry used, .the radial distribution function was considered as a function
of distance from the spherocylinder surface while the intermolecular
potential was distributed along the spherocylinder axis.
4. Correlation of solution properties for the gaseous aliphatic
hydrocarbons (C^C^) is quite good and predicted trends for the liquid
hydrocarbons,for which little data are available, are reasonable. The
results are quite sensitive to the chain segment length L which has
CH2
a stronger temperature dependence than the hard sphere diameter of the
spherical solutes. This leads to a large interaction entropy contribution
which actually dominates the cavity contribution at higher temperatures.
5. A partial model for the thermodynamic properties of ionic
micellization was developed based on the aqueous solubility model.
Contributions to the thermodynamic process which were not modeled as well
as present inadequacies in modeled effects resulted in poor quantitative
agreement with experiment. The desired result is a small difference of
large contributions and thus good agreement may not be obtainable. The
large entropy increase due to loss of monomerwater interactions upon
micelle formation suggests that the excluded volume effect is not the only
i .
I
significant entropy driving force for micellization.
6. A limited experimental investigation, on the potassium
oleate system, into the effect of temperature, pressure and dissolved
gas on the isotropicanisotropic transition for lyotropic liquid crystals
I
j
was conducted. Tentative results show a twophase region between 40 and
170
80C and little or no effect of pressure or dissolved gas on the isotropic
anisotropic transition except possibly for transients.
APPENDIX A
PROGRAM FOR CORRELATION OF SPHERICAL GAS
SOLUBILITY PROPERTIES
172
FORTRAN IV G LEVEL 21
MAIN
DATE
79108
00/49/39
000 1
0002
0003
0004
0005
0006
0007
0008
C PROGRAM TO FIT AQUEOUS SPHERICAL GAS SOLUBILITY DATA TO A FUNCTIONAL
C F CRM FOR THE CURVATURE DEPENDENCE PARAMETER DEL
C
C
C EXPLANATION OF INPUT DATA
C CV CHARACTERISTIC VOLUME OF SOLUTE CC/GMOL
C CT CHARACTERISTIC TEMPERATURE OF SOLUTE K
C ESI CHARACTERISTIC ENERGY PARAMETER FOR SOLUTE K
C CVW CHARACTERISTIC VOLUME FOR WATER CC/GMOL
C CTW CHARACTERISTIC TEMPERATURE FOR WATER K
C TT TEMPERATURES OF INPUT EXPERIMENTAL SOLUBILITY DATA K
C EXG DIMENSIONLESS EXPERIMENTAL GIBBS FREE ENERGY OF SOLUTION
C EXH DIMENSIONLESS EXPERIMENTAL ENTHALPY CF SOLUTION
C EXS DIMENSIONLESS EXPERIMENTAL ENTROPY OF SOLUTION
C EXCP DIMENSIONLESS EXPERIMENTAL HEAT CAPACITY CF SOLUTION
C NN NUMBER OF SOLUTES USED IN FITTING, MINE IN THIS CASE
C N NUMBER OF UNKNOWN VARIABLES TO eE DETERMINED
C M NUMBER OF PIECES OF DATA FIT
C SXCl,EXC2EXC3 COEFFICIENTS IN EQUATION FOR ARGON SOLUBILITY DATA
C
C ARGON WAS USED AS A REFERENCE SOLUTE
C
IMPLICIT REAL* 8 ( AH 0Z )
DIMENS ION CVI9) ,CT{9),EXS(9,4),EXH{9,4),EXG( 9,4) ,EXCP(9,4),GC(9 ,4)
2 GGI (9,4) ,HC(9 ,4) ,HI (9,4) SC{94),S I (9,4 ), CP CI9,4),CPI(9,4),TT(4) ,
3CSSI 9*4)CGS(9,4) CHS(94) ,CCPS(9,4) ,X(3),F(36),W(1000) .EPS I (9)
COMMON/A/CV,CT,EXS,EXH.EXG,EXCP.EPSI,TT,EXC1.EXC2.EXC3
CCMMON/B/CSS.CGS.CHS,CCPS
CCMMON/C/GC.GGI,HC,HI,SC,SI,CPC,CPI
CCMMON/D/CVW, CT W
M = 36
N=3
FORTRAN Â£ V G LEVEL
21
MAIN
79 1 oa
00/49/39
DATE =
0009
00 1 O
00 1 1
0012
00 1 3
00 1 4
00 1 Â£
00 l 6
001 7
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
003 1
0032
0032
0034
003 5
0036
NN=9
READ (5,1) ((EPSI (I ).CV(I) ,CT{ I ) ), I = t,9)
1 FORMAT (3F10.4)
READ (5,2) CVW.CTW, EXCl ,EXC2, EXC3
2 FORMAT (5E14.6)
00 3 K=1,4
READ (5 ,4 ) TT(K)
4 FORMAT (FIO.4)
DC 5 1=1,NN
READ (5,6) EXS(I K) ,EXH(I K),EX G(IK)* EX CP(I K)
FORMAT (4F10.4)
5 CCNT I NU c
3 CONTINUE
C INITIAL ESTIMATES OF PARAMETERS TO BE DETERMINED
X(l )=8.06
X ( 2) =2.53
X(3) = 19.16
C VA05AD IS A NONLINEAR FITTING ROUTINE FROM HARWELL
CALL VA05AD(M,N,F,X,1.0D03,l0.0,1.00,100,10,W)
CALL CALFUN(M,N,F,X)
WRITE (6,10)
10 FORMAT Cl*,///////////)
DC 7 L=i ,3
DO 3 K=l 4
WRITE (6.9) TT(K),GC(L,K),GGI(L,K) ,CGS(L,K),EXG(L,K) ,SC(L ,K) SI
2K),CSSL,K) ,EXS(L.K)
9 FORMAT (/,10X,F6.2.3X.F5.2.2X,F6.2,2X,F5.2,2XF6 .3,8X,F6.2,2X,F5.2
2.22X.F6.2))
8 CONTINUE
7 CONTINUE
WRITE (6,11)
11 FORMAT Cl',///////////)
FORTRAN IV G LEVEL 2i
MAIN
CATE
79 108
00/49/39
0037
0038
0039
0040
004 1
0042
0043
0044
0045
0046
0047
0048
0 04 9
0050
005 1
0052
0053
0054
0055
0056
005 7
0058
0059
0060
006 I
0062
0063
OC 12 L = 4,6
DC 13 K 1 ,4
WRITE (6,9) TT(K) ,GC(L,K) GGI (L.K)
2K),CSS(LK), EXS (L K )
13 CONTINUE
12 CONTINUE
WRITE (6,14)
14 FORMAT (* 1 ,///////////)
DO 15 L=7, 9
CGS(L.K),EXG( L K ) ,SC(L,K),SI(L.
DC 16 K=1.4
WRITE (6.9) TT(K) iGC(LiK) GGI (LK) CGS L K),EXG(L,K)SC(L,K),SI(L*
1K ) CSS (L K } EX S (L K )
16 CONTINUE
15 CONTINUE
WRITE (6i17)
17 FCRMAT* 1* ///////////)
DO 13 L=1,3
DO 19 K=l,4
WRITE (6,20) TT(K),HC(L.K),HIL.K).CHS(L.K),EXH(L,K),CPC(L,K) CP I (
2L,K)CCPS(L,K),EXCPL,KJ
20 FORMAT /,10X,F6.2,3X,F5.2,2X,F6.2.2X.F.2,2X.F7.3,aX,FS.2.2X,F6.2
2.2X,F5.2.2X.F5.2)
19 CONTINUE
18 CONTINUE
WRITE 16,21)
21 FORMAT {1,//////////}
DO 22 L4,6
DO 23 K=l,4
WRITE (6,20) TT(K),HC(L.K),HI(L,K),CHS(L,K),EXH(L,K),CPC(L,K),CPI
2
23 CONTINUE
22 CONTINUE
FORTRAN IV G LEVEL 21
MAIN
DATE
79i oa
00/49/39
0064
WRITS (6,24)
0065
24
FORMAT (i'.///////////)
0066
DO 25 L=7 9
0067
DO 26 K=1,4
0068
WRITE (6.20) TT(K),HC ,CPI(
2L.K) .CCPS(L.K) .EXCP(L.K)
0069
26
CCNTINUS
0070
25
CONTINUE
007 1
STOP
0072
END
175
FORTRAN IV G LEVEL 21
CALFUN
79103
00/49/39
DATE =
0001
0002
0003
0004
0005
0006
0007
0008
0009
001 0
00 11
00 12
00 13
001 4
0015
00 16
001 7
001 8
0019
002 0
0021
SUBROUTINE C AL FUN ( M N F X )
C OBJECTIVE CALCULATE VALUES WHOSE SUM SQUARE ERRCR IS MINIMIZED
C BY VA05AC TO OBTAIN DESIRED PARAMETER VALUES
C ENTROPY AND GIBES FREE ENERGY OF SOLUTION ARE FIT FOR LAST 6 SOLUTES AT
C 277, 293 *323 K
C ARGON IS USED AS REFERENCE SOLUTE HELIUM AND NcCN ARE NCT FIT
IMPLICIT REAL*8(AH.OZ)
D IMENSION HSDG(9) T 7 )Â£P SI(9)DH SDG(9)CV(9) CT(9) DCWG(9) ,DDCWG
2(9) ,X{3),F(36) EXG(9.4),EXS(9,4).EXH< 9,41,GI{7,9) ,GGI(9* 4) ,SC(9,4)
3 HC( 5*4) HI (7*9) HHI(94) H 11 C7) ,GC(94) .CPC(9,4).CPI (9,4),DEN GR(
49),TRG(9),SI(7,9),DHH{7 ),W( 10 00) ,HHSOW(4) ,HHSDG(9,4) ,AREAC(9,4) ,
5TT(4)EXCP(9,4) ,CSS(9 ,4),WAC(7,9),HAC(7,9),HH(7),SS I (9,4),SAC( 7,9)
6, DHI I( 7) ,CCPS( 9,4) ,CGS( 9,4) .CHS (9 ,4)
COMMON/A/CV.CT,EXS,EXH.EXG.EXCP.EPSI.TT.EXCl.EXC2.EXC3
COMMON/8/CSS.CGS.CHS,CCPS
COMMON/C/GC,GGI ,HC,HHI,SC,SSI ,CPC,CPI
CCMMCN/D/CV W. CTW
P1 = 3 141592C5D 0
XK=1.33066
RK=1.987
C COEFFICIENTS IN EQUATION FOR HARD SPHERE DIAMETERS
A1=0 .54008832
A2=l .2669302
A3=0 .051323 55
A4=2.9107424
A5=2.5167259
A6=2.1595955
A7=0.64269552
A8=0.17565885
A9=0.1 8874824
A10=l7.952388
A t 1=0. 48197123
176
FORTRAN IV G LEVEL 21
CALFUN
date = 79ioa
00/49/39
0022
0023
0024
0025
0026
0027
002 8
0029
0030
003 I
0032
0033
0034
0035
0036
0037
0038
0 039
A12=0.76696099
A13=0.76631363
A 14= C. 809657804
AI5=0.24062863
DC 4 K I .4
DO I 1 I=1,9
DC 10 J=1*7
T{ J) =TTCK)+C J4>* 0.500
TC=T(J)273.15
STW=li6.200.1477*TCJ)
D STW=0.1477
C SECTION TO CALCULATE WATER AND SOLUTE HARD SPHERE DIAMETERS
DENW=0.033433*(0.99984252 + 16*945227D03*TC7,987064 ID06*{TC**2)4
26 .1 7C6D09*C TC**3) + 10 5. S6334D12*( TC**4)280 .54337015* CTC**S ) )/ C 1
3.000+16.87985003*TC)
DDENW=0.033433*C16.945227D0315.9741282D06+TC133.5118009*(TC**
22) +422.25336D 12*( TC**3 ) 1 402.7 1685D15*( TC**4) ) / ( 1.00 + 1 6.87985D0
33*TC)DENW*(16.87985003)/(1.0+16.879585003*TC)
TRW=T(J)/CTW
DENWR=DENW*CVW/{ 0.6023)
DDENWR=DDENW*CVW/(0.6023)
HSDW = ((3.0*CVW/C2.0*PI* 0.6023))*(A7/(TRW**A8)+A2/(DEXP(A4*((DENWR +
2A 1*TRW)**2.0)) )A3/CDEXPCA5*({DENWR+A1*TRWA6)**2)))+A9/CDEXPCA 10*
3CCCTRWA13)**2)+AIl*C(DENWRA12)**2))))))**0.333333
DHSDW=i .00 00*C C3.0*CVW/C2.0*PI*0.6023) )*CA7*A8/CCTW*CTRW*CA8+1 .0
2) ))2.*A2*A4* C 0ENWR + A1*TRW)*CDDENWR+A1/CTW)/C DEXPC A4*C CDENWR+A1*TR
3W )**2.0)) ) + 2.0*A3*A5* CDENWR+A1*TRWA6)*C DDENWR + A1/CTW)/CDEXP C A5*C C
4DEN W R + A 1 *TR WA6 ) **2 .0 .) ) )2.0*A9*A 1 0* C C TRWA 1 3 ) /CTW + A 1 1 C DENWRA 1 2)
5*DDENWR)/CDEXP{Al 0* C CCTRWA13)* *2)+A11 CC DENW RA 12 ) **2 )))>))/{C HSD
6W**2.0)*3.00)
TRGC I)=TCJ)/CTC I)
TRAR=TCJ)/CTC3)
0040
004 1
FORTRAN IV G LEVEL 21
CALFUN
DATE
79 108
00/49/39
0042
004 3
0044
0 04 5
0046
0047
004 8
0049
0050
005 1
0052
0053
0054
0055
0056
005 7
0058
DENGR( I }=DENVR
IF (CT(I).LT.lOO) GO TO T9
HSDG{I ) = ( (3.0*CV(I )/( 2 O* PI 0 6 023) )* C A7/ (TRGI ) * A8)+A2/
2( {DENGR+Al* TRG(I)A6>**
32 ) ) ) +A9/(DEXP( A 10* ( ( (TRG ( I A13 ) **2 ) +A11 { ( DENGR ( I ) A l 2 ) **2 ) ) ) ) ) ) *
4*0.333333
HSDAR (3.0 *CVC 3)/(2.0*PI*0.6023) )*(A7/(TRAR + *A8)) )**0.3333
DHSDGCI )=1.0000*((3.0*CV(I )/(2.0*PI*0.6023)) *(A7* A3/ ( CT ( I) (TRGl I
2) **( A 8 + 1.0) ) )2.0*A2*A4*( DENWR+A 1 *TRG ( I ) ) (DDENWR+ Al /CT ( I ) ) / (DEXP (
3A 4* ( (DENWR + A1 *TRG( I ) ) *2 .0 ) ) ) +2.0 *A 3*A 5* (DEN W R+ A T TR G( I )A6 ) *( DDEN
4WR + A 1/C T I) )/(DEXP{A5*( {DENWR+A1*TRG(I )A6)**2.0) ) )2.0*A9*A10*((T
5RG { I )A 13 ) / CT ( I )+A1 1 *( DENWRA 1 2) *DDENWR) / (DE XP( A 1 0* ( ( { TRG (I >A13) *
6*2> + Ai 1 *<(DENWRA12)**2))))))/((HSDG(I)**2.0 )*3.00 )
DHSDAR=1.00+( (3.0*CVC 3)/{2.0*PI*0.6023))*(A7*A8/(CT(3)*
2 + 1 .0 )))).)/( (HSDAR** 2.0 ) *3.00)
GO TC 20
19 HSDG {I) =( <3 .0*CV( I )/(2 .O*PI*0.6023))*{A7/(TPG(I)**A8) ) )**0.3333
DHSD G( I ) = l .000 0*( (3.0*CV ( I)/(2.0*PI*0.6023)) (A 7* A8/( CT( I ) *( TRG{ I
2) **< A8+ 1.0) ) ) ) ) /( (HSDGC I ) **2.0) *3.0 0)
HSDAR={ (3.0* CV (3)/(2.0*PI *0.6023) ) ( A7/( TP. AR **A 3 ) ) ) * 0.3333
D HSD AR= 1. 00* { (3.0*CV<3)/(2.0*PI *0.60 23 )) *(A7*A8/( CT ( 3 ) ( TR AR ** ( A 3
2+1.0 )))))/{(HSDAP**2.0)*3.00)
C CAVITY DIAMTSR IS SUM OF SCLUTE AND WATER DIAMETERS
20 DCWG( I)=HSD W+H SDG(I )
D CW G (3 ) = HS DW +HS DAR
DDCWG(I)=0HSDW+CHSDG{I)
DDC WGC 3 ) =DHSDW +DH SOAR
C SECTION TO CALCULATE INTERACTION PROPERTIES FOR ARGON
GIAP=EPSI (3)*(6.7202D+ 004.9540+03/(T J) )+6.5480D+05/(T{J )**2)0.7
2600D+00*HS CAR 0.79250 +0 0*(HSDAR **2) ) / ( T ( J) )
SIAR=EPSI(3)*(4.954D+03/(T(J)**2)l3.096D+05/{T(J)**3)0.76D+00*D
2H SD AR1.595D + 0 0*H SOAR*DHSOAR)
FORTRAN
0059
0060
006 1
0062
0063
0064
0065
0066
0067
0068
0069
0070
007 1
0072
0073
0074
0075
0076
0077
0 078
IV G LEVEL 21 CALFUN DATE = 79108 00/49/39
HIAR = GIAR+S IAR
C SECTION TO CALCULATE INTERACTION PROPERTIES FOR OTHER SOLUTES
G K J,I )=EPSI{I )*< 6.720 24.954D + 03/C T J))+6.548D + 05/(T{J)**2)0.760
2D+00 *HSDG(I )0.7925D + 00*(HSDG1 I >**2) )/{T{J ) }
SI {J ,1 )=EPSI (I)*(4.954D+03/(T{J)**2)l3.096D+05/(T(J)**3)0.76D+0
20*DHSDG(I)1 535D + 0 0 THSDG(I)*DHSDG(I))
H I ( J I ) = G I ( J I ) +S I ( J I ).
C SECTION TO CALCULATE EXPERIMENTAL PROPERTIES FOR ARGON
GEX AR {EX Cl+EXC2*DL0G(T{J ) )+EXC3*(DLOGtT{J) )**2) )
HEXA R=E XC2+2.0*EXC3*DL0G(T(J))
SEXAR=(HEXARGEXAR)
C CAVITY PROPERTIES OBTAINED BY DIFFERENCE
WAC ( J.3)=GE XARGIA P
HAC(Ji3) = HÂ£XAR HIAR
SAC( J.3 > =SEXARSI AR
I F( I .EQ.3) GO TO 12
C SECTION TO CALCULATE CAVITY PROPERTIES FOR OTHER SOLUTES
DEL=X Cl ) + l .OD+03*X(2)/T(J)X{3)*l.00+04/{T(J )T*2 )
DDEL= l .0D+03*X(2)/(T( J)**2) +2. 0*1.00+ 04*X (3 ) /( T ( J ) **3 >
W AC( J. I ) = (PI*RK/XK J *{STW* ( ( DCWG( I ) * 2 ) ( DC WG {3)* *2) ) 4. 0* ( S TV*DEL*
2(DCWG(I)DCWG(3))))/(RK*T
S AC(J I )=(PI/XK)*{STW*{2.0+DCWGl I)*ODCWG{ I)2. 0*0CWG< 3)*D0CWG<3 > )
2 + DST fc* ( DCWG I ) **2 ) { DCVi G (3 ) **2 ) )4.0*ST W *DEL *( DDCXG( I )DDCWG( 3) )
34.0*DSTW*DEL*{0CWG( I )DCWG( 3) )4.0*ST**DDEL*(OCwG(I )DCWG{3) ))
4 + SAC(J,3)
HAC{J,I)=WAC{J,I)+SAC(J,I)
C SECTION TO OBTAIN HARD SPHERE DIAMETERS AND CAVITY AREAS AT INPUT
C DATA TEMPERATURES
12 IF(J.EQ.4) GO TO 35
GO TO 10
35 HHSDM K)=HSDW
HHSDG I K)=HSDG(I )
FORTRAN IV
0079
0030
0081
0082
0083
0084
0085
0086
0087
0088
0089
0090
0091
0092
0093
0094
0095
0096
0097
0098
0099
0100
0101
0 102
G LEVEL 21
CALFUN
DATE = 79108 00/49/39
P
ARE 4C( I ,K )=P I*( (HHSDWt K MHHSDGI I,K ) ) **2>
10 CGNTINUE
C NUMERICAL DIFFERENTIATION SECTION TO O ET AIN HEAT CAPACITIES FRGM
C TEMPERATURE DEPENDENCE OF ENTHALPIES
C DDE T 5 IS A SCIENTIFIC SUBROUTINES NUMERICAL DIFFERENTIATION
C RGUTINE
DC 15 J = l,7
HH{ J)=HAC( J ,1 )*T( J)
HII J)=HI(J, I)*T( J)
15 CONTINUE
CALL DDET5( C.5,HH ,DHH,7 ,1 ER)
CALL DDET5X 0.5, HI I DHI I ,7. IER)
C SECTION TO CONVERT VALUES AT MEAN TEMPERATURE FOR COMMON TO MAIN
S SI( I K)=SI{4,1 )
HHI(I,K)=HI(4,I>
GC{I,K)=WAC(4,1)
HC
SC(I,K)=SAC(4,I)
GGI ( I K )=GI (4,1 )
CPC( I.K ) = DHF(4)
CPI(I,K)=DH11(4)
C SECTION TO OBTAIN TOTAL PPOPERITES
CCPS(I.K)=CFI(I,K)+CPCI,K)
CGS(I,K)=GC(I,K)+GGI(I,K)
CHS ( I.K )=HC ( I, K )+HH I( I, K )
CSS( I,K)=SC(I,K)+SSI(I. K)
11 CONTINUE
4 CONTINUE
C DEFINING FUNCTIONS FOR ERROR TO BE MINIMIZED eY VA05AD. PERCENTAGE
C ERROR IN THIS CASE
DO 6 1=4,9
DO 3 K=1,3
180
FOR TRAN
0 103
0104
0105
0106
0 107
o ioa
0 109
IV G LEVEL 21 CALFUN CATE = 79108 00/49/39 o
J=K+3*(I4)
F ( J) =(EXS( I ,K)CSS { I.K ) )/ (EXS ( I ,K ) ) 100.00
F(JF 18) = (EXGI .K)CGS(I K) )/(EXG(I .K))*100.00
3 CCNTINUE
6 CONTINUE
RETURN
END
181
182
Two sets of watersolute interaction energy parameters ewere
investigated. The first data list which follows corresponds to the
results of Table 45, while the second corresponds to Table Al.
The two resulting functions for the curvature parameter 6 are:
Table 45:
6 = 8.3194896D+00 + 2.6052103EH03/T18.993069D+04/T2. (Al)
Table Al:
= 6.22467929EH00 + 2.1484507D+03/T 14.40244IH04/T2. (A2)
18 APRIL 1979
DATA LIST FOR SPHERICAL GAS SOLUBILITY
23.5820
50.0000
39.0000
c
78. 8007
60.4800
45.2000
c
189.0161
74.9000
150.8000
c
235.1349
88.4600
209.4000
c
282. 9748
l 14. 4600
289.7000
c
232.5133
96.0300
190.6000
c
256. 5033
147. 0000
227.6000
c
295.0468
203.9400
318.7000
c
333.7771
312.0900
433.8000
c
0.464000D + 02 0.438700D+03 0
.4883300+03
c
277.1500
c
13.2008
1.3993
1 1.8020
14.6855
c
l4.5949
3. 0504
1 1.5432
18.8223
c
17.1263
6.9688
10.1588
23.9104
c
18.1832
Â£. 72 74
9.4568
26.4419
c
19.4716
 1 0. 6963
8.7749
31.4142
c
16.0574
7.9536
10.1038
26.7086
c
21.9930
1 0. 1 033
1 1.8910
48.4399
c
25.0780
13.5313
1 1.5480
65.8078
c
27.6598
17.2758
10.3998
69.7333
c
298.1500
c
12.1490
0.2898
11.8631
14.1772
c
1 3. 2511
1.53 91
11.7099
18.1278
c
15.4152
4. 8253
l0.5884
23.4826
c
16.2859
6.2829
10.0036
25.5964
c
l 7. 2169
7.76 92
9.4476
30.5637
c
16.1449
5.54 87
10.5956
25.5511
c
18.5304
6.05 17
12.4784
46.5476
c
20.3774
8. 0395
12.3302
63.0146
c
22.7479
l1.3061
11.4413
65.5313
c
3 23. 1 500
c
11.0216
0. 8177
11.8412
13.8450
c
l1.8067
0.0367
11.7726
17.6950
c
13. 5330
2.6438
l0.8829
23.1 102
c
14.2476
3.8401
10.4096
25.0780
c
14.7811
4.8290
9.9530
29.9799
0.1703 06D+03
0145218D +02
183
18 APRIL 1979
DATA LIST FOR SPHERICAL GAS SOLUBILITY
c
14.1369
3. 1880
10.9457
24.4841
c
14.8364
2.0323
12.8013
45.4404
c
l5.3800
2.6157
12.7637
61 .3689
c
17.6095
 5. 49 89
1 2. 1 1 42
63.4675
c
358.1500
c
9. 61 75
2.0736
1 1 .6912
13 .3166
c
10.010l
1.6784
11.6891
17. 1263
c
11.1676
0.1375
11 .0307
22.5264
c
1 l. 6910
1.03 1 0
10.6574
24.6553
c
11.7212
 1.4444
10.2723
29.4162
c
11.6608
0.5255
11.1334
23.8450
c
10.2063
2. 5628
12.7693
44.68 04
c
9.1344
3.5724
12.7081
60.2365
c
1 l. 1 877
1.1515
12.3357
62 .3251
<781
18 APRIL 1979
DATA LIST FDR SPHERICAL GAS SOLUBILITY
143.2744
50.0000
39.0000
c
175. 0504
6 C. 4800
45.2000
c
245.2719
74 .9000
150.8000
c
273.6779
88.4600
209.4000
c
300.4175
114.4600
289.7000
c
266.2096
96.0300
190.6000
c
261.7464
147. 0000
227.6000
c
282.5186
203.9400
318.7000
c
305.62 08
312.0900
433 .8000
c
0. 464000D+02 0.438700D + 03 i
c
277.1500
c
13.2008
1.3993
11.8020
c
14.5949
3. 0504
ll.5432
c
17.1263
6.96 88
10.1588
c
18.1832
8. 72 74
9.4568
c
19.4716
 10. 6963
8.7749
c
18.0574
7.9536
10.1038
c
2 1. 9930
1 0. 1 033
11.8910
c
25.0780
13.5313
11.5480
c
27.6598
17.2758
10.3998
c
298.1500
c
12.1490
0.2898
11 .3631
c
13. 2511
1.53 91
1l.7099
c
15.4152
4. 8253
10.5884
c
16.2859
6.2829
10.0036
c
17.2169
7.76 92
9.4476
c
16.1449
5.5487
10.5956
c
18.5304
6.05 17
12.4784
c
20.3774
e. 0395
12.33 02
c
22.7479
11.3061
11.4413
c
323. 1 500
c
11.0216
0.8177
11.8412
c
11.8067
0.0367
11.7726
c
13. 5330
2.6438
10.8829
c
14.2476
3.8401
10.4096
c
14.7811
4.8290
9.9530
488330D+03 0.1 7 03 06D+03 0.1452180+02
14.6855
18.3223
23.9104
26.4419
31.4142
26.7086
48.4399
65.80 78
69.7333
14.1772
18.1278
23.4826
25.5964
30.5637
25.5511
46.5476
63.0146
65.5813
13.8450
17.6950
23.1102
25.0780
29.9799
185
18 APRIL 1979
DATA LIST FOR 3PHER ICAL GAS SOLUBILITY
C 14.1369
3. iaeo
1 0. 9457
24.4841
C 14.8364
2.0323
12.3013
45.4404
C 15.3800
2.6157
12.7637
61 .3689
C 17.6095
5. 4989
12.1142
63.4675
C 358.1500
C 9.6175
2.0736
1 1 .6912
13.3166
C 10.0101
1. 6784
1 t.6891
17.1263
C 1 1.1676
0.1375
1 1.0307
22.5264
C 1 1.6910
1.0310
10.6574
24.6553
C 11.7212
 1. 4444
1 0.2 723
29.4162
C 11.6608
0.5255
1 1.1334
23.8450
C 10.2063
2.5628
12.7693
44.6804
C 9.1344
3.5724
12.7081
60.2365
C 11.1877
1.15 15
12.3357
62.3251
Table Ala
Contributions to Free Energy and Entropy of Solution
Solute
T(K)
AG
c
RT
ag
1
RT
AGcal
RT
AG
exp
RT
AS
c
R
AS
i
R
AÂ§cal
R
AS
. exP.
R
He
277.15
17.00
5.21
1 1.79
1 1.8 02
+.001
12.48
0.82
13 .30

13.20
+ .
298.15
16.63
4 .77
11 .86
11 .863
+.001
11.01
1.27
 12.27

12.15
+ .
323.15
16.13
4.29
11.85
11 .841
+.002
9.48
 1. 57
l1.05

11 02
+ .
358.15
15.39
3. 71
11.88
1 1 .691
+.005
7.5 0
1.74
9 .24
9 .62
+ .
Ne
277.15
18.54
6 .99
11.55
11.543
+.001
13.42
1.07
14.49

14.59
+ .
298.15
18.11
6.40
11.71
11 .7 10
+.001
1 1 .65
1.61
13.26

13.25
+ .
323. 15
17. 54
5.77
11.77
11.773
+.002
9 .8 1
1.97
11 .78

11.81
+ .
358.15
16.67
5.00
11.67
11.689
+.035
7.45
2. 18
9.63

10. Cl
+
Ar
277.15
22.07
11.91
10. 16
10.159
+.003
15.38
1.70
17.08

17.13
+ .
298. 15
21.51
10.92
1 0.59
10.588
+.002
12 .95
2.45
15.40

15.4 2
+ .
3 23.15
20.75
9.87
10.88
10.883
+. 009
 1 0.42
2. 95
13.37

13.53
+ .
358.15
19.58
8.60
1 0. 99
11.031
+ .068
7.26
3.23
10.49
11.17
+ .
08
02
09
41
11
03
13
57
08
04
30
96
187
Table Ala (Continued)
Solute
T(K)
AG
e
RT
AG
i
RT
iEal
RT
AG
exp
RT
AS
c
R
AS?
X
R
iÂ§cal
R
AS
exp
R
Kr
277.15
24 .22
14.79
9. 43
9.457
+.001
16.58
1.99
13.58
18.18
+ .10
298.15
23.58
13.58
l 0. 00
10.004
+ .001
13 .72
2.85
16.57
16.25
+ .04
323.15
22.7 0
12.28
10 .42
10.410
+.004
 10.76
3. 40
 1 4. 1 6
14.25
+ .18
358.15
21.35
10.73
10.62
10.657
+.048
7.11
3.71
10.82
1 1 .69
.79
Xe
277.15
27. 65
18.95
8.70
8.775
* 001
18.22
2.4C
20.62
15.47
. 3.2
298.15
26.86
17.41
9.45
9.448
+ .001
14.72
2. 36
18.C9
17.22
+ .04
323.15
25. 76
15.78
10. CO
9.953
+.006
11.13
3.97
15.10
14.78
+.24
358.15
24.13
13.83
10.30
10.272
+ .062
6.74
4. 3 1
11.05
11.72
+ .99
CH.
4
277.15
24.9 2
14.87
10.05
10.104
+.005
16.88
2. C 1
ie.es
18.06
+ .30
298.15
24.24
13.65
1 0. 60
10.596
+.004
13.9 l
2.8 3
16.74
16.14
+ .04
323.15
23.32
12 .35
10.97
10.946
+ .003
10.83
3. 3 6
14.19
 14. 14
+ .07
358.15
21.9 1
10.80
11.11
11.133
+.008
7.04
3.65
10.69
11 .66
+ .29
188
Table Ala (Continued)
Solute
T(K)
AG
c
RT
AG
i
RT
AG
cal
RT
AG
exp
RT
AS
c
R
AS
i
R
Alcal
R
AS
exp
R
CF4
277.15
30 .33
18.43
11.91
11.891
+ 003
19.25
2. 22
21.57
21.99
+ .29
298.15
29. 4 1
16.93
12.48
12 .478
+ .002.
15 .35
3.13
18.48
18.52
+ .06
323.15
28.16
15.36
12.81
12.80 I
+.005
11 .32
3.64
14.96
14.84
+ .30
358.15
26 .27
13.49
12. 79
12.769
+ .083
6.42
3. 90
10.32
10.21
+1.38
SF6
277. 15
35. 94
24.36
11.38
11.548
+. 008
2 l .3 0
2.89
24.19
25.08
+ .30
298.15
34.73
22.40
12.33
12.330
+.010
16 .49
3.78
20.2 7
20.38
+ .08
323.15
33.12
20 .35
12. 77
12.764
+ .005
11.51
4.34
15.85
15.38
+ .36
358. 15
30. 71
17.92
12.79
12.708
+. 101
5.53
4.62
10.15
9.13
+1.74
nC,.
277.15
44 .66
34.25
10.41
10.400
+.087
23.68
3. 92
2 7. 6 0
27.66
+1.94
298.15
42. 96
31.53
11.44
l1.441
+.010
17.56
4.92
22.48
22.75
+ .58
323.15
40.7 5
28 .68
12.07
12.114
+ .015
1 1 .26
5. 33
16.78
 17.61
+; .22
353.15
37.49
25 .32
12. 17
12.336
+.045
3.7 1
5. 84
9.55
11.19
+1.30
Table Alb
Contributions to Enthalpy and Heat Capacity of Solution
AH
AH?
Solute
T(K)
c
1
RT
RT
He
277.15
4 .52
6 .03
298.15
5.62
 6 C3
322.15
6.66
5.85
358.15
7.89
5.45
Ne
277. 15
5. 1 1
8. 06
298.15
6.46
8.0 1
323.15
7.74
7.74
358. 15
9.22
7.18
Ar
277.15
6.69
13.61
298.15
8.56
13 .38
323.15
1 0.33
12.82
358.15
12.33
11.83
AH
AH
ACp
cal
exp
Q
RT
RT
R
1 .50
1.399
.152
21.21
*4
<*
o
1
0.290
.020
19.35
0 .80
0 .8 18
.091
18.87
2 .44
2.074
.387
19. 96
2.94
3.050
.111
25.42
l .55
l .539
.026
23.40
1
o
o
o
0.037
.133
22.65
2.0 4
1 .678
.533
23.35
6.92
6.969
.077
34.62
4.81
4.825
.034
32.22
2 .48
2 .644
.288
30.91
0 .50
0.138
.893
30. 74
ACp
R
o
4Cpcal
R
ACp
exp
R
7.53
13.68
14.69
2.1
4.37
14 .48
14.18
0.6
2.72
16.14
13.85
2.2
0.86
19.09
13.32
3.5
9.12
16 .30
18.82
2.9
5.88
17.52
18.13
0.9
3.26
19.39
17.70
3.0
0 .99
22.35
17.13
4.6
 12.75
21.87
23. 9 1
3.0
3.1 1
24.11
23.48
2.0
l
.
P
O
26 .50
23.11
4.8
 1.22
29.52
22.53
7.0
Table Alb (Continued)
Solute
T(K)
AH
c
RT
ah
1
RT
cal
RT
AH
exp
RT
Kr
277.15
7.64
16 .78
9.15
8.727
+
298.15
9.86
 16.42
6.56
6.283
.
323.15
11.94
15.69
3.74
3.840
358.15
14.24
14.44
0.20
 1 .031
.
Xe
277.15
9.43
21 .35
11.92
 10.696
.
298. 1 5
12. 1 3
20.77
8 .64
7.769
+
323.15
14.65
19.75
5.10
4.829
+
358.15
17. 39
18. 14
 0.75
1 .444
+
CH4
277.15
8.03
16.88
8 .8 4
7 .954
+
298.15
10.33
16.48
6. 14
5.549
323.15
12.43
15.71
3.22
3.188
.
358.15
14.37
14.45
0 .42
0 .526
+
ACp
c
R
ACp
i
R
cal
R
ACp
exp
R
40.83
t 4.56
26.28
26. 44
3.4
37. 81
9.15
28.66
25 .60
0.9
35.39
4.92
31 .07
25.08
4.0
35. 22
1.32
33.9 C
24.66
7.1
4 9. 73
16.44
33.29
31.4 1
4.2
46.06
10.18
35.88
30.56
1.3
43. 2
5. 42
38.20
29.98
5.1
42.04
 1 .45
40.53
29 .42
8.6
42.36
 13.98
28.3 8
26. 7 1
5.7
3 9.32
3.78
30 .55
25.55
2.5
3 7.42
4 .69
32.73
24.48
1.5
36.51
 1.21
35.29
23. as
5.1
099
035
173
734
125
040
231
926
30
05
07
28
191
Table Alb CContinuijd)
Solute
T CK)
AH
c
AH
i
cal
AH
exp
ACp
c
ACp?
X
ACp
exp
RT
RT
RT
RT
R
R
R
R
CF4
277.i5
11.08
2 0.75
9.66
10.103
.
271
55.39
13.85
41.54
48.44 7
298.15
14.06
20.06
6 .00
6.052
+ .
059
5 1.57
8.50
43.08
46.55 1
323.15
16. 85
19.00
2.15
2.032
+
300
48.79
4.40
44.39
45.44 7
358.15
19.85
17.38
2 .47
2.563
1.
29
46.64
0.96
45.6 9
44.68 13
SF6
277.15
14.64
27.25
 12.6 1
13.531
.
298
68. 45
15.56
52.39
65.31 8
298. 15
18.25
26.19
7.94
8 .040
.
074
63.69
9.34
54.35
63.01 1
323.15
21.6 1
24.69
3.03
2.616
+
352
60. 00
4.76
55.23
61 .37 9
358.15
25. 1 S
22. 54
2.64
3.572
1.
64
56.67
1 .05
55 .61
60 .24 17
n~C5
277.15
20.98
38.18
17.19
 17.276
2.
0
86.89
17.57
69. 32
69.73 22
293.15
25.4 0
36.45
l 1 .05
 11 .306
0.
6
80. 89
10.28
70.62
65.58 2.4
323. 15
29. 4 9
34.21
4.72
5.499
0.
23
7 5.95
5.1 9
70.76
63.47 g
358. 15
33.77
31.16
2 .61
1.152
1.
26
7 0.96
1.29
69. 67
62.33 17
,3
,7
2
5
9
5
0
6
6
2
4
5
192
APPENDIX B
HELMHOLTZ FREE ENERGY OF INTERACTION BETWEEN
A SPHEROCYLINDRICAL SOLUTE AND SPHERICAL SOLVENT
APPENDIX B
HELMHOLTZ FREE ENERGY OF INTERACTION BETWEEN
A SPHEROCYLINDRICAL SOLUTE AND SPHERICAL SOLVENT
Integration of the Components of the Helmholtz
Free Energy of Interaction
Consider the case of a potential fixed at y = 0 interacting with
molecular centers in the region 0 < y < L and 0 < z < 00. The expression
for the Helmholtz free energy of interaction (equation 58) is
..A
M.
x
8tt e p
ws w
L
r CO
r i2
6 1
dy
dz' z ghs(z')
a
ws
a
WS
. 2 2,6
, 2 2,3
'o J
b ws
W
(y +z )
(y +z )
(Bl)
We must now evaluate analytically:
L
r i2
6 1
a
a
dy
WS
ws
. 2 2,6
2 2,3
'o
(y +z )
(y +z )
(B2)
This integral is
of the form
dÂ£
, it 2. m+1
(a+by )
which is available in standard
integral tables. Using these tables
L
dy
*0
r i2
a
WS
6
a
ws
12
126L a
ws
i
, 2, 2,6
(y +z )
2 2 3
(y +zV
2
z
512z8(L2+z2)
l
i
1
960z4(L2+z2)3 1120z2(L2+z2)4 1260 (L2+Z2)5
6/t 2 2. 2
768z (L +z )
1 t 1L
r tan
9t z
512z L
194
195
306 L
ws
2 2 2
8z (L +z )
2 2 2
12(L +z )
3
8z L
1 L
tan
(B3)
Combining equations (B3) and (Bl) yields equation (59).
Consider the case of a potential fixed at y = 0 interacting with
molecular centers in the region y > L. The expression for the Helmholtz
free energy of interaction (equation 516) is
r i
. B2
M. = p
i w
d cos 0
a
dwf 8ir w^ eD g^s(w')
w
ws
2 2 3
(L +w +2Lw cos 0)
ws ws
12
7
WS
2 2 6
(L +w +2Lw cos 0)
(B4)
In this case we must evaluate
d cos 0
12
7
ws
(L^+w^+2Lw cos 0)^
0
ws
(L^4w^+2Lw cos 0)^
(B5)
This integral is of the form
1
dx 2 2
where x = cos 0, a = L + w and b = 2Lw.
.m
Jo (a+bx)
From standard tables this integrates to:
ws
lOLw
1 1
6
a
i ws
1 1
2 2 5 2 2 5
_(L 4w ) (L +w +2Lw)_
4Lw
2 2 2 2 2 2
_(L +w +2Lw) (L+v)_
(B6)
Upon rearrangement
1
r 12
6 i
12
0
0
0
d cos 0
0
ws
ws
ws
(L^+w^+2Lw cos 0) ^
(L^4w^+2Lw cos 0)^
lOLw
196
1
6
a
WS
1 1
?
O
1
*
4Lw
4 2 2 2
(L+w; (L +w )
Combining equations (B7) and (B5) yields equation (517).
Consider now the case of a differential potential dd> (r)
ws
.continuously distributed along the spherocylinder axis from y = 0 to
(B7)
y = L interacting with molecular centers in 0 < y < L and 0 < z < 00.
The expression for the Helmholtz free energy of interaction (equation
520) is
r 8ir e p
MC = ws_w
x L
ws
[L
f L
. 00
dx
0
dy
dz z
a
0
w
12
ws
(z +(yx) )
21 6
(z2+(yx)2)3
We must now evaluate analytically the double integral
CL
L
12
6
dx
^0
dy
0
ws
ws
(z2+(yx)2)6
(z2+(yx)2)3
Substituting q = y x, equation (B9) becomes
L
Lx
dx
dq
*0 ^
x
,12
ws
ws
, 2, 2,6 2 2,3
(z +q ) (z +q )
The integral
Lx
dq
12
ws
ws
x
, 2. 2,6 2 2,3
(z +q ) (z +q )
is of the form
(B8)
(B9)
(B10)
dx
(a+bx2)"^1
which can be evaluated from standard tables. The result is
197
' Lx
r 12
6 1
1 2
a
a
126(Lx)a
dq
ws
ws
ws
, 2 2. 6 _
,2 2.3
2
'x
(z +q )
(z +q )
z
512z((Lx)2+z2)
+
+
768z^ [(Lx)2+z2) 960z4((Lx)2+z2)3 1120z2((Lx)2+z2)4
+
1260((Lx)2+z?)3 512zy(Lx)
+ ^ tan"1 Sk2^L
126x a
+
12
ws
_10 8. 2 2.
512z (x +z )
+ . + , ,+ .V f 1
768z6(x2+z2)2 960z4(x2+z2)3 1120z2;(x2+z2) 4 1260(x2+z2)5
1 1 x
tan ~
512z x
3a (Lx)
ws
8z2((Lx)2+z2)2 12((Lx)2+z2)2
. 1 1 (Lx)
H r tan
8z (Lx) z
, 6
3a x
ws
1 + 1
2 2 2 2 2 2
8z (x +z ) 12(xZ+z )Z
1 lx
5 tan
3 z
(Bll)
8z x
We must now evaluate
L
dx INT, where INT denotes the integral
of equations (Bll). This can be written as
f L
dx INT =
r L
dx INT(x) +
dx INT(Lx)
(B12)
where INT(x) denotes terms in x, INT(Lx) terms in Lx.
Let s = Lx, then ds = dx and
ds INT(s).
L
L r
dx INT =
dx INT(x) 
'0 '
0 h
(B13)
198
The terms of equation (B13) are readily evaluated from integral
tables.
o
3a8 s
ds
WS
2
4
z
L J
6
3 a
ws
L + i + _L_ tani jl
a 2, 2 2, + 2 2,2 ^ 3 Can z
8z (s +z ) 12(s +z ) 8z s
L 1 L
r tan
3 z
2 2 2
24(L +z ) 24z
8z
(B14)
f 0 12 
126s cr ^
ws
1 + 1
1 1
c,0 2. 2^ nua 2o. V oiin 2. 2V3 non 2t 2j. M
512z (s +z ) 768z (s +z ) 960z (s +z ) 1120z (s +z )
H ^r i q tan ^
1260(s +z ) 512z s Z
ds
126a
12
ws
1 + 1
1 + 1
1536z8 1536z6(L+z2) 3840z8 3840z4(L2+z2)2
1 + 1
6720z2(l2+z2)3 6720z8 10,080(L2+z2)4 10,080z8
512z
L 1 L
9 tan z
(B15)
dx
3a8 x
ws
1 + 1
8z2(x2+z2) 12(x^+z)i' 8z x
1 1 ^x
2. 2,2 3 tan z
3a
ws
24z2 24(L2+z2) 8z
L 1 L
~ tan
3 z
(B16)
199
dx
r126x 12
ws
+
+
C1. 8, 2, 2. 6, 2 2.2 4. 2^ 2.3' 110ri 2. 2, 2.4
512z (x +z ) 768z (x +z ) 960z (x +z ) 1120z (x +z )
1260(x2+z2)5 512z"x
1 X
9~ tan T
126a
12
ws
1536z6(L2+z2)
+
1536z8 3840z4(L2+z2)2 3840z8 6720z2(L2+z2)3 6720z8
+
10,080(L2+z2)4 10,080z8 512z
L 1 L
9 tan T
(B17)
Combining equations (B8), (B14), (B15), (B06) and (B17)
yields equation (521).
Consider finally the case of a differential potential, d<> (r)
continuously distributed along the spherocylinder axis interacting with
molecular centers in y < 0 and y > L. The expression for the Helmholtz
free energy of interaction (equation 525) is
a12
ws
c
fL
r 1
i
_ 167T p e
,.D w ws
dx
d cos 0
* 2 hs *
dw w g (w ^
a/i. 
x L
o ^
o J
aw
(x2+w2+2xw cos @)8
ws
(x2+w2+2xw cos 0)3
(B18)
Again a double integral must be evaluated analytically,
evaluate the integral
First
f 1
d cos 0
12
7
WS
o
ws
(x2+w2+2xw cos 0)8 (x2+w2+2xw cos 0)3
200
Again this can be evaluated with integral tables to yield
1
d cos 0
12
7
ws
6
7
WS
2 2 6 2 2 3
(x +w +2xw cos 0) (x +w +2xw cos 0)
12
a
ws
2xw
5(x2+w2)5 5(x+w)10
+
ws
2xw
1 .
A 2 2 2
2(x+w) 2(x +w )
(B19)
Four different integrals over x must now be evaluated and
combined.
12 rL
a
ws
lOw
dx
0 x(x+w)
12
o
ws
10 lOw
10 ,T.v9 10 2, ',8 10 3 \ 2
w w(L+w) w w (L4w) w w (L+w)
9w^^(L+w)^ 8w^2(L+w)^ 7w^3(L+w)7
10 *,T.,x6 10 3, 5 10 6, .4 10 7, .3
+ w ~w (L+w) + w w (L+w) w w (L+w) + w w (L+w)
6w14(L+w)6 5w15(L+w)5 4w16(L+w)4 3w17(L+w)3
+
2w
L i
2w+L
10 w+L
2 + 1
10 10
w w
&n
W+X
(B20)
a12
ws
lOw
L
12
dx
a
ws
, 2, 2,5 lOw
x(x +w )
10 8, 2 V 10 6/T2 2,2 10 4, 2 2.3
w w (L +w ) w w (L +w ) w w (L +w )
18,2, 2, 16, 2 2,2 14, '^ 2,3
2w (L +w ) 4w (L +w ) 6w (L +w )
. w102 a2+uV 1
8w12(L2^2)4 210
Â£n
2. 2
x +w
Li
(B21)
rL
ws
4w
dx
, 2^ 2,2
0 x(x +w )
ws
8w
w
2 2
(L +w )
1 + Â£n
CM
L
2 2
x +w
o *
(B22)
a
ws
4w
dx
a
ws
o x(x+w)4 4w3
t 2
2w+L
w+L
+ W.3.~(J^.)3 2 +
3 3
3w (L+w)
*
L
n x
Jn ;
W+X
t J
o
(B23)
201
Combining equations (B18), (B20), (B21), (B22), and (B23)
yields equation (526).
Correlation of the Helmholtz Free Energy of Interaction
with CT^, L, and Temperature
The integral expression for the several contributions to the
Helmholtz free energy of interaction between a spherocylindrical solute
and a spherical solvent were numerically integrated as a function of
spherocylinder length L at several values of the solute diameter
o o o*
O (3.40 A, 3.60 A, 3.80 A) and temperature T (277.l5K, 298.15K,
s
323.15K, 358.15K). The resultant integral values were fit to various
functions of L at constant Oand T as noted in Chapter 5. The coeffi
cients in this fit at 277.15K, 298.15K and 323.15K were used to
solve for the coefficients in a quadratic temperature function.
C=A+BT+DT2.
n n n n
These parameters (A B D ) are given for the three a values in
n n n s
Tables Bla, Blb, and Blc.
(B24)
202
Table Bla
Parameters for Temperature Dependence of
Interaction Correlation Coefficients
a
s
= 3.40 A
Coefficient
A
n
B
n
D
n
V
0.341711DQ1
0.119371D01
0.209290D04
C2
0.218037D+01
0.118486D01
0.186620D04
C3
0.104553EH01
0.556065D02
0.903900D05
V
0.141446D+00
0.814235D04
0.131700D05
C5
0.148324D+01
0.297438D02
0.623100D05
C6
0.532135D+Q0
0.485514D02
0.815600D05
C7
0.100441D+02
0.755970D01
0.124682D03
C8
0.543575D+02
0.493098D+00
0.813850D03
C9
0.103527D+03
0.4 53733D+00
0.688320D03
C10
0.471167D+01
0.00 D+00
0.00 D+00
C11
0.196545D01
0.00 D+00
0.00 D+00
C12
0.387170D04
0.00 D+00
0.00 D+00
C13
0.135525D+01
0.266010D02
0.572700D05
C14
0.480870D+00
0.174845D03
0.522000D06
C15
0.385621IHO0
0.115702D03
0.190000D06
203
Table Bla (Continued)
Coefficient
a =
A
n
o
= 3.40 A
B
n
D
n
C16
0.257968D+00
0.924807D03
0.173800D05
C17
0.654435D01
0.560267D03
0.105100D05
n
i1
OO
0.436910D02
0.960170D04
0.178000D06
C19
0.130702D+00
0.101761D01
0.173590D04
C20
0.172373D+01
0.944589D02
0.140810D04
C21
0.800159D+00
0.451760D02
0.684800D05
C22
0.107662D+00
0.639490D03
0.979000D06
C23
0.213548D+01
0.269035D02
0.565300D05
C24
0.226508D+01
0.226790D02
0.516900D05
C25
0.460063D+01
0.626760D02
0.354270D04
C26
0.390806D+02
0.145317D+00
0.116600D03
C27
0.923215D+02
0.478891D+00
0.574200D03
C28
0.104253D+02
0.354899D01
. 0.717490D04
C29
0.416220D+00
0.406758D02
0.308600D05
o
o
0.892476D+00
0.910517D03
0.878000D06
C31
0.166259D+00
0.274624D03
0.152000D06
C32
0.221053IW02
0.719806D01
0.157688D03
C33
0.311827D+01
0. 752350D01
0.135211D03
C34
0.543894D+02
0.1013810+01
0.180780D02
0.326974DK) 3
0.264588D+01
0.447786D02
204
Table Blb
Parameters for Temperature Dependence of
Interaction Correlation Coefficients
O
a = 3.60 A
s
Coefficient
A
n
B
n
D
n
C1
0.14Q151D+00
0.133659D01
0.235170D04
C2
0.246585D+01
0.137794D01
0.222010D04
C3
0.116605EH01
0.657885D02
0.107410D04
C4
0.157925D+00
0.944590D03
0.155900D05
C5
0.156608D+01
0.299701D02
0.625300D05
C6
0.696151D+0Q
0.609446D02
0.103910D04
C7
0.143505D+02
0.106288D+00
0.178138D03
C8
0.832188D+02
0.706721D+00
0.118478D02
C9
0.191270D+03
0.151845D+01
0.254194D02
C10
0.547606IH01
0.00 D+00
0.00 D+00
C11
0.189400D01
0.00 D+00
0.00 D+00
C12
0.378880D04
0.00 D+00
0.00 D+00
C13
0.145017D+Q1
0.270027D02
0.577000D05
C14
0.473159D+00
0.276299D03
0.665000D06
C15
0.332932D+00
0.108019D03
0.192000D06
205
Table Blb (Continued)
a
s
O
= 3.60 A
Coefficient
A
n
B
n
D
n
C16
0.208121D+00
0.119422D02
0.209200D05
C17
0.278107D02
0.931600D03
0.159900D05
C18
0.117020D01
0.186010D03
0.314000D06
C19
0.677770D01
0.122746D01
0.210890D04
C20
0.190840D+01
0.108925D01
0.167870D04
C21
0.104334D+01
0.626399D02
0.989200D05
C22
0.143316D+00
0.895200D03
0.142300D05
C23
0.219251D+01
O.28865OD02
0.597000D05
C24
0.709688D+00
0.888180D02
0.138890D04
C25
0.282467D+02
0.154860D+00
0.240000D03
C26
0.152080D+03
0.918331D+00
0.143775D02
C27
0.280279D+03
0.177179D+01
0.287610D02
C28
0.107466D+02
0.424579D01
0.844100D04
C29
0.886122D+00
0.484657D03
0.300900D05
C30
0.150783M00
0.470217D02
0.102950D04
C31
0.119407D01
0.841499D03
0.171300D05
C32
0.241907IH02
0.849652D01
0.183420D03
C33
0.301193D+02
0.138118D+00
0.221850D03
C34
0.261866D+03
0.975024D+00
0.150355D02
C35
0.458463D+03
0.248906D+01
0.408856D02
206
Table Blc
Parameters for Temperature Dependence of
Interaction Correlation Coefficients
Coefficient
0
s
A
n .
O
= 3.80 A
B
n
D
n
C1
0.849988D+00
0.722813D02
0.13Q440D04
C2
0.104030D+01
0.401482D02
0.543200D05
C3
0.533194D+00
0.223701D02
0.327500D05
C4
0.726711D01
0.362763D03
0.557000D06
C5
0.165575D+01
0.294656D02
0.614200D05
C6
0.590079D+00
0.515182D02
0.838100D05
C7
0.113939D+02
0.825450D01
0.131290D03
C8
0.525518Df02
0.487378D+00
0.775848D03
C9
0.110752D+03
0.953354D+00
0.151580D02
C10
0.577680D+01
0.000 D+00
0.000 IH00
C11
0.214413D01
0.000 D+00
0.000 D+00
C12
0.422360D04
0.000 D+00
0.000 D+00
C13
0.133347D+01
0.409142D02
0.798500D05
C14
0.392375D+00
0.886390D04
0.9100Q0D07
C,c
0.288964D+00
0.266120D03
0.433000D06
207
Table Blc (Continued)
a
s
O
= 3.80 A
Coefficient
A
n
B
n
D
n
C16
0.384936D+00
0.682190D04
0.343000D06
C17
0.154952D+00
0.124761D03
0.900000D07
C18
0.195367D01
0.237170D04
0.220000D07
C19
0.123251D+01
0.207174D01
0.352720D04
C20
0.437732D+01
0.274843D01
0.446380D04
C21
0.211111IH01
0.134720D01
0.220160D04
C22
0.296671D+00
0.193113D02
0.316500D05
C23
0.216043D+01
0.365620D02
0.724300D05
C24
0.442062D+01
0.437975D01
0.724830D04
C25
0.985060D+02
0.624044D+00
0.102510D02
C26
0.504996D+03
Q.327473D+01
0.537615D02
C27
. 0.873619D+03
0.573767D+01
0.941360D02
C28
0.108286D+02
0.509862D01
0.989470D04
C29
0.199940D+01
0.185842D01
0.259000D04
C30
0.187161D+01
0.583554D02
0.670900D05
C31
0.272927D+00
0.780400D03
0.920000D06
C32
0.279471D+02
0.890740D01
0.194720DQ3
C33
0.836984D+00
0.689649D01
0.122525D03
C34
0.247125EH02
0.768669D+00
0.141372D02
C35
0.132752D+03
0.162651EH01
0.276010D02
APPENDIX C
PROGRAMS FOR GAS AND LIQUID HYDROCARBON
SOLUBILITY PROPERTIES
FORTRAN IV G LEVEL 21
MAIN
DATE
79 109
22/43/30
N3
O
VO
000 l
0002
C OBJECTIVE TO MODEL SOLUBILITY PROPERTIES OF GASEOUS HYDROCARBONS
C EXPLANATION OF INPUT DATA
C FDF EXPLANATION CONSULT LIQUID HYDROCARBONS PROGRAM LIST
C
IMPLICIT REAL*a(AH0Z)
DIMENSION S XS( 4,4) ,EXH(4,4) ,EXG(4,4) ,EXCP(4,4),GC(4,4),GGI(4,4),
2HC(4 ,4 ) HI(4.4),SC(44).SI(44) CP C ( 4,4),CPI (4,4) ,HSD W{4) ,
3AREAC(4,4) CSS(4,4) ,CGS(4,4) .CHS(4,4 ), CCP3 (4 4),TT(4 ) .F{ 13), X(3),
4W(500),H SD G(4,4),CPR V(4,4) ,GRV( 4,4) ,HRV(4,4) ,SRV(4,4) ,HSDCH4(4),
5UL4 ) T C { 9 9 ) ,EPS 1(4),C(4)
0003
C OMMON/A/EXS,EXH,E XG,EXCP
000 4
COMMON/ B/GC GGI HC HI, SC, SI .CPC ,C=>I HSDW HSDCH4 UL
0005
CCMMON/C/CSS,CGS,CHS,CCPS,TT
0006
C CMM OM/D/GR V HRV.SPV.CPRV
0007
C CMMON/E/TC.EPSI,C,CVW,CTW,EXCl,EXC2.EXC3,CVCH4,EPSIAP ,CTCH4
0008
CCMMON/F/CVAR,CTAP
0009
READ (5,1) (T C(K) ,K= 1,99)
00 l 0
1
FORMAT (6E13.6)
00 1 1
READ (5,2) ICPSI(L),L = l ,4)
00 12
2
FORMAT (4F10.4)
0013
READ (5,3) { C{ I ) ,1=1 ,4)
0014
3
FORMAT (4F10.4)
001 5
READ (5,4) CVW,CTW, CVCH4, EPS IAR,CTCH4,CVAR,CTAR
0016
4
FORMAT (7F1 0.4)
00 17
READ (5,7) EXC1,EXC2,EXC3
00 18
7
FORMAT (3E15.6)
0019
DO 17 K=l,4
002 0
READ (5,6) TT(K)
002 l
6
FORMAT (FI 0.4)
0022
DO 9 1= 1,4
0023
READ (5.10) EXS(I,K),EX H(IK),EXG(I,K).EXCP( I,K)
0024
10
FORMAT (4F10.4)
FORTRAN IV G LEVEL 21 MAIN DATE = 79109 22/43/30
0025
9
C CNT INUE
0026
17
CONTINUE
0027
X c 1) =1.9133
0028
X (2) =5.187
0029
X{3> =3.395
0030
CALL VA05AC{18,3,F,X, 1.0003, 10.00. 1.00, 100, 10,W)
003 1
CALL CALF UN (18 ,3,F X )
0032
WRITE (6.100)
0033
100
FORMAT (1'.///////////)
0034
DO 101 1=1,3
0035
DO 102 K= 1,4
0036
WRITE (6,103) TT(K),GC(I.K),GGI(I,K),GRV( I K ),CGS( IK) ,E XG{ I K)
0037
1 03
FORMAT (/, 1 5X.F8.2,3X ,F6.2 .3X.F7.2 ,3X F6.2 ,3X ,F6 .2,3X, F7 .3)
0038
102
CONTINUE
0039
l 01
CONTINUE
0040
WRITE ( 6,1 00)
0041
DO 104 1=4,4
0042
DO 1 05 K=1 ,4
0043
WRITE (6,103) TT(K) ,GC(I.K) ,GGI 11,K) ,GRV( I,K),CGS(I.K).EXG(I,K)
0044
105
continue
0045
104
continue
0046
WRITE (6.100)
0047
DO l 06 1=1 ,3
0048
DO 107 K=1,4
0049
WRIT2(6,103) TT CK ) HC ( I ,< ) .HI ( I ,K ) ,HRV(I K ) CHS( I K) ,E XH{ I ,K)
0050
1 07
CONTINUE
005 1
106
CONTINUE
0052
WRITE (6,100)
0053
00 108 1=4,4
0054
DO 109 K=1,4
0055
WRIT5(6 l03 ) TT(K) ,HC(I ,K),HI(I,K),HRV (I,K ). CHS{ I K).EXH{ I,K)
0056
1 09
CONTINUE
210
OR TR
0057
0058
0059
0060
006 1
0062
0063
0064
0065
0066
0 06 7
0068
0069
0070
0071
0072
0073
0074
0075
0076
0077
0078
0079
0 08 C
008 1
0082
0033
LEVEL 21 MAIN DATE = 79109 22/43/30 P
108 CONT INU ~
WPI TE (6,100 >
DO 110 1=1,3
DC 111 K=1*4
WRITE(6 ,103 ) TT(K} SC ( I K ) ,S I ( I K ) S RV ( I K ) CSS( I,K),EXS( I,K)
111 CONTINUE
110 CONTINUE
WRITE (6.100)
DD 112 14,4
DO 113 K=l,4
WRITS(6,103) TT (K).SC(I,K).SI(I.K).SRV{I,K), CSS( I,K) ,EXS(I,K)
1 13 C CNTINUE
112 CONTINUE
WRITE (6.100)
DO 1141=1,3
DO 115 K=1,4
WRIT= (6.1 03) TT(K ) CP C (I.K),CPI(I,K ), CPRV ( I K ) CCPS( I.K ) ,EXCP{ I ,K
2)
115 CONTINUE
114 CONTINUE
WRITE (6,100)
DO 1 IS 1=4,4
DC 117 K=1 ,4
WRITE (6,103) TT(K) ,CPC(I ,K) ,CPI(I ,K) ,CPRV(I ,K) ,CCPS(I ,K) ,EXCP(I.K
2 )
1 17 CONTINUE
116 CONTINUE
STO
Â£ ND
ho
I1
FORTRAN
000 I
0002
0003
0004
0005
0006
0007
oooa
0009
00 1 o
00 l 1
00 l 2
00 L 3
001 4
0015
00 1 6
00 1 7
00 1 8
00 19
0020
002 1
0022
0023
0024
0025
IV G LEVEL 21 CAL FUN DATE = 79 109 22/43/30
SUBROUTINE C ALFUN ( M N F ,X )
IMPLICIT FE AL*8( AH *0Z )
DIMZNSION C(4) ,T(7), TT(4),DCH4(4),SRV(7,4 >,HRV( 7,4),GRV(7,4) ,UL(7)
2 *'AC (7 *4) S AC(7,4) H AC (7,4)X(3)GI(74)SI(74)HI(74) ,UUL (4) ,
3HHSDW 4),HH(7),HI I{ 7> SSI (4,4) HH I ( 4,4) ,GC(4 ,4) .HC(4,4} SC(4 4) ,
4GGI(4.4).CPC(4.4)*CPI(4,4),CCPS(4,4),CGS(4,4).CHS(4,4),F(18),HARV(
54,4) ,GAFV( 4,4) SAR V( 4 ,4 ) CP AR V(4 ,4) DHHH ( 7 ) HHH ( 7) ,TDCH4 (4 ) OHH( 7 )
6,DHII(7 ),CSS(4 4),EXG(4,4),EXS( 4,4) ,EXH(4,4) ,EXCP(4.4) ,XTC(99) ,
75PSI (4)
C CM MON/A/EXS.EXH,E XG,EXCP
COMMON/B/GC,GGI,HC.HHI,SC.SSI,CPC.CP I.HHSDW.DCH4.UUL
C CMMCN/C/CSS.CGS.CHS,CCPS,TT
C CMMON/0/GAFV.HAPV,SARV,CPARV
CCMMON/E/XTC.EPS I,C,CVW,CTW.EXC1,EXC2,EXC3,CVCH4,Â£PSIA9,CTCH4
CC.MMCN/F/C VAP ,CTAF
P 1=3 I 4 l 59265D0
XK = l ,38066
P K= 1,93 7
C PARAMETER IN EXPRESSIONS FOR HARD SPHERE DIAMETER
A 1 = 0.54006832
A 2= 1 .2669302
A3=0 .05132355
A4=2.9107424
A5= 2 .5167259
A6=2.1595955
A 7=0 .64269552
AS=0.17565885
A 9=0.1 3874624
A 10= 17.952388
Aii=0.48197123
A12 = 0. 76696099
A 13=0.76631363
212
FORTRAN IV G LEVEL 21
CALFUM
DAT E
79109
22/43/30
0026
0027
0028
0029
0030
0031
003 2
0033
0034
0035
003 6
0037
0033
0039
0040
004 1
0042
0043
A 140.8 0965 7804
A1 5 = 0.2 4 062 863
DO 4 K=1,4
DO l1 I=1,4
DO 10 J=l,7
T = TT (K)+(J4)*0.500
TC=* { J) 273.15
SEC"rION TO CALCULATE SOLUTE AND WATER HARD SPHERE DIAMETERS
D EN W = 0.033433*(0.9998 42 52+169 4522 7D 03* TC 798 7 0641D 06*(TC* *2)4
16.1 7 06 D09* ( TC**3 ) +105.56 334D1 2* ( T C ** 4 )280.54 33 70 1 5*( TC**5))/( l
2.000+16.3798 5D 03* TC )
DDENW=0.033433*{16.945227D0315.9741282D06*TC 138.5 l 13D09*{ TC**
22) + 422.25336C12* TC**3)1 402.71585P1 5*(TC**4) )/(l.000+l6.3798SD
303*TC>DENW*<16.87985D03)/{ (1.000+16.87935D03*TC))
TRW=T(J)/CTW
DENWR=DENW*CVW/(0.6023)
DDENWR=DDENW*CVW/(0.6023)
HSDW=( (3.0*CVW/(2.0*PI*0.6023) ) ( A 7/ (T RW * A8 ) + A 2/< DEXP ( A 4*( ( DENWR +
2A 1*TRW)* *2.0)) )A3/(DEXP< A5* {{DENWR+Ai*TRWA6)**2 ) ) )+A9/{ DEXP(A10*
3( {(T RWA13)**2)+AIl*( (DENWRA12)**2) ) ) )) )**0. 333333
DHSD W=l 00 0 0* ( (3.0*CVW/ (2 .0*PI*0.6023) ) ( A7 A8/ ( CTW ( TR W **' ( A8 + 1.0
2) ) )2.*A2*A4*{DENWR+Al*TRW)*( DOENWR + A 1/CTW)/{DEXP(A4*((0ENWR + A1*TR
3W)^2.0)))+2.0*A3*A5*(DENWR+Al*TRWA 6)*(DDENWR +A 1/CTW)/(DEXP(A5*( {
4DENWR+A 1*TP WA6)**2.0) ) )2 O* A9* A1 0 ( ( TR W A1 3 )/CT W +A1 1 C DENWR A 12 )
5 DDENW R)/< DEXP(A10 *( ( (TRWA 13)**2)+A 11*{{DENWRA12)**2))))>)/((HSO
6W**2.0)*3.00)
TRCH4=T{J)/CTCH4
T R A R =T ( J ) / CT A R
HSDCH4={3.0*CVCH4/(2.0*PI*0.6 023) ) *(A7/CT RC H4** A8)+A2/ DEXP(A4*({
2DENW R+A1*TR CH4)**2) ) )A3/{DEXP{A5*< (DENWP+A 1 *TPCH4A6)**2)))+A9/(D
3EXP(A10*(((TRCH4A13)**2)+Al 1 *( (DENWRA12)**2)) ))) )**0.333333
HSD4R=( { 3. 0*CVAR/(2.0*PI*0.6 023) ) C 4 7/ (TRAP** A8.) +A2/ (DEXP (A4*( (DEN
213
FOP TRAN IV G LE VEL 21
CALFUN
DATE
79 109
22/43/30
0044
004 5
0046
004 7
0048
0049
0050
005 1
0052
0053
0054
0055
0056
2W RVA1*T PAR ) **2 ) ) ) A 3/( DEXP { A 5*{ C DENWR4 A 1 *TRARA ) * 2) ) ) + A9/( DEXP( A
31 0+{ ((TRARA13)*2)+AI1 *( < DENWRA12)**2)))))>**0.3 3333
dcwar=hsdar+hsdw
DCWCH4 = HSDCH44HSDW
DHSCH4= ( ( 3. 0*CVCH4/(2.0*PI*0.6023 ) ) (A7 A8/ ( CT C H4 ( TRCH4 ** { A8 4 l 0
2 ) ))2.0 *A2* A4*( DEN WR 4A l *TPCH4 )*{ DDENWR 4A 1/CTCH4) /( DEXP ( A4* ( ( DENWR4
3 A1*'PCH4 ) **2.0 ) ) )+2 .0 *A3*A5*( DENWR4A l*TRCH4 A6) *(DDENWR 4A 1/CTCH4) /
4(DEXO(A5*( ( DENWR + A1 TP.CH4 A6 )** 20) ) )2. O* A9* Al 0* ( (TRCH4A13 J/CTCH
5 4 4A 1 1*< DENWRA12 >*DDENWR)/(DEXP(A10*(((TRCH4A13)**2)+A1 l*{(D5NWR
6A12)**2))))))/( (HSD CH4**2.0)*3.0)
TDCH4I J )=HSDCH4
DHSDAR={(3.0*CVAR/(2.0TPI*0.6023))*{A7*A8/(CTAR*{TRAR**(A8+1.0)>)
22. 0* A 2* A4* DE NWF. 4A 1 TR AR )*{ DDENWR 4A 1 / CT AR )/ ( DEX P ( A4* ( ( OENWR+A 1 *TR
3AR) **2.0) ) ) 42.0*A3*A5*( DEN WP 4 A 1 *TR ARA 6) ( DDENWR 4A l/CTA ) /{DEXP( A5
4* ( (DENWR+Al *TRAFA6 ) **2 .0 ) ) )2 O* A9 A l 0* ( ( TP ARA 13 )/C TAR 4A 1 1 +( DEN W
5P A 1 2)* DDENWR) /(DE XP( A1 0* ( { (TRARA1 3 )**2 ) 4A1 1 *{ ( DENWRA12 )**2 ) ) ) ) )
6 ) / ( ( FS D AP **2.0 ) *3 0 )
DCC WG4 = DHSD W4DHSCH4
DDCWGA = DHSD W4DHSDAR
C SECTION ~0 CALCULATE INTERACTION CONTRIBUTION TO ARGON PROPERTIES
GIAP=EPSIAP (6. 72 02D + 0 04.954 D 40 3/ ( T { J) ) 46.5480D 405/C T ( J )**2)0.7
26 00 D 40 0 *HSDAR0 .79 2 5D 40 0*( HSDAR **2))/(T(J))
S IA R =5 P S IA R *(4.9540403/ (T(J)**2)t 3.09 6D405/ T ( J ) **3 )0 .76D400*D
2HSDAR1.585D40 0*HSDAP*DH SD Ap )
H I A R =GI AR 4S IA R
C SECTION TO CALCULATE EXPERIMENTAL ARGON SOLUBILITY PROPERTIES
GEXA R= (EXC14EXC2*DL0G( T( J ) ) 4EXC3M DLOG( T( J) ) **2) )
HEX AR=EXC2 42.0* EX C3*DLCG(T (J ) )
SE XA R=HE XAR GE XAP
C ARGON CAVITY PROPERTIES ARE OBTAINED BY DIFFERENCE
WACARGEXARGIAR
0057
CALFUN
79 109
FOR TRAN
0059
0060
006 l
0062
0063
0064
0065
0066
0067
0063
0 06 9
0070
0071
0072
0073
0074
0075
0076
IV G LEVEL 21
DA
22/43/30
5ACAR=SEXARSIAP
C SECTION TO CALCULATE ROTATIONAL AND VIBRATIONAL PROPERTIES FROM
C PERTURBED HAROCHAIN THEORY
VBARW=4 .9355D02/DENV.
D VBA PW=4. 93550 02*DDENW/(DENW**2 )
GRV J, I )=( C( I )l ) *{ ( 3.0* ( ( 0. 7405/VB AR W) **2) { 4. 0*0. 740 5/
2VBARW) )/ ( (1 .00 .7405/VBARW )**2) )
SP V( J, I ) = G R V ( J ,1 ) + T( J) *( C ( I > l )*DVSARW* ( ( ~6.0* 0.7405 *2/ {
2 V BARW**3 ) +4 .0 *0 .7405/1 VBAPW**2) ) / ( ( 1.00. 7405/VBARW) **2)+(3.0*((0.
274 05/VBARW)**2)(4.0*0.7405/V3ARW)}*(2.0*0.7405/1 VBARW**2 > )/(( 1.0
30.7405/VBARW)**3))
HRV(J)I)=GRV{J I)+ SRV(J I )
C SECTION TO CALCULATE CAVITY CONTRIBUTION TG PROPERTIES
STW= 1. 162D+02 1 .4 77D01 *T( J )
DSTWl .477 00 1
D Â£L = S.3194 896D + 0 0+26052 103D+03/T J)13.993 0690 +04/(T(J ) *2 )
DCEL = 2.6052103D+03/T( J)**2)+2.0 18.993069D+04/{ T(J)**3)
UL(J)=X(1 ) X(2)*1 O D + 02/T(J)+X(3)*1 .00+04 /(T(J)**2)
DUL = X( 2 ) *1.0D+02/ T(J)**2)2.0*X(3)*l.0Q + 04/(T(J)**3)
WAC(J,I) =
2DCWAR* *2) )4.0*DEL*STW*(DCWCH4DCWAR)STW*DEL*2.0*UL(J )*( 11) )+WA
3CAR
S AC ( J.I ) = ( PI/XK)* ( DUL* (11 ) *STW*DCW CH4+UL ( J ) *( I l > *DSTW*DC WCH 4
2+UL ( J) ( I 1 )*STW*DDC WG4+DSTW* { ( DCWCH4**2 ) (DCW A R* *2 ) ) +ST W *(2.0 *DCW
3 C H4 DDCW G42.0*DCWAR*DDCW GA)4.0*DDEL*STW*(DCWCH 4DCWAR)4.0*DEL*0
4 STW*(DC WCH4DC WAP)4.0*DEL*STW *(DDCW G4 DDCWGA )0STW*DEL*2.0 *UL ( J)*
5(11 ) STW*DDEL *2. 0*UL( J ) ( I 1 ) ST W *DEL *2.0DUL* ( I l ) ) + SACAR
HAC ( J. I )=WAC( J D+SAC1J I )
IF(I .EQ. 1) GO TO 2
GO T0 1
2 G ICH4 = EPSI (1)*(6.7202 4.954D+03/T(J ))+6.548D+05/ T(J )**2)0.76D +
2 0 0* HSDC H4 0 792 50+ 0 0* ( HSDCH4 * 2 ) )/T{ J)
215
FORTRAN IV G LEVEL 21
CAL FUN
DA
79 109
22/43/30
0077
0078
0079
0080
008 1
0082
0083
0084
0085
0086
0087
0088
0089
0090
009 i
0092
0093
0094
0095
0096
0097
0098
0099
C
c
r
C
C
GI(J,1)=GICH4
SICH4=EPSI(1)*(4.954D+03/(T(J)**2)13.096D+05/(T(J)**3)0.76CD+OO
2*DHSCH4l .585D+00*HSDCH4*DHSCH4>
SI{J,l)=SICF4
GO TO 7
i UL J ) = UL ( J ) *( 11 )
DUL=DUL*(11)
CF=(EPSI<1>l.5*EPSI(l ) >/(EPSI( I))
DF=1 .OD + OOCF
DDF = 0*00
DCF= 0.0 0
IF(UL(J ) .LT .3.600) GO TO 5
LARGE UL VALUE SECTION
CONTINUOUS DISTRIBUTION SECTION
0
GCl=XTC(l)+XTC(2)*T(J)+XTC(3)*T(J)**2
GC2=XT C(4)+X7C(5)*T(J}+XTC(6)*T(J)**2
GC3=XTC(7)+XTC(8)*T ( J )+XTC(9 )*T(J)**2
GC4=XTC( 1 O)+XTC(1l)*T(J) + XTC 12)*T(J)**2
GC5 = XT C(13)+X~ C 14)*T{J )+XTCC 15)*T (J)**2
GCOLLI= DEX P(GC1 + GC2/UL(J)+GC3/(UL(J )* *2 )+GC4/UL(J)**3 > +GC5/(UL(J
2)**4 ) )
SCI=XTC(2)+2.0*XTC(3)*T(J)
SC2XTC ( 5) +2. 0*XTC (6) *T< J)
SC3=XTC(3>+2.0*XTC(9}*T(J)
SC4=XTC ( 1 1 ) +2.0*XTC( 12) *T ( J )
SC5=XTC(14)+2.0*XTC{15)*T(J)
S COLL1 = GCOLLI*(SC 1+SC2/UL J)GC2*0UL/(UL(J) **2> +SC3/**2)2.
20*GC3*D UL/(UL(J)**3)+SC4/{UL{J)*3)3.0*GC4*DUL/(UL(J)**4)+SC5/{UL
3(J)**4)4.0*GC5*DUL/(UL(J)** 5))
DISCRETE DISTRIBUTION SECTION
0
216
FORTRAN IV G LEVEL 21
CALFUN
DATE
79109
22/43/30
0 100
0101
0102
0 103
0104
0105
0106
0 10 7
0108
0109
0 110
01 1 1
01 l 2
0 113
0 114
0 115
0116
0 117
one
0119
0120
012 1
0 122
GC1= XTC{ 16)+XTC(17)?T{J)+XTC{ 13)*T(J ) *2
GC2 =XT C(19)+XTC(20)*T(J)+XTC(21 )*T( J )**2
GC3=XTC( 22)+XTC(23)*T( J ) <XTC ( 24 ) T ( J)**2
GC4=XTC(25)+XTC(26)*T(J)FXTC(27)*T{J)**2
GC5=XTC(23)+XTC(29)*T( J)+XTC(30 )*T(J )**2
GD0LLI=DEXP(GC 1+GC2/UL ( J ) +GC 3/ ( L ( J )**2 ) 4GC4/( UL ( J)**3 ) +GC5/ (ULl J
2)**4>)
SCI =XTC( 17 M2.0 *XTC ( 18 ) *T ( J)
SC2=XTC( 20)+2.0*XTC(21)*T( J)
S C3XTC ( 23 ) +2 0 *XTC ( 24 ) *T ( J )
SC4 = XTC(26)+2 0*XTC(27)*T(J)
SC5=XTC(29)+2.0*XTC(30)*T{ J)
SDOLLI=GDOLL I (SCI +SC2/UL ( J )GC2+DUL/ ( UL ( J ) **2) +SC3/ UL ( J ) **2)2.
20*GC 3*D UL/( UL( J )*3) + sC 4/( UL( J) **3) 3.0*GC4*DLL/(UL( J 1**4 > *S C5/ (UL
2 ( J ) *4 ) 4.0 *GC5*DUL/ ( UL ( J ) *'*5 ) )
C CONTINUOUS DISTRIBUTION
C OUTSIDF REGION Y LT. O Y GT. UL
GCl=X''C (67 ) +XTC (68) *T ( J MX TC ( 69 ) *T ( J ) *2
GC2=XTC(70)+XTC71 )*T( J)+XTC(72)*T{ j)k*2
GC3=XTC(73)+XTC(74)*T(J)+XTC(75)*(J)**2
GC4=XTC(76)+XTC(77)*T(J)+XTC(76)*T(J )**2
GCGL L I = ( GC 1 /UL { J) +GC2/( UL( J)**2) <CC3/ (UL(J)'**3) +GC4/ (UL ( J ) **4 ) )
SC1=XTC (68 ) +2.0 *XTC{ 69) *T J )
SC 2 = XTC(71 )+20 *X TC{72)(J)
SC3=XTC(74)+2.0*XTC(75)T(J)
SC4=XTC (77 > +2.0*XTC{78 ) t ( J )
SCGLLI=(SCI/UL(J)GC1*OUL/(UL(J)**2)+SC2/(UL(J)**2)2.0YGC2*DUL/(U
2L(J)**3)+SC3/(UL(J ) **3)3.0*DUL *GC 3/(UL(J)**4)+SC4/(UL(J)**4>4.0*
3DUL* GC4 / { UL ( J > **5 ) )
C DISCRETE DISTRIBUTION
C OUTSIDE REGION
GC1 =XTC (82)+XTC(33)*T(J)+XTC(84)*T (J ) *2
217
FORTRAN
0 123
O 124
0125
O 126
0127
0128
0129
0 130
0 131
0132
0133
0134
0135
0 136
0137
0138
0 139
014 O
014 1
0142
0 143
0 144
IV G LZVcL 21 CALFUN DATE = 79109 22/43/30 R
GC2=XTC{ 85) +XTC {86)*T{'j )+XTC(87 )*T< J )**2
GC3=XTC [ 33)+XTC( 39 ) *~( J MXTC ( 90 > *T( j ) **2
GC4 = XTC(91 )+ XTC(92)*T{J)fXTC{93 ) *TCJ )**2
GC 5= XT C(94)+XTC(95)*T(J) + XTC 96)*T(J )**2
GC6=XTC(97>+XTC{98)*T(j )+XTC(99)*T(J )**2
GEE=DEXP( G C1 + GC2* DLQG (UL( J) )+GC3*DLCG (UL ( J ) ) **2+ GC4 DLOG(UL (J))**
23+GC5*DL0G(UL ( J ) ) **4+GC 6 *D L3G ( UL ( J ) ) ** 5)
GDE=GEE+0.50*(XTC(79)+XTC{80)*T(J)+XTC{3l)*T(J)**2)
SCi=XTCC33)+2.0*XTC(84)*T(J)
SC2=XTC (86 >+2.0 *XTC( 87) *T { J )
SC3 = XTC{89)+2 0*XTC(90)*T(J)
SC4=XTC(92J+2.0*XTC(93)*T(J)
SC5 = XTC (95 )+2.0 *XTC (96 ) *T ( J )
SC6=XTC(98)+2.O*XTC(99)*T(J)
S DE=GEE*(SCI+ S C2*0L0 G(UL(J) )+ GC2*DUL/UL(J)+ SC3*DLOG(UL(J))**2 + 2.*
2 GC3 DLOG(UL(J) )*DUL/UL(J)+SC4*DLQG(UL(J) )**3 + 3.O*GC4*DL0G(UL{J >)**
32*DUL/UL(J)+SC5*DLCG(UL(J)>**4+4.0*GC5*DL0G(UL(J))+*3*QUL/UL(J)+SC
4 6 *DLOGCUL(J ))**5 + 5.O*GC6*DL0G(UL { J ) )**4*DUL/UL(J) )+0.5 0*(XTC(80)
52 O* XTC ( 8.1 ) *T( J ) )
G I (J I)=(DF*EPSI(I)*(GDOLLI+GD E)+CF*EPSI( I )* (GCOLL I + GCGLLI ) )/(T(J)
2)
SI(J .1 )=DF*EPSI(I)*(SDOLLI + SDE)+CF+EPSI(I )*{SCOLLI+SCGLLI )0DF*EP5
1 I ( I )*{ GDOLL I + GDE)DCF*EPS I ( I )*( GCOLL I + GCGLLI >
GO TO 7
C SMALL UL VALUES
C CONTINUOUS DISTRIBUTION
C 0
5 GC1 = XTC(31)+X7C(32)*T(J)+XTC(33)* T{ J)* *2
GC2=X'r C (34 ) +XT C (35 ) *T(J )+XTC(36 ) *+( J )**2
GC3=XTC(37)+XT C(38)*T(J) + XTC(39)*T(J)**2
GC4= XT C(40)+XTC{41)*T{J)+XTC( 42)*T(J)**2
GCOL SI=(GC1*UL(J)+GC2*UL(J)**2+GC3*UL(J )**3+GC4*UL(J )**4 )
218
FORTRAN IV G LEVEL 21
CAL FUN
DATE = 79109
22/43/30
0 145
0 146
0 147
0 148
0149
0150
0 15 1
0 152
0 153
01 54
0155
0 156
0157
0158
0159
0160
016 1
O 162
0 163
016 4
0165
0166
016 7
0168
0169
SC 1= XTC(32)+2.0*XTC{33)*T( J)
SC2=X7C(35)+2.0*XTC(36>*T(j)
SC3 = XTC(38)+2O *XTC C39)*T(J}
SC4=XTC{ 4D+2. 0*XTC( 42)*T( J)
SC0LSI=(SC1 *UL( J) +GC1*DUL+SC2*UL(J)**2+2.O*GC2*UL(J)*D UL+SC3*UL(J)
l**3+3. 07GC3*DUL+UL( J)**2+4.0*GC4*DUL*UL(J )**3+SC4*UL(J )**4)
C DISCRETE DISTRIBUTION
C 0
GC1=XTC{43)+XTC(44)*T(J)+XTC(45)*T(J>**2
GC2= XT C ( 46 ) +XT C(47)*T( J)+XTC( 48) *T ( J J *2
GC3= XT C(4 9)+XTC(5 0) *T ( J)+XTC(5l)*T(J)**2
GC4=XTC ( 52+XTC (53) *T( J)+XTC( 54)*T( J)**2
GD0LSI=(GC1*UL(J)+GC2*UL J)**2+GC3*UL(J)**3+GC4*UL(J)**4)
SC1=XTC{44)+2.O*XTC(45)*T(J)
SC2=XTC(47)+2.0+XTC(48)*T(J)
SC3 = XTC(50> +2.0*XTC (51 ) *T ( J )
SC4=XTC ( 53) +2. 0*XTC (54) *T{ J)
SCOL S I = (SCI TUL ( J) +GC1 *DUL +SC2*UL(J ) *2 +2.0*GC 2* UL ( J> *DUL + SC3*UL( J)
1**343.0*GC.3*DUL*UL(J )**2 + 4.0*GC4*DUL*UL(J)**3+SC4*UL(J)**4)
C CONTINUOUS DISTRIBUTION
C OUTSIDE REGION
GC1=XTC(55)+XTC(56)*T(J)+XTC(57)*T(J)**2
GC2= XT C ( 58 ) +XT C { 59 ) *T ( J ) + .XTC.C 60)*T{ J ) **2
GC3 XTC (61 ) +XTC (62 ) *T ( J )+XTC( 63 ) *T { J 1**2
GC4=XTC ( 64) +XTC (65) *T( J) + XTC (66 )*T ( J )**2
GCGLSI = ( GC1+GC2*UL ( J )+GC3*UL ( J )*72+GC4*UL ( J ) ** 3)
SC1=XTC(56)+2.0*XTC(57)*T(J)
S C2= XT C( 59) +2.0*XTC( 60) *T{ J)
S C3 = XT C ( 62 ) +2 0*XTC(63)*T (J)
SC4=XTC { 65) +2.0*XTC (66) *T( J)
SCGL SI = (SCI +SC2 MJL ( J ) +GC 2*DUL + SC 3*UL { J )**2+ 2. 0*GC 3*0 UL *UL ( J ) +3.0*G
1C4*UL(J)**2*DUL+SC4*UL(J)**3)
otro
0171
0172
01 73
0 174
0175
0176
0177
0178
0179
0180
0 18 1
0 132
0 183
0184
0 18 5
0186
0187
0 183
0189
0190
O 19 1
0192
O n
IV G LEVEL. 21 CALFUN DATE = 79 109 22/43/30
D I SC PETE DISTRIBUTION
OUTSIDE REGION
GC1=XTC{32)+ XTC(33)*T(J)+XTC(84)*T(J)**2
GC2=XTC (85)+XTC(86)*T(J)+XTC 87)*T< J )**2.
GC3 = X~C (83) + XTC (89) *T { J ) + XTCI 90 )*+(J )**2
GC4 = XTC ( 91 ) + X^C (92 ) *T( J ) +XTC(93 )*T { J )**2
GC5=XTC(94)+XTC(95)*T{ J ) + XTC( 96)*'r( J)**2
GC6=XTC(97)+XTC(98)*T(J)+XTC(99)*T(J)**2
GEE = DEXP GC1+GC2*DL0G(UL(J))+GC3*DL0G(UL(J) )**2+GC4*0L0G(UL(J))**
13+GC5*DL0G(UL(J))**4+GC6*DL0G(UL(J))**5)
GDE =GEE + 0.50*{XTC(79)+X7C(30)*T(J)+ XTC (3 1 )*T {J)**2 )
SC1 = XTC ( 33) +2. 0*XTC ( 84) *T( J)
5 C2=XTC (86 )+2.0 *XTC (87) *T ( J )
SC3=XTC(89)+2.0+XTC(90)*T{j)
SC4=XTC (92 ) +2.0*XTC( 93 ) *T( J )
SC5 = XTC (95 ) +2.0*XTC (96 ) *T ( J )
SC6=XTC ( 98) +2. 0*XTC ( 99)*T( J)
3 CZ=GEZ*(S Cl+5C2*DL0G( UL(J))+ GC2*DUL/UL(J)+ SC3*DLCG(UL(J))**2 + 2.*
IGC3*0L0G(UL(J) )*DUL/UL(3)+SC4*DL0G(UL( J) )**3+3.0 *GC4*DL0G(UL(J ))*7
12DL/UL ( J ) +SC 5+DLQG UL { J ) ) **4+4. 0*GC5*DLOG( UL( J) ) **3*DUL/UL ( J) +SC
16*DLDG(UL(J))**5+5.0 GC6* DLO G(UL(J ) )**4*DUL/UL( J) )+0.50*(XTC( 80)
12.0* XTC( 81)7T( J ) )
GI(J.I)=(DF*EPSI( I ) *( GDOL S I+GDE )+CF *Â£P SI ( I ) ( GC GL S I+GC OLSI ) ) /T( J)
SI ( J .1 ) OF*cPSI ( I )* (3D0LS I+SDE)+CF*Â£PS I ( I ) *( 3CGL 5 I +SC0LS I )DOF*EPS
1 I ( I ) *( GDOLS I+GPE) + (DCtr)*EPSI ( I ) (GC GL 3I+GC0L3I )
7 HI(J,I)=GI(J.I)+SI(J,I)
SECTION FOR CALCULATING HEAT CAPACITIES
10 CONTINUE
DC 15 J=1 ,7
HH{J)=HAC{J .1 )*T{J)
H I I ( J >=H I ( J I ) *T( J )
HHH(J)=HRV(J,I)*T(J)
220
FORTRAN IV G LEVEL 21
CALFUN
DATE
79 I 09
22/43/30
0 193
0 194
0195
0196
01 97
0198
0199
0200
020 1
0202
0203
0204
0205
0206
0207
0208
0209
0210
021 1
0212
021 3
021 4
021 5
0216
0217
0218
021 9
0220
0 22 1
0222
15 CC'NtinuE
H=C.5000
MN=7
CALL DPET5{ H ,HH ,DHH ,MN, IER)
CALL DDET5( 6,HI I.DHII.MN, IER)
CALL DDET51H.HHH.DHHH.MN,IER)
C SECTION FOR CALCULATING PROPERTIES AT DESIRED TEMPERATURES
SSI { I K ) =S 1(4* I )
HHICI,K)=HI(4,I)
GC{T,K)=WAC(4,I)
HCCI ,K)=HAC(4,I)
SC(I ,K) = SAC<4.I}
GGI ( I, K ) = GI (4, I 1
SAPV(IK)SRV(4,I)
GARV{I,K)=GRV(4,1)
HARV ( I ,K)=HRV(4, I )
CPC(I.K)=DHH(4)
CPI { I,K ) = DH II( 4 )
C PAR VC I *K) = DHHH(4)
C SECTION FOR CALCULA TI NG TOTAL PROPERTIES
CCPS
CGS(I ,K)=GC {I K)+GGI( I,K)+GARV( I,K)
C HS( I K)=HC(I,K1+HHI(I.K)+HARV I,K)
CSS I I,K)=SC(I,K)+SS I ( I,K)+SARV( I.K )
11 CONTINUE
4 CONTINUE
DO 12 1=2,4
DO 3 K=1,3
J=6 ( 12 ) +K
F(J)=(E XG(I K)CGS(I.K))/(EXG{I,K))*100.00
F (J +3)= (EXS ( I ,K )CSS( I. K) )/( EXS< I K) >* 100. 00
3 C ONT I NU E
CAL FUN
DATE
79109
22/43/30
FORTRAN IV G LEVEL
0223 12
0224
0225
21
continue
RETURN
END
222
223
For the gaseous hydrocarbon (C^ C^) solution property correla
tions, two sets of interaction energy parameters Â£ were investigated.
The sets of Â£ values are listed at the top of the data lists which
ws
follow. The first list corresponds to the results of Table 53, the
second to Table Cl.
The values of the curvature parameter 6 utilized and chain
segment length L which results as output are:
CH ^
Table 53:
6 = 8.3149896D+00 + 2.6052103D+02/T 18.993069D+04/T2. (Cl)
L = 0.52778918 + 3.194678D+02/T 2.8715619D+04/T2. (C2)
Lh2
Table Cl:
6 = 6.2246729 + 2.1484507D+03/T 14.40244D+04/T .
L = 0.49403381 + 3.0775424EH02/T 2.3198407D+04/T2.
vtl ^
(C3)
(C4)
For the prediction of liquid hydrocarbon (C^ solution
properties two parallel sets of Â£ were also investigated. These values
WS
are shown in Table 54 and Table C3.
19 APRIL 1979
232.5113
398.4875
570.6245
c
l. 0000
1.2527
1 .4362
c
46.4000
438.7000
96.0300
c
0.438330D+03 0.
1703 06D+03
c
277. 1500
c
18.0574
7.9537
10.1038
c
21.0 418
11.44 07
9.6012
c
22.9945
 13. 2840
9.7076
c
24.635l
14.8241
9.8088
c
298.1500
c
16.1449
5.5487
10.5956
c
18.2587
7.95 20
10.3071
c
 19. 5773
S. 0519
10.5210
c
21.1827
 10.45 18
10.7294
c
323.1500
c
 14. 1369
3. 1 880
l0.9457
c
15.3397
4.5351
10.8068
c
15.9889
4.9093
1 1.08 14
c
17.5591
6. 1633
1 1.3945
c
358.1500
c
1 1.6608
0. 52 58
1 1.1340
c
11.7514
0. 6834
1 1.071S
c
11.5903
0.2480
1 1 .3421
c
13.1002
 1.32 52
11.7750
DATA LIST FOR GASEOUS HYDROCARBONS
749.9852
1 .6058
189.0161 190.6000
0.145218D+02
26.7086
38.7771
47.7000
48.1329
25.551 1
37.0911
45.5913
46.0594
24.4841
35.4957
43.5984
44.1168
23.8450
34.57 98
42.4157
42.9894
74.9000 150.7000
19 APRIL 1979
266.2096
403.3456
.543.2209
c
l. 000 0
1.2527
1.4362
c
46.4000
438.7000
96.0300
c
0.4883300+03 0.
l7030 6 D + 03
c
277.1500
c
18.0574
7.9537
10.1038
c
21. 0418
11.4407
9.6012
c
22.9945
 13. 2840
9.7076
c
24.6351
14.8241
9.8088
c
c
298. 1500
16.1449
5.54 87
10.5956
c
18.2587
7.9520
10.3071
c
19. 5773
S. 0519
10.5210
c
21.1827
 10.45 18
10.7294
c
c
323.1500
14. 1369
3. 1880
10.9457
c
15.3397
4.5351
10.8063
c
15.9889
4.9093
1 1.0814
c
17.5591
6. 1633
1 1.3945
c
c
358.1500
11.6608
0.5258
1 1.1340
c
11.7514
0. 6834
11.0715
c
1l.5903
0.2480
11.3421
c
13. 1002
1.32 52
11.7750
DATA LIST FOR GASEOUS HYDROCARBONS
686.1968
l.6058
245.2719 190.6000 74.9000 150.7000
O. 145218D+02
26.7086
38.7771
47. 7000
48.1329
25.5511
37.0911
45.5913
46.0594
24.4841
35.4957
43.5984
44.1168
23.8450
34.5798
42.4157
42.9894
225
Table Cla
Contributions to Free Energy of Solution of Gaseous Hydrocarbons
Solute
T(K)
AG
c
RT
AG
i
RT
AG
r>v
RT
cal
RT
AG
exp
RT
CH,
4
277.15
24.92
 14.37
0.0
1 0. 05
10.10 4
+.005
298. 15
24. 24
13.65
0.0
10 .60
10.596
+.004
323.15
23.32
12.35
0.0
10.97
10.946
+.003
358.15
21.91
 10.80
0. 0
11.12
1 1.134
+.008
C2H6
277. 1 5
3 0.72
22.36
1.27
9.62
9.601
+.008
298.15
29 .65
20 .6 2
1.26
10.30
10.30 7
+ .006
323.15
28.26
 18.72
1.24
10.77
1 0.30 7
+.003
358. 1 5
26 .21
16.32
1.13
11.07
11.072
+. 010
C3H8
277.15
36.51
23.98
2.20
9. 73
9.708
+.009
298. 15
35.06
26.72
2.18
10 .52
10 .52 1
+ .007
323.15
33.19
24.25
2.14
11.08
11.081
+. 006
358.15
30.50
21.10
2. 04
11.44
11 .342
+.010
Table Cla (Continued)
Solute
T(K)
AG
c
RT
AG
i
RT
AG
r,v
RT
C4H10
277.15
42.31
35.57
3.05
298. 1 5
40.46
32.77
3.03
323.15
38.13
29 .72
2.97
358.15
34.80
25.94
2.83
AG 
AG
cal
exp
RT
RT
9. 79
9. 809
+ .011
10.73
10.729
+.006
1 l .38
11.39 5
+.005
11.69
11.775
+. Oil
227
Table Clb
Contributions to Enthalpy
AH
AH
Solute
T(K)
c
RT
c
RT
CH4
277.15
8.06
16 .
298.15
10 .34
 16.
323.15
12. 49
15.
358.15
14.86
14 .
C2H6
277.15
13.10
24.
298. 1 5
15.95
23 .
323.15
1 8.54
23.
358. 15
2 1.21
23.
C3H8
277. 1 5
18.14
31 .
298.15
21.57
30 .
323. 15
24.59
30.
358.15
27.56
30 .
Solution of Gaseous Hydrocarbons
AH
r,v
RT
cal
RT
AH
exp
RT
0.0
8.82
7.954
+ .30
0.0
6. 14
5.549
+ .05
0.0
3 .22
3.188
+.07
0.0
0.41
0.526
+ .28
0. 0 0
l 0.99
11.441
+ .43
0.23
7.53
7.952
+ .07
0.43
4.47
4.53S
+ .10
0.66
1.33
0.683
.4!
0 .00
13.01
13.234
+ .53
0.40
8. 88
9.052
+.08
0.75
5 .29
4 .909
+.13
1.13
1.85
0.248
+.50
of
3.8
48
71
45
09
72
45
25
15
0 5
64
54
Table Clb (Continued)
AH
AH?
AH
Solute
T(K)
c
1
r,v
RT
RT
RT
C4H10
277. 1 5
23.10
38 .72
0.00
298.15
27.13
33.13
0.56
3 23. 15
30. 65
37.76
1.04
358. 15
33.91
39.40
1.57
i5al
RT
AH
exp
RT
15.53
14.824
+.54
10.40
 10.452
+.08
6 .07
6.163
.13
3.91
1.325
+.52
229
Table Clc
Contributions to the Entropy of Solution of Gaseous Hydrocarbons
AS AS? AS AS AS
c i r,v cal e:
Solute
T(K)
c
R
i
R
r.v
R
cal
R
exÂ£
R
CH4
277. 15
16.86
2.01
0.0
18.87
18 .057
+.30
298.15
l3.90
2 .83
0.0
16.73
 16.145
+.04
323.1 5
10.83
3.36
0.0
14.20
14.137
+.07
358. 15
7. 05
3.65
0.0
10.70
11 .66l
+.29
C2H6
277.15
17.61
1 .72
1.27
20.61
21.042
+.42
238.15
13.70
3. 10
1.03
1 7. 83
18.259
+. 06
323.15
9.72
4.72
0.80
15.24
15.340
+.n
358.15
5.00
6.93
n
in
.
o
1
 12.45
11.751
+. 42
C3H8
277.15
13.37
2. 17
2.20
22.74
22.995
+ .52
238. 1 5
13.49
4.13
1.73
19.40
19.577
+ .08
323. 1 5
8.60
6.39
1.39
16.38
 15.989
+.13
358. 15
2.94
9.44
0.91
13.29
11.590
+.51
230
Table Clc (Continued)
Solute
T(.K)
As
c
R
As?
i
R
AS
r,v
R
C4H10
277.15
19.12
3. 14
3.05
298. 15
13 .29
5.37
2.47
323.15
7.48
1
CD
.
o
(>
1.92
358. 15
0.89
13.46
 1 .26
AS
cal
R
AS
exp
R
25.32
24.635
.53
21.12
21.183
+.08
 17.45
 17.559
+.13
15.60
13.100
+.52
231
Table Cld
Contributions to Heat Capacity of Solution of Gaseous Hydrocarbons
Solute
T(K)
ACp
c
ACp?
ACp
r,v
ASpcal
ACp
exp
R
R
R
R
R
CH4
277.15
42.05
 13 .98
0.0
28. 07
26.709
+5.7
298. 15
39.14
8.78
0.0
30 .36
25 .55 1
2.5
323.15
37 .32
4.69
0.0
32.63
24.484
+1.5
358.15
36.47
1.21
0. 0
35.26
23.845
+5.1
C2H6
277. 15
56.05
13.18
3.77
41.64
38 .777
+8.3
293.15
51 .43
 19.50
2.99
34. 92
37.091
+3.6
323.15
47.66
20.73
2.74
29.61
35 .496
+2 .i
35a. 1 5
44 .40
22 .07
2.69
25.03
34.580
+7.4
C3H8
277.15
70 .05
26.25
6.50
50.30
47. 70 0 +1Q.2
298. 15
53.71
27.43
5.17
4 1.46
45.591
+4.4
323.15
58 .00
28.77
4.72
33.95
43.598
+2.6
358.15
52.33
30.58
4.65
26.41
42.416
+9.0
232
Table Cld
Solute
T(K)
ACP
R
ACp
R
C4H10
277. 15
84.06
29.18
298. 15
76.00
31.71
323.15
68.33
34.79
358.15
60 .26
46.30
(Continued)
ACp
r,v
R
9.03
7.18
6.56
6.46
A^cal
R
ACp
exp
R
63. 91
48. 133
+10.4
51 .47
46.059
45
40.11
44. 117
+ 2.6
20. 42
42.989
+ 9.2
233
FORTRAN IV G LEVEL 21
MAIN
DATE
79 109
22/46/19
000 1
0002
0003
0004
0005
0006
0007
C OBJECTIVE CALCULATE PREDICTIONS OF AQUEOUS SOLUBILITY PROPERTIES
C OR LIQUID HYDROCARBONS
r
C
C EXPLANATION OF INPUT DATA
C CVW CHAPACTcR STIC VOLUME FOP. WATER CC/GMCL
C CTW CHARACTERISTIC TEMPERATURE FOR WATER K
C CVCH4 CHARACTERISTIC VOLUME C0F METHANE CC/GMOL
C CTCH4 CHARACTERISTIC TEMPERATURE FOR METHANE K
C CVAP CHARACTERISTIC VOLUME FOR ARGON CC/GMOL
C CTAR CHARACTERISTIC TEMPERATURE FOR ARGON K
C EPSIAR INTERACTION ENERGY PARAMETER FOR ARGONWATER K
C FPSICl INTERACTION ENERGY PARAMETER FOR METHANEWATEP K
C EPS I INTERACTION ENERGY PARAMETER FOR HYDRCCAR30N 50LUTE
C WATER K
C TC COEFFICIENTS IN CORRELATION EQUATIONS FOR VARIOUS
C CONTRIBUTIONS TO THE WATERHYDROCARBON INTERACTION
C C CHAIN LENGTH PARAMETER OF PERTURBED HARDCHAIN THEORY
C FROM GMEHLING 1978
C TT TEMPERATURES AT WHICH SO_JBILITY PROPERTIES ARE PREDICTED K
C E XC 1 EXC 2 E XC3 COEFFICIENT IN EQUATIONS TO CALCULATE
C SOLUBILITY PROPERTIES FOP ARGON,WHICH IS USED AS A REFERENCE SOLUTE
IMPLICIT REAL*8(H,0Z)
D IMENS I ON GC(10,4) ,GGI(10,4),HC(10,4),HI(10,4),SC(10,4),SI(10,4),
2CPC ( 10.4 ) CPI ( 1 0,4 ) HSDW (4 ) CSS ( 10,4 ), CGS< 10,4) CHS 10,4) ,CCPS( 10,
34 ) TT< 4) ,GR V( 10,4) HP V{ 1 0 ,4 ) SR VC 1 0 ,4) ,HSDCH4 (4),UL(4),TCC99),
4EPSI (10 ), C( 10),CPPV( 10,4),X( 3)
commcn/a/exs.exh,exg,excp,cvw,ctw
CCMMON/B/GC ,GG I ,HC.HI ,SC,SI .CPC ,CPI ,HSDW,HSDCH4 ,UL
CGMMCN/C/CSS,CG5,CHS,CCPS.TT
C CMMON/D/GR V,HRV,SRV,CPRV
COMMCN/E/TC ,EPSI ,C,EPSI AR. ,EPSIC1,CVCH4,CTCH4,CVAR,CTAR
NJ
OJ
4^
00 08
000S
0010
001 1
0012
00 13
00 1 4
00 I 5
00 l 6
00 1 7
00 1 8
001 9
0020
0021
0022
0023
0024
0025
0026
0 02 7
0028
0029
0030
003 1
0032
0033
0034
0035
G LEVEL
21
MAI N
DATE =79109
22/46/19
COMM ON/5/ EX Cl EXC.2 EXC3
READ (5*1) (TC(K) ,K = 1 ,99)
1 FORMAT (6E13.6)
READ (5,2) (C(L),EPS I (L) ) ,L=1 10)
2 FORMAT (8=10.4)
READ' (5,3) CVCH4.CTCH4, =PSIC 1 ,C VAR .CTAR.EPSI AR.CTW ,CVW
3 FORMAT (3F10.4)
READ (5,4) EXC1 ,EXC2 ,EXC3
4 FORMAT (3E15.6)
D O 1 7 K = l ,4
READ(5,6) TT(K)
6 FORMAT (Fi 0.4)
17 CONTINUE
X(1)=0.52778910
X(2) =3.I 946780
X( 3) =2.371 561 9
CALL CALFUN(X)
WRITE (6,19)
19 FORMAT ('1',///////////)
D O 2 0 1=1,3
DO 21 K=1,4
WRITE (6,22) TT (K ) GC ( I ,K ) ,GGI ( I ,K) ,GRV( I K) CGS{ I ,K) SC ( I ,K) ,
2 SI(I ,K) ,SR V(I,K) ,CSS( I K )
22 FORMAT ( / 1 OX,F 7. 2.3X ,F6.2,2X ,F7. 2 ,2X,3{ F6.2 ,2X ) F7.2,2X ,2( F6 .2,2X
2) )
21 CONTINUE
20 CONTINUE
WRITE (6,19)
DO 2 3 1=4,6
DO 24 K=1,4
WRITE (6,22 ) TT(K),GCCI,<),GGI( I,K), GRV( I,K),CGS(I,K) ,SC(I .K) ,
2SI(I,K),SRV(I,K),CSS
K>
OJ
FORTRAN
0037
0 03 8
0035
0040
004 1
0042
0043
0044
004 5
0046
0047
0048
0049
0050
005 1
0052
0053
0054
0055
0056
0057
0058
0 05 9
006 0
0061
0062
0063
0064
IV G LE VEL 21 MAIN DAT H = 79109 22/46/19
24 CONTINUE
23 CONTINUE
WRI TE { 6,19)
DO 25 1=7,9
DO 26 K = i ,4
WPITE 16,22) TT(K) ,GC I ,K) ,GGI { I ,K) GRV( I ,K) ,CGS ( I ,K),SC
2SI ( I K),S P V ( I K),CSS( IK)
26 CONTINUE
25 C QNTINUE
WRITE (6,19)
DC 27 1=10,10
DO 28 K=1,4
WRITE (6,22) TT{K) ,GC(I K) ,GGI( I.K) GRV( I.K ) ,CGS( I,<),SC( I ,K),
2SKI.K), SR V(I ,K) ,CSS( I K)
28 CONTINUE
27 CONTINUE
wpiE.ig)
DO 29 I=1 ,3
DO 30 K=1,4
WRITE(6,22) T7(K),HC(I,K),HI(I,K),HRV(I.K).CHS(I,K).CPC(I.K),
2CPI(IK),CPRV(I,K),CCPS(I.K)
30 CONTINUE
29 CONTINUE
WRITE(6,19)
DO 31 1=4,6
DO 32 K=1,4
WRITÂ£(6,22) TT(K) HC(I K ),HI(I.K), HRV( I ,K),CHS( I.K),CPC{I.K),
2CPK I,K),CPRV(I,K),CCPS(l,K)
32 CONTINUE
31 CONTINUE
WRITE6 19)
DO 33 1=7,9
236
FORTRAN IV G LEVEL 21
MAIN
79109
22/46/19
DATE =
0065
0 066
0067
0068
006 9
0070
0 071
0072
0072
0074
0075
0076
DO 34 K = 1,4
WRITEI6.22) TT(K),HCII,K),HI(I,K),HRV(T,K) ,CH5{ I ,K) ,CPC(I K) .
2CPItCCPS(I,K)
34 CONTINUE
33 CONTINUE
WPT TE(619)
00351=10,10
DC 36 K = 1 ,4
WPI^E{ 6,22) TT(K) ,HC(I K),H! (I K) ,HRV( I K ) ,CHS( I .K),CPC CI,K),
2CPI ( I, K ) CPRVC I ,K ) CCPS( I K )
36 CONTINUE
35 CONTINUE
STOP
END
237
FORTRAN IV
000 1
0002
0003
000 4
0005
oooe
0007
oooa
0009
001 0
00 l l
00 I 2
0013
00 1 4
0015
00 1 6
00 1 7
00 l 8
0019
0020
0021
0022
0023
0024
0025
0026
G LEVEL 21 CALFUN DATE = 79109 22/46/19
SUBROUTINE CALFUN(X)
IMPLICIT REAL*8{AH,0Z)
DI ME NS ICN C t10) ,T(7) ,TT(4 ) ,DCH4 (4 ) ,SRV(10,4) ,HRV(10,4).GRV(10,4),
2UL{7),W4C1710)SAC(7 10)HAC(710) X(3) ,GI(710) SI(7t10).HI (7*10
3) UUL(4 ) HHSDW(4) HH(7) *HII(7),SSI(10 4 ) HHI ( 10,4),GC( 10,4),HC( 10,
44 ) SCI 10,4), GG II.10*4) ,CPC ( 10,4) ,CPI ( 10 ,4 ) ,CCPS( 10,4),CGS (10,4),CHS
5(10,4),HARV(10,4),GARVI 10,4),SARV( l0,4>,TQCH4(4),0HH(7) ,OHI I (7) ,CS
6S ( 1 0 ,4) XTC I 99 ) ,EPSI ( 10 ) ,HHH(7 ) DHHH (7 ), CP ARV { l 0,4 ) CPRV ( 7. 10 )
C0MM0N/A/EX5,E XH,EXG,EXCP ,CV'W.CTW
C CMMON/G/GC,GGI,HC,HHI,SC,SSI,CPC,CP I.HHSDW,DCH4,UUL
CCMMON/C/CSS,CGS,CHS,CCPS,TT
CCMMON/D/GARV,HARV,SARV,CPARV
C OMMON/E/XTC,EPSI,C,EPSIAR,EPS IC1,CV CH4,C TCH 4,CVAR,CTAR
C CM M ON/F/E XC 1 E XC 2 E XC3
P 1=3.1 4159265D0
XK=1.38066
RK= 1 .987
C PARAMETER IN EXPRESSIONS FOR HARD SPHERE DIAMETER
A 1=0.54008332
A2=1.2669302
A3=0.05132355
A4=2.9107424
A5=2.5167259
4 6=2.1595955
A 7= 0 .64269552
A8=0.17565385
A 9=0.18874824
A10=17.952388
A11=0.48197123
4 12=0.76696099
A13=0 .76631363
A 14=0.309657804
238
FORTRAN
0027
0028
0 023
0030
0031
0032
0 0 33
0034
0035
0036
0037
0038
0039
004 O
0041
0042
0043
IV G LEVEL 21
CAL FU N
DATE = 79109 22/46/19
A 15=0.24062863
DO 4 K= 1 ,4
DO 11 1=1.10
DO 10 J=1.7
T(J)=TT(K)+{J4)*0.500
TC=T ( J )273.1 5
C SECTIONTO CALCULATE SOLUTE AND WATER HARD SPHERE DIAMETERS
DENW=0.033 433 0.9998 4252+16.9452270 0 3*T C 798 70641D 06*{TC**2)4
16.1706D09*(TC**3)+105.56334D12*(TC**4)280.54337Dl5*(TC**5))/(1
2.000+16.87985D03*TC)
DDENW=0.033433*(16.9452 27D0315.97 41282D06 + TC138.51l8D09*(TC**
22)+422.25336D12*( TC**3)140 2.71635D15*(TC**4> )/( 1.000+16.37985D
30 3 + T C) DE.NW* (16 .87985D03 )/ ( ( 1 000 + 1 6.37935D03 *TC ) >
TRW=T(J)/CTW
DENWP=DENW*CVW/(0 .6023)
DDENWR=DDENW*CVW/(0 .6023)
HSDW=((3.0*CVW/(2.0*PI*0.6023)) (A7/(TRW**A8)+A2/(DEXP(A4*( (DENWR+
2 A 1*7 RW)**2.0) ) )A3/ (DEXP{A5*((DENWP +Al*TRWA6)**2) ))+A9/(DEXP(Al0*
3( ( ( r R W A l 3) * 2 ) + A1 1 ( (DENWRA12)**2 )))))) **0 .333333
DHSDW=1 .00 0 0*( (3.0*CVW/(2.0*PI*0. 6023) )*(A7* A8/(CTW*(TRW**(A8 + 1 .0
2 ) ) )2.*A2*A4*(DENWR+Al*TRW ) *(DDENWR+A1/CTW )/ ( DEXP ( A 4 ( DSNWR +Al*TR
3W )*=*2.0) ) ) + 2. 0*A3*A5* ( DENWR + A 1* TP WA6) (DDENWR + A1/CT'*)/( DEXP (A5*( (
4DENW R + A1*TRWA6)**2.0)) )2.0*A9*A10*((TRWA 13)/CTW+A11 *(DENWRA 12)
5* DDE NWR) /( DEXP ( Al 0* ( ( (TRWA13 )**2 ) +A 1 1 ( ( DENWR A l 2 ) **2 ))))))/(( HSD
6W **2 .0) *3.00)
TRCH4=T { J)/CTCH4
TRAP =T(J)/CTAR
HSDC H4= ( (3.0*CVCH4/ { 2 .0 *p I *0.6023) ) ( A 7/( TRC H 4* A 8 ) + A2/( DEXP(A4*((
2DENWP+A1*TRCH4)**2))>A3/(OEXP(A5*((DENWR+A1*TRCH4A6)**2)))+A9/(D
3E XP ( A 1 0 ( ( ( TRCH4A 13 ) ** 2 ) + A l 1 ( (DENWRA1 2 ) **2 ) ) ) ) ) )**0 .333333
HSDAR= ( (3.0 *CVAR/ (2.0*P 1*0 .60 23 ) ) *( A 7/ ( TP AP. * A3 ) + 4 2/ ( DEXP ( A 4* (.(DEN
2WF + A 1*TAR)**2) ))A3/(DEXP(A5 *( (DENWR + A1=TPARA6)* *2) ) ) +A9/(DEXP{A
NJ
U)
FORTRAN IV
0044
0045
0046
004 7
0048
0049
0050
005 i
0052
0052
0054
0055
0056
0057
0058
0059
G LEVEL 21 CALFUN DATE = 79109 22/46/19
310*{ ((TRAPA 13)**2)+A1l*( (DENWRA12)**2) ) ) )) )**0.33333
DCWAR=HSDAR+HSDW
DCWCH4 = HSDCH4+HSDW
DHSCH4= ( (3. 0*CVCH4/( 2.0*P 1*0.6023) ) *( A7*A3/< CTCH4* ( TRCH4*'* ( A 8+1 O
2 ) ))2.0*A2*A4*( DENWP.FA1 TRCH4 ) *( DDENWR+A 1/CTCH4) /( DEXP ( A 4* ( ( DENWR+
3A 1 TR C H 4)* 2O) )) + 2.O*A3*A5*(DENWRFA 1*TRCH4A6)*(DDENWR+A1/CT CH4)/
4(DÂ£XP(A5*( (DENWP+A1*TPCH4A6)**2.0)) )2.O*A9*A 1 O*{(TRCH4A13)/CTCH
54+Al 1 *(DENWRA12)*DDENW R)/(DEXP(Al O*(( {TRCH4A13 ) *2 )+A 1 l *< {DENWR
6A 12) **2 ) ) ) ) ) ) /(
TDCH4 J) =HS0CH4
DHSDAR= ( ( 3. 0*CVAR/t 2.0*P 1*0.6023) )*(A7* A8/t CTAR* (TRAR**(A8 + 1.0>)>
22.0 *A2*A4* (DENWR +A 1*TRAR ) *{ O DENWR 4A 1/CTAR ) / ( DE XP ( A4* { DENWR+A1 *TR
3A R )* *2O) ) )+2.0*A3*A5*(DENWR4A1*TRARA6) *(DDENWRFAl/CTAR )/(DEXP(A5
4*((D EN WF FA 1*TR A RA 6)**2.0)) )2.O*A 9*Al O*( (TRARA13)/CTARFAl1*(DENW
5 RAl2)*DDENWR)/(DEXP(A 1O ( ((TRARA13)**2)FA11*( (DENWRA i 2)**2)) ) ) )
6 ) /( ( HSD AR**20 )'*3. O)
DDCWG4=DHSDW+DHSCH4
DDCWGA=DHSDW+DH3DAR
C SECTION to CALCULATE INTERACTION CONTRIBUTION TO ARGON PROPERTIES
GIAR = EPSIAR *(6.7202D +0 0 4954D + 03/( T( J) )F6. 5480D+05/{T(J)**2)0.7
26 OOD+0 0*HSDAR0.7925D+0 0*(HSOAR **2) )/(T(J ) )
SIAR =EPSIAR *(4.954C+ 03/(T(J)**2)l3.0960 F05/(T(J)**3)0.760+00*D
2HSDAR1 585 D+0 O *HS DAR *D HSD AR )
HIAR=GIAP+SIAR
C SECTION TO CALCULATE EXERIMZNTAL APGON S0LU8ILITY PROPERTIES
GEXAR= (EXC1 +5XC2*0L0G(T ( J ) ) FEXC3*( DLOGC T ( J ) ) **2> )
HEXA R=EXC2 + 2 0*E XC3*DL0G ( T ( J) )
SEX AR= HEXARGEXAR
C ARGON CAVITY PROPERTIES ARE OBTAINED BY DIFFERENCE
WACAP = GEXARGI AP.
HACAR=HEXARHIAR
SACA R=SEXAR SI A R
240
FORTRAN IV G LEVEL 21
CALFUN
DATZ
79 109
22/46/19
0060
00 6 1
0062
0063
0064
0065
0066
0 06 7
0068
0059
0070
007 1
0072
0073
0074
0075
0076
0077
C SECTION TO CALCULATE ROTATIONAL ANO VIBRATIONAL PROPERTIES FROM
C PERTURBED HARDCHAIN THEORY
VBAR W=4. 9355D02/DENW
DVBARW=4.9355D02*DDENW/(DENW**2)
GRV(J,I)=(C(I)1>*((3.0*{{0.740 5/VBARW)**2)<4.0*0.74 05/
2VBARW> } /( ( 1.0 0.7405/VBARW)**2) )
SRV { J, I )=GRV( J I ) +T( j ) *( c C I ) 1 ) +DVBAR W*{ 6.0*0.7405**2/{
2VBAPW**3)+4.0*0.7405/+ < 3.0 *( ( 0.
2740 5/VBAR W)**2)(4.0* 0.7405/VBARW))*( 2.0*0.7405/(VBA R W* *2))/((1.0
3 0.7405/V BARW)* *3) )
HPV( J, I )=GRV< J. I) + SFV( J ,1 )
C SECTION TO CALCULATE CAVITY CONTRIBUTION TO PROPERTIES
S T W= 1.1620 + 021 .477 D0 1 *T ( J )
DSTW=1.477D01
DEL=O.31948960+00+2.6052103D+03/7(J)13.993069D+04/(T(J)**2)
DCEL = 2. 60 521 03D+03/( T(J)**2) +2.0*18 .9930690+04 / (T (J )* *3 )
UL J ) = X( 1 )X2) *1.0D+02/T{ J ) .+ X( 3) *1 0D+04/(T( J) * 2)
DUL=X{ 2 ) l OD+O2/(T(J ) *2 )2.0 *X ( 3 > 1 OD + 04/( T( J ) *3 )
WAC( J.I)=(PI/( XK* T{ J) ) } *( UL( J)* ( I +3 >*STW*DCWCH4+STW*({DCWCH4**2)(
2DCWAR**2)>4.0DEL*STW*(0CWCH4DCWAR) STW*DEL*2. 0*UL(J)*
3CAR
SAC( J. I ) = ( PI/XK)*{DUL*{I+3)*STW*DCWCH4+UL(J)*( 1+3)*DSTW*0CWCH4
2 +UL(J) *(I+3)*ST W*DDCW G4 +DSTW*( (DCWCH4**2)(DCWAR**2))+STW*(2.0*DCW
3CH4* DDCWG42.O+DCWAF*DDCWGA)4.0*DDÂ£L*STW*(DCWCH4OCWAR)4.0 +CEL *D
4STW*(DCWCH4DCWAP )4.0 *DEL*STW*(DOCWG4DDCWGA)DSTW+DEL*20 UL(J)*
5(1+3) STW*DDEL *2 0 *UL(J)*(I+3)STW*DEL*2.0*DUL ( 1+3) J+SACAR
HAC( J, I )=WAC( J I ) + SAC J .1 )
C SECTION TO CALCULATE INTERACTION CONTRIBUTIONS
UL(J)=UL{J)*(I+3)
DUL=DUL *( I+3)
CF(EPS I(I)l.5*EPSICl)/{EPS I( I ))
DF=1.OD+OOCF
FORTRAN IV G LEVEL 21
CALFUN
DATS
79 109
22/46/19
0078
0 079
0080
008 I
0082
0083
0084
0085
0086
0087
0088
0089
0090
0091
0092
0093
0094
0095
0096
0097
0098
0099
0100
DDF= 0 OD +00
DCF =0.0 0 D +0 0
C LARGE UL VALUE SECTION
C CONTINUOUS DISTRIBUTION SECTION
C 0< Y
GC t= XTC ( 1 ) + XTC { 2) *T( J) + XTC (3 ) T ( J) **2
GC2= XT C(4)+XTC(5)*T(J )+XTC(6)T(J)**2
GC3=XTC(7)+XTC(8)*T(J)+XTC(9)*T(J)**2
GC4=XTC(10)+XTC(11)*T(J)+XTC(12)*T(J)**2
GC5=XTC(13)+XTC(14)*T
GCOLLI=OEXP(GC1+GC2/UL(J}+GC3/(UL(J)**2)+GC4/(UL(J ) **3)+GC5/(UL(J
2)**4 ) )
SC1=XTC(2)+2.0*XTC(3)*T(J)
SC2= XTC ( 5) + 2. 0* XTC ( 6 > T ( J )
SC3=XTC(8)+2.0*XTC(9)*T (j)
SC4 = XTC ( 1 1 ) +2.0 *XTC( 1 2 ) *T ( J )
SC5=XTC( 14)+2.0*X7C(15)*T(J)
S CO L LI=GCO LLI *{S C1+S C2/UL(J)GC2*DUL/CUL { J) **2)+SC3/C UL(J)** 2}2.
20*GC 3TDUL/ UL(J)**3)*SC4/(UL(J)**3)3.0*GC4*DUL/ (UL(J ) **4)+SC5/(UL
31J)~T4)4.0*GC5*0UL/{UL(J)**5))
C DISCRETE DISTRIBUTION SECTION
C 0
GC1=XTC(16)+XTC(17)*T{J)+XTC{18)*T(J)*T2
GC2=XTC(19)+XTC(20)*T(J)+XTC(21) *T(J>**2
GC 3= XTC ( 22) +XTC (23) *T(J)+XTC(24)*T(J)**2
GC4=XTC(25)+XTC{26J *T(J)+XTC(27)*T( J )**2
GC5=XTC(28)+XTC(29)*T(J)+XTC(30)*T(J)**2
GDOL LI = DEXPGC HGC2/ULJ) +GC3/(UL(J)**2)FGC4/(UL(J)**3)+GC5/(U L(J
2)**4))
SC 1=XTC( 17) +2. 0*XTC ( l 8) *T< J)
SC2=XTC (20 ) +2.0 *XTC( 2 1 ) *T< J )
SC3 = XTC(23)+20 *X T C(24 )+T ( J )
FORTRAN IV
010 1
0 102
0103
0104
0105
0 106
0107
0108
01 09
0110
0 111
0 112
0 113
01 14
0 115
0116
0117
0118
0119
0120
0 12 1
0 122
0 123
G LEVEL 21 CAL FUN DA TE = 79 109 22/46/19
SC4=XC ( 26) 42.0*XTC (27) T{ J)
SC5=XTC ( 29 ) +2 O *XT C(30)*T(j)
SDOLLI =GD OLL I (5C1 SC2/U L ( J )GC2 DUL/ (UL ( J ) **2 ) +SC3/1 UL( J ) **2)2.
20 *GC 3*DUL/{ UL{ J )**3) + SC 4/( UL ( J ) **3 ) 3. 0*GC4*OUL/ (ULU)**4)*S C5/ (UL
2( JlT^A)4.O *GC5 *DUL/(UL ( J ) *T5 ))
C CCNTINUCUS DISTRIBUTION
C OUTSIDE REGION Y LT. O Y GT. UL
GCl=XTC ( 67 ). + XTC (68 ) *T ( J ) 4X TC ( 69 ) *T ( J ) *2
GC2=XTC ( 70)+XTC( 71 ) *T{ J)+XTC (72)*T( J )**2
GC3= XT C ( 73 )+XTC74)*T(J)+XTC(75)*T(J)**2
GC4=XTC(76)+XTC(77)*T(J )+XTC (78 ).*T {J)**2
GCGLLI=(GC 1/UL ( J ) +GC2/ ( UL (J)** 2) +GC3/(UL( J) **3 ) + GC4/{ UL ( J ) * 4 ) )
S Cl =XT C ( 68 )+2 O *XTC ( 69 ) *T ( J )
SC2=XTC(71)+2.0*XTC(72)*T(J)
SC3=XTC(74)+2.0*XTC(75)*T(J)
SC4=XTC(77)+2.0*XTC(78 ) *T { J)
SCGL LI = ( SC 1/UL ( J) GC 1 *DUL/( UL ( J ) * 2 ) +SC2/ (UL ( J)**2 ) 2 O *GC2 DUL/ ( U
2L (J ) **3 ) S C3/ ( UL ( J ) **3)3.0*DUL *GC3/(UL( J ) ** 4) + SC4/( UL ( J ) **4)4. O*
3DUL*GC4/(UL( J)**5) )
C DISCRETE DISTRIBUTION
C OUTS IOE REGION
GCl=XTC(82)+XTC(83)*T(J)+XTC84)*T(J)**2
GC2=XTC(85)+XTC86)*T(J)+X^C(87)*T(J)**2
GC3=XTC(88)+XTC(09)*T(J)+XTC90)*T(J)**2
GC4=XTC(91)+XTC(92)*T(J)+XTC(93)*T(J)**2
GC5=XTC(94)+XTC(95)*T{J)+XTC(96)*T(J)**2
GC6 = XTC ( 97) +XTC (98) *T( J ) +XTC99 )*T ( J )**2
GEE=DEXP( GCHGC2*DL0G( UL ( J ) ) + GC3 *0 L OG ( UL ( J) ) ** 2+GC4*DL0G ( UL ( J) )
2 3 + GC5* DLCG{UL(J) )**4 + GC6 *DLO G(UL(J ) )**5)
GDE=GEE+0. 5 0M XTC(79)+XTC(80)*T(J)+XTC(8l )*T { J)**2 )
SC1 = XTC(83) +2 O *XTC( 84) *T ( j )
SC2=XTC ( 06 )+2.0*XTC (87 ) *T ( J)
fo
O
CALFUN
79 109
22/46/19
FORTRAN IV G LEVEL 21
DATE =
0 124
0 125
0 126
0127
0128
0129
0 130
0 13 1
0 132
0 133
0134
0 135
0136
0137
013 8
0139
014 0
0 141
0 142
0 143
0144
0 145
0 146
0147
SC3=XTC(89>+2.0*XTCC90)*T(J)
SC4=XTC(92 ) +2.0 "XTCt 93 ) *T ( J )
SC 5 = XTC{95)+20*XTC(96)*T(J)
SC6=XTC{98)+2.0*XTC(99)*T(J)
SDE=GEE*(S C1+SC2*DL0G(UL{J) )+GC2*DUL/UL(J ) +SC3*DL0G(UL(J))**2+2.*
2G C3*tDLQG ( UL ( J ) ) *D UL/UL ( J ) + SC4 *DLOG ( UL { J) )**3 +3.0 GC4 CLO G (UL { J ) ) *
3 2* DU L/UL(J)+SC5 *DL OG(UL(J) )**4+4.0*GC5*DL0G(UL( J) )**3*DUL/UL{J) + SC
4 6*DLCG(UL(J))**5F50*GC6DLOG(UL{J) )**4 *DUL/UL{J >)+0.50*(XTC(80)
52 .0*XTC { 81 ) *T( J ) )
GKJ.I ) ={DF*EPS I ( I ) ( GDOLL I + GDE ) +CF *EP SI ( I ) *(GCOLLI+GCGLLI ))/CT(J)
2)
SUJ.I ) = DF *EP S I (I ) *( SDOLL I + SDE )+CF *EPSI ( I ) *( SCOLL HSCGLLI lDDF+EPS
1 I { I > (GDOLLI+GDE)DCF*EPSI(I)*( GCOLLI + GCGLLI )
HI ( J I)=GI( J I )+SI(J I)
C SECTION FOR CALCULATING HEAT CAPACITIES
10 CONTINUE
DO 15 J=l,7
HH(J}=HAC(J,I)*T(J)
H I I ( J) =Ht ( J .1 ) *T( J)
HHH{ J ) = HP. V { J I ) *T ( J )
15 CONTINUE
H = 0. 5000
MN=7
CALL DDET5{H,HH,DHH,MN,IER)
CALL D0ET5 H.HI I ,DHI I ,MN, IER)
CALL DDET5(H,HHH.DHHH.MN,IER)
C SECTION FOR CALCULATING PROPERTIES AT DESIRED TEMPERATURES
SS I { I K ) = SI (4,1)
HHI(I.K)=HI(4.I)
GC(I .K) =WAC(4,I)
HC(I,K)=HAC(4,I)
SC(I .K) =SAC(4,1)
244
FORTRAN IV G LEVEL 21
CALFUN
DATE
79 109
22/46/19
0 148
0149
0150
0 15 1
0 152
0 153
0154
0155
0 156
0157
0158
0159
0160
0 16 1
0162
GGI(IiK)=G1(4*1)
SARV (I,K)=SRV(4.I)
GARV(I,K)=GRV(4,I)
HAR V( I iK) =HRV( 4,1)
CPC( I,K ) = DHH(4 )
CPI(I,K)=DHII(4)
CPAPV I ,K)=DHHHC4)
C SECTION for CALCULATING TGTAL PROPERTIES
CCPStI ,K)=C PI( I *K)+CPC(I,K)+CPARVI ,K)
CGSCI,K)=GC(I,K)+GGI(I,K)+GARV(I,K)
C HS( I,K)=HC(I ,K)+HHI( I K)+HARV( I.K)
C SS I K)SC(I ,K) + SSI( IK)+SARV(I,K)
11 CONTINUE
4 CONTINUE
RETURN
END
1C APPIL 1979 DATA LIST FOR LIQUID HYDROCARBON PROGRAM
C X(l) = 0.0527789 IS .
C X ( 2 13 1946760
C X(3)2.8715619
c
DEL=
8.31948960+00+2.605
2103D + 03/T(J) 1 8
. 9 9306 5D+ 04/( T{ J)i
** 2)
c
1.8018
9 2 5.0SC6
1.9521
1098.3725
2.
1379
1231.3616
2.3022
1455.9570
c
2.5190
1 645. 9275
2.6969
1 823. 3736
2.
7987
2020.1494
2.9004
2199.8172
c
3.0752
2401.1257
.3.2500
2532.3034
s
96. 0300
1 SC. 60 00
232.5113
74.9000
150.
7000
1 89 .0 1 6 1
433.7000
46.40 00
c
0 .4 8333
OD +03 0.
1 70 3060+03
0. 14 5216D +02
C 277.1500
C 298. I 500
C 323.1500
C 358.1500
246
nnnnnnnnnnnrtn
19 APRIL 1979
DATA LIST FOR LIQUID HYDROCARBON PROGRAM
X ( 1 ) =0 .4940330l
X{ 2)=30775424
X(3) = 2 .3138407
DEL=6 .2246729 + 2.1484507D+03/T(J1 14
1.8010 824. 1700 1.9521 560.3200 .
2.5190 1395.1208 2.6969 1535.7769
3. 0752 1 958.2831 3.2500
96.0300 190.60C0 266.2056
0.48 03 3 0 D + 0 3 0.1703060 + 03
40244D+04/IT(J )* *2)
2.1379 1 1 05.931 1
2.7987 1693.3008
14 1 .7155
74.9000 150.7000
0.145218D +02
245 .2719
277.1500
298.1500
323.1500
353. 1500
2.3022
2.9004
438.7000
1242.7756
I 835.2233
46 .4000
247
Table C2a
Contributions to Free Energy and Entropy of Solution of Liquid Hydrocarbons
Solute
T(K)
AG
c
RT
AG?
i
RT
AG
RT
AGcal
RT
AS
c
R
AS?
1
R
AS
r>v
R
AS
cal
R
C5H12
2 77.15
48.10
42.06
4.04
10.08
19.83
3.97
4. 04
27. 88
293.15
45.87
38. 73
4.01
11.15
13. 08
6.5 9
3.27
22 .94
323.15
43.06
35.10
3.93
l 1.39
6.36
9.65
2.55
18.56
358.15
39 .09
30.53
3.75
12.31
1.17
 1 3. 7 7
1.6 7
14.27
C6H14
2 77.1 5
53.90
48. 39
4.80
10.31
20.63
4. 69
4.80
30.1 l
298.15
5 l .23
44.55
4.76
11.49
12.87
7.74
3. 89
24. 50
323.15
48 .00
40.36
4.66
12.30
5. 25
 1 1. 29
3. 03
19.56
358.15
43.39
35.OS
4.45
12.76
3.22
16.02
l .98
 14.78
C7H16
277.15
59 .70
55. 14
5.73
10.29
21.39
5.48
5.73
32. 55
293.15
56 .68
50.75
5.69
11.63
12.67
8. 98
4. 64
26.29
323.15
52.93
45. 57
5.57
12 .54
4.13
13.04
3.62
20.79
353.15
47 .63
39.94
5.32
13.07
5.28
18.45
2. 37
 15.53
248
Table C2a (Continued)
Solute
T(K)
AG
c
RT
AG
X
RT
AG
rv
RT
AGcal
RT
AS
c
R
AS?
i
R
AS
r, v
R
AScal
R
C8H18
277.15
65.49
61.44
6.56
10.6 1
22. 1 4
6. 25
6. 56
34.95
298.15
62. 09
56.54
6.52
12.07
12.46
10. 16
 5.32
27.93
323.15
57.87
51.20
6 .38
13.04
3.0 1
14.69
4.14
21.84
358.15
51.98
44.46
6.09
13.60
7. 33
20. 70
2.71
16.08
C9H20
277.15
7 1.2 3
68.47
1 0.42
1 3.24
22.89
7.11
7.65
37.65
298.15
67 .49
62.99
10.12
14.63
12 .25
11.47
6. 20
29.92
323.15
62.31
57. C3
9. 79
1 5. 56
1.90
16.51
4.83
23.23
358.15
56.28
49.52
9.38
16.14
9.39
23. 19
3.16
 16.96
C10H22
277.15
77 .09
74.92
8.55
10.72
23.65
7. 9 l
8.55
40.11
298.15
72 .90
68.92
8.49
12.47
12.05
12.68
6. 93
31.65
323.15
67.74
62.39
8. 31
1 3.66
0. 78
18.19
5 .39
24.36
358.15
60.57
54.15
7.93
1 4.35
11.44
25.46
3.53
17.55
Table C2a (Continued)
AG
ag
1
AG
Solute
T(K)
c
r,v
RT
RT
RT
C11H24
277.15
82.88
82. 16
9. 06
298.15
78 .31
75.57
9.00
323.15
72 .68
68.40
8.81
356.15
64.87
59.36
8.41
C12H26
277.15
33 .68
83.65
9.58
298.15
83.71
81.53
9.51
323. 1 5
77.61
73.79
9.30
358.15
69.16
64.03
8.88
C13H28
277.15
94.47
96.13
1 0.46
298.15
89.12
88.40
l 0 .3e
323.15
82.55
80.0
10. 16
358.15
73.46
69.41
9.70
AG
AS
AS?
O
ic/:
<
AS
cal
c
1
r,v
cal
RT
R
R
R
R
9.78
24 .40
8.81
9.06
42.27
11.74
11.84
14.03
7.34
33.22
13.08
0.34
20.07
5.72
25.44
13.91
13.50
28.0 l
3.74
 18.25
9.60
25.16
9.62
9.57
44.36
1 1 .69
11.63
15.26
7.76
34.65
13.13
1.46
21.76
6. 04
26.34
14.01
15.55
30. 29
3.95
18.70
8.80
25.91
10.55
10.45
46 .9 1
1 1 10
ll.43
 16.66
8.47
36.56
12.70
2.57
23.69
6. 59
27.72
13.74
l 7.6 1
32.92
4.32
19.63
Table C2a (Continued)
AG
AG
i
AG
Solute
T(K)
c
r,v
RT
RT
RT
C14H30
277.15
100.27
 102.9
1 1 .34
298.15
94 .53
94.42
11.26
323.15
87.43
85.44
11.02
358.15
77.75
74.12
1 0. 51
AGcal
. AS
c
AS.
i
t>
col
il o
<
AS ,
cal
RT
R
R
R
R
8.92
26.67
11.36
11 .33
49.37
11.36
11.22
 17.89
9.18
38. 30
13.06
3. 69
25. 40
7. 1 5
28.86
14.15
19.66
35.22
4.68
20.24
251
Table C2b
Contributions to Enthalpy and Heat Capacity of Solution of Liquid Hydrocarbons
Solute
T(K)
AH
c
AH?
X
AH
r,v
AH
cal
ACp
c
ACp?
ACp
r, v
A^cal
RT
RT
RT
RT
R
R
R
R
C5H12
277.15
28 .23
46.03
0.00
17.80
98. 06
34.89
11.95
75.11
298.15
32.79
45.32
0.74
11.79
88.29
36.95
9.50
60.84
323.15
36 .70
44.75
1.38
6.67
78.67
38.97
8. 68
48. 38
35e .15
40 .26
44.30
2. 08
 1.95
68. 1 5
41.05
3.55
35.68
C6H14
277.15
33.27
53.08
0.00
19.81
112.06
40.67
14.19
85.53
298.15
38 .40
52.29
0.88
13.01
100. 58
42. 98
11.28
68. 89
323.15
42.75
51.65
1.64
7.26
89.00
45.05
10.31
54 .26
358.15
46.61
51.10
2 .47
2.02
76. 12
46.76
10.15
39.5 1
C7H16
277.15
38.31
60.62
0.00
22.31
126.06
46. 6 1
16.96
96.41
298.15
44.02
59. 73
1.05
14.67
112.87
49.25
13.48
77.10
323.15
48 .80
59.01
l .96
8.25
99.34
51.56
12.32
60. i C
358.15
52 .96
58.38
2. 95
2.47
84. 04
53.23
12.13
42.94
252
Table C2b (Continued)
Solute
T(K)
AH
c
AH
i
AH
r, v
i5cal
ACp
c
ACp
i
ACp
r ,v
RT
RT
RT
RT
R
R
R
R
C8H18
277.15
43.35
67.69
0.00
24.34
140.06
52. 03
19.40
107.44
298.15
49.63
66. 69
1.20
15.87
125.16
54.95
15.43
85 .64
323.15
54.86
65.89
2 .24
8.79
109.68
57.46
14.10
66.32
3 58.15
59 .31
65.16
3. 38
2.47
91.97
59. 1 1
13.88
46.74
C9H20
277.15
48.39
75. 58
0. 00
27.18
154.06
58.01
22.63
1 18.69
298.15
55.24
74.46
1.40
17.32
137.45
61.22
1 8.00
94. 23
323.15
60 .91
73. 54
2. 6 1
 10.02
120.01
63.91
16.45
72.55
358.15
65.66
72.70
3.94
3.10
99.90
65.53
16.19
50.56
C10H22
277.15
53 .44
32.83
0.00
29.40
168.07
63.48
25.29
129.87
298.15
60.85
81.60
1.56
 19. 18
149.73
66.91
20.11
102.93
323. 1 5
66.96
80.58
2.92
10.70
130.35
69.74
18.37
78.9 8
358.15
72 .0 1
79.62
4.40
3.21
107.83
71. 27
16.09
54.64
Table C2b (Continued)
Solute
T(K)
AH
c
AH
X
AH
r ,v
ARal
>
m
O O
ACp?
i
ACp
r,v
iECi
RT
RT
RT
RT
R
R
R
R
C11H24
277.15
58 .43
90.97
0.00
32.49
182.07
69. 63
26. 80
139.24
298.15
66.47
39.60
1.66
21.43
162.02
73.31
21.31
110.03
323.15
73.01
36.47
3.09
12.36
140.68
76. 28
1 9. 48
83.89
358.15
78 .36
87.37
4.67
 4.34
115.76
77. 69
19.17
57.24
C12H26
277.15
63.52
98. 28
0. 00
34.75
196.07
75.15
28.32
149.23
298.15
72 .08
96.79
1.75
22.96
174.31
79.03
22. 52
117.80
323.15
79 .0 7
95.55
3. 27
 13.21
151.02
82. 09
20.58
89.51
358.15
34.71
94.32
4.93
4.68
123.68
83.35
20.25
60.59
C13H28
277.15
68 .56
106.68
15.63
22.49
210.07
8 1.55
9.55
138.06
298.15
77 .69
 105.06
15. 18
12.19
186.60
85.66
9.06
1 10 .00
323.15
85.12
103.70
14.69
3.89
161.36
88.83
3.59
81.12
358.15
9 1 .06
 102.33
14.07
2.30
131.6 1
89. 93
8. 1 0
49. 78
254
Table C2b (Continued)
AH
ah
o
ffll
o
Solute
T)K)
c
X
r,v
RT
RT
RT
C14H30
277.15
73 .60
114.05
0.00
298.15
83 .30
 112.32
O
.
Cl
323.15
91.17
110.84
3.87
358.15
97 .4 1
 109.35
5.84
i5cal
>
Ol
o o
ACp?
i
ACp
r,v
RT
R
R
R
R
40.44
224.07
87. 16
3 3.53
170.43
26.94
198.89
91.46
26.66
134.09
15.80
171 .69
94.72
2 4.36
101.34
6. 10
139.54
95.62
23.98
67.90
Table C3
Energy Parameter Values and Length Function
Hydrocarbon
e /k
ws
Methane
266.2
Ethane
403.4
Propane
543.2
Butane
686.2
Pentane
824.2
Hexane
960.3
Heptane
1105.9
Octane
1242.8
Nonane
1395.1
Decane
1535.8
Undecane
1693.3
Dodecane
1835.2
Tridecane
1998.3
Tetradecane
2141.7
Lou = 0.49403381 + 3.07754241H02/T
 2.3198407D+04/T with T in K.
APPENDIX D
PROGRAM FOR CALCULATION OF THERMODYNAMIC
PROPERTIES OF MICELLIZATION
258
FORTRAN IV G LEVEL 21
MAIN
DATE = 70108
01 /OI/23
C
C
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
OBJECTIVE CALCULATE SCME CONTRIBUTIONS TO THE THERMODYNAMICS OF
MICELLIZATION FROM THE SOLUBILITY MODEL AND CTHER CONSIDERATIONS
SURF ACT ANTS STUDIED ARE SODIUM OCTYL SULFATE.DECYL SULFATE
AND D0D5CYL SULFATE
EXPLANATION OF INPUT DATA
CVW CHARACTERISTIC VOLUME OF WATER CC/GMCL
CTW CHARACTERISTIC TEMPERATURE OF WATER K
EXC1 ,EXC2.EXC3 COEFFICIENTS IN EQUATION FOR ARGON SOLUBILITY
PROPERTIES
EPSIAR INTERACTION ENERGY PARAMETER FOR ARGON K
C VAR CHARACTERISTIC VOLUME OF ARGON CC/GMOL
CHARACTERISTIC TEMPERATURE FOR ARGON K
C TAR CHARACTERISTIC TEMPERATURE FCR ARGON K
CVCH4 CHARACTERISTIC VOLUME FOR METHANE CC/GMOL
CTCH4 CHARACTERISTIC TEMPERATURE FOR METHANE K
EXMS DIMENSIONLESS ENTROPY OF MICELLIZATION AT 298 K
EXMH DIMENSIONLESS ENTHALPY OF MI CELLIZATI ON AT 298 K
EXMG DIMENSIONLESS FREE ENERGY OF MI CELL IZAT ION AT 298 <
PMVM SURFACTANT PARTIAL MOLAR VOLUME IN MICELLE AT 298 K
XMC MOLE FRACTION OF SURFACTANT IN SOLUTION AT CMC
V STAR CHARACTERISTIC VOLUME FOR HYDROCAREON OF EQUAL CHAIN
LENGTH TO SURFACTANT FROM GMEHLING 1978 L/GMOL
C CHAIN LENGTH PARAMETER FROM GMEHLING 1978
T STAR CHARACTERISTIC TEMPERATURE FOR EQUIVALENT HYDROCARBON
FROM GMEHLING 1978 K
A NM PARAMETERS FOR PERTURBATION CONTRIBUTION TO INTERMOLECULAR
POTENTIAL FROM GMEHLING 1978
RHG SURFACTANT HEAD GROUP RADIUS A
RCl COUNTERION RADIUS A
FBC FRACTION OF COUNTERIONS BOUND TO MICELLE
CC/GMOL
P
000 i
0002
0002
0004
0005
0006
0007
0008
OOOS
00 10
001 l
0012
00 13
001 4
0015
0016
0017
LEVEL 21 MAIN DATE = 79108 01/01/23 PA<
C NC SURFACTANT CHAIN CARBON NUMBER
C NAGN MICELLE AGGREGATION NUMBER
C T TMMPERATURE OF MODEL K
C EPSI INTERACTION ENERGY PARAMETER FOR WATERMONOMER SAME AS
C FOP EQUIVALENT HYDROCARBONS CAL/GMCLE
C XT C COEFFICIENTS IN FUNCTIONS FOR CONTRIBUTIONS TO
C WATERMONOMER INTERACTIONS. SAME AS FOR EQUIVALENT HYDROCARBONS
C
c
IMPLICIT REAL*8(AH0Z)
DIMENSION XL{4) ,NC(3) ,DXL(4) ,T(4) ,WMON(3.4),SMON3,4),HMON(3,4),
lNAGN(3)PMVM(3),WMIC(3,4J,SMIC(3,4),HMIC(3,4),WCI(4),SCI(4),HCI(4)
1 PHO M { 3 ) ,RHCS(3 ) ,SMC (3 ,4 ) W MC (3,4 ) HMC (3.4), VSTAR 3), WRV( 3.4 ) ,
1HRV 3,4),SRVC 3,4) ,WCA(3,4) .SCAI3.4) .HCA(3,4),EXMS(3),EXMH(3),EXMG(
13 ) ,C(3).EPS I 3),XTC(99),TSTAR(3),ANM(2,5),PPMVM{3) .UL(4) .XCMC(3)
1WMMI(3.4) ,HMMI (3,4) .SMMI(3,4) ,WT(3.4),HT(3,4),ST(3,4 ) ,WMW(3,4 ) ,SMW
1(3.4), HMW (3,4 ) WMCC 3,4), SMCC ( 3.4 ) H MCC ( 3,4)
PI=3.141592 65D 0
XK=1.38066
RK=1 .987
A 1 = 0.54008832
A 2= 1 .2669802
A3=0.05132355
44=2.9107424
A5=2.5167259
A6=2.1595955
A 7=0.64269552
A3=0.17565885
A 9=018874824
A 10=17.952388
A11=0.48197123
A 12 = 0. 76696 099
259
FORTRAN IV
oo i a
0019
0020
0021
0 022
0023
002 4
0025
0026
0 02 7
0028
0 02 9
0030
003 1
0032
0033
0034
003 5
0036
0037
0038
0039
0040
004 1
G LEVEL 21 MAIN CATE =79108 01/01/23 PA
A 13=0.76631363
A 14=0 .809657804
A 15=0.24062863
C VW= 46 4
CTW=438.7
C READ EXPERIMENTAL MI CELL IZATIQN PROPERTY VALUES
READ (5.1) ((EXMS(I).EXMH(I).EXMG(I))1=1.3)
1 FORMAT (8F10.4)
C READ MICELLE PARTIAL MOLAR VOLUME AND XCMC AT 298K
READ (5,2) ((PMVM( I ).XCMC( I )) .1=1.3)
2 FORMAT (6E12.4)
C READ PARAMETERS FOR ROTATIONAL AND VIBRATIONAL EFFECTS
READ (5,3) ( (VSTAR(I), C( I )), 1=1,3)
3 FORMAT (6F1C.4)
C READ PARAMETERS FOR MONOMERMONOMER INTERACTION IN MICELLE
READ (5,7) (TSTAR(I),1=1,3)
7 FORMAT (3F1C.4)
DO 10 N=1,2
READ (5,9) (ANM(N.M),M=1,5)
9 FORMAT (5F10.5)
10 CONTINUE
C READ HEAD GROUP RADIUS.COUNTERI ON RADIUS AND FRACTION OF 80UND COUNTERIONS
READ (5,8) RHG RCI FBC
8 FORM AT (3F1 0 .4 )
C READ CARBON NUMBER OF SURFACTANT AND AGGREGATION NUMBER OF MICELLE
READ(5.6) ( (NC( I),NAGN( I)),1=1,3)
6 FORMAT (614)
C READ METHANE AND ARGON CHARACTERISTIC VOLUMES AND TEMPERATURES
READ (5,14) CVCH4,CTCH4.CVAR,CTAR
14 FORMAT (4F10.4)
C READ ARGON ENERGY PARAMETER AND COEFFICIENTS FOR ARGON SOLUBILITY
READ (5,15) EPSIAR.EXCl.EXC2.EXC3
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6W**2.0) *3.00
0055 TRCH4=T{JJ/CTCH4
FORTRAN IV
0060
006 1
0062
0062
0064
006 5
0066
0067
006
0069
0070
007 l
0072
G LEVEL 21 MAIN DA TE = 79108 01/01/23 P
TRAR =T(J)/CTAR
HSDC H4= ({3.0*CVCF4/(2.0*P1*0.6023)) *(A7/(TRCH4**Ae)+A2/(DEXP(A4*((
2DENW R+A l TRCH4 )**2 ) ) )A3/ (CSXP(A5* { { DENWR +AI TRCH 4A 6 ) **2 ) ) ) +A 9/( D
3EXP( A 1 0*( { ( TRCH 4A 13) ** 2) + A1 l { ( DE NW R Al 2 ) **2 > ) ))))**0 .333333
HSDAR=( (3.0*CVAR/(2.0*PI10.6023) ) *( A7/(TRAR**A8) +A2/(DEXP(A4*( (DEN
2WR+A 1*TRAR)**2) ))A3/(DEXP(A5*( (DENWR + A1*TRARA6)**2) ))+A9/(DEXP(A
310* ( ({TRARA 13) **2) +A1 1 *( ( DENWRA 12) **2) ) ) ) )** 0.33333
DHSCH4={ (3.0*CVCH4/(2 .0 *P I *0.60 23 ) ) *( A7 A8/(CTCH4*(TRCH4**{A 8+1.O
2)))2.0*A2*A4*(DENWR+Al*TRCH4)*(DDENWR+Al/CTCH4)/(DEXP(A4*((DENWR+
3A 1*T RCH4 ) **2 .0 ) ) ) + 2 .0 *A3*A5*( DENWR+A l*TRCH4A6) *( DDE N WR + A1 /C TCH4 ) /
4(DE XP(A5*((DENWR+A1*TRCH4A6)**2.O) ) )2.O*A9*Al O*{(TRCH4A 13)/CTCH
54+A 11*(DENWRA12)*DDENWR)/(DEXP(A 1 O* C( (TRCH4Al 3)**2)+A11*((DENWR
6A 12) **2 '))))))/( (HSDCH4**2.0) *3.0 )
DHSD AR=((3.0*CV AR/(2O *P1*0.6023) )*{A7*A8/(CTAR*(TRAR**{A8 + 1 .0)) )
22.0*A2*A4*(DENWR +A1*TRAR )*(DDE NWR + A1/CTAR)/(DEXP(A4* ((DENWR+Al*TR
3AR)* *2O) ) )+2.0*A3*A5*( DENWR + Al*TRARA6)*(DDENWR+Al/CTAR)/(DEXP(A5
4*({DENWR+Al*TRARA6}**2.O)) )2.O*A9*Al O*((TR ARA13)/CTAR+ A11 *(DENW
5 RAl 2)* CCENWR)/(DEXP(AlO *( ((TRARA13)**2 )+A11*( (DENWRAl2)**2)> ) ) )
6) /( {HSD AR**2O)*3.O )
C CAVITY DIAMETER EQUALS SUM OF WATER AND SOLUTE DIAMETER
DCWAR=HSDAR+HSDW
DCWGARDC WAR
DCWCH4 = +SDCH4 + HSDW
DDCWG4=DHSDW+DHSCH4
DDCWGA= DHSCW+DHSDAR
C SECTION TO CALCULATE INTERACTION PROPERTIES FOR ARGON
G IAR =EP SI AR *( 6.7202D + 004.9540+03/( T( J) )+6.548OD+05/(T{J )**2)0.7
2600D+00+HSDAR0 .7925D+0 0* ( HSDAR **2) ) / ( T( J) )
SIAREPSIAR *(4.9540+03/ ( T ( J ) **2 ) 1 3.096 0+05/ ( T(J ) **3 )0.76D+ 00 *D
2HSDAR1.585D+0 0*HSDAR*DHSDAR)
HIAR=GIAR+SIAR
C SECTION TO CALCULATE EXPERIMENTAL PROPERTIES FOR ARGON
262
FORTRAN IV G LEVEL 21
MAI N
DATS
79108
01/01/23
0073
0074
0075
0 07 6
0077
007e
0079
0080
008 1
0082
0083
0064
0085
0086
0087
0088
0089
0090
0091
0092
0093
GEXAR=(EXCi+EXC2*DL0G(T(J))+EXC3*(OLOGT J ) )**2))
HEX A REX C2+2. O *EXC 3*DL0G T(J))
SEXAR=HEXARGE XAR
C SECTION TO CALCULATE ARGON CAVITY PROPERTIES BY DIFFERENCE
WACAR=G2XARGIAR
HACA R=HEXARHI A R
SACA R=SEXARS TAR
C CALCULATE PARTIAL MOLAR VOLUME IN MICELLE AS A FUNCTION OF TEMPERATURE
PPMVMI >=PMVM ( I )*( 1 .0+( T{ J >293.15)/700. 00)
DPMV M=FMVM C I )/700
C CALCULATE SURFACTANT DENSITY IN AQUEOUS SOLUTION AND IN MICELLE
RHOM(I} = 0 6 023/PPMVM( I )
RFOS
C CALCULATE SURFACE TENSION CURVATURE PARAMETER AND SEGMENTAL LENGTH
STW=1.162D+021 .477D01*T(J)
D STW=1. 477D01
DEL=3.3241071D+00+2.6065614D+03/T(J)19.001953D+04/(T(J)**2)
DDEL=2.6065614D+03/T(J)**2)+2.0*19.001953D + 04/ T(J )**3 )
XL( J >=0.527 78918+3. 19 46 78D+ 02/T ( J ) 2.871 56 1 9 D +0 4 / ( T { J )**2>
DXL{J)=3.1946780+02/(T{J)**2)+2.0*2.87l5619D+04/{T(J)**3)
C SECTION TO CALCULATE MONOMER CAVITY PROPERTIES
XMONL=XL(J>*{NC(I>1)
DXMONL=CXL (J)*(NC(I)1 )
W MO N(I J)={PI/(XK* T J) ) >*(XM0NL*STW*DCWCH4+STW*( (CCWCH4**2)(DCWA
2R **2 ) ) 4 0 C EL *STW ( DC WCH 4DC WAR) ST W*DEL* 2. 0*XMONL)WACAR
SMON ( I J ) = ( PI/XK ) ( DX MCNL*ST W *DCW CH4 + XM0NL DST W +DCWCH4+XMQNL AST V*
2DDCWG4+DSTW+(OCWCH4**2)(DCWAR**2) )+3TW*(2.0*DCwCH4*CGCWG42.0 *DC
3 W AR* DDCW GA )4.0 +DOEL *ST W* ( DCW CH 4DC WAR )4 .0*0 EL *D S TV* ( DC WCH4DCWR
4) 4.0*DEL*STW+ ( DDCWG4 DDCWG A ) DSTW*DEL *2 .0 *X MONLST rf*DDEL *2 O +XMO N
5LST W *DEL *2.0*D XMONL)SAC AR
H MON (I J)=WMCN{ ItJ) +SMON{ I J )
C SECTION TO CALCULATE COUNTERION CAVITY PROPERTIES
PA
263
FORTRAN IV
0094
0095
0096
0097
0098
0099
0100
0 10 I
0102
01 03
010 4
0105
01 06
0107
0108
0109
0110
0111
0112
0 113
0114
0 115
G LEVEL 21 MAIN CATE = 79108 01/01/23 P
HSDC1=2.O+RCI
DCWCI=HSDCI+HSDW
D DC WC I = DH SD Vk
WCI ( J )=FBC*( (P I/( XK*T( J ) ) ) *( STW*(DCWCI**2DCV>AR**2) 4.0*STW*DEL*
2(DCWCIDCWAR))+WACAR)
SCI { J) =FBC*( (PI /XK ) ( DSTW* ( DC WC I **2DC W AR* *2 ) tSTviK (2 D CW C I ODC'W
2CI2 .0*DCWAR*DOCWGA )4.0 *D STW *DEL* ( DC WCIDC WAR) 4.0* ST**DOEL* (
3DCWC IDCWAR)4.0*STW*DEL*(DDCWCIDDCWGA))+SACAR )
HCI J) = WCT ( J) + SCI C J)
C SECTION TO CALCULATE TOTAL OF MONOMER AND COUNTERION CAVITY PROPERTIES
WMCC(I,J)=WMON(IJ)+WCI {J)
SMCCI, J) = SMON( I, JJ+SCKJ l
H MCC(I J)=H VON ( I*J> +HCI (J)
C SECTION TO CALCULATE MICELLE CAVITY PROPERTIES CN A PER MONOMER 8ASIS
HSDM=(6.0*NAGN( I )*PPMVM{I)/PI )**0.333
DHSDV=0.3333*(6.0*NAGN(I)*DPMVM/PI )/ (HSDM**2 )
DCWM=HSDM+HSDW
DDCW M=OHSDW+DHS CM
WMIC(I *J) = ( (PI/(XK*T(J) ))*(STW*(DCWM**2DCWAR**2 )4.0*STW*DEL*(DCW
2MDCWAR ) )+WACAR )/NAGN( I )
SMIC(IJ)=((PI/XK)*(DSTW*(DCWM**2DCWAR**2)+STW*(2.0*DCWM*D0CWM
22 0* DCWAR* DDC WGA)4.0*0 ST W*DEL*(DCWMDCWAR)4.0*STW*DDEL*(DCWMDCW
3AR)4.0 *ST W *DEL *(DDCWM CDCW GA) )+SACAR)/NAGN( I )
HMI C (I J) = WVTC ( I J) +SMI C( I J)
C SECTION TO CALCULATE MONOMERWATER INTERACTIONS
UL(J)=X MCNL
DUL=DXMCNL
CF= ( EPS I ( I)1.5*4.62D+02)/(EPSI (I ) )
DF = 1 .OD+OOCF
DDF=0.00
D CF= 0.0 0
C LARGE UL VALUE SECTION
264
FORTRAN IV
0116
01 1 7
o i a
o i g
O 120
0121
0 122
0 122
0124
0125
0126
0127
0128
0129
0 130
0131
0132
0133
0134
0135
0136
0137
O 138
0139
G LEVEL 21 MAIN OATE = 79108 01/01/23 P
C CONTINUOUS DISTRIBUTION SECTION
C O
GC1= XTC(1>+ XTC(2)*T(J)+XTC<3)*T(J)**2
GC2=XTCC4J+XTCC5)*T
G C3=XT C (7 )+ XT C( 8 ) *T(J)+XTC(9)*T(J) * 2
GC4=XTC( 10) +XTC (11 )*T
GC5=XTC(13)+XTC{ 14)*T(J)+XTC( 15)*T( J)**2
GCOLLI=DEXF(GC1+GC2/UL{J)*GC3/(UL {J)**2)+GC4/(UL(J)**3)+GC5/< UL(J
2)*4 ) )
SC1 = XTC(2)+2.0*XTC( 3>*T( J )
SC2=XTC(5)+2.0*XTC(6)*T{J)
SC3=XTC( 8)*2.0*XTC( 9)*T< J)
S C4 =XT C { 11 )+2 O *XT C ( i 2 ) *T ( J )
SC5=XTCC14)+2,0+XTC(15)*T(J)
SC0LLI=GCOLLI*(SC1+SC2/UL{J)GC2*DUL/(UL(J)**2)+SC3/{UL(J)**2)2.
2 0*GC3*DLL/(UL(J)**3)+SC4/(UL(J)**3)3.0 + GC4*DUL/(UL{J ) **4 )+ SC5/(UL
3(J )**4)4.0*GC5*DUL/(UL(J)**5))
C DISCRETE DISTRIBUTION SECTION
C 0< Y
GC1= XTC(16)+XTC (17)*T(J)+XTC( 13)*T(J)* *2
GC2 =XT C(19)*XTC(20)*T C J)FXTC(21 )*T(J > **2
GC3= XTC(22)+XTC(23)*T< J)FXTC(24)*T(J)**2
GC4= XT C{25)+XTC(26) *T{ J)+XTC(27)*T(J)**2
GC5=XTC(28)+XTC(29)*T{J)+XTC30)*T{J)**2
GD OLLI = DE XP(G C1+GC2/UL(J)+GC3/UL( J)**2)+ GC4/(UL(J)**3)+GC5/(ULC J
2)**4) )
SCI=XTC(17)+2.0*XTC(18)*TJ)
SC2= XTC(20)+2.0 *XTC(21)*T< J)
SC3=XTC(23)+2.0*XTC(24)*T(j)
SC4= XTC(26)+2 O *XTC(27)*T{J)
S C5= XT C{29)+2O *XTC{30)*T(J)
SDOLL I=GDOLLI*(SC1+SC2/UL(J)GC2*DUL/(UL(J)**2 ) +SC3/(UL(J))2.
265
FORTRAN IV G LEVEL 21
MAIN
DATE
79 l OS
01/01/23
0140
0 14 1
0 142
0143
0 144
0 145
0146
0147
0148
0149
0150
0151
0152
0 153
0 154
0 155
0 156
0 157
0 158
0159
0160
0 16 1
0162
20 *GC 3*D LL/( UL J ) **3 ) + SC4/ ( LL( J) **3 ) 3.0*GC4* CUL/ (UL(J)**4) +SC5/(UL
3< J)**4)4.0*GC5*DUL/( UL ( J )**5) )
C CCNTINUCUS DISTRIBUTION
C OUTSIDE REGION Y LT O Y GT.L
GCl=XT C{67)+XT C(68)+T(J)+XTC(69)*T(J)**2
GC2=XTC(70) + XTC (71 ) *T( J)+XTC(72 )*T{ J )**2
GC3=XTC(73)+XTC(74)*T{J)+XTC(75)*T( J)**2
GC4=XTC(76)+XTC(77)*T(J)+XTC(78}*T(J)**2
GCGLLI = ( GCi/UL J)+GC2/(UL(J)**2)+GC3/ (UL{ J)* ) +GC4/
SCl=XTC(68>+2.O*XTCC69)*T(J)
SC2 = XTC(71 ) +2.0*XTC(72) *T { J)
SC3= XTC(74)+2O+XTC(75)*T(J)
SC4=XTC( 77 1+2.0 XTC(78) +T(J )
SCGLLI = (SCI /UL ( J)GC1 *DUL/(UL (J )**2.) +S C2/ (UL ( J ) **2 >2 O*GC2*DUL/( U
2L(J)**3)+SC3/(UL(J)**3)3.0*DUL*GC3/{UL(J)+*4) + SC4/(UL(J)**4)4.O *
2DUL*GC4/(UL(J)**5) )
C DISCRETE DISTRIBUTION
C OUTSIDE REGION Y LT.O Y GT.L
GC1=XTC(82)+XTC(83)*T(J)+XTC(84)*T{J )**2
GC2= XTC ( 85)+XTC(36) *T( J)+XTC ( 37) *T ( J )T*2
GC3=XTC (88) +XTC (89 ) *T ( J >+XTC( 90 )*T( J )**2
GC4=XTC(91)+XTC(92)*T(J )+XTC(93)*T(J)**2
GC5= XT C(94)+XTC(95)*T(J) + XTC(96)*T(J>**2
GC6=XTC (97 ) +XTC (98 ) *T ( J )+XTC(99 )*T ( J )**2
G EE=DE XP(GC1+GC2TDLQG(UL(J) )+GC3*DLCG(UL(J) )**2+ GC4*DLOG(UL(J) >**
23 +GC5*DL0G(UL(J))**4 + GC6 +DLOG(UL(J ) )**S)
GDE=GEE+0.50*(XTC(79)+XTC(Q0)*T(J)+XTC(81) *T{J)**2)
SC1=XTC(83)+2.0+XTC(84)*T(J)
S C2 =XT C(86) +2.0 *XT C(37)*T(J)
SC3=XTC(89)+2.OTXTC(90)*T(J)
SC4=XTC(92)+2.0 4XTC( 93)+T( J)
SC5=XTC(95)+20+XTC(96)*T(J)
FORTRAN IV G LEVEL 21
MAIN
DATE
79 108
01/01/23
0163
0 164
0165
0166
0 167
0 168
0169
0170
0 17 1
0172
0173
0 174
0 175
0176
0 177
0 178
SC6=XTC(98)+2.0*XTC(99)*T(J)
SCE=GEE*(S C1+SC2*DL0G( UL ( J ) ) +GC2*DUL/UL ( J > + SC3*DL0G ( UL ( J) ) **2 + 2. *
2GC 3* DLOG(UL(J) )*DUL/UL(J) +SC4 *DLQG(UL(J) )**3+3.0*GC4*CLOG(UL(J) )**
32+DUL/UL(J)+SC5*DL0G(UL(J))**4+4.0*GC5*DL0G{UL(J))**3*0UL/UL(J)+SC
46*DLCG
52. 0*XTC ( 81 ) *T( J) )
C SECTION TO CALCULATE TOTAL WATERMONOMER INTERACTION PROPERTIES
WMW( I,J)=(OF*EPS I(I)*(GDOLLI + GDE)+CF+EPSI{I ) *(GCOLLI+GCGLLI ) )/(RK
2*T( J ) )
SMW(I.J)=IDF*EPSI(I)*(SDOLLI+SOE)+CF*EP3I(I)*(SCOLLI+SCGLLI)DDF*E
2PSICI)*(GDOLLI+GDE)DCF*EPST(I)*(GCOLLI+GCGLLI))/(RK)
HMW(I.J)=WMW(I.J)+ SMWI,J)
C SECTION TO CALCULATE ROTATIONAL AND VIBRATIONAL CONTRIBUTIONS
VBAR =4. 9355D G2/0ENW
DV8ARW= 4.93550 02 *DDENW/{DENW**2)
VBARM=PPMVW(I)*1.000003/VSTAR(I)
D VBA RM=DPMVM*1000003/VSTAR(I)
*RV(I, J )=C ( I ) *( (3.0* ( (0.7405/VBARM )**2)4.0*0. 74 0 5/VBARM) / ( Cl. 0 t
20.7405/VBARM) )**2) )+(C(I)l)*((3.0*((0.7405/VBARW)**2)4 .0*0.7405/
3VBARW)/( ( 1.0{ 0.7405/VBAR VI ) ) **2) )
S RV ( I J ) =WRV( I t J) +T IJ ) *C ( I ) { DVB ARM **(( 6.0*0.7 40 5**2/( VBARM**3) +
24.0* 0.7 405/(VBAPM**2))/{(1.00.7405/V3ARM)**2) + (3.0*( (0.7405/VBARM
3)**2)(4.0*0.7405/VBARM)> *<2.0*0. 7405/(VBARM**2) )/(( i.0C.74 05/VB
4 ARM) **3 )))(C(I)l) *DVBARW* ( (6 .0*0.7405**2/ ( V8ARVn**3 ) +4 C*0.7405/
5( VBARW**2) ) /( ( 1.00.740 5/VBAR Vi) **2) + (3.0* ( ( 0 .7405/V8ARW)**2)(4.0*
60.7405/VBARW))*(2.0*0.7405/(VSARW**2))/((1.00.7405/VBAR*)**3>)
HRV( I J)=WRV( I J) + SRV( I J )
C SECTION TQ CALCULATE MO NO ME R / MO NO ME R INTERACTION IN MICELLE
TBAR=T(J)/TSTAR(I)
GSUM=0.CO
SSUM=0.00
DC 17 M=1 ,5
267
FORTRAN
0179
0 180
0181
0182
0183
0 184
0185
0186
0187
O 188
0189
O 190
0 19 1
0192
0192
01 94
0 195
O 196
019 7
0198
0199
0200
0201
0202
0203
IV G LEVEL 21 MAIN OATE = 79108 01/01/23 PA
DO 18 N=l,2
GSUMGSUM+M*ANM{N M)/ { { TB AR **N } *( VB ARM* *M ) }
SSUM=SSUM+M*ANM(N,M)/( ((TBAR**{N + 1 ) ) ( V B ARM *M ) )*TSTAR( I } )+ANN< N.M
2)*M*DVBARM/{(TBAR** N)* C VBARM**(M + l ) } )
18 CONTINUE
17 CONTINUE
WMMI ( I, J) = C { I )*GSUM
SMMI(I,J)=WMMI(I,J)+T{J)*C(I)*SSUM
HMMI{I.J)=WMMI{I,J)+SMMI{I,J)
C SECTION TO CALCULATE COUNTER1QN ADSORPTION PROPERTIES
RH0CM=2.00*C,75/( (185.0NAGN(I ) )*2.718 )
RHOCS= < i .OFBC) *RHOS( I )
WCA
SCA ( I J ) = WCA ( I J )
HCA( I, J )=O.OOD+O0
C SECTION TO CALCULATE TOTAL PROPERTIES
WT{ I J )=WMON{ I J) + WMI C( I J) + WCI (J) tWRV I J)+WCA{ I J) +
2WMW( I, Jl+WMVK I,J )
HT( I J) =HMCM I J) +HMI C I J )+HC I (J ) 4HRV ( I J )+FCA { I, J ) +
2HMW( I,J l+HMMI (I ,J)
STC I J) =SMCM I J) +SMIC I. J J+SCI (J ) +SRV { I. J )+SCA ( I J ) +
2SMW(I,JJ+SMMI{I.J)
11 CONTINUE
4 CONTINUE
WRITE (6,100
100 FORMAT { 1 ',///////////>
DO 1 01 1=1 ,3
DO 102 J=1,4
IF(J.EQ.2) GO TO 117
WRITE (6,104) T(J) ,WMW(I,J) ,WMCCI,J) ,WMIC(I,J),WRV ( I J ) ,WMMI (I,J )
2,WCA(I,J),WT(I,J)
104 FORMAT {/,1 OX,F7.2,3X ,F6.2.2X,F7.2,2X5(F6.2,2X) )
268
FORTRAN IV G LEVEL 21
MAIN
DATE
79 loa
01/01/23
0204
0205
0206
0207
0206
0209
021 0
021 1
0212
02 13
0214
0215
0216
0217
0218
0219
0 22 0
0221
0222
0223
0224
0 22 5
0226
022 7
0228
0229
0230
GO TO 102
117 WRITE (6*118) T(J ),WMW( I, J),WMCC{ I J) *WMIC(I ,J) ,WRV( I J) ,WMMI( I J)
2.WCA{I,J),WT(I.J),EXMG(I)
118 FORMAT { / 1 CX*F7. 2 ,3X .F6.2.2X *F 72.2X* 6(F6 .2 ,2X ) )
102 CONTINUE
1C1 CONTINUE
WRITE (6.105)
105 FORMAT (1*///////////)
DC 106 1=1.3
DO 107 J=1,4
IFJ.EQ.2) GO TO 119
WRITE (6.104) T(J) .HMW{I .J) ,H MCC{I ,J) HMIC(I .J) ,HRV( I J),HMMI( I,J )
2 FCA ( I J ) HT ( I J )
GO TO 107
119 WRITE6 118) T(J) HMW( I.J) ,HMCC(I,J)HyiC(I,J).HRV(I,J).HVMl(I,J),
2HCA( I,J).HT(I J),EXMH{ I )
107 CONTINUE .
106 CONTINUE
WRITE (6,103)
108 FORMAT (1///////////)
DO 109 1=1,3
DO 110 J=l .4
IF (J.E0.2) GO TO 120
WRITE (6,111) T(J),SMW{I,J),SMCC( I,J ) SMICI J) ,SRV{ I J) ,SMMI {I J )
2. SCA ( I J) ST( I J)
111 FORMAT (/. 10X.F7.2,3X,7(F6.2,2X))
GO TO 110
120 WRI TEC 6,121 ) T( J) SMW( I J) .SMCC (I J) ,SVIC{ I, J) SRV ( I, J) ,SMMI ( I, J ),
2SCA( I ,J)ST( I ,J ) ,EX MS ( I)
121 FORMAT (/, l OX ,F7.2 ,3X.d (F6.2,2X ) )
110 CONTINUE
109 CONTINUE
FORTRAN IV G LEVEL 21
0231 STOP
0232 END
MAIN
DATE = 79 1 OS
01/01/23
27Q
18 APRIL 1979
DATA LIST FOR MICELLE PROGRAM
C
C
c
c
c
c
c
c
c
c
c
c
13.QQQQ
Q.OOQQ
10.6QQ 0.05QQ 10.6500
15.3700
1.8470D+02 2.3600D03 2.1920D+02 5.9900D04 2.5310D+02
13.QQQQ
0.0853 2.3022 0. 1023
350.5600 365.5400 385.8700
7.04677 7.22636 3.16538
3.56999 11.35209 10.85375
2.5200 1.0100 0.7500
8 27 10 41 12 64
96.0300 190.6000 74.9000
0l89016D+02 0.488330D+03
277. 1500 298. 1500 323. I 500
2.6969
14.34352
3. 61310
0.1237
1 .26227
7. 34334
2 .9004
150.7000
0.1703060 + 03
358.1500
0 14521 8D + 02
C2893.0066 3624.0368 4371.0368
15.50QQ 0.13QQ
1.4600004
ro
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BIOGRAPHICAL SKETCH
Robert Brugman was born on April 15, 1952, in Jamestown, North
Dakota. He graduated from Jamestown High School in 1970 and received
his Bachelor of Science in Chemical Engineering from the University of
North Dakota in May 1974. He commenced graduate studies in the Depart
ment of Chemical Engineering, University of Florida, in September of 1974,
and recieved the Master of Science degree in August, 1975.
280
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
// / P
o y.
jhn P. O'Connell, Chairman
Professor of Chemical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Robert D. Walker
Professor of Chemical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Jopd 6. Biery
Professor of Chemical/fengineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of .
Doctor of Philosophy.
orge x. O'
George x. Ohoda
Professor of Materials Science
and Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Federico A.Vilallonga
Professor of Pharmacy
This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate Council, and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
June 1979
J'
Dean, College of Engineering
Dean, Graduate School
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TITLE: A molecular thermodynamic model for aqueous solutions of nonpolar
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201
Combining equations (B18), (B20), (B21), (B22), and (B23)
yields equation (526).
Correlation of the Helmholtz Free Energy of Interaction
with CT^, L, and Temperature
The integral expression for the several contributions to the
Helmholtz free energy of interaction between a spherocylindrical solute
and a spherical solvent were numerically integrated as a function of
spherocylinder length L at several values of the solute diameter
o o o*
O (3.40 A, 3.60 A, 3.80 A) and temperature T (277.l5K, 298.15K,
s
323.15K, 358.15K). The resultant integral values were fit to various
functions of L at constant Oand T as noted in Chapter 5. The coeffi
cients in this fit at 277.15K, 298.15K and 323.15K were used to
solve for the coefficients in a quadratic temperature function.
C=A+BT+DT2.
n n n n
These parameters (A B D ) are given for the three a values in
n n n s
Tables Bla, Blb, and Blc.
(B24)
1C APPIL 1979 DATA LIST FOR LIQUID HYDROCARBON PROGRAM
C X(l) = 0.0527789 IS .
C X ( 2 13 1946760
C X(3)2.8715619
c
DEL=
8.31948960+00+2.605
2103D + 03/T(J) 1 8
. 9 9306 5D+ 04/( T{ J)i
** 2)
c
1.8018
9 2 5.0SC6
1.9521
1098.3725
2.
1379
1231.3616
2.3022
1455.9570
c
2.5190
1 645. 9275
2.6969
1 823. 3736
2.
7987
2020.1494
2.9004
2199.8172
c
3.0752
2401.1257
.3.2500
2532.3034
s
96. 0300
1 SC. 60 00
232.5113
74.9000
150.
7000
1 89 .0 1 6 1
433.7000
46.40 00
c
0 .4 8333
OD +03 0.
1 70 3060+03
0. 14 5216D +02
C 277.1500
C 298. I 500
C 323.1500
C 358.1500
246
144
d3. = 6n V /it (617)
mic mxc
where n is the mean micelle aggregation number and V ^ is the partial
molar volume of surfactant monomers in the micelle. Table 63 lists
values and literature sources for V and n at 298.15K. The aggrega
mic
tion number was assumed to be temperature independent. The partial molar
volume was given a temperature dependence modeled after the results of
Shinoda and Soda (1963) for sodium tetradecyl sulfate.
V (T) = V (298.15)
mic mic
1 +
T 298.15
(
700
(618)
The Gibbs free energy of formation of the micelle cavity on a per monomer
basis is designated AG. in Table 62a.
mic
B. WaterMonomer Interactions
For step 1 of the thermodynamic process we calculated the water
monomer interacions which are eliminated upon removal of the monomer in
the same manner as the hydrocarbon solubility model. (The waterhead
group interactions are assumed unchanged upon micellization.) The inter
action free energy is designated as AG^/RT in Table 62a.
C. Counterion Adsorption
Interaction of bound sodium counterions with the water was
assumed to be unchanged upon micellization. This is equivalent to
assuming that they remain hydrated upon micellization as assumed by
Stigter (1964).
Adsorption of sodium counterions to the micelle (step 5) results
in a considerable decrease of entropy. Since the translation entropy is
proportional to Zn V where V is the specific molar volume, the entropy
decrease can be modeled as
Table 45b (Continued)
Solute
T(K)
AH
c
RT
AH?
i
RT
AH
cal
RT
AH
exp
RT
CF4
277.15
10 .69
20.33
9.65
 10.103
298.15
13. 5c
19.66
6.1 0
6.052
323.15
16.37
18.61
2.25
2.032
358.15
19.52
17.04
2.48
2. 563
SF6
277. 1 5
15. 92
28.45
12.53
13.531
298.15
19.34
27.35
8.01
3.040
323.15
22.66
25.79
3.13
2.616
358. 15
26.33
23.54
2 .79
3.572
nC^
277.15
24.7 2
41.69
16.97
 17.276
298.15
28. 80
39.80
11.00
11 .306
323.15
32.75
37.36
4.61
5.499
358.15
37.07
34.03
3.04
1. 152
R
ACp?
i
R
R
ACp
e*P
R
.271
52. 63
13.57
39.06
48.44
+ 7.3
.059
50 .57
8.33
42 .25
46.55
1.7
.300
49.20
 4.31
44.89
45. 44
7.2
1.29
48.1 9
0 .94
47 .25
44 .68
13.5
.298
65.81
16.25
49.56
65.8 1
8.9
.074
63. 25
9. 75
53.50
63.01
1.5
.352
61.32
4.98
56 .35
61 .37
9.0
1.64
5 9.39
1.10
58.29
60.24
17.6
2.0
84. 28
19.19
65.09
69.73
22.6
0.6
8 1.20
l l .23
69.97
65.58
14.2
0.23
78.57
5.67
72.90
63.47
8.4
1.26
75.37
1.40
73 .96
62.33
17.5
o
FORTRAN IV G LEVEL 21
MAIN
DATE
79 108
01/01/23
0163
0 164
0165
0166
0 167
0 168
0169
0170
0 17 1
0172
0173
0 174
0 175
0176
0 177
0 178
SC6=XTC(98)+2.0*XTC(99)*T(J)
SCE=GEE*(S C1+SC2*DL0G( UL ( J ) ) +GC2*DUL/UL ( J > + SC3*DL0G ( UL ( J) ) **2 + 2. *
2GC 3* DLOG(UL(J) )*DUL/UL(J) +SC4 *DLQG(UL(J) )**3+3.0*GC4*CLOG(UL(J) )**
32+DUL/UL(J)+SC5*DL0G(UL(J))**4+4.0*GC5*DL0G{UL(J))**3*0UL/UL(J)+SC
46*DLCG
52. 0*XTC ( 81 ) *T( J) )
C SECTION TO CALCULATE TOTAL WATERMONOMER INTERACTION PROPERTIES
WMW( I,J)=(OF*EPS I(I)*(GDOLLI + GDE)+CF+EPSI{I ) *(GCOLLI+GCGLLI ) )/(RK
2*T( J ) )
SMW(I.J)=IDF*EPSI(I)*(SDOLLI+SOE)+CF*EP3I(I)*(SCOLLI+SCGLLI)DDF*E
2PSICI)*(GDOLLI+GDE)DCF*EPST(I)*(GCOLLI+GCGLLI))/(RK)
HMW(I.J)=WMW(I.J)+ SMWI,J)
C SECTION TO CALCULATE ROTATIONAL AND VIBRATIONAL CONTRIBUTIONS
VBAR =4. 9355D G2/0ENW
DV8ARW= 4.93550 02 *DDENW/{DENW**2)
VBARM=PPMVW(I)*1.000003/VSTAR(I)
D VBA RM=DPMVM*1000003/VSTAR(I)
*RV(I, J )=C ( I ) *( (3.0* ( (0.7405/VBARM )**2)4.0*0. 74 0 5/VBARM) / ( Cl. 0 t
20.7405/VBARM) )**2) )+(C(I)l)*((3.0*((0.7405/VBARW)**2)4 .0*0.7405/
3VBARW)/( ( 1.0{ 0.7405/VBAR VI ) ) **2) )
S RV ( I J ) =WRV( I t J) +T IJ ) *C ( I ) { DVB ARM **(( 6.0*0.7 40 5**2/( VBARM**3) +
24.0* 0.7 405/(VBAPM**2))/{(1.00.7405/V3ARM)**2) + (3.0*( (0.7405/VBARM
3)**2)(4.0*0.7405/VBARM)> *<2.0*0. 7405/(VBARM**2) )/(( i.0C.74 05/VB
4 ARM) **3 )))(C(I)l) *DVBARW* ( (6 .0*0.7405**2/ ( V8ARVn**3 ) +4 C*0.7405/
5( VBARW**2) ) /( ( 1.00.740 5/VBAR Vi) **2) + (3.0* ( ( 0 .7405/V8ARW)**2)(4.0*
60.7405/VBARW))*(2.0*0.7405/(VSARW**2))/((1.00.7405/VBAR*)**3>)
HRV( I J)=WRV( I J) + SRV( I J )
C SECTION TQ CALCULATE MO NO ME R / MO NO ME R INTERACTION IN MICELLE
TBAR=T(J)/TSTAR(I)
GSUM=0.CO
SSUM=0.00
DC 17 M=1 ,5
267
Table Cld
Solute
T(K)
ACP
R
ACp
R
C4H10
277. 15
84.06
29.18
298. 15
76.00
31.71
323.15
68.33
34.79
358.15
60 .26
46.30
(Continued)
ACp
r,v
R
9.03
7.18
6.56
6.46
A^cal
R
ACp
exp
R
63. 91
48. 133
+10.4
51 .47
46.059
45
40.11
44. 117
+ 2.6
20. 42
42.989
+ 9.2
233
40
ice have length approximately 2.76 A, the smallest sphere which could
0
enclose these triangles would have radius r = 2.25 A. Thus, for ice
nothing beyond the pair (n=2) terms in P(r) and G(r) would be required,
O
provided r does not exceed 2.25 A.
Certainly the hydrogen bond pattern present in ice is severely
distorted upon melting. However, the coordination number in the liquid
remains low and thus it seems reasonable that the tendency toward tetra
hedral bonding persists in the liquid (Narten and Levy, 1969). Stillinger
assumed that for cold liquid water triplets of oxygen nuclei are seldom
distorted into a more compact configuration than would result from reduc
ing 0 to 90. The resulting triplet will fit into a sphere of radius
r = 1.95 A.
The repressions for G(r) are thus the following from equations
(34) and (314):
G(r) = [1 (4TT/3)pr3] 1
while from equations (34) and (315)
(2x
TT p
(0 < r < 1.20 A)
(328)
1 +
dt g(2)(t)t2(t2r)
G(r) =
o
, 4tt 3 n2
1 pr + (up)
( 2r
, (2) 2,1 3 2 8 3.
dt gv '(t)t (^ t 2r t+r )
(329)
(1.20 1 r < 1.95 A).
O
In order to specify G(r) beyond r 1.95 A in terms of correla
tion functions, knowledge of g^3^,g^\... would be required. In the
absence of such knowledge Stillinger relied on the conventional series
expressions for G(r) (equation 319) truncated after the fourth term
P 2^Â£v ^2 ^4
G(r)" m+ ^+ i +f
r r
(1.95 A < r < ) .
(330)
Table Alb CContinuijd)
Solute
T CK)
AH
c
AH
i
cal
AH
exp
ACp
c
ACp?
X
ACp
exp
RT
RT
RT
RT
R
R
R
R
CF4
277.i5
11.08
2 0.75
9.66
10.103
.
271
55.39
13.85
41.54
48.44 7
298.15
14.06
20.06
6 .00
6.052
+ .
059
5 1.57
8.50
43.08
46.55 1
323.15
16. 85
19.00
2.15
2.032
+
300
48.79
4.40
44.39
45.44 7
358.15
19.85
17.38
2 .47
2.563
1.
29
46.64
0.96
45.6 9
44.68 13
SF6
277.15
14.64
27.25
 12.6 1
13.531
.
298
68. 45
15.56
52.39
65.31 8
298. 15
18.25
26.19
7.94
8 .040
.
074
63.69
9.34
54.35
63.01 1
323.15
21.6 1
24.69
3.03
2.616
+
352
60. 00
4.76
55.23
61 .37 9
358.15
25. 1 S
22. 54
2.64
3.572
1.
64
56.67
1 .05
55 .61
60 .24 17
n~C5
277.15
20.98
38.18
17.19
 17.276
2.
0
86.89
17.57
69. 32
69.73 22
293.15
25.4 0
36.45
l 1 .05
 11 .306
0.
6
80. 89
10.28
70.62
65.58 2.4
323. 15
29. 4 9
34.21
4.72
5.499
0.
23
7 5.95
5.1 9
70.76
63.47 g
358. 15
33.77
31.16
2 .61
1.152
1.
26
7 0.96
1.29
69. 67
62.33 17
,3
,7
2
5
9
5
0
6
6
2
4
5
192
FORTRAN IV
0079
0030
0081
0082
0083
0084
0085
0086
0087
0088
0089
0090
0091
0092
0093
0094
0095
0096
0097
0098
0099
0100
0101
0 102
G LEVEL 21
CALFUN
DATE = 79108 00/49/39
P
ARE 4C( I ,K )=P I*( (HHSDWt K MHHSDGI I,K ) ) **2>
10 CGNTINUE
C NUMERICAL DIFFERENTIATION SECTION TO O ET AIN HEAT CAPACITIES FRGM
C TEMPERATURE DEPENDENCE OF ENTHALPIES
C DDE T 5 IS A SCIENTIFIC SUBROUTINES NUMERICAL DIFFERENTIATION
C RGUTINE
DC 15 J = l,7
HH{ J)=HAC( J ,1 )*T( J)
HII J)=HI(J, I)*T( J)
15 CONTINUE
CALL DDET5( C.5,HH ,DHH,7 ,1 ER)
CALL DDET5X 0.5, HI I DHI I ,7. IER)
C SECTION TO CONVERT VALUES AT MEAN TEMPERATURE FOR COMMON TO MAIN
S SI( I K)=SI{4,1 )
HHI(I,K)=HI(4,I>
GC{I,K)=WAC(4,1)
HC
SC(I,K)=SAC(4,I)
GGI ( I K )=GI (4,1 )
CPC( I.K ) = DHF(4)
CPI(I,K)=DH11(4)
C SECTION TO OBTAIN TOTAL PPOPERITES
CCPS(I.K)=CFI(I,K)+CPCI,K)
CGS(I,K)=GC(I,K)+GGI(I,K)
CHS ( I.K )=HC ( I, K )+HH I( I, K )
CSS( I,K)=SC(I,K)+SSI(I. K)
11 CONTINUE
4 CONTINUE
C DEFINING FUNCTIONS FOR ERROR TO BE MINIMIZED eY VA05AD. PERCENTAGE
C ERROR IN THIS CASE
DO 6 1=4,9
DO 3 K=1,3
180
170
80C and little or no effect of pressure or dissolved gas on the isotropic
anisotropic transition except possibly for transients.
BIOGRAPHICAL SKETCH
Robert Brugman was born on April 15, 1952, in Jamestown, North
Dakota. He graduated from Jamestown High School in 1970 and received
his Bachelor of Science in Chemical Engineering from the University of
North Dakota in May 1974. He commenced graduate studies in the Depart
ment of Chemical Engineering, University of Florida, in September of 1974,
and recieved the Master of Science degree in August, 1975.
280
FORTRAN
0037
0 03 8
0035
0040
004 1
0042
0043
0044
004 5
0046
0047
0048
0049
0050
005 1
0052
0053
0054
0055
0056
0057
0058
0 05 9
006 0
0061
0062
0063
0064
IV G LE VEL 21 MAIN DAT H = 79109 22/46/19
24 CONTINUE
23 CONTINUE
WRI TE { 6,19)
DO 25 1=7,9
DO 26 K = i ,4
WPITE 16,22) TT(K) ,GC I ,K) ,GGI { I ,K) GRV( I ,K) ,CGS ( I ,K),SC
2SI ( I K),S P V ( I K),CSS( IK)
26 CONTINUE
25 C QNTINUE
WRITE (6,19)
DC 27 1=10,10
DO 28 K=1,4
WRITE (6,22) TT{K) ,GC(I K) ,GGI( I.K) GRV( I.K ) ,CGS( I,<),SC( I ,K),
2SKI.K), SR V(I ,K) ,CSS( I K)
28 CONTINUE
27 CONTINUE
wpiE.ig)
DO 29 I=1 ,3
DO 30 K=1,4
WRITE(6,22) T7(K),HC(I,K),HI(I,K),HRV(I.K).CHS(I,K).CPC(I.K),
2CPI(IK),CPRV(I,K),CCPS(I.K)
30 CONTINUE
29 CONTINUE
WRITE(6,19)
DO 31 1=4,6
DO 32 K=1,4
WRITÂ£(6,22) TT(K) HC(I K ),HI(I.K), HRV( I ,K),CHS( I.K),CPC{I.K),
2CPK I,K),CPRV(I,K),CCPS(l,K)
32 CONTINUE
31 CONTINUE
WRITE6 19)
DO 33 1=7,9
236
275
Kirkwood, J. G. and F. P. Buff, "The Statistical Mechanical Theory of
Surface Tension," J. Chem. Phys., 17, 338 (1949).
Kitahara, A., "Micelle Formation of Cationic Surfactants in Nonagueous
Media" in Cationic Surfactants, Ed., E. Jungermann, Marcel Dekker,
New York, N. Y. (1970).
Koenig, F. 0., "On the Thermodynamic Relation Between Surface Tension
and Curvature," J. Chem. Phys., _18, 449 (1950).
Lebowitz, J. L., E. Helfand and E. Praestgaard, "Scaled Particle Theory
of Fluid Mixtures," J. Chem. Phys., 43, 774 (1965).
Leduc, P. A., J. L. Fortier and J. E. Desnoyers, "Apparent Modal Volumes,
Heat Capacities, and Excess Enthalpies of nAlkylamine Hydrobromides
in Water as a Function of Temperature," J. Phys. Chem., J78, 1217
(1974).
de Ligny, C. L. and N. G. van der Veen, "A Test of Pierotti's Theory for
the Solubility of Gases in Liquids, by Means of Literature Data on
Solubility and Entropy of Solution," Chem. Eng. Sci., 27, 391 (1972)
Lin, I. J. and P. Somasundaran, "FreeEnergy Changes on Transfer of
Surface Active Agents Between Various Colloidal and Interfacial
States," J. Colloid Interface Sci., 37, 731 (1971).
Lovett, R. A., Statistical Mechanical Theories of Fluid Interfaces, Ph.D.
Dissertation, University of Rochester (1966).
McAuliffe, C., "Solubility in Water of Paraffin, CycloParaffin, Olefin,
Acetylene, CycloOlefin and Aromatic Hydrocarbons," J. Phys. Chem.,
70, 1267 (1966).
Mathias, P., Thermodynamic Properties of HighPressure Liquid Mixtures
Containing Supercritical Components, Ph.D. Dissertation, University
of Florida (1978).
Melrose, J. C., "Thermodynamics of Surface Phenomena," Pure Applied Chem.
22, 273 (1970).
Mijnlieff, P. F., "Thermodynamics of Micellar Equilibrium for Ionic
Detergents. An Alternative Description and Some Relations Derived
from It," J. Colloid Interface Sci., 37, 255 (1970).
Molyneaux, P. and C. T. Rhodes, "Calculation of the Thermodynamic Param
eters Controlling Micellization, Micellar Binding and Solubiliza
tion," KolloidZ.u.Z. Polymere, 250, 886 (1972).
Moroi, Y., N. Nishikido, H. Uehara and R. Matuura, "An Interrelationship
Between Heat of Micelle Formation and Critical Micelle Concentra
tion," J. Colloid Interface Sci., M), 254 (1975).
129
Comparison with Infinite Dilution
Properties of Surfactants
Since the ultimate objective of this work was to lay the founda
tions for a molecular theory for thermodynamic properties of micelliza
tion, a comparison between properties of surfactant solutions at infinite
dilution and the present model as a function of carbon number would be
instructive. Unfortunately only infinite dilution heat capacity data
were found in the literature.
Table 56 presents a comparison hetween infinite dilution heat
capacities for nalkylamine hydrobromides and the present model at
298.15K. The surfactant data were obtained, using calorimetry, by
Leduc et al. (1974). The absolute values should differ in the two cases
due to the amine hydrobromide group on the surfactant. However the
incremental change with carbon number should agree. For the hydrobromides
AC p/R is over 10 for each additional carbon, whereas the model predicts
a value of less than 9 for the longer chains with considerable differ
ence for short chains. Since Cp is a second temperature derivative, it
will strongly reflect model inadequacies. These seem to be present to a
certain extent.
Suggestions for Future Work
Beyond suggestions made in Chapter 4 concerning aspects common
to the spherical gas model, several possibilities exist for interesting
future work. Of a short term nature would be combination of the several
correlations used for the contributions to AA. into one each for discrete
i
69
Pierotti's final expression for Â£n K from equations (410),
(414) (418) is
8 C. TTp _
Â£n K + G /RT Â£n (RT/V ) =
. J o
6 kT a
12
%
 (11.17p/T)(e /k)'2 (e /k)"2 a?. .
W Z .Z
(420)
Pierotti determined E /k from the best linear fit of the left
w
3
hand side of equation (420) as a function of (c^/k)z A reasonably
straight line is obtained, insuring a good fit of the experimental K val
ues. However, the value of e^/k (= 85.3) obtained seems unreasonably low
when compared to that resulting from fitting other thermodynamic property
data (e.g., Rigby et al. (1969)).
Table 41 presents the results of Pierotti's model at 298.15K
and 323.15K. Under the assumptions of his model AS_^ = 0 and ACp^ = 0.
The experimental values as previously discussed are included for compar
ison. Note that terms arising from the term Â£n (RT/V^) in equation (410)
have been lumped with the cavity terms in Table 41. Considering that
no fitting of solute parameters was done, the results are quite good
except for the heat capacities.
Table Clc
Contributions to the Entropy of Solution of Gaseous Hydrocarbons
AS AS? AS AS AS
c i r,v cal e:
Solute
T(K)
c
R
i
R
r.v
R
cal
R
exÂ£
R
CH4
277. 15
16.86
2.01
0.0
18.87
18 .057
+.30
298.15
l3.90
2 .83
0.0
16.73
 16.145
+.04
323.1 5
10.83
3.36
0.0
14.20
14.137
+.07
358. 15
7. 05
3.65
0.0
10.70
11 .66l
+.29
C2H6
277.15
17.61
1 .72
1.27
20.61
21.042
+.42
238.15
13.70
3. 10
1.03
1 7. 83
18.259
+. 06
323.15
9.72
4.72
0.80
15.24
15.340
+.n
358.15
5.00
6.93
n
in
.
o
1
 12.45
11.751
+. 42
C3H8
277.15
13.37
2. 17
2.20
22.74
22.995
+ .52
238. 1 5
13.49
4.13
1.73
19.40
19.577
+ .08
323. 1 5
8.60
6.39
1.39
16.38
 15.989
+.13
358. 15
2.94
9.44
0.91
13.29
11.590
+.51
230
43
Curvature Dependence of Surface Tension
The expression for the curvature dependence of the surface
tension (equation 318) is an approximation to the rigorous relation of
Tolman (1949) and Koenig (1950). From the Gibbs theory of surface
tension Koenig developed general equations for the change of surface
tension with curvature for systems having an arbitrary number of com
ponents. For the special case of a spherical surface Koenig's expres
sion reduces by an appropriate choice of Gibbs surfaces to a form iden
tical with that of Tolman for a onecomponent system.
9 &n y
3(6q)
T
 2(l + 6q+i <$2q2
l+26q(l+<5q + j <5^q2)
(331)
where q = 1/r. For 6q 1 equation (331) reduces to equation (318).
<5 is a measure of the distance from the surface of tension to a Gibbs
auxiliary surface measured from the spherical phase outward. It is
reasonable to assume that both the surface of tension and the auxiliary
surface lie within or very near to the interface layer and consequently
<5 is of the order of magnitude of the thickness of the interface layer.
Kirkwood and Buff (1949) performed approximate calculations for liquid
argon based upon statistical mechanics which suggested that 6 is of the
O
order of 3 A.
However, Lovett (1966) in his analysis of fluctuations about the
mean position of the interface shows that 6 should be related to the
third moment of the fluctuations, whereas the interfacial thickness is
related to the second moment. Because of this uncertainty, we will view
6 as a parameter in the study of aqueous solutions.
18
entropic at low temperature (T 10C) and becomes increasingly enthalpic
with increasing temperature as reflected in a large heat capacity (Pierotti,
1965). As previously noted, the thermodynamic properties of micellization
follow this same temperature trend.
Early approaches beginning with Debye (1949) tended to view micelli
zation as an enthalpic rather than entropic process. Debye ascribed micelle
formation to the opposition between hydrocarbonchain attraction and ionic
repulsion. The more complex approach of Poland and Scheraga (1965, 1966)
attributed the solvent contribution to micelle formation to the theory of
hydrophobic bonding of Nemethy and Scheraga (1962ab). This theory attrib
utes hydrophobic bonding to an increase in:the amount of hydrogen bonding
of the water near a solute over its average value in pure water and is
similar to the concept of: "iceberg formation" around nonpolar solutes of
Frank and Evans (1945). It should be noted that there are both strong
entropy and enthalpy contributions in this approach. The calculated stan
dard free.energies, enthalpies and entropies of solution for alkanes and
benzene homologs agree well with experimental values within the temperature
range of 0 to 70C.
The principal difference between this approach and that of Ben
Naim (1971) is that scaled particle theory is applicable, to any solvent
and does not rely explicitly on microscopic details concerning change in
water structure or conformation upon inclusion of the solute. These effects
may well be implicit in the details of scaled particle theory (particularly
with the modifications discussed in Chapter 3).
The hydrophobic free energy change Ay /RT associated with a sphere
rib
of diameter O being inserted into a solvent of diameter is given by
157
which an isotropic low viscosity system may be transformed into a high
viscosity anisotropic system.
The gases were investigated since considerable methane may be
present in wells which are not depleted of their natural pressure and
carbon dioxide is of considerable interest as a potential alternative to
surfactant systems for tertiary oil recovery.
Description of Experimental Apparatus
Figure 71 illustrates schematically the experimental apparatus
used. Pressure limitation is 10,000 psi; the limit for the High Pres
sure Equipment 30 cc capacity pressure intensifier. The pressurizing
medium was mercury except above HP2 where hydraulic oil was used to
protect the 15,000 psi temperature compensated Heise gauge. The gauge
has 20 psi graduations with an estimated reading accuracy of 10 psi.
A mercury reservoir and mercury level indicator were used to
control the mercury level and refill the pressure intensifier with the
aid of compressed air.
The sample under investigation was contained in an Aminco high
pressure optical absorption cell equipped with quartz windows. The cell
pressure rating is 50,000 psi at 400K, 75,000 psi at room temperature.
Total sample volume is about 9.5 cc with about 35% of this total visible
through the windows. The optical cell was housed in an insulated box
and heated with a high temperature heating tape.
Due to pressure requirements a thermocouple well was drilled at
one end of the cell. Thus considerable time (~ 57 hrs) was required
Table 55b (Continued)
Solute
T(K)
AH
c
ah?
1
AH
r,v
AH
cal
>
Ol
O O
ACp?
i
ACp
r,v
A^cal
RT
RT
RT
RT
R
R
R
R
C11H24
277.15
75 .86
 108.08
0. 00
32.22
1 91.88
82.28
26.80
136.40
298.15
83 .46
 106.43
1 .66
21.30
175.75
36.67
2 1.31
110.41
323.15
39 .85
105.04
3.09
12.11
156.26
90. 14
19. 48
85. 60
358.15
95.08
 103.69
4.67
3.95
131.1 6
91.61
19.17
58.72
C12H26
277. 15
82.85
ll7.32
0.00
34.47
207.13
89.26
28.32
146.19
298.15
90 .99
115.52
1.75
22.78
189.57
93. 90
22.52
118.19
323.1.5
97.79
 1 1 4. 00
3.27
12.95
168.20
97.49
20.58
91.29
358.15
103.29
112.48
4 .93
4.27
140.59
98.75
20.25
62. 10
C13H28
277.15
89.85
 127.69
15. 68
22.17
222.3 8
97. 12
10.82
136.09
298.15
98.52
 125.72
15.32
1l.89
203.38
102.05
10.21
111.54
323.15
1 05 .73
124.05
14.90
3.43
180.14
 105.77
9.63
84. 0 0
358.15
111 .49
122.36
14. 35
3. 49
150.02
1 06.82
9.02
52.23
127
9
el
The value of y^ is determined from some expression such as that
of DebyeHuckel theory leading to
el
y1 = + RT Â¡L n y1
(219)
el
The relationship of the counterions to y^ is one of equilibrium between
those in bulk solution and those in the Stern (bound) layer and the Gouy
Chapman (diffuse) layer (Stigter, 1964)
y (solution) = y(s) + yf^ + RT in a (s)
c c 1 c
el
y (solution) = y (micelle) = y(m) 1^ ac^m^
c c c I
or
el
(220)
(221)
(222)
p(m) i(s) + A ET
CCS
el el
Substituting for y ^ Ny^ in equation (18) and combining the standard
state chemical potentials yield
Ja+ a+(m)
y N(y + y) + RT Un > = 0
m + +,
(a a (s) ]
1 c
(223)
Assuming that we can replace a^ and ac(ra) by unity (micelles) and a^
and a*(s) by mole fraction (solution), using the definition of Ag from
equation (21) where all the species are uncharged gives
AG
NRT
= Â£n x, x
1 c
(224)
which for no added salt (x+ = x*) is
c 1
AGm +
Z = 2 Â£n x
NRT 1
(225)
FOP TRAN IV G LE VEL 21
CALFUN
DATE
79 109
22/43/30
0044
004 5
0046
004 7
0048
0049
0050
005 1
0052
0053
0054
0055
0056
2W RVA1*T PAR ) **2 ) ) ) A 3/( DEXP { A 5*{ C DENWR4 A 1 *TRARA ) * 2) ) ) + A9/( DEXP( A
31 0+{ ((TRARA13)*2)+AI1 *( < DENWRA12)**2)))))>**0.3 3333
dcwar=hsdar+hsdw
DCWCH4 = HSDCH44HSDW
DHSCH4= ( ( 3. 0*CVCH4/(2.0*PI*0.6023 ) ) (A7 A8/ ( CT C H4 ( TRCH4 ** { A8 4 l 0
2 ) ))2.0 *A2* A4*( DEN WR 4A l *TPCH4 )*{ DDENWR 4A 1/CTCH4) /( DEXP ( A4* ( ( DENWR4
3 A1*'PCH4 ) **2.0 ) ) )+2 .0 *A3*A5*( DENWR4A l*TRCH4 A6) *(DDENWR 4A 1/CTCH4) /
4(DEXO(A5*( ( DENWR + A1 TP.CH4 A6 )** 20) ) )2. O* A9* Al 0* ( (TRCH4A13 J/CTCH
5 4 4A 1 1*< DENWRA12 >*DDENWR)/(DEXP(A10*(((TRCH4A13)**2)+A1 l*{(D5NWR
6A12)**2))))))/( (HSD CH4**2.0)*3.0)
TDCH4I J )=HSDCH4
DHSDAR={(3.0*CVAR/(2.0TPI*0.6023))*{A7*A8/(CTAR*{TRAR**(A8+1.0)>)
22. 0* A 2* A4* DE NWF. 4A 1 TR AR )*{ DDENWR 4A 1 / CT AR )/ ( DEX P ( A4* ( ( OENWR+A 1 *TR
3AR) **2.0) ) ) 42.0*A3*A5*( DEN WP 4 A 1 *TR ARA 6) ( DDENWR 4A l/CTA ) /{DEXP( A5
4* ( (DENWR+Al *TRAFA6 ) **2 .0 ) ) )2 O* A9 A l 0* ( ( TP ARA 13 )/C TAR 4A 1 1 +( DEN W
5P A 1 2)* DDENWR) /(DE XP( A1 0* ( { (TRARA1 3 )**2 ) 4A1 1 *{ ( DENWRA12 )**2 ) ) ) ) )
6 ) / ( ( FS D AP **2.0 ) *3 0 )
DCC WG4 = DHSD W4DHSCH4
DDCWGA = DHSD W4DHSDAR
C SECTION ~0 CALCULATE INTERACTION CONTRIBUTION TO ARGON PROPERTIES
GIAP=EPSIAP (6. 72 02D + 0 04.954 D 40 3/ ( T { J) ) 46.5480D 405/C T ( J )**2)0.7
26 00 D 40 0 *HSDAR0 .79 2 5D 40 0*( HSDAR **2))/(T(J))
S IA R =5 P S IA R *(4.9540403/ (T(J)**2)t 3.09 6D405/ T ( J ) **3 )0 .76D400*D
2HSDAR1.585D40 0*HSDAP*DH SD Ap )
H I A R =GI AR 4S IA R
C SECTION TO CALCULATE EXPERIMENTAL ARGON SOLUBILITY PROPERTIES
GEXA R= (EXC14EXC2*DL0G( T( J ) ) 4EXC3M DLOG( T( J) ) **2) )
HEX AR=EXC2 42.0* EX C3*DLCG(T (J ) )
SE XA R=HE XAR GE XAP
C ARGON CAVITY PROPERTIES ARE OBTAINED BY DIFFERENCE
WACARGEXARGIAR
0057
FORTRAN IV
0094
0095
0096
0097
0098
0099
0100
0 10 I
0102
01 03
010 4
0105
01 06
0107
0108
0109
0110
0111
0112
0 113
0114
0 115
G LEVEL 21 MAIN CATE = 79108 01/01/23 P
HSDC1=2.O+RCI
DCWCI=HSDCI+HSDW
D DC WC I = DH SD Vk
WCI ( J )=FBC*( (P I/( XK*T( J ) ) ) *( STW*(DCWCI**2DCV>AR**2) 4.0*STW*DEL*
2(DCWCIDCWAR))+WACAR)
SCI { J) =FBC*( (PI /XK ) ( DSTW* ( DC WC I **2DC W AR* *2 ) tSTviK (2 D CW C I ODC'W
2CI2 .0*DCWAR*DOCWGA )4.0 *D STW *DEL* ( DC WCIDC WAR) 4.0* ST**DOEL* (
3DCWC IDCWAR)4.0*STW*DEL*(DDCWCIDDCWGA))+SACAR )
HCI J) = WCT ( J) + SCI C J)
C SECTION TO CALCULATE TOTAL OF MONOMER AND COUNTERION CAVITY PROPERTIES
WMCC(I,J)=WMON(IJ)+WCI {J)
SMCCI, J) = SMON( I, JJ+SCKJ l
H MCC(I J)=H VON ( I*J> +HCI (J)
C SECTION TO CALCULATE MICELLE CAVITY PROPERTIES CN A PER MONOMER 8ASIS
HSDM=(6.0*NAGN( I )*PPMVM{I)/PI )**0.333
DHSDV=0.3333*(6.0*NAGN(I)*DPMVM/PI )/ (HSDM**2 )
DCWM=HSDM+HSDW
DDCW M=OHSDW+DHS CM
WMIC(I *J) = ( (PI/(XK*T(J) ))*(STW*(DCWM**2DCWAR**2 )4.0*STW*DEL*(DCW
2MDCWAR ) )+WACAR )/NAGN( I )
SMIC(IJ)=((PI/XK)*(DSTW*(DCWM**2DCWAR**2)+STW*(2.0*DCWM*D0CWM
22 0* DCWAR* DDC WGA)4.0*0 ST W*DEL*(DCWMDCWAR)4.0*STW*DDEL*(DCWMDCW
3AR)4.0 *ST W *DEL *(DDCWM CDCW GA) )+SACAR)/NAGN( I )
HMI C (I J) = WVTC ( I J) +SMI C( I J)
C SECTION TO CALCULATE MONOMERWATER INTERACTIONS
UL(J)=X MCNL
DUL=DXMCNL
CF= ( EPS I ( I)1.5*4.62D+02)/(EPSI (I ) )
DF = 1 .OD+OOCF
DDF=0.00
D CF= 0.0 0
C LARGE UL VALUE SECTION
264
72
Contributions to the Thermodynamic Properties
of Solution from Cavity Formation
Consider the origin of the series approximation (equation 330)
to the contact correlation function G(r) when the cavity radius r is
large. As noted in Chapter 3, the exact expression for G(r) derived
from Koenig's (1950) expression for the curvature dependence of the
surface tension (339) should allow a more rigorous calculation of G(r)
than the series result of Stillinger. However, equation (341) shows
that G(r) cannot pass through an extremum (dG/dr = 0) for finite values
of 6. Thus, the extremum value of G(r) seems a natural dividing point
between microscopic and macroscopic regimes.
Now, previous expressions (329) for G(r) in the region where
r contains two water molecular centers can pass through an extremum and
be used to match equation (339). However, we have found that the assump
tion of constant 6 severely restricts the range of r values in which the
two functions can be matched. Coupled with a lack of knowledge concern
ing the value of r at which the triplet correlation function becomes sig
nificant, this casts considerable doubt on the rigor of the expressions.
While the contact correlation function must be continuous with r, the
form is truly unknown in the region of the maximum.
Considering these difficulties, it was decided to abandon
efforts at linking microscopic and macroscopic approaches and simply
use a reference solute. The experimental solution properties of the
reference solute encompass the microscopic detail and allow use of a
macroscopic expression for differences between the reference and other
solutes. Argon was chosen as the reference solute for this work.
198
The terms of equation (B13) are readily evaluated from integral
tables.
o
3a8 s
ds
WS
2
4
z
L J
6
3 a
ws
L + i + _L_ tani jl
a 2, 2 2, + 2 2,2 ^ 3 Can z
8z (s +z ) 12(s +z ) 8z s
L 1 L
r tan
3 z
2 2 2
24(L +z ) 24z
8z
(B14)
f 0 12 
126s cr ^
ws
1 + 1
1 1
c,0 2. 2^ nua 2o. V oiin 2. 2V3 non 2t 2j. M
512z (s +z ) 768z (s +z ) 960z (s +z ) 1120z (s +z )
H ^r i q tan ^
1260(s +z ) 512z s Z
ds
126a
12
ws
1 + 1
1 + 1
1536z8 1536z6(L+z2) 3840z8 3840z4(L2+z2)2
1 + 1
6720z2(l2+z2)3 6720z8 10,080(L2+z2)4 10,080z8
512z
L 1 L
9 tan z
(B15)
dx
3a8 x
ws
1 + 1
8z2(x2+z2) 12(x^+z)i' 8z x
1 1 ^x
2. 2,2 3 tan z
3a
ws
24z2 24(L2+z2) 8z
L 1 L
~ tan
3 z
(B16)
151
interaction result in a large negative enthalpy change. A significant
negative entropy change arises from monomermonomer interactions involv
ing both density and energy effects. Monomermonomer interaction contri
butions are denoted by subscript mmi in Table 62.
The compression process also involves changes in rotational and
vibrational freedom when the surfactant is transferred from the reduced
water density to the reduced micelle density. Step lb in the thermo
dynamic process involves an increase in rotational and vibrational
freedom since the reduced density decreases sharply from the aqueous
solution state to the ideal gas state. The reverse is true of step 3
where the monomer is compressed to the micelle state. The results in
Table 62 denoted by subscript rv are the combination of these two
steps. The net result is that expected since the reduced density of the
monomers is less in the micelle than in aqueous solution. However, the
extremum as a function of chain length is unexpected.
Step 4, the creation of a micelle cavity, was originally
hypothesized to involve a negative entropy change and a positive enthalpy
change. The signs of the changes are found, but the enthalpy change
dominates the small entropy change. This arises from the spherical gas
solubility model where the curvature dependence of the surface tension,
6, dominates. The temperature dependence of the micelle partial molar
volume also results in a significant temperature dependence for the
micelle diameter. Note also that the. temperature dependence of the
aggregation number has not been included, although it does vary with
chain length. The micelle cavity contribution is denoted by subscript
mic in Table 62.
FORTRAN IV G LEVEL 21
MAIN
79109
22/46/19
DATE =
0065
0 066
0067
0068
006 9
0070
0 071
0072
0072
0074
0075
0076
DO 34 K = 1,4
WRITEI6.22) TT(K),HCII,K),HI(I,K),HRV(T,K) ,CH5{ I ,K) ,CPC(I K) .
2CPItCCPS(I,K)
34 CONTINUE
33 CONTINUE
WPT TE(619)
00351=10,10
DC 36 K = 1 ,4
WPI^E{ 6,22) TT(K) ,HC(I K),H! (I K) ,HRV( I K ) ,CHS( I .K),CPC CI,K),
2CPI ( I, K ) CPRVC I ,K ) CCPS( I K )
36 CONTINUE
35 CONTINUE
STOP
END
237
14
Finally, this may be rearranged to give
Â£n(x^/x^) = Jin
K1 + x2
fo
L xl ..
j \j
MQ % Q o
+ 
N
 y2
N
RT
 ^ Jin [x (x* + x )]. (244)
N
For the correlation of equation (236) to hold, the form of
the standard state chemical potential must be
U U
^MQ ^MO C) o
n ho _
N N N Q
: ; = ^ Jin x +
RT N
Q
L
+ 1 + K.' Jin (x{ + x^)
 (1 + K') Jin x
+o
(245)
Mijnlieff shows that the reciprocity relation
9n
9n,
T,P,n
T,P,nr
(246)
leads to
Q =
.
(1 + Kf)
2 + (lKf)x^/x2
< 0.
In the limit x2/x^ K< ^
Q (1+K'>*2
(1 K')
+
(247)
(248)
This equals zero when x2 = 0. In the limit 1, Q/N = 0.16 for
anionic amphiphiles and 0.21 for cationics. The fact that it is
constant, but different for the charge types must be of significance.
CHAPTER 6
MODELING OF THE THERMODYNAMIC PROPERTIES
OF MICELLIZATION
Introduction and Review of Thermodynamic
Process for Micelle Formation
A brief review of a thermodynamic process for micelle formation
(Figure 61) would be appropriate at this point. This process was devel
oped along three parallel branches for surfactant monomers, counterions
and water. The first step involves removal of counterions and surfac
tant monomers from cavities in solution to a gas state of the same density.
Steps 2 and 4 involve aggregation of these dispersed cavities into a
micelle size cavity, while step 3 involves compression of the gas phase
monomers to the density of the micelle interior with restriction of the
monomer head groups to the micelle surface. Step 5 places the compressed
monomer aggregate and counterions into the micelle cavity. A detailed
discussion of the thermodynamic property changes for each step of this
process was presented in Chapter 2.
The objective of this chapter is to present and discuss results
of modeling several steps of this process using the previously discussed
solubility models and models available in the literature. The ultimate
goal is a unified theory for aqueous solutions including micelle formation.
The initial section of this chapter discusses rigid body equations
of state and presents thermodynamically consistent expressions for the
132
FORTRAN IV G LEVEL 21
CALFUN
79103
00/49/39
DATE =
0001
0002
0003
0004
0005
0006
0007
0008
0009
001 0
00 11
00 12
00 13
001 4
0015
00 16
001 7
001 8
0019
002 0
0021
SUBROUTINE C AL FUN ( M N F X )
C OBJECTIVE CALCULATE VALUES WHOSE SUM SQUARE ERRCR IS MINIMIZED
C BY VA05AC TO OBTAIN DESIRED PARAMETER VALUES
C ENTROPY AND GIBES FREE ENERGY OF SOLUTION ARE FIT FOR LAST 6 SOLUTES AT
C 277, 293 *323 K
C ARGON IS USED AS REFERENCE SOLUTE HELIUM AND NcCN ARE NCT FIT
IMPLICIT REAL*8(AH.OZ)
D IMENSION HSDG(9) T 7 )Â£P SI(9)DH SDG(9)CV(9) CT(9) DCWG(9) ,DDCWG
2(9) ,X{3),F(36) EXG(9.4),EXS(9,4).EXH< 9,41,GI{7,9) ,GGI(9* 4) ,SC(9,4)
3 HC( 5*4) HI (7*9) HHI(94) H 11 C7) ,GC(94) .CPC(9,4).CPI (9,4),DEN GR(
49),TRG(9),SI(7,9),DHH{7 ),W( 10 00) ,HHSOW(4) ,HHSDG(9,4) ,AREAC(9,4) ,
5TT(4)EXCP(9,4) ,CSS(9 ,4),WAC(7,9),HAC(7,9),HH(7),SS I (9,4),SAC( 7,9)
6, DHI I( 7) ,CCPS( 9,4) ,CGS( 9,4) .CHS (9 ,4)
COMMON/A/CV.CT,EXS,EXH.EXG.EXCP.EPSI.TT.EXCl.EXC2.EXC3
COMMON/8/CSS.CGS.CHS,CCPS
COMMON/C/GC,GGI ,HC,HHI,SC,SSI ,CPC,CPI
CCMMCN/D/CV W. CTW
P1 = 3 141592C5D 0
XK=1.33066
RK=1.987
C COEFFICIENTS IN EQUATION FOR HARD SPHERE DIAMETERS
A1=0 .54008832
A2=l .2669302
A3=0 .051323 55
A4=2.9107424
A5=2.5167259
A6=2.1595955
A7=0.64269552
A8=0.17565885
A9=0.1 8874824
A10=l7.952388
A t 1=0. 48197123
176
Table C2a (Continued)
AG
AG
i
AG
Solute
T(K)
c
r,v
RT
RT
RT
C14H30
277.15
100.27
 102.9
1 1 .34
298.15
94 .53
94.42
11.26
323.15
87.43
85.44
11.02
358.15
77.75
74.12
1 0. 51
AGcal
. AS
c
AS.
i
t>
col
il o
<
AS ,
cal
RT
R
R
R
R
8.92
26.67
11.36
11 .33
49.37
11.36
11.22
 17.89
9.18
38. 30
13.06
3. 69
25. 40
7. 1 5
28.86
14.15
19.66
35.22
4.68
20.24
251
g = gas phase
el = electrostatic quantity
hs = hard sphere quantity
L = liquid phase
ref = refers to reference solute
V = vapor phase
Overline
= denotes partial molar, average, or reduced property
FORTRAN IV G LEVEL 2i
MAIN
CATE
79 108
00/49/39
0037
0038
0039
0040
004 1
0042
0043
0044
0045
0046
0047
0048
0 04 9
0050
005 1
0052
0053
0054
0055
0056
005 7
0058
0059
0060
006 I
0062
0063
OC 12 L = 4,6
DC 13 K 1 ,4
WRITE (6,9) TT(K) ,GC(L,K) GGI (L.K)
2K),CSS(LK), EXS (L K )
13 CONTINUE
12 CONTINUE
WRITE (6,14)
14 FORMAT (* 1 ,///////////)
DO 15 L=7, 9
CGS(L.K),EXG( L K ) ,SC(L,K),SI(L.
DC 16 K=1.4
WRITE (6.9) TT(K) iGC(LiK) GGI (LK) CGS L K),EXG(L,K)SC(L,K),SI(L*
1K ) CSS (L K } EX S (L K )
16 CONTINUE
15 CONTINUE
WRITE (6i17)
17 FCRMAT* 1* ///////////)
DO 13 L=1,3
DO 19 K=l,4
WRITE (6,20) TT(K),HC(L.K),HIL.K).CHS(L.K),EXH(L,K),CPC(L,K) CP I (
2L,K)CCPS(L,K),EXCPL,KJ
20 FORMAT /,10X,F6.2,3X,F5.2,2X,F6.2.2X.F.2,2X.F7.3,aX,FS.2.2X,F6.2
2.2X,F5.2.2X.F5.2)
19 CONTINUE
18 CONTINUE
WRITE 16,21)
21 FORMAT {1,//////////}
DO 22 L4,6
DO 23 K=l,4
WRITE (6,20) TT(K),HC(L.K),HI(L,K),CHS(L,K),EXH(L,K),CPC(L,K),CPI
2
23 CONTINUE
22 CONTINUE
A MOLECULAR THERMODYNAMIC MODEL FOR AQUEOUS SOLUTIONS
OF NONPOLAR COMPOUNDS AND MICELLE FORMATION
By
ROBERT J. BRUGMAN
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1979
76
Table 42
Universal Correlation for the Reduced Hard Sphere Diameter
Reduced Temperature: T = T/T*
Reduced Density:
a
p = p/V*
T 2 0.73: f = a_/T
s 7
8
T Â£ 0.73: f = a,, exp [aic T]
s 14 1j
2 3
ttN dJ ?
= fs + a2/exp [a4(p+ alT) ]
2 2
a^/exp [a^(p+a^T ag) ] + ag/exp UiotCT al3) +
an(p a12) }]
(427)
(428)
(429)
a = 0.54008832
a2 = 1.2669802
a3 = 0.05132355
a. = 2.9107424
4
ac = 2.5167259
5
a = 2.1595955
6
a? = 0.64269552
aD = 0.17565885
8
ag = 0.18874824
a1Q = 17.952388
a n = 0.48197123
a12 = 0.76696099
a13 = 0.76631363
a., = 0.809657804
a15 = 0.24062863
73
For macroscopic properties, Melrose (1970) showed that for two
phases, a and 3, in contact
dU = TdS + E y. dN. PadVa P^dV^ + vda +
ill
where J is the arithmetic mean curvature.
lx
dJ
add
(421)
J.L + L
R1 R2
where and are the principal radii of curvature.
For an isothermal constant composition process, the Helmholtz
Free Energy (work) is
c a 5v'
(422)
dA = PadVa P^dV^ + yda +
IX
9J
adJ.
Since our cavity creation process is constant pressure with
dVa = dV6
d4 = yda +
lx
l9JJ
adJ .
(423)
Using the one term macroscopic approximation
y = y (16 J)
IX _
dJ
= y 6.
(424)
The work of changing the cavity from that of the reference
solute (argon) is then
Ab 'ref Y V^ref1 Y 6[
OQ
= y (a a r)
1 s ref
1 
6(a J a J ,.)
s s ref ref
(a a r)
s ref
Since for a sphere
, 2
a 4ttR and J = 2/R
26
A A c = 4tt y
s ref
2 2
R R r
s ref
(R +R J ,
s ref J
(425)
(426)
CHAPTER 7
EXPERIMENTAL INVESTIGATION OF PHASE BEHAVIOR AND TRANSITIONS
FOR CONCENTRATED SURFACTANT SOLUTIONS
Introduction
The initial section of this chapter discusses the objectives of,
and justification for, this experimental investigation into phase behav
ior and transitions for concentrated surfactant solutions. This is fol
lowed by a description of the experimental apparatus as well as general
operating procedures. Pertinent calculations for dissolved gas experi
ments are also included.
The final section of this chapter discusses experimental results
with a view towards suggestions for future research.
Experimental Objectives
The objective of this experimental study was to ascertain the
effect of temperature, pressure and dissolved gas (CH^ CO^) on the phase
behavior of surfactant formulations of interest in tertiary oil recovery,
particularly with regard to the isotropicanisotropic transition common
to microemulsion and lyotropic liquid crystal systems. Such a transition
is particularly important since the anisotropic phase may exhibit suffi
cient viscosity to seriously damage an oil well undergoing tertiary
recovery. It would be highly desirable to ascertain the conditions under
156
75
b) At high densities the hard sphere diameter is a function of
reduced temperature only.
c) The first two exponential terms were chosen to represent the
minimum in the isothermal change of the hard sphere diameter with density.
The particular form was chosen since the position of the minimum seems
to shift linearly with temperature.
d) The last exponential term is used for changes in the hard
sphere diameter required to obtain an accurate representation in the
critical region.
nnnnnnnnnnnrtn
19 APRIL 1979
DATA LIST FOR LIQUID HYDROCARBON PROGRAM
X ( 1 ) =0 .4940330l
X{ 2)=30775424
X(3) = 2 .3138407
DEL=6 .2246729 + 2.1484507D+03/T(J1 14
1.8010 824. 1700 1.9521 560.3200 .
2.5190 1395.1208 2.6969 1535.7769
3. 0752 1 958.2831 3.2500
96.0300 190.60C0 266.2056
0.48 03 3 0 D + 0 3 0.1703060 + 03
40244D+04/IT(J )* *2)
2.1379 1 1 05.931 1
2.7987 1693.3008
14 1 .7155
74.9000 150.7000
0.145218D +02
245 .2719
277.1500
298.1500
323.1500
353. 1500
2.3022
2.9004
438.7000
1242.7756
I 835.2233
46 .4000
247
otro
0171
0172
01 73
0 174
0175
0176
0177
0178
0179
0180
0 18 1
0 132
0 183
0184
0 18 5
0186
0187
0 183
0189
0190
O 19 1
0192
O n
IV G LEVEL. 21 CALFUN DATE = 79 109 22/43/30
D I SC PETE DISTRIBUTION
OUTSIDE REGION
GC1=XTC{32)+ XTC(33)*T(J)+XTC(84)*T(J)**2
GC2=XTC (85)+XTC(86)*T(J)+XTC 87)*T< J )**2.
GC3 = X~C (83) + XTC (89) *T { J ) + XTCI 90 )*+(J )**2
GC4 = XTC ( 91 ) + X^C (92 ) *T( J ) +XTC(93 )*T { J )**2
GC5=XTC(94)+XTC(95)*T{ J ) + XTC( 96)*'r( J)**2
GC6=XTC(97)+XTC(98)*T(J)+XTC(99)*T(J)**2
GEE = DEXP GC1+GC2*DL0G(UL(J))+GC3*DL0G(UL(J) )**2+GC4*0L0G(UL(J))**
13+GC5*DL0G(UL(J))**4+GC6*DL0G(UL(J))**5)
GDE =GEE + 0.50*{XTC(79)+X7C(30)*T(J)+ XTC (3 1 )*T {J)**2 )
SC1 = XTC ( 33) +2. 0*XTC ( 84) *T( J)
5 C2=XTC (86 )+2.0 *XTC (87) *T ( J )
SC3=XTC(89)+2.0+XTC(90)*T{j)
SC4=XTC (92 ) +2.0*XTC( 93 ) *T( J )
SC5 = XTC (95 ) +2.0*XTC (96 ) *T ( J )
SC6=XTC ( 98) +2. 0*XTC ( 99)*T( J)
3 CZ=GEZ*(S Cl+5C2*DL0G( UL(J))+ GC2*DUL/UL(J)+ SC3*DLCG(UL(J))**2 + 2.*
IGC3*0L0G(UL(J) )*DUL/UL(3)+SC4*DL0G(UL( J) )**3+3.0 *GC4*DL0G(UL(J ))*7
12DL/UL ( J ) +SC 5+DLQG UL { J ) ) **4+4. 0*GC5*DLOG( UL( J) ) **3*DUL/UL ( J) +SC
16*DLDG(UL(J))**5+5.0 GC6* DLO G(UL(J ) )**4*DUL/UL( J) )+0.50*(XTC( 80)
12.0* XTC( 81)7T( J ) )
GI(J.I)=(DF*EPSI( I ) *( GDOL S I+GDE )+CF *Â£P SI ( I ) ( GC GL S I+GC OLSI ) ) /T( J)
SI ( J .1 ) OF*cPSI ( I )* (3D0LS I+SDE)+CF*Â£PS I ( I ) *( 3CGL 5 I +SC0LS I )DOF*EPS
1 I ( I ) *( GDOLS I+GPE) + (DCtr)*EPSI ( I ) (GC GL 3I+GC0L3I )
7 HI(J,I)=GI(J.I)+SI(J,I)
SECTION FOR CALCULATING HEAT CAPACITIES
10 CONTINUE
DC 15 J=1 ,7
HH{J)=HAC{J .1 )*T{J)
H I I ( J >=H I ( J I ) *T( J )
HHH(J)=HRV(J,I)*T(J)
220
105
other molecules fixed and 0/2 is the intermolecular potential energy of
one molecule due to the presence of all other molecules.
For a polyatomic molecule with n atoms, there are a total of
3n degrees of freedom. However, many vibrations are of such small ampli
tude and high frequency that at normal densities they do not affect inter
molecular interactions. Prigogine (1957) postulated that for each mole
cule there are 3c external degrees of freedom which affect intermolecular
interactions. For argon (or methane), c = 1; for more complex molecules,
c > 1. For all molecules, c/n <1.
The rotational and vibrational partition function can be factored
into external (densitydependent) and internal (densityindependent)
terms.
(q ) = (q ) (q ) .
Hr,v Hr,v ext r,v int
(531)
Beret and Prausnitz (1975), following Prigogine, assumed that
contributions to the partition function from rotational and vibrational
motions may be calculated as contributions from equivalent translational
motions. Each translational degree of freedom contributes
r 1/3
V,
T 6XP 2kT
to the nonideal part of the equation of state. Treating the external
rotational and vibrational motions similarly, Beret and Prausnitz obtained
3(cl)
(q )
t,v ext
0
3
2kT
(532)
FORTRAN IV G LEVEL 21
CAL FUN
DA
79 109
22/43/30
0077
0078
0079
0080
008 1
0082
0083
0084
0085
0086
0087
0088
0089
0090
009 i
0092
0093
0094
0095
0096
0097
0098
0099
C
c
r
C
C
GI(J,1)=GICH4
SICH4=EPSI(1)*(4.954D+03/(T(J)**2)13.096D+05/(T(J)**3)0.76CD+OO
2*DHSCH4l .585D+00*HSDCH4*DHSCH4>
SI{J,l)=SICF4
GO TO 7
i UL J ) = UL ( J ) *( 11 )
DUL=DUL*(11)
CF=(EPSI<1>l.5*EPSI(l ) >/(EPSI( I))
DF=1 .OD + OOCF
DDF = 0*00
DCF= 0.0 0
IF(UL(J ) .LT .3.600) GO TO 5
LARGE UL VALUE SECTION
CONTINUOUS DISTRIBUTION SECTION
0
GCl=XTC(l)+XTC(2)*T(J)+XTC(3)*T(J)**2
GC2=XT C(4)+X7C(5)*T(J}+XTC(6)*T(J)**2
GC3=XTC(7)+XTC(8)*T ( J )+XTC(9 )*T(J)**2
GC4=XTC( 1 O)+XTC(1l)*T(J) + XTC 12)*T(J)**2
GC5 = XT C(13)+X~ C 14)*T{J )+XTCC 15)*T (J)**2
GCOLLI= DEX P(GC1 + GC2/UL(J)+GC3/(UL(J )* *2 )+GC4/UL(J)**3 > +GC5/(UL(J
2)**4 ) )
SCI=XTC(2)+2.0*XTC(3)*T(J)
SC2XTC ( 5) +2. 0*XTC (6) *T< J)
SC3=XTC(3>+2.0*XTC(9}*T(J)
SC4=XTC ( 1 1 ) +2.0*XTC( 12) *T ( J )
SC5=XTC(14)+2.0*XTC{15)*T(J)
S COLL1 = GCOLLI*(SC 1+SC2/UL J)GC2*0UL/(UL(J) **2> +SC3/**2)2.
20*GC3*D UL/(UL(J)**3)+SC4/{UL{J)*3)3.0*GC4*DUL/(UL(J)**4)+SC5/{UL
3(J)**4)4.0*GC5*DUL/(UL(J)** 5))
DISCRETE DISTRIBUTION SECTION
0
216
FORTRAN IV G LEVEL 21
CALFUN
DATZ
79 109
22/46/19
0060
00 6 1
0062
0063
0064
0065
0066
0 06 7
0068
0059
0070
007 1
0072
0073
0074
0075
0076
0077
C SECTION TO CALCULATE ROTATIONAL ANO VIBRATIONAL PROPERTIES FROM
C PERTURBED HARDCHAIN THEORY
VBAR W=4. 9355D02/DENW
DVBARW=4.9355D02*DDENW/(DENW**2)
GRV(J,I)=(C(I)1>*((3.0*{{0.740 5/VBARW)**2)<4.0*0.74 05/
2VBARW> } /( ( 1.0 0.7405/VBARW)**2) )
SRV { J, I )=GRV( J I ) +T( j ) *( c C I ) 1 ) +DVBAR W*{ 6.0*0.7405**2/{
2VBAPW**3)+4.0*0.7405/+ < 3.0 *( ( 0.
2740 5/VBAR W)**2)(4.0* 0.7405/VBARW))*( 2.0*0.7405/(VBA R W* *2))/((1.0
3 0.7405/V BARW)* *3) )
HPV( J, I )=GRV< J. I) + SFV( J ,1 )
C SECTION TO CALCULATE CAVITY CONTRIBUTION TO PROPERTIES
S T W= 1.1620 + 021 .477 D0 1 *T ( J )
DSTW=1.477D01
DEL=O.31948960+00+2.6052103D+03/7(J)13.993069D+04/(T(J)**2)
DCEL = 2. 60 521 03D+03/( T(J)**2) +2.0*18 .9930690+04 / (T (J )* *3 )
UL J ) = X( 1 )X2) *1.0D+02/T{ J ) .+ X( 3) *1 0D+04/(T( J) * 2)
DUL=X{ 2 ) l OD+O2/(T(J ) *2 )2.0 *X ( 3 > 1 OD + 04/( T( J ) *3 )
WAC( J.I)=(PI/( XK* T{ J) ) } *( UL( J)* ( I +3 >*STW*DCWCH4+STW*({DCWCH4**2)(
2DCWAR**2)>4.0DEL*STW*(0CWCH4DCWAR) STW*DEL*2. 0*UL(J)*
3CAR
SAC( J. I ) = ( PI/XK)*{DUL*{I+3)*STW*DCWCH4+UL(J)*( 1+3)*DSTW*0CWCH4
2 +UL(J) *(I+3)*ST W*DDCW G4 +DSTW*( (DCWCH4**2)(DCWAR**2))+STW*(2.0*DCW
3CH4* DDCWG42.O+DCWAF*DDCWGA)4.0*DDÂ£L*STW*(DCWCH4OCWAR)4.0 +CEL *D
4STW*(DCWCH4DCWAP )4.0 *DEL*STW*(DOCWG4DDCWGA)DSTW+DEL*20 UL(J)*
5(1+3) STW*DDEL *2 0 *UL(J)*(I+3)STW*DEL*2.0*DUL ( 1+3) J+SACAR
HAC( J, I )=WAC( J I ) + SAC J .1 )
C SECTION TO CALCULATE INTERACTION CONTRIBUTIONS
UL(J)=UL{J)*(I+3)
DUL=DUL *( I+3)
CF(EPS I(I)l.5*EPSICl)/{EPS I( I ))
DF=1.OD+OOCF
Table 62c
Contributions to Entropy of Micellization
Surf.
T(K)
AS
wm
R
AS
mcc
R
AS.
mic
R
SOS
277.15
7 .49
29.28
3.4 2
298.15 .
12.02
18.76
2. 1 1
223.15
1 7.29
8.21
0.6 1
358.15
24 .27
3.79
1 .34
SDS
277.15
9.60
30. 3C
2.2 8
298.15
15 .20
17.66
1.17
323.15
21 .69
5. 02
0. 09
358.15
30.24
9.17
1.75
SDDS
277.15
1 1 .75
31 .32
1 .34
298.15
18 .45
16.55
0. 43
323.15
26.17
1.84
0.62
358.15
36 .30
14.56
1 .99
AS
rv
R
AS .
mmi
R
AS
ca
R
AS
cal
R
AS
exp
R
0.87
5.65
5.18
23.39
1.12
5.39
5.18
19.23
1 0 .60
l .33
5.10
5.13
15.94
1.60
4. 71
5. 18
13.54
l .76
6.93
6.64
25.80
2.05
6.62
6.64
20. 47
1 3. 00
2.27
6.2 7
6.64
16.17
2.51
5.79
6.64
12.89
1.64
8.0 1
8.23
27. 15
1.99
7. 69
8.23
20.65
15.50
2.24
7.31
8.23
15.33
2.40
6. 79
3. 23
11.11
143
56
including water with diverse intermolecular forces and orientational
effects. The reduced isothermal compressibility can be related to the
direct correlation function by
1 = 3P/RT
PRTKT 3p
1 4irp
2
r
c(r)dr
(347)
Figure 32 and Table 35 represent an attempt to extend this
investigation to a microscopic level using a reduced direct correlation
function. The reducing parameters obtained by Mathias (1978) by fitting
pure component compression data are utilized
&
Argon : V =75.4 cc/gmole T = 150.7K
Water : V =46.4 cc/gmole T = 438.7K.
The ratios of these values are similar to those found by Gubbins
and O'Connell (1974) although the temperature ratio is lower.
The reduced direct correlation function (integrand in equation
*1/3 2 *1/3
(347) 4irpV r c(r) as a function of reduced distance r = r/V at
several temperatures is presented in Table 35. Figure 32 shows the
reduced direct correlation function for water compared with two states of
liquid argon whose reduced density bound that of water. (Argon data of
Yarnell et al., 1973, and Smelser, 1969).
Of particular interest in Figure 32 are the unusual extremum in
the reduced correlation function for water at r*x0.8 and the shift of
the maximum in the water results to considerably larger distances (r* = 1.3)
than that of argon (r = 0.9). While any interpretation of these results
is highly speculative, the dual nature of liquid water as proposed by
Narten and Levy (1969) may offer some assistance since it suggests the use
64
For the dissolved gas, the standard state at any chosen temperature is
taken to be the hypothetical state found by extrapolation of the line
representing Henrys law on an f^ vs graph to unit mole fraction for
the dissolved gas solute. The fugacity in this standard state is K, the
Henry's constant. This choice makes the activity, ^2^2 = ^2^ t^e
dissolved gas approach the mole fraction of the dissolved gas in very
dilute solutions where Henry's law is valid. Thus, for dilute solutions,
equation (41) becomes
diss o,diss
(j (j0
+ RT n x.
For equilibrium
and
or
diss g
G2 G2
Tro 770,diss 770,g _
AG
AG = RT in K
 G^,b = RT(Â£n x2 in ip
(43)
(44)
(45)
(46)
where Henry's law has been used.
The other standard state thermodynamic properties of solution
can be obtained from the appropriate temperature derivatives.
Ao
Ah = T
2 3(AG/t)
3t
AS = 
3t
and
*770 __ 3(AH)
ACP
(47)
The sources of the Henry's constant data used in this work are listed
below.
Gas
Helium, Neon, Argon, Krypton, Xenon
Carbon Tetrafluoride, Sulfur Hexa
fluoride
Methane
Neope ntane
Henry's Constant Data Source
Benson and Krause (1976)
Ashton et al. (1968)
Wilhelm et al. (1977)
Wetlaufer et al. (1964),
Shoor et al. (1969)
Table 53d (Continued)
Solute
T(K)
ACPc
R
ACp
i
R
ACp
r,
R
C4H10
277.15
85.14
31.78
9.03
298. 15
79.13
34.46
7.18
323.15
72.70
37.54
6.56
358.15
65.15
53.11
6.46
A^cal
R
ACp
R
62. 39
48.133
+10.4
51.85
46.059
4.5
41.72
44.117
+ 2.6
18. 50
42.989
+ 9.2
118
Table Cla
Contributions to Free Energy of Solution of Gaseous Hydrocarbons
Solute
T(K)
AG
c
RT
AG
i
RT
AG
r>v
RT
cal
RT
AG
exp
RT
CH,
4
277.15
24.92
 14.37
0.0
1 0. 05
10.10 4
+.005
298. 15
24. 24
13.65
0.0
10 .60
10.596
+.004
323.15
23.32
12.35
0.0
10.97
10.946
+.003
358.15
21.91
 10.80
0. 0
11.12
1 1.134
+.008
C2H6
277. 1 5
3 0.72
22.36
1.27
9.62
9.601
+.008
298.15
29 .65
20 .6 2
1.26
10.30
10.30 7
+ .006
323.15
28.26
 18.72
1.24
10.77
1 0.30 7
+.003
358. 1 5
26 .21
16.32
1.13
11.07
11.072
+. 010
C3H8
277.15
36.51
23.98
2.20
9. 73
9.708
+.009
298. 15
35.06
26.72
2.18
10 .52
10 .52 1
+ .007
323.15
33.19
24.25
2.14
11.08
11.081
+. 006
358.15
30.50
21.10
2. 04
11.44
11 .342
+.010
140
removal of the monomer cavities from solution were modeled identically
to those for the equal chain length hydrocarbon of Chapter 5. For
example, the octyl sulfate has the same chain length as octane. This
represents the assumption that the surfactant headgroup is not included
in the micelle. The Gibbs free energy expression for formation of a
monomer cavity in aqueous solution is
AG AG
Rf ET 2,r Y (V5) L + Y Wd 86Y< Wf > (614>
The sodium counterion cavity properties were modeled as equiva
lent to those for a spherical nonpolar molecule with a temperature inde
O
pendent diameter of 2.02 A. This is the bare sodium ion diameter. The
Gibbs free energy expression for formation of a counterion cavity in
solution is
AG. AG ,
Cl rpf oo oo
er = y (a a ,) 8y 6(r .r ,)
1 ci ref 1 Cl ref
(615)
aRT RT v~ci
where a is the fraction of counterions bound in the Stern layer of the
micelle. In this model a = 0.75 as suggested by the experiments of
Evans et al. (1978).
In order to alleviate overcrowding in Table 62, the monomer and
counterion cavity results were combined to yield
AG
me
RT
I_AG
m
ag;
ci
RT
RT
(616)
the negative sign indicating that the cavities have been removed from
solution.
The micelle cavity properties were modeled as equivalent to
those of a spherical solute with diameter
172
FORTRAN IV G LEVEL 21
MAIN
DATE
79108
00/49/39
000 1
0002
0003
0004
0005
0006
0007
0008
C PROGRAM TO FIT AQUEOUS SPHERICAL GAS SOLUBILITY DATA TO A FUNCTIONAL
C F CRM FOR THE CURVATURE DEPENDENCE PARAMETER DEL
C
C
C EXPLANATION OF INPUT DATA
C CV CHARACTERISTIC VOLUME OF SOLUTE CC/GMOL
C CT CHARACTERISTIC TEMPERATURE OF SOLUTE K
C ESI CHARACTERISTIC ENERGY PARAMETER FOR SOLUTE K
C CVW CHARACTERISTIC VOLUME FOR WATER CC/GMOL
C CTW CHARACTERISTIC TEMPERATURE FOR WATER K
C TT TEMPERATURES OF INPUT EXPERIMENTAL SOLUBILITY DATA K
C EXG DIMENSIONLESS EXPERIMENTAL GIBBS FREE ENERGY OF SOLUTION
C EXH DIMENSIONLESS EXPERIMENTAL ENTHALPY CF SOLUTION
C EXS DIMENSIONLESS EXPERIMENTAL ENTROPY OF SOLUTION
C EXCP DIMENSIONLESS EXPERIMENTAL HEAT CAPACITY CF SOLUTION
C NN NUMBER OF SOLUTES USED IN FITTING, MINE IN THIS CASE
C N NUMBER OF UNKNOWN VARIABLES TO eE DETERMINED
C M NUMBER OF PIECES OF DATA FIT
C SXCl,EXC2EXC3 COEFFICIENTS IN EQUATION FOR ARGON SOLUBILITY DATA
C
C ARGON WAS USED AS A REFERENCE SOLUTE
C
IMPLICIT REAL* 8 ( AH 0Z )
DIMENS ION CVI9) ,CT{9),EXS(9,4),EXH{9,4),EXG( 9,4) ,EXCP(9,4),GC(9 ,4)
2 GGI (9,4) ,HC(9 ,4) ,HI (9,4) SC{94),S I (9,4 ), CP CI9,4),CPI(9,4),TT(4) ,
3CSSI 9*4)CGS(9,4) CHS(94) ,CCPS(9,4) ,X(3),F(36),W(1000) .EPS I (9)
COMMON/A/CV,CT,EXS,EXH.EXG,EXCP.EPSI,TT,EXC1.EXC2.EXC3
CCMMON/B/CSS.CGS.CHS,CCPS
CCMMON/C/GC.GGI,HC,HI,SC,SI,CPC,CPI
CCMMON/D/CVW, CT W
M = 36
N=3
5
AG \i
m m
RT RT
P1
Â£n x
RT L
1
&n x
m
(21)
At the critical micelle concentration (CMC), relatively abrupt
changes in properties are observed because the concentration of monomeric
species begins to change very little with the mole fraction of added
amphiphile, xq. (micelles are being replicated)
x = x,. + N x .
o 1 m
(22)
Around the CMC the value of 9x.,/9x _ falls rapidly from near
1 o ' T,r
unity to near zero. The CMC definition of Phillips (1955), explored by
Hall (1972), is
lim
\ +
x * x
o o
T,P
(23)
where xq is the CMC and c is an "ideal colligative property" which depends
only on the number of solute species (monomers and micelles). As noted
by Chung and Heilweil (1970), this definition is difficult to interpret
unambiguously in terms of the experiments done with micellar solutions.
As an alternative, the expression of Hall and Pethica (1970) can
be used
lim
9(x +x )
1 m
, 9 x
^ + o
X > X
o o
= 0.5
(24)
T,P
which is essentially equivalent to equation (23) for sharp CMC points.
This expression can be used to obtain (Hall and Pethica (1970))
+ 2 +
x = (N 2N)x .
1 m
(25)
CHAPTER 5
AQUEOUS SOLUBILITY OF ALIPHATIC HYDROCARBONS
Introduction
The subject of this chapter is development of a thermodynamic
model for the aqueous solubility of aliphatic hydrocarbons. The initial
section presents modifications to the expressions of Chapter 4 for the
thermodynamic properties of cavity formation to allow calculations for
a spherocylindrical solute using a spherical reference.
Modeling of the thermodynamic properties of interaction between
a spherocylindrical solute and a spherical solvent is discussed in the
second section. Derivations of the four contributions to the total
Helmholtz free energy of interaction are presented along with correlations
of the results to facilitate use in a computer program to model hydro
carbon solubility.
The third section presents a model for the thermodynamic property
changes associated with changes in rotational and vibrational motions of
the hydrocarbons upon solution. This model is based on the perturbed
hard chain theory of Beret and Prausnitz (1975).
Analysis of the experimental solubility data is discussed in
section four with emphasis on a unified approach to both gas and liquid
hydrocarbons. The hydrocarbon vapor pressure is used to convert the
92
ACKNOWLEDGMENTS
I wish to express my deepest appreciation to Dr. John OConnell
for his interest and enthusiastic guidance throughout my graduate studies.
Working with him has been a truly valuable and enjoyable experience.
I also wish to thank Drs. J. C. Biery, G. Y. Onoda and F. A.
Vilallonga and. Prof. R. D. Walker for serving on my supervisory committee
It is a pleasure to thank the faculty and students of the Depart
ment of Chemical Engineering for providing an enjoyable, pressurefree
environment in which I could pursue my research.
I am extremely grateful to Mrs. Thomas Larrick for her excellent
typing despite the author's poor handwriting and the highly technical
nature of the task.
Finally, I am grateful to the Department of Energy and the members
of the Enhanced Oil Recovery Consortium at the University of Florida
who provided financial support for this work.
ii
30
and applicability of the theory to aqueous solutions. Several more exten
sive reviews are available (Reiss, 1965, 1977).
Scaled particle theory was originally developed and used for the
study of hard sphere fluids (Reiss et al., 1959). Application was also
made to one and twodimensional systems [rods and rigid disks] (Helfand
et al., 1961; Cotter and Martire, 1970 ab; Cotter and Stillinger, 1972)
as well as to mixtures of disks and spheres (Lebowitz et al., ,1965). An
attempt was also made to rigorously extend the formalism to real fluids
(Helfand et al., I960) along with some application to simple fluids such
as the inert gases in the liquid state (Yosim and Owens, 1963, 1964).
More recently scaled particle theory has been applied to aqueous solutions
(Pierotti, 1965; Stillinger, 1973).
The starting point of scaled particle theory is consideration of
the work of creating a cavity at some fixed position in the fluid. In a
fluid consisting of hard spheres of diameter a, a cavity of radius r cen
tered at R^ is the same as a requirement that no centers of. the hard
spheres can be found in a sphere of radius r at Rq. Thus, creation of a
cavity of radius r at Rq is equivalent to placing a hard sphere solute of
diameter b at R^ such that r = (a + b)/2. Hence the work required to create
such a cavity is also the work required to introduce a hard sphere solute
at Rq. This work is computed by using a continuous process of "building up"
the solute in the solvent. Hence the name "scaled particle theory."
It is important to note that a cavity is considered "empty" in
scaled particle theory if no centers of particles are found in it. Also it
is worth noting that a hard sphere of zero diameter produces a cavity of
radius a/2 in the system, whereas a cavity of zero radius is equivalent to
placing a hard sphere of negative diameter b = a in the system.
CHAPTER 3
SCALED PARTICLE THEORY MODIFIED
FOR AQUEOUS SOLUTIONS
Introduction
The initial section of this chapter provides a brief summary of
scaled particle theory with particular emphasis on aspects important in
extension of the theory to aqueous solutions. Previous efforts at such
an extension (Pierotti, 1965; Stillinger, 1973) are discussed critically,
particularly with regard to treatment of the contact correlation function
G(r) at macroscopic r values.
The expression of Koenig (1950) for the curvature dependence of
the surface tension is then utilized to derive an exact relation for G(r)
in the macroscopic region.
Finally, structural aspects of liquid water are investigated
through calculation of the radial distribution function and direct corre
lation function from Xray diffraction data of Narten and Levy (1971).
Structural features are particularly apparent when comparing the direct
correlation function with that of liquid argon at similar density.
Basis of Scaled Particle Theory
The intent of this section is to provide a brief survey of scaled
particle theory with particular emphasis on the assumptions involved and
29
Table 55a (Continued)
Solute
T(K)
AG
c
RT
AG
i
RT
AG
r>v
RT
AG 1
cal
RT
AS
c
R
AS?
X
R
AS
r,v
R
A5oal
R
C14H30
277.15
120.55
123.06
1 1 .34
0.83
23.71
13.94
11.33
48.98
298.15
1 13.13
113.13
11.26
11.25
7. 08
21. 75
9. 1 8
38.01
323.15
104.27
 102.35
11.02
12.93
9.40
30.72
7.15
28.47
358.15
92.23
88.76
10.5 1
13.98
27.47
42.45
4. 68
 19.6
124
Table 55a (Continued)
Solute
T(K)
AG
c
RT
AG
i
RT
AG
r v
RT
C11H24
277.15
93.05
97.36
9.06
298.15
92 .22
89.53
9.00
323.15
85.23
81.02
8. 81
358.15
75 .69
70.29
8.41
C12H26
277.15
105 .55
 105.57
9. 58
293.15
99.1 9
97.07
9.51
323.15
91 .57
37.33
9.30
358.15
81.20
76. 19
8. 88
C13H28
277.15
1 1 3 .05
114.79
10.46
298.15
106.16
105.54
10 .38
323.15
97.92
95. 49
10.16
358.15
36 .72
82.82
9.70
AG
AS
AS?
o
AS
AS
cal
c
i
r,v
cal
RT
R
R
R
R
9.75
22.19
10.72
9.06
41.97
11.69
3. 75
 16. 90
7.34
32.99
13.01
4 .62
24.03
5.72
25.12
13.31
19.39
33. 40
3. 74
17. 75
9. 55
22.70
11.75
9.57
44.03
1 1 .63
8.2 0
18.45
7.76
34.40
13.05
6. 22
26. 17
6. 04
25.99
1 3.90
22.08
36.30
3.95
18.16
8.7 1
23.20
 12.90
 10.45
46. 56
11.00
7. 64
20. 18
8.47
36.29
12.59
7.8 1
23.56
6.59
27.35
13.60
24.78
39.54
4. 32
 19. 08
123
164
Table 71
Temperature Dependence of Two Phase Region
Temperature
> ~ 65C
62 65C
59 62C
56 59C
53 56C
A) High Pressure Apparatus
P = 600 psia
Anisotropic Fraction Estimated Fraction
of Visible Sample of Whole Sample
0.00 0.350
0.400.450
0.43 0.50
0.45 0.55
0.55 0.65
0.00
0.15 0.35
.0.25 0.50
0.30 0.60
0.60 1.00
Temperature
> 65C
60 65C
56 60C
50 56C
45 50C
40 45C
< 38C
B) Constant Temperature Oven
P = 14.7 psia
Anisotropic Fraction
of Sample
0.00
0.00 0.20
0.15 0.25
0.25 0.40
0.40 0.67
0.67 0.75
1.00
279
Tartar, H. V., "A Theory of the Structure of the Micelles of Normal
Paraffin Chain Salts in Aqueous Solution," J. Phys. Chem., 5J9, 1195
(1955).
Tartar, H. V., "On the Micellar Weights of Normal Paraffin Chain Salts
in Dilute Aqueous Solutions," J. Colloid Sci., 14, 115 (1959).
Tenne, R. and A. BenNaim, "Application of the Scaled Particle Theory
to the Problem of Hydrophobic Interaction," J. Chem. Phys., 67,
627 (1977).
Thomas, G. F. and W. J. Meath, "Van der Waals Constants for Hydrogen
and Light Alkane Pair Interactions," AICHE J., 25_, 352 (1979).
Throop, G. J. and R. J. Bearman, "Radial Distribution Functions for
Mixtures of Hard Spheres," J. Chem. Phys., 42, 2838 (1965).
Tolman, R. C., "The Effect of Droplet Size on Surface Tension," J.
Chem. Phys., 17, 333 (1949).
TullySmith, D. M. and H. Reiss, "Further Development of Scaled Particle
Theory of Rigid Sphere Fluids," J. Chem. Phys., JI3, 4015 (1970).
Wetlaufer, D. B., S. K. Malik, L. Stoller, and R. L. Coffin, "Nonpolar
Group Participation in the Denaturation of Proteins by Urea and
Guanidinium Salts. Model Compound Studies," J. Amer. Chem. Soc.,
86, 508 (1964).
White, H. J., "Absorption of Cationic Surfactants by Cellulosic Sub
strates," in Cationic Surfactants, Ed. E. Jungermann, Marcel Dekker,
New York, N. Y. (1970).
Wilhelm, E., R. Battino and R. J. Wilcock, "LowPressure Solubility of
Gases in Liquid Water," Chem. Rev., 77, 219 (1977).
Yarnell, J. L., M. J. Katz, R. G. Wenzel and S. H. Koenig, "Structure
Factor and Radial Distribution Function for Liquid Argon at 85K,"
Phys. Rev. A, 7, 2130 (1973).
Yosim, S. J. and B. B. Owens, "Calculation of Heats of Vaporization and
Fusion of Nonionic Liquids from the Rigid Sphere Equation of State,"
J. Chem. Phys., 39, 2222 (1963),
Yosim, S. J. and B. B. Owens, "Calculation of Thermodynamic Properties
of Fused Salts from a Rigid Sphere Equation of State," J. Chem.
Phys., 41, 2032 (1964).
Table Ala
Contributions to Free Energy and Entropy of Solution
Solute
T(K)
AG
c
RT
ag
1
RT
AGcal
RT
AG
exp
RT
AS
c
R
AS
i
R
AÂ§cal
R
AS
. exP.
R
He
277.15
17.00
5.21
1 1.79
1 1.8 02
+.001
12.48
0.82
13 .30

13.20
+ .
298.15
16.63
4 .77
11 .86
11 .863
+.001
11.01
1.27
 12.27

12.15
+ .
323.15
16.13
4.29
11.85
11 .841
+.002
9.48
 1. 57
l1.05

11 02
+ .
358.15
15.39
3. 71
11.88
1 1 .691
+.005
7.5 0
1.74
9 .24
9 .62
+ .
Ne
277.15
18.54
6 .99
11.55
11.543
+.001
13.42
1.07
14.49

14.59
+ .
298.15
18.11
6.40
11.71
11 .7 10
+.001
1 1 .65
1.61
13.26

13.25
+ .
323. 15
17. 54
5.77
11.77
11.773
+.002
9 .8 1
1.97
11 .78

11.81
+ .
358.15
16.67
5.00
11.67
11.689
+.035
7.45
2. 18
9.63

10. Cl
+
Ar
277.15
22.07
11.91
10. 16
10.159
+.003
15.38
1.70
17.08

17.13
+ .
298. 15
21.51
10.92
1 0.59
10.588
+.002
12 .95
2.45
15.40

15.4 2
+ .
3 23.15
20.75
9.87
10.88
10.883
+. 009
 1 0.42
2. 95
13.37

13.53
+ .
358.15
19.58
8.60
1 0. 99
11.031
+ .068
7.26
3.23
10.49
11.17
+ .
08
02
09
41
11
03
13
57
08
04
30
96
187
FORTRAN IV
000 1
0002
0003
000 4
0005
oooe
0007
oooa
0009
001 0
00 l l
00 I 2
0013
00 1 4
0015
00 1 6
00 1 7
00 l 8
0019
0020
0021
0022
0023
0024
0025
0026
G LEVEL 21 CALFUN DATE = 79109 22/46/19
SUBROUTINE CALFUN(X)
IMPLICIT REAL*8{AH,0Z)
DI ME NS ICN C t10) ,T(7) ,TT(4 ) ,DCH4 (4 ) ,SRV(10,4) ,HRV(10,4).GRV(10,4),
2UL{7),W4C1710)SAC(7 10)HAC(710) X(3) ,GI(710) SI(7t10).HI (7*10
3) UUL(4 ) HHSDW(4) HH(7) *HII(7),SSI(10 4 ) HHI ( 10,4),GC( 10,4),HC( 10,
44 ) SCI 10,4), GG II.10*4) ,CPC ( 10,4) ,CPI ( 10 ,4 ) ,CCPS( 10,4),CGS (10,4),CHS
5(10,4),HARV(10,4),GARVI 10,4),SARV( l0,4>,TQCH4(4),0HH(7) ,OHI I (7) ,CS
6S ( 1 0 ,4) XTC I 99 ) ,EPSI ( 10 ) ,HHH(7 ) DHHH (7 ), CP ARV { l 0,4 ) CPRV ( 7. 10 )
C0MM0N/A/EX5,E XH,EXG,EXCP ,CV'W.CTW
C CMMON/G/GC,GGI,HC,HHI,SC,SSI,CPC,CP I.HHSDW,DCH4,UUL
CCMMON/C/CSS,CGS,CHS,CCPS,TT
CCMMON/D/GARV,HARV,SARV,CPARV
C OMMON/E/XTC,EPSI,C,EPSIAR,EPS IC1,CV CH4,C TCH 4,CVAR,CTAR
C CM M ON/F/E XC 1 E XC 2 E XC3
P 1=3.1 4159265D0
XK=1.38066
RK= 1 .987
C PARAMETER IN EXPRESSIONS FOR HARD SPHERE DIAMETER
A 1=0.54008332
A2=1.2669302
A3=0.05132355
A4=2.9107424
A5=2.5167259
4 6=2.1595955
A 7= 0 .64269552
A8=0.17565385
A 9=0.18874824
A10=17.952388
A11=0.48197123
4 12=0.76696099
A13=0 .76631363
A 14=0.309657804
238
APPENDIX B
HELMHOLTZ FREE ENERGY OF INTERACTION BETWEEN
A SPHEROCYLINDRICAL SOLUTE AND SPHERICAL SOLVENT
Integration of the Components of the Helmholtz
Free Energy of Interaction
Consider the case of a potential fixed at y = 0 interacting with
molecular centers in the region 0 < y < L and 0 < z < 00. The expression
for the Helmholtz free energy of interaction (equation 58) is
..A
M.
x
8tt e p
ws w
L
r CO
r i2
6 1
dy
dz' z ghs(z')
a
ws
a
WS
. 2 2,6
, 2 2,3
'o J
b ws
W
(y +z )
(y +z )
(Bl)
We must now evaluate analytically:
L
r i2
6 1
a
a
dy
WS
ws
. 2 2,6
2 2,3
'o
(y +z )
(y +z )
(B2)
This integral is
of the form
dÂ£
, it 2. m+1
(a+by )
which is available in standard
integral tables. Using these tables
L
dy
*0
r i2
a
WS
6
a
ws
12
126L a
ws
i
, 2, 2,6
(y +z )
2 2 3
(y +zV
2
z
512z8(L2+z2)
l
i
1
960z4(L2+z2)3 1120z2(L2+z2)4 1260 (L2+Z2)5
6/t 2 2. 2
768z (L +z )
1 t 1L
r tan
9t z
512z L
194
36
G(r) = A + B/r + C/r2
(323)
The coefficient A in equation (323) was determined from
equation (322). Expressions for B and C were determined by matching
values and derivatives of equations (319) and (320) when r equalled
a/2. If P is the experimental value the expressions lead to
^ A
Y 3i_kT _1_ 3
Y 2 Uy 2 2
ira ^ } (1y)
pkT
1 +
3y
(324)
(325)
2+y2(ly) (P/pkT)^
where y = Trpa3/6.
The lower solid curve in Figure 31 shows the resulting G(r)
O
function. Its most distinctive feature is the maximum at r ~ 2.0 A.
Similar maxima occur for other temperatures, but always at r = 46 in
the Pierotti approximation.
Integration of equation (311) with this expression for G(r)
yields
W(r) = K + K,r + K.r2 + K0r3 (326)
o 1 2 3
where the coefficients are
K = kT[ Â£n(ly) + 4.5z2] \ ir Pa3 (327a)
o 6
K = (kT/a) (6z + 18z2) + TT Pa2 (327b)
K2 = (kT/a2) (12z + 18z2) 2tt Pa (327c)
K3 = 4ir P/3 (327d)
where z = y/(1y).
Thus, an approximate expression is obtained for the work required
to create a cavity of radius r in a hard sphere fluid of diameter a.
2
excluded volume effects associated with creating and destroying the appro
priate sized cavities in water.
Interest in this excluded volume effect leads to a discussion
in Chapter 3 of the application of scaled particle theory to aqueous
solutions of nonpolar gases as originally done by Pierotti (1965), with
modifications suggested by Stillinger (1973) to account for the special
effect of the hydrogen bond structure of water. Fundamental difficulties
arise when considering the appropriate matching of the microscopic expres
sion of Stillinger for the work of cavity formation with the macroscopic
expression resulting from the relation of Koenig (1950) for the curvature
dependence of the surface tension. Rather than address this controversy
directly, we have chosen to use argon as a reference solute and use only
the macroscopic expressions to obtain properties for the other solutes
from those of argon.
Application of the model to aliphatic hydrocarbons is presented
in Chapter 5. To model the linear hydrocarbons as spherocylinders, an
expression is derived from perturbation theory for the total interaction
energy between a spherocylindrical solute and a spherical solvent. This
derivation considers the radial distribution function to be a function
of distance from the spherocylinder surface rather than a centertocenter
distance (spherical case) and distributed the intermolecular potential
along the spherocylinder axis.
The correlation for the gaseous hydrocarbons is quite good and
predicted trends in liquid hydrocarbon solubility properties seem reason
able. Sensitivity of the results to chain segment length is examined.
154
interaction enthalpy AH .. The difference between these two contribu
tions is uncomfortably large. A portion of this is due to using dif
ferent methods of calculating the two contributions. Perturbed hard
chain theory (used for monomermonomer interactions) should be used to
calculate the watermonomer interaction properties and compared with the
present watermonomer calculations using the approach of Chapter 5.
Also, since the parameter c of perturbed hardchain theory is a measure
of restriction on the degrees of freedom of the molecule, c should cer
tainly be larger than that for the equal chain length hydrocarbon (value
presently used) due to restriction of the monomer headgroups to the
micelle surface. A larger c will increase the monomermonomer interaction
properties, improving the final result. Also, some specific effects may
be important since the orientation of the monomers relative to each other
is more favorable to interaction along the entire length of the chain
than in a pure hydrocarbon liquid. Also an effort should be made within
the perturbed hardchain model to ascertain the validity of the maximum
of the rotationalvibrational contributions as a function of chain length.
Ultimately a lattice model may be necessary for the micelle inter
ior if perturbed hardchain theory proves incapable of dealing with the
special orientational features present.
Dealing with those contributions completely missing from the
present model should be the most challenging and potentially rewarding.
Clearly the area of the electrostatic interactions involved in micelle
formation for ionic surfactants is in need of clarification judging from
the multitude of possible models suggested by Stigter (1975b). Probably
one should deal first with a nonionic surfactant, data permitting.
Step lb
AS > 0,
'AH > 0
Step 3
AH < 0?
AS < 0
Dispersed Monomers
Compressed Monomers
Ah < o
Mo
@
Step la
AH = ?
Step 2
AH < 0
Step 4
AH > 0 .
r
Step 5a
Ah = ?
0
AJA
AS = *
AS > 0
AS < 0
V
J
AS = ?^
Yiy
Dispersed Monomers'
and Counterions in
Water
Water With
lispersed Cavities
Step lc
AH > 0
AS ~ 0 ?
Water
Micelle in Water
With Bound Counterions
Dispersed
Counterions
Fig. 61. A Thermodynamic Process for Micelle Formation
133
FORTRAN IV G LEVEL 21
CAL FUN
DATE = 79109
22/43/30
0 145
0 146
0 147
0 148
0149
0150
0 15 1
0 152
0 153
01 54
0155
0 156
0157
0158
0159
0160
016 1
O 162
0 163
016 4
0165
0166
016 7
0168
0169
SC 1= XTC(32)+2.0*XTC{33)*T( J)
SC2=X7C(35)+2.0*XTC(36>*T(j)
SC3 = XTC(38)+2O *XTC C39)*T(J}
SC4=XTC{ 4D+2. 0*XTC( 42)*T( J)
SC0LSI=(SC1 *UL( J) +GC1*DUL+SC2*UL(J)**2+2.O*GC2*UL(J)*D UL+SC3*UL(J)
l**3+3. 07GC3*DUL+UL( J)**2+4.0*GC4*DUL*UL(J )**3+SC4*UL(J )**4)
C DISCRETE DISTRIBUTION
C 0
GC1=XTC{43)+XTC(44)*T(J)+XTC(45)*T(J>**2
GC2= XT C ( 46 ) +XT C(47)*T( J)+XTC( 48) *T ( J J *2
GC3= XT C(4 9)+XTC(5 0) *T ( J)+XTC(5l)*T(J)**2
GC4=XTC ( 52+XTC (53) *T( J)+XTC( 54)*T( J)**2
GD0LSI=(GC1*UL(J)+GC2*UL J)**2+GC3*UL(J)**3+GC4*UL(J)**4)
SC1=XTC{44)+2.O*XTC(45)*T(J)
SC2=XTC(47)+2.0+XTC(48)*T(J)
SC3 = XTC(50> +2.0*XTC (51 ) *T ( J )
SC4=XTC ( 53) +2. 0*XTC (54) *T{ J)
SCOL S I = (SCI TUL ( J) +GC1 *DUL +SC2*UL(J ) *2 +2.0*GC 2* UL ( J> *DUL + SC3*UL( J)
1**343.0*GC.3*DUL*UL(J )**2 + 4.0*GC4*DUL*UL(J)**3+SC4*UL(J)**4)
C CONTINUOUS DISTRIBUTION
C OUTSIDE REGION
GC1=XTC(55)+XTC(56)*T(J)+XTC(57)*T(J)**2
GC2= XT C ( 58 ) +XT C { 59 ) *T ( J ) + .XTC.C 60)*T{ J ) **2
GC3 XTC (61 ) +XTC (62 ) *T ( J )+XTC( 63 ) *T { J 1**2
GC4=XTC ( 64) +XTC (65) *T( J) + XTC (66 )*T ( J )**2
GCGLSI = ( GC1+GC2*UL ( J )+GC3*UL ( J )*72+GC4*UL ( J ) ** 3)
SC1=XTC(56)+2.0*XTC(57)*T(J)
S C2= XT C( 59) +2.0*XTC( 60) *T{ J)
S C3 = XT C ( 62 ) +2 0*XTC(63)*T (J)
SC4=XTC { 65) +2.0*XTC (66) *T( J)
SCGL SI = (SCI +SC2 MJL ( J ) +GC 2*DUL + SC 3*UL { J )**2+ 2. 0*GC 3*0 UL *UL ( J ) +3.0*G
1C4*UL(J)**2*DUL+SC4*UL(J)**3)
63
The second step is modelled similarly to Pierotti but using
a full reference radial distribution function and a more appropriate
pair potential for water. The energy parameters for unlike interactions
were based on the approach of Rigby et. al. (1969) but allowed to vary to
obtain a highly accurate fit of the Henry's constant at 298.15K. Care
has been taken to provide accurate models for the temperature dependent
hard sphere diameters and mixture radial distribution function..
The possibility of contributions to the properties of solution
arising from a change in the interfacial tension upon introduction of
a solute into a cavity is discussed. Such a possible effect is shown to
be insignificant.
The last section of this chapter discusses the results of the
modeling from the viewpoint of possible inadequacies and thus provides
a basis for suggestions of future research.
Thermodynamic Properties of Solution from Experimental Data
Appropriate derivatives of the Henry's constant with respect to
temperature yield the enthalpy and entropy of solution and the difference
in heat capacity of the gas between the liquid and gaseous phases. For
states at the same temperature, the activity of the gas in each phase
is related to the partial molal free energy, or chemical potential by
C.2 = C? + RT An a2 (41)
If the standard state of the gas in the gaseous phase at any fixed
temperature is defined as that in which the gas has a fugacity of unit
pressure, the chemical potential for that phase is
Gg = G^g + RT In fg (42)
6
From the definition of x in equation (22)
o
+ + 2
x = x /(N N)
m o
+ +2 2
x = x (N 2N)/(N
1 o
N)
(26)
(27)
which then yields
AC + ,
m 0 + I
32 X'll x ~
NRT N
 Â£n x+ + Â£n + Â£n(l N/N^)
o
+ Â£n Q(12N/N2)/(1N/N2) 3
To assume that polydispersity is unimportant the righthand
(28)
side of equation (28) should be small enough to neglect, which means its
value is approximately 0.05 for 5% error in x+ (typical experimental
uncertainty). By taking
1 1/N N/N
(29)
equation (28) becomes, in first order approximation
AG Â£n x+ + 2Â£n 1
m + .. o
xn x
NRT N
(210)
The righthand side of equation (210) is always positive so
Â£n x+ represents a lower algebraic bound to AG/NRT. For the effect to
o m
be less than 5%, N should be greater than about 300 to 600. However, it
ranges upward from 10, so polydispersity is normally important and thermo
dynamic theories for AG/NRT should give an equation which can yield
m
2 _
values of N and N. Further, data analysis to obtain values of Ag from
m
CMC values must allow for these terms.
The division of the free energy change of micellization into
enthalpy and entropy components is accomplished by the relations
trends in solution properties for the liquid hydrocarbons were in
reasonable accord with the few experimental data available. The entropy
contribution from intermolecular interactions was unexpectedly large,
actually dominating the cavity (excluded volume) contribution at higher
temperatures.
A primitive model for the thermodynamic properties of ionic
micellization was tested using the aqueous solubility model and electro
static theories. Quantitative agreement with experiment was not found
due to contributions to the thermodynamic process which were omitted
as well as model inadequacies. Suggestions for improvement have been
made.
A limited experimental investigation was conducted into the
effect of temperature, pressure and dissolved gas on the isotropic
anisotropic transition for lyotropic liquid crystals. Tentative results
indicate a two phase region between 40 and 80C and little or no effect
of pressure or dissolved gas on the isotropicanisotropic transition,
except for transients.
xv
18 APRIL 1979
DATA LIST FOR SPHERICAL GAS SOLUBILITY
23.5820
50.0000
39.0000
c
78. 8007
60.4800
45.2000
c
189.0161
74.9000
150.8000
c
235.1349
88.4600
209.4000
c
282. 9748
l 14. 4600
289.7000
c
232.5133
96.0300
190.6000
c
256. 5033
147. 0000
227.6000
c
295.0468
203.9400
318.7000
c
333.7771
312.0900
433.8000
c
0.464000D + 02 0.438700D+03 0
.4883300+03
c
277.1500
c
13.2008
1.3993
1 1.8020
14.6855
c
l4.5949
3. 0504
1 1.5432
18.8223
c
17.1263
6.9688
10.1588
23.9104
c
18.1832
Â£. 72 74
9.4568
26.4419
c
19.4716
 1 0. 6963
8.7749
31.4142
c
16.0574
7.9536
10.1038
26.7086
c
21.9930
1 0. 1 033
1 1.8910
48.4399
c
25.0780
13.5313
1 1.5480
65.8078
c
27.6598
17.2758
10.3998
69.7333
c
298.1500
c
12.1490
0.2898
11.8631
14.1772
c
1 3. 2511
1.53 91
11.7099
18.1278
c
15.4152
4. 8253
l0.5884
23.4826
c
16.2859
6.2829
10.0036
25.5964
c
l 7. 2169
7.76 92
9.4476
30.5637
c
16.1449
5.54 87
10.5956
25.5511
c
18.5304
6.05 17
12.4784
46.5476
c
20.3774
8. 0395
12.3302
63.0146
c
22.7479
l1.3061
11.4413
65.5313
c
3 23. 1 500
c
11.0216
0. 8177
11.8412
13.8450
c
l1.8067
0.0367
11.7726
17.6950
c
13. 5330
2.6438
l0.8829
23.1 102
c
14.2476
3.8401
10.4096
25.0780
c
14.7811
4.8290
9.9530
29.9799
0.1703 06D+03
0145218D +02
183
FORTRAN IV G LEVEL 21
CALFUM
DAT E
79109
22/43/30
0026
0027
0028
0029
0030
0031
003 2
0033
0034
0035
003 6
0037
0033
0039
0040
004 1
0042
0043
A 140.8 0965 7804
A1 5 = 0.2 4 062 863
DO 4 K=1,4
DO l1 I=1,4
DO 10 J=l,7
T = TT (K)+(J4)*0.500
TC=* { J) 273.15
SEC"rION TO CALCULATE SOLUTE AND WATER HARD SPHERE DIAMETERS
D EN W = 0.033433*(0.9998 42 52+169 4522 7D 03* TC 798 7 0641D 06*(TC* *2)4
16.1 7 06 D09* ( TC**3 ) +105.56 334D1 2* ( T C ** 4 )280.54 33 70 1 5*( TC**5))/( l
2.000+16.3798 5D 03* TC )
DDENW=0.033433*{16.945227D0315.9741282D06*TC 138.5 l 13D09*{ TC**
22) + 422.25336C12* TC**3)1 402.71585P1 5*(TC**4) )/(l.000+l6.3798SD
303*TC>DENW*<16.87985D03)/{ (1.000+16.87935D03*TC))
TRW=T(J)/CTW
DENWR=DENW*CVW/(0.6023)
DDENWR=DDENW*CVW/(0.6023)
HSDW=( (3.0*CVW/(2.0*PI*0.6023) ) ( A 7/ (T RW * A8 ) + A 2/< DEXP ( A 4*( ( DENWR +
2A 1*TRW)* *2.0)) )A3/(DEXP< A5* {{DENWR+Ai*TRWA6)**2 ) ) )+A9/{ DEXP(A10*
3( {(T RWA13)**2)+AIl*( (DENWRA12)**2) ) ) )) )**0. 333333
DHSD W=l 00 0 0* ( (3.0*CVW/ (2 .0*PI*0.6023) ) ( A7 A8/ ( CTW ( TR W **' ( A8 + 1.0
2) ) )2.*A2*A4*{DENWR+Al*TRW)*( DOENWR + A 1/CTW)/{DEXP(A4*((0ENWR + A1*TR
3W)^2.0)))+2.0*A3*A5*(DENWR+Al*TRWA 6)*(DDENWR +A 1/CTW)/(DEXP(A5*( {
4DENWR+A 1*TP WA6)**2.0) ) )2 O* A9* A1 0 ( ( TR W A1 3 )/CT W +A1 1 C DENWR A 12 )
5 DDENW R)/< DEXP(A10 *( ( (TRWA 13)**2)+A 11*{{DENWRA12)**2))))>)/((HSO
6W**2.0)*3.00)
TRCH4=T{J)/CTCH4
T R A R =T ( J ) / CT A R
HSDCH4={3.0*CVCH4/(2.0*PI*0.6 023) ) *(A7/CT RC H4** A8)+A2/ DEXP(A4*({
2DENW R+A1*TR CH4)**2) ) )A3/{DEXP{A5*< (DENWP+A 1 *TPCH4A6)**2)))+A9/(D
3EXP(A10*(((TRCH4A13)**2)+Al 1 *( (DENWRA12)**2)) ))) )**0.333333
HSD4R=( { 3. 0*CVAR/(2.0*PI*0.6 023) ) C 4 7/ (TRAP** A8.) +A2/ (DEXP (A4*( (DEN
213
13
writes the reaction for the neutral species (M^ = Amphiphilic Salt,
= Added Salt, = Micelle)
Mx + Q S2 J Mmq (237)
and the mass action relation for amphiphile (1) and salt (2) as
V'l + Q u2 vm
(238)
Now for an ideal solution where the added salt has a common ion with the
amphiphilic salt
^MQ ^MQ
JJ = y + RT An xx
11 1 c
y = y + RT An x x
2 2 2 c
(239)
(240)
(241)
where xc = x^ + x^ is the mole fraction of counterion in the system from
both a 11 amphiphilic salt and a 11 added salt. The relations for other
salts would be similar in form but more complex in detail. The equilib
rium relation is then
AGm 4 N ^ Q **2
RT RT
In the limit x^ = 0, Q = 0
= Â£n^xfX + K2)<1+(>/'J.
o o o
yM0 N yi n +o
= 2 An x,
o 1
N RT
(242)
(243)
where N is the micelle number in the absence of added salt and the
standard state chemical potential of the micelle without salt y^ may
differ from that with salt, y^. Again, these are neutral species, not
charged.
Table C2a (Continued)
Solute
T(K)
AG
c
RT
AG
X
RT
AG
rv
RT
AGcal
RT
AS
c
R
AS?
i
R
AS
r, v
R
AScal
R
C8H18
277.15
65.49
61.44
6.56
10.6 1
22. 1 4
6. 25
6. 56
34.95
298.15
62. 09
56.54
6.52
12.07
12.46
10. 16
 5.32
27.93
323.15
57.87
51.20
6 .38
13.04
3.0 1
14.69
4.14
21.84
358.15
51.98
44.46
6.09
13.60
7. 33
20. 70
2.71
16.08
C9H20
277.15
7 1.2 3
68.47
1 0.42
1 3.24
22.89
7.11
7.65
37.65
298.15
67 .49
62.99
10.12
14.63
12 .25
11.47
6. 20
29.92
323.15
62.31
57. C3
9. 79
1 5. 56
1.90
16.51
4.83
23.23
358.15
56.28
49.52
9.38
16.14
9.39
23. 19
3.16
 16.96
C10H22
277.15
77 .09
74.92
8.55
10.72
23.65
7. 9 l
8.55
40.11
298.15
72 .90
68.92
8.49
12.47
12.05
12.68
6. 93
31.65
323.15
67.74
62.39
8. 31
1 3.66
0. 78
18.19
5 .39
24.36
358.15
60.57
54.15
7.93
1 4.35
11.44
25.46
3.53
17.55
19 APRIL 1979
232.5113
398.4875
570.6245
c
l. 0000
1.2527
1 .4362
c
46.4000
438.7000
96.0300
c
0.438330D+03 0.
1703 06D+03
c
277. 1500
c
18.0574
7.9537
10.1038
c
21.0 418
11.44 07
9.6012
c
22.9945
 13. 2840
9.7076
c
24.635l
14.8241
9.8088
c
298.1500
c
16.1449
5.5487
10.5956
c
18.2587
7.95 20
10.3071
c
 19. 5773
S. 0519
10.5210
c
21.1827
 10.45 18
10.7294
c
323.1500
c
 14. 1369
3. 1 880
l0.9457
c
15.3397
4.5351
10.8068
c
15.9889
4.9093
1 1.08 14
c
17.5591
6. 1633
1 1.3945
c
358.1500
c
1 1.6608
0. 52 58
1 1.1340
c
11.7514
0. 6834
1 1.071S
c
11.5903
0.2480
1 1 .3421
c
13.1002
 1.32 52
11.7750
DATA LIST FOR GASEOUS HYDROCARBONS
749.9852
1 .6058
189.0161 190.6000
0.145218D+02
26.7086
38.7771
47.7000
48.1329
25.551 1
37.0911
45.5913
46.0594
24.4841
35.4957
43.5984
44.1168
23.8450
34.57 98
42.4157
42.9894
74.9000 150.7000
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Federico A.Vilallonga
Professor of Pharmacy
This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate Council, and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
June 1979
J'
Dean, College of Engineering
Dean, Graduate School
150
negative temperature derivative of the surface tension and partly from
the variation of the curvature parameter 6 with temperature. The monomer
cavity entropy contribution is highly temperature sensitive; results at
277K show an increasingly negative entropy with increasing chain length,
whereas those at higher temperatures show an opposite trend.
A portion of step 1 of the thermodynamic process, the relaxation
of constraints on rotational and vibrational degrees of freedom of the
monomers by their aqueous environment, has been included in the calcula
tions for step 3 and will be discussed later.
Step 2, involving removal of monomer and counterion cavities from
the water, was viewed as the dominant driving step for micellization in
Chapter 2 with a large positive entropy change and a modest negative
enthalpy change being predicted. Indeed, the calculations agree. However,
the cavity enthalpy change is large, counterbalancing the large positive
enthalpy change from watermonomer interactions. Also, the entropy change
is unusual, the magnitude for the large monomers is less than that for the
much smaller counterions. This is a reflection of the decrease in entropy
with increasing length for the spherocylindrical cavities discussed above.
Micellization is an entropy driven process with large enthalpy effects
cancelling in the range of ambient temperature. Note that in Table 62,
the properties for the monomer and counterion cavities have been lumped
to alleviate crowding and are denoted with subscript mcc.
The compression of monomers to micelle density (step 3) was
hypothesized to have negative entropy and positive enthalpy changes.
However, the present model does not consider the electrostatic repulsions
at the micelle surface and thus the attractive forces of monomermonomer
FORTRAN IV G LEVEL 21
MAIN
DATE
79 109
22/46/19
000 1
0002
0003
0004
0005
0006
0007
C OBJECTIVE CALCULATE PREDICTIONS OF AQUEOUS SOLUBILITY PROPERTIES
C OR LIQUID HYDROCARBONS
r
C
C EXPLANATION OF INPUT DATA
C CVW CHAPACTcR STIC VOLUME FOP. WATER CC/GMCL
C CTW CHARACTERISTIC TEMPERATURE FOR WATER K
C CVCH4 CHARACTERISTIC VOLUME C0F METHANE CC/GMOL
C CTCH4 CHARACTERISTIC TEMPERATURE FOR METHANE K
C CVAP CHARACTERISTIC VOLUME FOR ARGON CC/GMOL
C CTAR CHARACTERISTIC TEMPERATURE FOR ARGON K
C EPSIAR INTERACTION ENERGY PARAMETER FOR ARGONWATER K
C FPSICl INTERACTION ENERGY PARAMETER FOR METHANEWATEP K
C EPS I INTERACTION ENERGY PARAMETER FOR HYDRCCAR30N 50LUTE
C WATER K
C TC COEFFICIENTS IN CORRELATION EQUATIONS FOR VARIOUS
C CONTRIBUTIONS TO THE WATERHYDROCARBON INTERACTION
C C CHAIN LENGTH PARAMETER OF PERTURBED HARDCHAIN THEORY
C FROM GMEHLING 1978
C TT TEMPERATURES AT WHICH SO_JBILITY PROPERTIES ARE PREDICTED K
C E XC 1 EXC 2 E XC3 COEFFICIENT IN EQUATIONS TO CALCULATE
C SOLUBILITY PROPERTIES FOP ARGON,WHICH IS USED AS A REFERENCE SOLUTE
IMPLICIT REAL*8(H,0Z)
D IMENS I ON GC(10,4) ,GGI(10,4),HC(10,4),HI(10,4),SC(10,4),SI(10,4),
2CPC ( 10.4 ) CPI ( 1 0,4 ) HSDW (4 ) CSS ( 10,4 ), CGS< 10,4) CHS 10,4) ,CCPS( 10,
34 ) TT< 4) ,GR V( 10,4) HP V{ 1 0 ,4 ) SR VC 1 0 ,4) ,HSDCH4 (4),UL(4),TCC99),
4EPSI (10 ), C( 10),CPPV( 10,4),X( 3)
commcn/a/exs.exh,exg,excp,cvw,ctw
CCMMON/B/GC ,GG I ,HC.HI ,SC,SI .CPC ,CPI ,HSDW,HSDCH4 ,UL
CGMMCN/C/CSS,CG5,CHS,CCPS.TT
C CMMON/D/GR V,HRV,SRV,CPRV
COMMCN/E/TC ,EPSI ,C,EPSI AR. ,EPSIC1,CVCH4,CTCH4,CVAR,CTAR
NJ
OJ
4^
49
Table 32 (Continued)
r(A) g (r)
m
Temperature (C)
4
20
25
50
75
100
7. 10
1.0 5
1. 04
1.06
1.05
1.04
1.03
7.20
1.04
1.05
1.05
1.04
1.04
1.04
7. 30
1.03
1 .04
1.03
1.03
1.03
1.03
7.40
1.02
1.01
1. 02
1.02
1.02
1. 02
7. 50
1.01
0.99
1.01
1.01
1.02
1.00
7.60
1.00
0.98
1.00
1.no
1.00
0.99
7. 70
0.98
0.98
0. 98
0.99
0.98
0.98
7. 80
0.98
0.98
0. 98
0. 99
0.98
0. 98
7.90
0.98
0.98
0. 98
0. 99
0.98
0.93
8.00
0.98
C. 99
0. 98
0.98
0. 99
0.98
3. 10
0. 98
0.98
0.99
0. 97
0.98
0.99
8.20
0.97
0.97
0. 99
0. 97
0.98
0.99
8. 30
0.99
0.98
0.98
0. 98
0.98
0.99
8.40
0.99
1.00
0. 99
0. 99
0. 99
0.99
8. 50
0.99
0.99
0. 99
0. 99
0.99
0.99
8.60
0.99
0. 98
0. 99
0. 99
1.00
0. 99
8. 70
1.00
0.99
0. 99
1.00
1.00
0. 99
8. 80
1.0 1
1.01
1.00
1.00
0. 99
0.99
8. 90
1.01
1.01
1.00
1.00
1.00
1.00
9.00
1.00
1.00
1. 00
1.00
1.00
1.01
9. 10
1.00
1 .00
1.00
1.00
1.01
1.01
9.20
1.01
1.01
1.00
1. 0 1
1. 00
1.02
9. 30
1.00
1.01
1.01
1.01
1.00
1.02
9.40
0.99
1.00
1.00
1.00
1.00
1.00
9.50
0.99
0.99
1.00
1,00
1.01
0.99
9.60
1.00
1.00
1.00
1.00
1.00
0.99
9.70
0.99
1 .00
1.00
1.00
1.00
1.00
9.80
0.99
1.00
1.00
1.00
1.00
1.00
9.90
0. 99
1 .00
1.00
1.01
1.00
1.00
10.00
1.00
1.00
1.00
1.0 1
1.00
1.00
FORTRAN IV
010 1
0 102
0103
0104
0105
0 106
0107
0108
01 09
0110
0 111
0 112
0 113
01 14
0 115
0116
0117
0118
0119
0120
0 12 1
0 122
0 123
G LEVEL 21 CAL FUN DA TE = 79 109 22/46/19
SC4=XC ( 26) 42.0*XTC (27) T{ J)
SC5=XTC ( 29 ) +2 O *XT C(30)*T(j)
SDOLLI =GD OLL I (5C1 SC2/U L ( J )GC2 DUL/ (UL ( J ) **2 ) +SC3/1 UL( J ) **2)2.
20 *GC 3*DUL/{ UL{ J )**3) + SC 4/( UL ( J ) **3 ) 3. 0*GC4*OUL/ (ULU)**4)*S C5/ (UL
2( JlT^A)4.O *GC5 *DUL/(UL ( J ) *T5 ))
C CCNTINUCUS DISTRIBUTION
C OUTSIDE REGION Y LT. O Y GT. UL
GCl=XTC ( 67 ). + XTC (68 ) *T ( J ) 4X TC ( 69 ) *T ( J ) *2
GC2=XTC ( 70)+XTC( 71 ) *T{ J)+XTC (72)*T( J )**2
GC3= XT C ( 73 )+XTC74)*T(J)+XTC(75)*T(J)**2
GC4=XTC(76)+XTC(77)*T(J )+XTC (78 ).*T {J)**2
GCGLLI=(GC 1/UL ( J ) +GC2/ ( UL (J)** 2) +GC3/(UL( J) **3 ) + GC4/{ UL ( J ) * 4 ) )
S Cl =XT C ( 68 )+2 O *XTC ( 69 ) *T ( J )
SC2=XTC(71)+2.0*XTC(72)*T(J)
SC3=XTC(74)+2.0*XTC(75)*T(J)
SC4=XTC(77)+2.0*XTC(78 ) *T { J)
SCGL LI = ( SC 1/UL ( J) GC 1 *DUL/( UL ( J ) * 2 ) +SC2/ (UL ( J)**2 ) 2 O *GC2 DUL/ ( U
2L (J ) **3 ) S C3/ ( UL ( J ) **3)3.0*DUL *GC3/(UL( J ) ** 4) + SC4/( UL ( J ) **4)4. O*
3DUL*GC4/(UL( J)**5) )
C DISCRETE DISTRIBUTION
C OUTS IOE REGION
GCl=XTC(82)+XTC(83)*T(J)+XTC84)*T(J)**2
GC2=XTC(85)+XTC86)*T(J)+X^C(87)*T(J)**2
GC3=XTC(88)+XTC(09)*T(J)+XTC90)*T(J)**2
GC4=XTC(91)+XTC(92)*T(J)+XTC(93)*T(J)**2
GC5=XTC(94)+XTC(95)*T{J)+XTC(96)*T(J)**2
GC6 = XTC ( 97) +XTC (98) *T( J ) +XTC99 )*T ( J )**2
GEE=DEXP( GCHGC2*DL0G( UL ( J ) ) + GC3 *0 L OG ( UL ( J) ) ** 2+GC4*DL0G ( UL ( J) )
2 3 + GC5* DLCG{UL(J) )**4 + GC6 *DLO G(UL(J ) )**5)
GDE=GEE+0. 5 0M XTC(79)+XTC(80)*T(J)+XTC(8l )*T { J)**2 )
SC1 = XTC(83) +2 O *XTC( 84) *T ( j )
SC2=XTC ( 06 )+2.0*XTC (87 ) *T ( J)
fo
O
KEY TO SYMBOLS
A
A
a
a
c(r)
Cdis
Cp
d
f
g(r)
G
G(r)
H
H (s)
m
J
K
L
lch2
P
P
o
Q
= Helmholtz free energy
= Helmholtz free energy
= activity
= cavity surface area
= direct correlation function
= dispersion coefficient in intermolecular potential
= heat capacity
= hard sphere diameter
= fugacity
= radial distribution function
= Gibbs free energy
= contact correlation function
= enthalpy
= scattering structure function
= arithmetic mean curvature
= Henry's constant
= spherocylinder length
= segmental length
= average micelle aggregation number
= pressure
= probability of an empty cavity
= canonical partition function
x
204
Table Blb
Parameters for Temperature Dependence of
Interaction Correlation Coefficients
O
a = 3.60 A
s
Coefficient
A
n
B
n
D
n
C1
0.14Q151D+00
0.133659D01
0.235170D04
C2
0.246585D+01
0.137794D01
0.222010D04
C3
0.116605EH01
0.657885D02
0.107410D04
C4
0.157925D+00
0.944590D03
0.155900D05
C5
0.156608D+01
0.299701D02
0.625300D05
C6
0.696151D+0Q
0.609446D02
0.103910D04
C7
0.143505D+02
0.106288D+00
0.178138D03
C8
0.832188D+02
0.706721D+00
0.118478D02
C9
0.191270D+03
0.151845D+01
0.254194D02
C10
0.547606IH01
0.00 D+00
0.00 D+00
C11
0.189400D01
0.00 D+00
0.00 D+00
C12
0.378880D04
0.00 D+00
0.00 D+00
C13
0.145017D+Q1
0.270027D02
0.577000D05
C14
0.473159D+00
0.276299D03
0.665000D06
C15
0.332932D+00
0.108019D03
0.192000D06
153
insufficient, when combined with the above considerations, to counter
balance the positive total enthalpy of the present model. The consequent
negative entropy change would improve agreement with experiment but
would not be likely to vary with chain length.
The overall effect of missing contributions is likely to lead
to a small (possibly negative) enthalpy contribution with increased
monomermonomer interaction in the micelle and watermicelle interac
tions cancelling to a large extent the electrostatic repulsion between
head groups and loss of hydration of counterions. The overall entropy
change should be negative due to restriction of monomer headgroups to
the micelle surface and watermicelle interaction.
If these missing contributions are insufficient to lead to an
accurate model of the properties of micellization, what aspects of the
present model are questionable and what alterations could be recommended?
As noted from Table 62, the total enthalpy is a small difference
of large numbers, the two largest contributions being the enthalpy of
monomer and counterion cavity removal AH and watermonomer interaction
mcc
AH^. However, the relative difference between the monomercavity enthalpy
and watermonomer interaction enthalpy is severely restricted since they
are the overwhelming contributions to the total hydrocarbon solubility
enthalpy. The only conceivable change would be shortening of the portion
of the hydrocarbon chain considered to be included in the micelle. From
results of the hydrocarbon solubility model this would yield improved
results and is justifiable since water may penetrate the micelle to some
extent.
Also easily noted from Table 62 is the large magnitude of the
watermonomer interaction enthalpy relative to the monomermonomer
101
C. Consider the case of a differential potential, d (r) =
ws
C dx
ws(r) T* continuously distributed along the spherocylinder axis
from y = 0 to y = L interacting with molecular centers in 0 < y < L and
0 < z < co as shown Figure 5lc.
The volume element dV = 2tt z dy dz and from the geometry of
2 2 2
Figure 5lc, r = z + (yx) .
4C
i
L
o
 v
dy
dz
o J
0
o
dx C
2it z 4e
L ws
12
a
ws
(z't'+(y,x) )
2t 6
ws
hs, \
Ss(z >
(z2+(yx)2)3
Substituting z' z + O O yields
w ws
(519)
m: =
8tt e p
ws w
L
C L
dx
dy
'0 J
0
w
dz' Z ghS(z')
(z2f(yx)2)6
12
J
ws
ws
2t 3
(z +(yx)")
(520)
Upon analytical integration over x and y, equation (520) becomes
4
8tt EC p
MC = p_w
x L
a.
dz' ghs(z')
ws
126a12
ws
w
n 2 2. 2 ,_2. 2 2
(L +z ) z + (L +Z )
+
(L2+z2)3 Z6 (L2+zÂ¥ z8 L
3360(L2+z2)3 5040(L2+z2)4 512z
9 a 768(L2tz2) 1920(L2+z2)2
1 L 1 U
xt, i M
tan tan
z z 1
3a. .
ij
f 2 2 2
(L +z )z
3
z
2 2
[l2(l +z )
, L t. 1 L tan L*
8z ^ z z J
(521)
131
and continuous potential distributions. This would reduce correlating
inaccuracy and computation time. Also the possibility of using a tem
peraturedependent spherocylinder radius should be studied with the goal
of determining the effect of this change on the temperature dependence of
the length parameter L. Also a deeper literature search, particularly in
foreign sources, for both long chain hydrocarbon solubility data at sev
eral temperatures and infinite dilution surfactant properties may be very
helpful in determining the validity of model predictions for long chain
hydrocarbons.
A more fundamental effect would be correlation of the sperocyl
inder length L with other thermodynamic property data. The work of
Bienkowski and Chao (1975) concerning the hard cores of normal fluids
may provide an initial guide. An appropriate property to model would be
the solute partial molar volume.
Table 45b
Contributions to Enthalpy and
Solute
T(K)
AH
c
ah
X
AH
cal
AH
exp
RT
RT
RT
RT
He
277.15
0. 52
0.59
1 .51
1 .3 99
.
298.15
0.77
0.99
0.23
0 .290
323.15
2.02
0.96
1 .05
0.8 18
.
358.15
3. 54
0.90
2.65
2.074
.
Ne
277.15
0.68
3.63
2.95
3.050
m
298.15
2.17
3.6 1
 1.44
 1.535
.
323.15
3.63
3.48
0.15
0 .0 37
.
358.15
5.38
3.23
2.15
1.678
+ .
Ar
277.15
3.56
10.49
6.93
6.965
298. 15
5.49
10.31
4 .82
4.825
.
323.15
7.39
9.88
2 .48
2.644
+ #
358. 15
9. 6 1
.9.11
0.50
0.138
+
Capacity of Solution
o
ACPC
AcP
A^cal
ACp
0
exp
R
R
R
R
i a. 56
1.24
17.33
14 .69
2.1
17.10
o.ao
16.80
14.18
0.6
16. 59
0.45
16.55
13.85
2.2
1 8.49
0.14
18.35
13.32
3.5
22.71
4.11
18.60
18.82
2.9
21.28
2.65
18.63
18.13
0.9
21.05
 1 .47
19.58
17.70
3.0
22. 26
 0. 45
21.81
17.13
4.6
31.75
9.82
21 .92
23.9 1
3.Q
30.38
6.25
24.14
23.4 8
2.0
29. 51
3. 39
26.52
23.11
4.8
30.47
0 .94
29 .52
22.53
7.0
. Heat
152
020
091
387
111
Q26
133
533
077
034
288
893
99
B. Consider now the case of a potential fixed at y =0 inter
acting with molecular centers in the regions y < 0 and y > L as shown
in Figure 5lb.
2
The volume element dV = 2ttw sin 0 dw d0. For the region Y < 0,
r = w and
. .B1
M. = p
w
tt/2
d0
jo 2 D hs.
dr 2it r sm 0 4e g (r')
ws ws
12 6
j o
ws ws
.12.
(511)
MB1 = 87T p eD
1 w ws
, 2 hs, ,.
dr r 8s(r,)
w
12 'rr6
a a
ws ws
r12 r6
(512)
where rf = r + a a
w ws
,B1
Upon integration of equation (512), A/L was fit to the follow
ing function of temperature.
MB1= +eD (Cin + C., T + C19 T2).
i ws 10 11 12
(513)
Expressions for the coefficients in equation (513) can be found
in Appendix B.
Consider now the region Y > L. From the geometry of Figure 5lb,
2 2 2
r = (L + w cos 0) + (w sin 0)
= L2 + w2 + 2Lw cos 0.
M52 = p
x w
,.B2
M. = p
x w
/tt/2
00
f 12 6
d0
dw 2ttw2 sin 0 4Â£B g^S(wf)
a a
ws ws
'o J
ws ws
0
r12 r6
Tf/2
d0
jo 2 . D hs, ..
dw 8tt w sin 0 Â£ g (w')
ws ws
,12
WS
(514)
(L2+w2+2Lw cos 0)^
ws
(1? +w^+2Lw cos 0)B
(515)
o
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o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
ro
U)
N>
ro
ro
to
>
tn
tn
Ov
o
'O
o>
cv
tn
4>
4^
to
ro
h1
o
o
o
o
o
O
a
i
o
1
o
1
o
1
o
1
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
O
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
O
o
o
o
o
O
o
o
o
o
o
tn
M
o
{*
h
o
o
o
o
ro
ro
ro
to
to
to
to
ro
ro
ro
ro
K*
to
to
4>
4>*
4>
tn
o
1
o
o
1
o
1
o
1
o
o
o
o
1
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
O
O
o
H*
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
O
o
o
o
o
O
o
O
ro
ro
ro
to
to
M
o
o
o
o
(1
o
o
!1
ro
ro
o
o
ro
to
4>
CTv
Crv
o>
Table 33 (Continued)
81
hard sphere fluid whose hard sphere diameter is d and is at a reduced
aw
n 3
density of r) = E p. d., then our approximation is
i=l 1 1
hs ..
8s(r) 1
hs ,, ,
g (d ) 1
ws ws
hs P ,
8 (dav> 1
'8hS P '
(437)
This form was deduced from the results of Throop and Bearman
(1965) who numerically evaluated the radial distribution function for
various hard sphere binaries using the PercusYevick equation.
We evaluated equation (433) for several R values over the
temperature range considered in this work and obtained the following
accurate correlation.
G = (6. 7202 4.954 xl03/T + 6.548 xl05/T2 1.52R 3.17R2) (438)
The other interaction contributions to the thermodynamic
properties H_^, S_^, Cp^ can be obtained from the appropriate temperature
derivatives of equation (438).
Analysis of Spreading Pressure of
a Solute Occupying a Cavity
There is the possibility of a contribution to the thermodynamic
properties of solution due to a change in the interfacial tension upon
introduction of a solute into a cavity. This section will however illus
trate that the magnitude of such an effect is insignificant.
For an ideal dilute solution, the thermodynamic and monolayer
analysis of Chapters 7 and 12 of Defay et al. (1966), respectively, lead
to the same result
109
CL V rAQ ...
= 2 = f2 which can be rewritten as
L s
~ K *2 ^or negligible water solubility
and an ideal aqueous solution.
K = ?S2/*2 (540)
Thus the Henry's constant can be calculated from the mole frac
tion of hydrocarbon in solution (McAuliffe, 1966) and the hydrocarbon
vapor pressure at the desired temperature. Table 52 lists the mole
s
fraction x2, the vapor pressure and Â£n ^ for pentane through decarte
at 298.15K. Unfortunately, this is the only temperature at which data
are available, preventing determination of thermodynamic properties other
than A Gxp. The vapor pressure was calculated using the Antoine equa
tion as described in Reid et al. (1977). Beyond decane the accuracy of
existing data appears insufficient to warrant analysis.
Table 52
Properties
Required to Analyze
Liquid Hydrocarbon Solubility
Hydrocarbon
Aqueous
Mole Fraction
Vapor
Pressure
(k Pa)
Â£n (K/P
atm'
Pentane
0.961D05
0.6840D+02
0.1116D+02
Hexane
0.199D05
0.2016D+02
0.1151D+Q2
Heptane
0.527D06
0.6080D+01
0.1164D+02
Octane
0.104D06
0.1864D+01
0.1208D+02
Nonane
0.309D07
0.5715D+00
o.mim02
Decane
0.659D08
0.1733D01
0.1247D+02
24
interface relative to the head group position (Tanford, 1972; Stigter,
1975ab) and thus whether the head group should be considered hydrated
upon micellization.
As noted in Figure 21, significant entropy and enthalpy
(relative importance is dependent on temperature as previously noted)
contributions to micellization are due to a change in the volume excluded
to the water molecules upon micelle formation. This effect can be readily
calculated using scaled particle theory, which will be discussed in
Chapter 3, both in the original form and modified for unusual structural
properties of water.
In principle, all of these effects should be considered in
development of a theory for micellization. However, such a develop
ment would lead to models containing too many parameters with exces
sively complex expressions. Empirically, it has been observed that
the entropy change upon micellization is large and positive at low tem
peratures and decreases, eventually becoming negative at higher temper
atures. The enthalpy change is positive at low temperatures, changes
sign at approximately 25C and becomes quite negative at higher temper
atures. Thus micellization switches from an entropy to an enthalpy
driven process with increasing temperature. This temperature behavior
is closely followed by both the water structure approach of Nemethy. and
Scheraga (1962ab) and the excluded volume concept of BenNaim (1971).
As noted previously this latter approach may macroscopically utilize
some of the microscopic detail of the former.
For micellization, the exluded volume which is dispersed with
monomers is coalesced when micelles are present. While it may seem
unusual for coalescence to lead to increased entropy, it should be
48
Table 32 (Continued)
r (A) 8m(r)
Temperature (C)
4
20
25
50
75
100
3.60
0.84
0.87
0.89
1.04
1.03
0.97
3.70
0.92
0.94
0.95
0.99
0.98
0.93
3.80
0.94
0.95
0.96
0.95
0.97
0.91
3.90
0.92
0.94
0.94
0.95
0.97
0.91
4.00
0.96
0.98
0.94
0.96
0.97
0.95
4.10
1.03
1.03
0.99
0.97
0.97
0.99
4.20
1.10
1.07
1.05
0.99
0.99
1.02
4.30
1.12
. 1.09
1.08
1.02
1.00
1.03
4.40
1.13
1.10
1.10
1.03
0.99
1.02
4.50
1.15
1.11
1.12
1.03
0.98
1.02
4.60
1.17
1.13
1.13
1.06
1.02
1.01
4.70
1.17
1.15
1.13
1.11
1.05
1.02
4.80
1.14
1.14
1.10
1.11
1.04
1.02
4.90
1.09
1.07
1.07
1.06
1.01
1.01
5.00
1.05
1.00
1.04
1.00
1.00
1.00
5.10
1.01
0.97
1.00
0.98
1.00
0.99
5.20
0.96
0.96
0.96
0.96
0.98
0.99
5.30
0.91
0.92
0.93
0.93
0.95
0.98
5.40
0.88
0.88
0.91
0.91
0.95
0.96
5.50
0.86
0.87
0.89
0.91
0.96
0.95
5.60
0.86
0.89
0.87
0.92
0.95
0.94
5.70
0.86
0.89
0.86
0.92
0.94
0.92
5.80
0.86
0.87
0.87
0.92
0.94
0.92
5.90
0.89
0.89
0.93
0.95
0.95
0.94
6.00
0.92
0.93
0.93
0.95
0.96
0.98
6.10
0.95
0.96
0.96
0.95
0.96
1.01
6.20
0.97
0.98
0.98
0.96
0.97
1.02
6.30
0.99
1.01
0.99
0.98
1.00
1.03
6.40
1.02
1.04
1.01
1.00
1.01
1.03
6.50
1.04
1.05
1.04
1.02
1.02
1.03
6.60
1.06
1.05
1.06
1.04
1.03
1.02
6.70
1.07
1.06
1.06
1.06
1.04
1.01
6.80
1.07
1.07
1.05
1.07
1.04
1.02
6.90
1.06
1.07
1.05
1.06
1.04
1.03
7.00
1.06
1.05
1.06
1.05
1.04
1.03
78
Contributions to the Thermodynamic Properties
of Solution from Intermolecular Forces
From perturbation theory (Reed and Gubbins, 1973) the configura
tion integral L for a mixture can be related to the intermolecular
potential by
In L = In Lref Z I p pD V
kT a=a g=a 3
P ref 2
TaB aB
(430)
ref P
where L is the reference configuration integral, (J)^ is the differ
ence between the real pair, potential and the reference state pair poten
ref
tial, and g^g is the reference mixture radial distribution function.
We have restricted the model to a firstorder perturbation theory.
A basic relation of statistical mechanics is
3 Â£n L
p = kT
Y
3N.
Y J
(431)
T> V, N,
B^Y .
With the hard sphere as the reference state, equations (430)
and (431) yield
p = p + 4ttp
Y Yhs Kw
*vs(r) Cs
(432)
where R is the distance from the center of the solute molecule to the
center of the nearest water molecule, d> (r) is the watersolute inter
hs
molecular potential and (r) is the watersolute hard sphere radial
distribution function. Since = p^
G. = 4ttp
1 w
(r) ghs(r) r2dr.
ws ws
(433)
200
Again this can be evaluated with integral tables to yield
1
d cos 0
12
7
ws
6
7
WS
2 2 6 2 2 3
(x +w +2xw cos 0) (x +w +2xw cos 0)
12
a
ws
2xw
5(x2+w2)5 5(x+w)10
+
ws
2xw
1 .
A 2 2 2
2(x+w) 2(x +w )
(B19)
Four different integrals over x must now be evaluated and
combined.
12 rL
a
ws
lOw
dx
0 x(x+w)
12
o
ws
10 lOw
10 ,T.v9 10 2, ',8 10 3 \ 2
w w(L+w) w w (L4w) w w (L+w)
9w^^(L+w)^ 8w^2(L+w)^ 7w^3(L+w)7
10 *,T.,x6 10 3, 5 10 6, .4 10 7, .3
+ w ~w (L+w) + w w (L+w) w w (L+w) + w w (L+w)
6w14(L+w)6 5w15(L+w)5 4w16(L+w)4 3w17(L+w)3
+
2w
L i
2w+L
10 w+L
2 + 1
10 10
w w
&n
W+X
(B20)
a12
ws
lOw
L
12
dx
a
ws
, 2, 2,5 lOw
x(x +w )
10 8, 2 V 10 6/T2 2,2 10 4, 2 2.3
w w (L +w ) w w (L +w ) w w (L +w )
18,2, 2, 16, 2 2,2 14, '^ 2,3
2w (L +w ) 4w (L +w ) 6w (L +w )
. w102 a2+uV 1
8w12(L2^2)4 210
Â£n
2. 2
x +w
Li
(B21)
rL
ws
4w
dx
, 2^ 2,2
0 x(x +w )
ws
8w
w
2 2
(L +w )
1 + Â£n
CM
L
2 2
x +w
o *
(B22)
a
ws
4w
dx
a
ws
o x(x+w)4 4w3
t 2
2w+L
w+L
+ W.3.~(J^.)3 2 +
3 3
3w (L+w)
*
L
n x
Jn ;
W+X
t J
o
(B23)
203
Table Bla (Continued)
Coefficient
a =
A
n
o
= 3.40 A
B
n
D
n
C16
0.257968D+00
0.924807D03
0.173800D05
C17
0.654435D01
0.560267D03
0.105100D05
n
i1
OO
0.436910D02
0.960170D04
0.178000D06
C19
0.130702D+00
0.101761D01
0.173590D04
C20
0.172373D+01
0.944589D02
0.140810D04
C21
0.800159D+00
0.451760D02
0.684800D05
C22
0.107662D+00
0.639490D03
0.979000D06
C23
0.213548D+01
0.269035D02
0.565300D05
C24
0.226508D+01
0.226790D02
0.516900D05
C25
0.460063D+01
0.626760D02
0.354270D04
C26
0.390806D+02
0.145317D+00
0.116600D03
C27
0.923215D+02
0.478891D+00
0.574200D03
C28
0.104253D+02
0.354899D01
. 0.717490D04
C29
0.416220D+00
0.406758D02
0.308600D05
o
o
0.892476D+00
0.910517D03
0.878000D06
C31
0.166259D+00
0.274624D03
0.152000D06
C32
0.221053IW02
0.719806D01
0.157688D03
C33
0.311827D+01
0. 752350D01
0.135211D03
C34
0.543894D+02
0.1013810+01
0.180780D02
0.326974DK) 3
0.264588D+01
0.447786D02
Table 53a
Contributions to Free Energy
AG
AG
Solute
T(K)
c
.1
RT
RT
CH.
4
277. 15
23.03
12
2 9 8. 1 5
22 .52
11
323.15
21.76
 10
3 5 8. 15
20. 55
9
C2H6
277.15
30.53
22
298.15
29.49
20
322.15
28.11
18
358.15
26 .07
16
C3H8
277.15
38.04
30
298.15
36.46
28
323.15
34.46
25
258. 15
31.58
"22
Solution of Gaseous Hydrocarbons
AG AG AG
r,v cal exp
RT
RT
RT
o
o
1 0. 05
10.104
+.005
0 .0
10.60
10.596
+.004
O
o
t o. ga
10.946
+ .003
o
o
11.12
11.134
+.008
l .27
9.62
9.60 1
+.008
1.26
1 0. 31
10.307
+.006
1 .24
10.79
10.807
+ .003
1.18
11.09
1 1.072
+ .010
2. 20
9. 73
9 .708
+ .009
2.18
10.52
10.521
+ .007
2. 14
11.09
11.081
+.006
2.04
11.45
11 .342
+ .010
of
99
92
79
43
18
44
55
16
5 1
12
51
1 7
111
195
306 L
ws
2 2 2
8z (L +z )
2 2 2
12(L +z )
3
8z L
1 L
tan
(B3)
Combining equations (B3) and (Bl) yields equation (59).
Consider the case of a potential fixed at y = 0 interacting with
molecular centers in the region y > L. The expression for the Helmholtz
free energy of interaction (equation 516) is
r i
. B2
M. = p
i w
d cos 0
a
dwf 8ir w^ eD g^s(w')
w
ws
2 2 3
(L +w +2Lw cos 0)
ws ws
12
7
WS
2 2 6
(L +w +2Lw cos 0)
(B4)
In this case we must evaluate
d cos 0
12
7
ws
(L^+w^+2Lw cos 0)^
0
ws
(L^4w^+2Lw cos 0)^
(B5)
This integral is of the form
1
dx 2 2
where x = cos 0, a = L + w and b = 2Lw.
.m
Jo (a+bx)
From standard tables this integrates to:
ws
lOLw
1 1
6
a
i ws
1 1
2 2 5 2 2 5
_(L 4w ) (L +w +2Lw)_
4Lw
2 2 2 2 2 2
_(L +w +2Lw) (L+v)_
(B6)
Upon rearrangement
1
r 12
6 i
12
0
0
0
d cos 0
0
ws
ws
ws
(L^+w^+2Lw cos 0) ^
(L^4w^+2Lw cos 0)^
lOLw
Table 55b
Contributions to Enthalpy and Heat Capacity of Solution of Liquid Hydrocarbons
Solute
T(K)
AH
c
ah
1
AH
r,v
45al
ACp
c
ACp?
ACp
r ,v
^cal
RT
RT
RT
RT
R
R
R
R
C5H12
27715
33. 89
51.56
0. 00
1 7.67
100.38
38.8 l
1 1.95
73.52
298.15
38 .30
50.75
0.74
l1.71
92.93
41.25
9. 50
61.18
323.15
42 .20
50. 1 1
i .38
6.53
84. 64
43. 59
8.68
4 9.73
358.15
45.84
49. 59
2.08
1.67
74.58
45.89
8.55
37.23
C6H14
277.15
40 .88
60.50
0.00
19.62
1 15.63
46.05
1 4. 19
83. 77
298.15
45 .83
59.58
0.83
12.87
1 06. 74
48.80
1 1 .28
69.22
323.15
50.15
58.85
1.64
7 .07
96.57
51.26
10.31
55.62
358.15
54 .05
58.22
2.47
 1.70
84. 0 1
53. 24
10.15
40.91
C7H16
2 77.15
47. 88
69. 95
0. 00
22.08
130.88
53.45
16.96
94.39
298. 1 5
53 .3 6
68.91
l .05
14.50
120.54
56.59
13.48
77.44
323.15
58 .09
68.07
1 .96
 8.02
108.5 1
59. 3 1
12.32
61.52
358.15
62.26
 6 7. 32
2.95
2.11
93.44
61.23
12.13
44 .33
125
276
Mukerjee, P., "Hydrophobic and Electrostatic Interactions in Ionic
Micelles. Problems in Calculating Monomer Contributions to the
Free Energy," J. Phys. Chem., 73>, 2054 (1969).
Mukerjee, P. and K. J. Mysels, Critical Micelle Concentrations of
Aqueous Surfactant Systems, Nat. Stand. Ref. Data Ser., Nat. Bur.
Stand., No. 36 (1971).
Narten, A. H., M. D. Danford and H. A. Levy, "Xray Diffraction Study
of Liquid Water in the Temperature Range 4200C," Discussions
Faraday Soc., 43, 97 (1967).
Narten, A. H. and H. A. Levy, "Liquid Water: Molecular Correlation
Functions from Xray Diffraction," J. Chem. Phys., 55, 2263 (1971).
Nemethy, G. and H. A. Scheraga, "Structure of Water and Hydrophobic
Bonding in Proteins. II. Model for the Thermodynamic Properties
of Aqueous Solutions of Hydrocarbons," J. Chem. Phys., 3j6, 3401
(1962a).
Nemethy, G. and H. A. Scheraga, "The Structure of Water and Hydrophobic
Bonding in Proteins. III. The Thermodynamic Properties of
Hydrophobic Bonds in Proteins," J. Phys. Chem., ^6, 1773 (1962b).
O'Connell, J. P., Thermodynamic and Transport Properties of Water Vapor
and Aqueous Vapor Mixtures, Ph.D. Dissertation, University of
California, Berkeley (1967).
O'Connell, J. P. and R. J. Brugman, "Some Thermodynamic Aspects of
Models of Micelles, Microemulsions and Liquid Crystals," in
Improved Oil Recovery by Surfactant and Polymer Flooring, Eds.,
D. 0. Shah and R. S. Schechter, Academic Press, New York, N.Y.
(1977).
Phillips, J. N., "The Energetics of Micelle Formation," Trans. Faraday
Soc., 51, 561 (1955).
Pierotti, R. A., "The Solubility of Gases in Liquids," J. Phys. Chem.,
67, 1840 (1963).
Pierotti, R. A., "Aqueous Solutions of Nonpolar Gases," J. Phys. Chem.,
69, 281 (1965).
Poland, D. C. and H. A. Scheraga, "Hydrophobic Bonding and Micelle
Stability," J. Phys. Chem., ^9, 2431 (1965).
Poland, D. C. and H. A. Scheraga, "Hydrophobic Bonding and Micelle
Stability; the Influence of Ionic Head Groups," J. Colloid Inter
face Sci., 21, 273 (1966).
Pratt, L. R. and D. Chandler, "Theory of the Hydrophobic Effect,"
J. Chem. Phys., 67_, 3683 (1977).
AH
ah
1
RT
AH
cal
RT
Solute
T(K)
c
RT
Kr
277.15
5.23
14.42
9. 19
296. 15
7.48
14.11
6.64
323.15
9.67
13.48
3.8 1
358.IS
12. 16
12.41
 C.23
Xe
277.15
3. 1 5
20.11
11 .96
298.15
10.82
19.56
8.75
323. 15
13. 40
18.60
5.20
358.15
16.32
17.08
0 .76
CH.
4
277.15
5.87
14.74
8.87
298.15
8.18
14.39
6.21
323.15
10.43
13.72
3 .29
358.15
13.0 1
 12.62
0.39
Table 45b
(Continued)
AH
exp
ACp
c
ACp
A^cal
ACp
rexp
RT
R
R
R
R
8.727
.099
38.19
12.51
25 .69
26.4 4
3.4
6.283
+ .035
36.30
7.86
28.45
25.60
0.9
3.840
.171
35. 43
4.23
31.21
25.03
4.0
1.031
.734
35.47
1 .13
34 .33
24 .66
7.1
10 .696
.125
4 7.20
15.49
31.72
31.4 1
4.2
7 .769
.040
44.95
9.59
35.36
30.56
1.3
4.829
.231
43.69
5.11
38.58
29.98
5.1
l .444
.926
43.09
 1.37
41.72
29.42
8.6
7.954
.30
39.64
12.21
27 .42
26.7 1
5.7
5 .549
.05
37.84
7.67
30.17
25.55
2.5
3.188
.07
36.96
4.10
32.86
24.48
1.5
0.526
.28
36. 89
1 .06
35 .34
23 .35
5.1
CO
FORTRAN
0027
0028
0 023
0030
0031
0032
0 0 33
0034
0035
0036
0037
0038
0039
004 O
0041
0042
0043
IV G LEVEL 21
CAL FU N
DATE = 79109 22/46/19
A 15=0.24062863
DO 4 K= 1 ,4
DO 11 1=1.10
DO 10 J=1.7
T(J)=TT(K)+{J4)*0.500
TC=T ( J )273.1 5
C SECTIONTO CALCULATE SOLUTE AND WATER HARD SPHERE DIAMETERS
DENW=0.033 433 0.9998 4252+16.9452270 0 3*T C 798 70641D 06*{TC**2)4
16.1706D09*(TC**3)+105.56334D12*(TC**4)280.54337Dl5*(TC**5))/(1
2.000+16.87985D03*TC)
DDENW=0.033433*(16.9452 27D0315.97 41282D06 + TC138.51l8D09*(TC**
22)+422.25336D12*( TC**3)140 2.71635D15*(TC**4> )/( 1.000+16.37985D
30 3 + T C) DE.NW* (16 .87985D03 )/ ( ( 1 000 + 1 6.37935D03 *TC ) >
TRW=T(J)/CTW
DENWP=DENW*CVW/(0 .6023)
DDENWR=DDENW*CVW/(0 .6023)
HSDW=((3.0*CVW/(2.0*PI*0.6023)) (A7/(TRW**A8)+A2/(DEXP(A4*( (DENWR+
2 A 1*7 RW)**2.0) ) )A3/ (DEXP{A5*((DENWP +Al*TRWA6)**2) ))+A9/(DEXP(Al0*
3( ( ( r R W A l 3) * 2 ) + A1 1 ( (DENWRA12)**2 )))))) **0 .333333
DHSDW=1 .00 0 0*( (3.0*CVW/(2.0*PI*0. 6023) )*(A7* A8/(CTW*(TRW**(A8 + 1 .0
2 ) ) )2.*A2*A4*(DENWR+Al*TRW ) *(DDENWR+A1/CTW )/ ( DEXP ( A 4 ( DSNWR +Al*TR
3W )*=*2.0) ) ) + 2. 0*A3*A5* ( DENWR + A 1* TP WA6) (DDENWR + A1/CT'*)/( DEXP (A5*( (
4DENW R + A1*TRWA6)**2.0)) )2.0*A9*A10*((TRWA 13)/CTW+A11 *(DENWRA 12)
5* DDE NWR) /( DEXP ( Al 0* ( ( (TRWA13 )**2 ) +A 1 1 ( ( DENWR A l 2 ) **2 ))))))/(( HSD
6W **2 .0) *3.00)
TRCH4=T { J)/CTCH4
TRAP =T(J)/CTAR
HSDC H4= ( (3.0*CVCH4/ { 2 .0 *p I *0.6023) ) ( A 7/( TRC H 4* A 8 ) + A2/( DEXP(A4*((
2DENWP+A1*TRCH4)**2))>A3/(OEXP(A5*((DENWR+A1*TRCH4A6)**2)))+A9/(D
3E XP ( A 1 0 ( ( ( TRCH4A 13 ) ** 2 ) + A l 1 ( (DENWRA1 2 ) **2 ) ) ) ) ) )**0 .333333
HSDAR= ( (3.0 *CVAR/ (2.0*P 1*0 .60 23 ) ) *( A 7/ ( TP AP. * A3 ) + 4 2/ ( DEXP ( A 4* (.(DEN
2WF + A 1*TAR)**2) ))A3/(DEXP(A5 *( (DENWR + A1=TPARA6)* *2) ) ) +A9/(DEXP{A
NJ
U)
93
mole fraction solubilized data for the liquid hydrocarbons into Henry's
constants.
The remainder of this chapter discusses results of the model of
hydrocarbon solubility and makes comparison with trends in infinite
dilution thermodynamic properties of surfactants as determined from calor
imetric data. The discussion emphasizes sensitivity to model parameters
and suggestions for future research.
Calculation of Thermodynamic Properties of Cavity
Formation for Aliphatic Hydrocarbons
The general relation (equation 425) for the Helmholtz free energy
change upon creation of a desired solute cavity from a reference cavity is
rewritten here for convenience.
A A c = y (a a r)
s ref s ref
6 (a J a ,.J ,)
s s ref ref
(a a c)
s ref
(51)
where J = 1/R^ + l/R^ with R^ being a principal radius of curvature.
For modeling purposes we chose to represent aliphatic hydrocarbons
as spherocylinders. For a spherical reference and spherocylindrical
solute
a r = 4tt R .
ref ref
J p = 2/R
ref ref
and
a = 2 7T R L
s s
= 4tt R
s
J = 1/R
s s
= 2/R
s
Combining these expressions yields
a J = 8 tt R +2ttL
s s s
a c J c
ref ref
8ir R
ref
(cylindrical portion)
(spherical caps)
(52)
and
(53)
d
> s
J> O
'JC'0>0'ff\0'0>t^O'0>0'Ui'JlUlU>UlUlUlUlUlUiÂ£'t'4>st't'4>64''WUWW
O'0(vJ0Un^UNHOv000MCJ\yi'UNIO')CviO'i'UNHO'lt''10>
oo. ooooooooooooooooooooooooooooooooo
ooooooooo
000 000000
lilil
oooooooo
oooooooo
WHNlnO'WiN)
ooooooooo
OMMMIjJUJWUJU)
P't'OvOViOP'^JCT't'O
I II I I
ooooooooo
NNHOHNWf'O'
siNOiiHNONO
P
I I I I I I II
ooooooooo
ooooooooo
CT'WWOJWOIOM
I I I I
oooooooo
oooooooo
OMP'PMMUJOO
OOOOOOOOO
HMUPUiO'O'O'
jomoohhcno
I I
OOOOOOOOO
UlUiMNNHOON
COUnOHvOM'JUiO'
M
O
OOOOOOOOO
OOOOOOOOO
MNiUPUiPHOO
I I i I I I I I
oooooooo
OOOOMOOO
HU!JivOOUiH
OOOOOOOOO
OHHNWUJWWW
Ul00P'ON54>U)O
I I I I I
ooooooooo
tOMHOHHNWLn
UiOMHHiOO"JP
I I I I I
OOOOOOOOO
OOOOOOOO!1
OMMOt'JC'OOOO
I I I I I I I I
OOOOOOOO
I1 I1 I1 I1 I1 I1 I* I1
OHNOiUMO
I
OOOOOOOOO
OOOHNWWMN
UVOWUOOU1H
I I I I I
OOOOOOOOO
HHOOOHNNW
UiOOHOiHWO
H
(t>
3
d
(D
d
U
n
C
d
ro
l
OOOOOOOOO
OOOOOOOOO
tnP'P'P'Ji'Lo t'ji'ro
I I I I I I I I
OOOOOOOO
OOOOOOOO
O O O 00 O O O o
O O !1 H1 !1 !* Ni !1 h1
NWKiPO'dOO'H
I I I I I I.
OOOOOOOOO
OOOOMMhoWOJ
dvIUOOWUOP
ooooooooo
ooooooooo
o'O'uiuisiojoiaivi
ooooooooo
OOOOOOOOO
UlNN^UNJOWt^
ooooooooo
OHNPUiUiUNO
I I I I I I
OOOOOOOO
OOOIN3U)tOP'
BJMjJUUOUIO
o
o
(_n
Table 33 (Continued)
258
FORTRAN IV G LEVEL 21
MAIN
DATE = 70108
01 /OI/23
C
C
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
OBJECTIVE CALCULATE SCME CONTRIBUTIONS TO THE THERMODYNAMICS OF
MICELLIZATION FROM THE SOLUBILITY MODEL AND CTHER CONSIDERATIONS
SURF ACT ANTS STUDIED ARE SODIUM OCTYL SULFATE.DECYL SULFATE
AND D0D5CYL SULFATE
EXPLANATION OF INPUT DATA
CVW CHARACTERISTIC VOLUME OF WATER CC/GMCL
CTW CHARACTERISTIC TEMPERATURE OF WATER K
EXC1 ,EXC2.EXC3 COEFFICIENTS IN EQUATION FOR ARGON SOLUBILITY
PROPERTIES
EPSIAR INTERACTION ENERGY PARAMETER FOR ARGON K
C VAR CHARACTERISTIC VOLUME OF ARGON CC/GMOL
CHARACTERISTIC TEMPERATURE FOR ARGON K
C TAR CHARACTERISTIC TEMPERATURE FCR ARGON K
CVCH4 CHARACTERISTIC VOLUME FOR METHANE CC/GMOL
CTCH4 CHARACTERISTIC TEMPERATURE FOR METHANE K
EXMS DIMENSIONLESS ENTROPY OF MICELLIZATION AT 298 K
EXMH DIMENSIONLESS ENTHALPY OF MI CELLIZATI ON AT 298 K
EXMG DIMENSIONLESS FREE ENERGY OF MI CELL IZAT ION AT 298 <
PMVM SURFACTANT PARTIAL MOLAR VOLUME IN MICELLE AT 298 K
XMC MOLE FRACTION OF SURFACTANT IN SOLUTION AT CMC
V STAR CHARACTERISTIC VOLUME FOR HYDROCAREON OF EQUAL CHAIN
LENGTH TO SURFACTANT FROM GMEHLING 1978 L/GMOL
C CHAIN LENGTH PARAMETER FROM GMEHLING 1978
T STAR CHARACTERISTIC TEMPERATURE FOR EQUIVALENT HYDROCARBON
FROM GMEHLING 1978 K
A NM PARAMETERS FOR PERTURBATION CONTRIBUTION TO INTERMOLECULAR
POTENTIAL FROM GMEHLING 1978
RHG SURFACTANT HEAD GROUP RADIUS A
RCl COUNTERION RADIUS A
FBC FRACTION OF COUNTERIONS BOUND TO MICELLE
CC/GMOL
P
CAL FUN
DATE
79109
22/43/30
FORTRAN IV G LEVEL
0223 12
0224
0225
21
continue
RETURN
END
222
130
Table 56
Infinite Dilution Heat Capacity of Surfactants
in Water at 298.15K
Chain
Carbon Number
1
2
3
4
5
6
7
Alkylamine Hydrobromides
Hydrocarbon Model
(Leduc et al.,
1974)
Cp/R
ACp/R
Cp/R
ACp/R
0.93
11.3
30.36
4.2
12.22
10.8
34.52
6.6
23.04
9.9
41.15
10.1
32.92
10.3
51.19
9.5
43.23
10.6
60.69
8.5
53.85
10.5
69.15
8.2
64.32
10.7
77.35
8.8
75.09
86.11
8
23
Enthalpic contributions due to properties of the amphiphile
species include repulsive interaction (AH > 0) between charged or polar
amphiphile head groups crowded at the micelle surface and attractive
(AH < 0) Van der Waals interactions between hydrocarbon chains in the
micelle interior. Theories of electrostatics such as Stigter's attempt
to model the repulsive interactions which have also been handled semi
empirically by Tanford. Unfortunately, the Van der Waals attractive
interactions are frequently considered as part of the "hydrophobic force"
(Tanford 1974a).
A significant entropy effect attributable to the amphiphile is
the change in constraints on the motion of the hydrocarbon chains in the
micelle compared to monomeric form. Considerable evidence indicates that,
at least near the polar head group, the hydrocarbon chains in micelles
exhibit a considerable degree of rigidity (Kalyanasundaram and Thomas,
1976; Roberts and Chachaty, 1973). Unfortunately, knowledge of the
configuration of the hydrocarbon chains in water is inconclusive.
A partial model of this effect is possible using a hardbody equation
of state (Gibbons, 1969; Boublik, 1975) to calculate the entropy change
associated with the change in hydrocarbon density from the monomer
solution to the micelle. More specific effects associated with rota
tional freedom of the hydrocarbon chains would best be modeled from
a lattice approach (Poland and Scheraga, 1965).
Enthalpic effects associated with the solvent include changes
in binding of the solvent to the amphiphile head groups and ions upon
micellization. A lack of knowledge concerning this effect is compounded
by considerable debate concerning the location of the micellesolvent
FORTRAN IV G LEVEL 21
0231 STOP
0232 END
MAIN
DATE = 79 1 OS
01/01/23
27Q
31
The fundamental distribution function in scaled particle theory
is PQ(r), the probability that no molecule has its center within the
spherical region of radius r centered at some fixed Rq in the system.
This function was originally introduced by Hill (1958).
Let PQ(r+dr) be the probability that the centers of all molecules
are excluded from the sphere of radius r + dr. Now the probability that
2
the spherical shell of thickness dr and volume 4irr dr contains a particle
2
center is 4irr pG(r)dr where G(r) is defined so that pG(r) measures the con
centration of molecular centers just outside the sphere. Thus G(r) mea
sures the conditional probability that the center of a molecule will be
found within the spherical shell at r when the region enclosed by the
shell is known to be empty. The probability that the spherical shell is
free of molecular centers is
1 4irr2pG(r)dr (31)
For the volume of radius r + dr to be devoid of centers, it is
necessary that the volume of radius r, and the shell of thickness dr, be
simultaneously free of centers. Thus the probability PQ(r+dr) is given by
PQ(r+dr) = Pq(r) [l47rr2pG(r)dr].
Expanding PQ(r+dr) to first order in dr yields
3P
P (r+dr) = P (r) + r^ dr + ..
o o 9r
Combining equations (32) and (33) yields
9 Â£n P 9
5 = 4irr pG(r).
dr
Upon integration
Po(r)
exp [
4'ITr^pG^r,)dr,
y o
(32)
(33)
(34)
(35)
Table 61
Comparison of Properties of Hard Spheres with Those
ofSome NonSpherical Particles
Particle
Sphere
Spherocylinder
Oblate
Ellipsoid
Prolate
Characteristic
Dimension
a.
i
R
(radius)
R, L = aR
(radius)(length)
1 + a/4
a = major axis
b = minor axis
, 2 .2. 2
Â£ = (a b )/a
1
2
(lÂ£2)
1/2
sin ^ Â£
a = major axis
b = minor axis
, 2 V2W 2
E = (a b )/a
1
2
1Â£
1/2
+
IkÂ£_I
2e
1/2
Jin
l+Â£ I
1Â£ I
b.
i
4tt
(4+2a)TT
1Â£ l+Â£
2rr < 1 + Jin
2Â£ 1Â£
2ir < 1 +
sin Â£
2.1/2
Â£(!Â£ )
c.
i
4tt/3
(4/3 +a)n
4u(lÂ£2)1/2/3
4it
3(1e2)1/2
137
197
' Lx
r 12
6 1
1 2
a
a
126(Lx)a
dq
ws
ws
ws
, 2 2. 6 _
,2 2.3
2
'x
(z +q )
(z +q )
z
512z((Lx)2+z2)
+
+
768z^ [(Lx)2+z2) 960z4((Lx)2+z2)3 1120z2((Lx)2+z2)4
+
1260((Lx)2+z?)3 512zy(Lx)
+ ^ tan"1 Sk2^L
126x a
+
12
ws
_10 8. 2 2.
512z (x +z )
+ . + , ,+ .V f 1
768z6(x2+z2)2 960z4(x2+z2)3 1120z2;(x2+z2) 4 1260(x2+z2)5
1 1 x
tan ~
512z x
3a (Lx)
ws
8z2((Lx)2+z2)2 12((Lx)2+z2)2
. 1 1 (Lx)
H r tan
8z (Lx) z
, 6
3a x
ws
1 + 1
2 2 2 2 2 2
8z (x +z ) 12(xZ+z )Z
1 lx
5 tan
3 z
(Bll)
8z x
We must now evaluate
L
dx INT, where INT denotes the integral
of equations (Bll). This can be written as
f L
dx INT =
r L
dx INT(x) +
dx INT(Lx)
(B12)
where INT(x) denotes terms in x, INT(Lx) terms in Lx.
Let s = Lx, then ds = dx and
ds INT(s).
L
L r
dx INT =
dx INT(x) 
'0 '
0 h
(B13)
273
Corrin, M. L. and D. H. Harkins, "The Effect of Salts on the Critical
Concentration for the Formation of Micelles in Colloidal Electrolytes,"
J. Am. Chem. Soc., 6j), 683 (1947).
Cotter, M. and D. E. Martire, "Statistical Mechanics of Rodlike Particles.
I. A Scaled Particle Treatment of a Fluid of Perfectly Aligned Rigid
Cylinders," J. Chem. Phys., 52,, 1902 (1970a).
Cotter, M. and D. E. Martire, "Statistical Mechanics of Rodlike Particles.
II. A Scaled Particle Investigation of the Aligned Isotropic Tran
sition in a Fluid of Rigid Spherocylinders," J. Chem. Phys., 52,
1909 (1970b).
Cotter, M. and F. H. Stillinger, "Extension of Scaled Particle Theory for
Rigid .Disks,"..J.. Chem. Phys., 57, 3356 (1972).
Debye, P., "Light Scattering in Soap Solutions," J. Phys. Chem., J53, 1
(19.49) .
Defay, R., I. Prigogine, A. Bellemans and D. H. Everett, Surface Tension
and Adsorption, John Wiley, New York, N. Y. (1966).
Desnoyers, J. E., R. De Lisi, C. Ostiguy, and G. Perron, "Thermodynamics
of Micellization," paper presented at 52nd Colloid and Surface Sci
ence Symposium, Knoxville, Tenn., June 1214, 1978.
Emerson, M. F. and A. Holtzer, "On the Ionic Strength Dependence of
Micelle Number," J. Phys. Chem., 69, 3718 (1965).
Emerson, M. F. and A. Holtzer, "On the Ionic Strength Dependence of
Micelle Numher. II," J. Phys. Chem., 71, 1898 (1967a).
Emerson, M. F. and A. Holtzer, "The Hydrophobic Bond in Micellar Systems.
Effects of Various Additives on the Stability of Micelles of Sodium
Dodecyl Sulfate and of nDodecyltrimethylammonium Bromide," J. Phys.
Chem., 71, 3320 (1967b).
Evans, D. F., K. Kale and E. L. Cussler, "Ion Surfactant Electrodes An
Unexploited Technique for Characterizing Micellar Solutions," paper
presented at 52nd Colloid and Surface Science Symposium, Knoxville,
Tenn., June 1214, 1978.
Fisher, M. E., "Correlation Functions and the Critical Region of Simple
Fluids," J. Math. Phys., J5, 944 (1964).
Frank, H. S. and M. W. Evans, "Free Volume and Entropy in Condensed
Systems. III. Entropy in Binary Liquid Mixtures; Partial Molal
Entropy in Dilute Solutions; Structure and Thermodynamics in Aqueous
Electrolytes," J. Chem. Phys., 13, 507 (1945).
Frank, H. S. and W. Wen, "III. IonSolvent Interaction. Structural
Aspects of IonSolvent Interaction in Aqueous Solutions: A Suggested
Picture of Water Structure," Discussions Faraday Soc., .24, 133 (1957).
Table 62b
Contributions to Enthalpy of Micellization
AH
AH
AH.
AH
AH .
AH
Ah
ah
wm
mcc
mic
rv
mint
ca
cal
exp
RT
RT
RT
RT
RT
RT
RT
RT
SOS
277.15
78.96
51.27
1 1 .70
0.00
19.48
0. 0
15.91
298.15
77.77
57. 80
12.13
0.01
18.05
0 .0
14.07
0.05
323. 1 5
76.82
63.46
12.63
0.05
16.54
0.0
5.50
358.15
75.94
68.65
13.25
0.24
14.70
0.0
e.cs
SDS
2 77. 15
97.95
65.26
11.62
0.00
23.90
0.0
20 .42
298.15
96 .46
72.85
11.87
0.0 1
22.17
0.0
13.32
C. 0
323.15
95 .23
79.34
12. 17
0.04
20.32
0.0
7 .77
358.15
94.05
85.06
12.55
0.20
18.08
0 .0
3 .66
SDDS
277.15
1 17 .32
79.24
11.13
0.00
27. 62
0. 0
21.64
298.15
115.52
87. 50
1 1.27
0.00
25.74
0.0
13.15
0.13
323.15
114.00
95.22
11.41
0 .00
23.70
0.0
6 .48
358.15
112 .48
101.48
1 l .59
0.04
21.20
0. 0
1.44
m
42
I'ig. 31. Contact Correlation Function; Comparison of
Different Models
FOR TRAN
0 103
0104
0105
0106
0 107
o ioa
0 109
IV G LEVEL 21 CALFUN CATE = 79108 00/49/39 o
J=K+3*(I4)
F ( J) =(EXS( I ,K)CSS { I.K ) )/ (EXS ( I ,K ) ) 100.00
F(JF 18) = (EXGI .K)CGS(I K) )/(EXG(I .K))*100.00
3 CCNTINUE
6 CONTINUE
RETURN
END
181
*
55
60
65
70
75
80
85
90
95
00
05
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
00
60
Table 35 (Continued)
, *1/3 2 ,
4tt p V r c (r)
m
Temperature (C)
4
20
25
50
75
100
1.94
0.18
2.87
2.06
0.54
0.57
0.95
1.96
1.33
2. 30
0.70
0.09
1.77
0.43
2.53
1.80
0.58
2. 15
1.62
1.10
1.68
1.62
0.55
1.07
1.51
0.37
1.61
1.42
0.49
1.63
0.54
0.75
1.06
1.70
0. 1 1
0.55
1.79
1.71
0.67
1.05
1.13
0.56
2.01
1.59
0. 50
1.51
0. 27
1. 12
0.45
1.57
0.14
0.60
0.47
0.20
0.60
0.85
0. 14
0.98
0.44
0. 11
0.55
1.47
0.55
1.39
0.07
0.64
0.04
0.75
0. 37
0.75
1.30
0.01
0.58
1.64
1.09
1.63
0.6 3
0.66
0.35
0.8 3
1. 15
1.45
0.42
0.25
1.19
1.43
1.53
0.91
1.24
1. 12
1.82
0.99
1. 28
1.34
1.37
2. 92
1.30
1.09
1.42
0.53
1.11
0.94
1.23
0.87
1. 42
0. 87
1. 33
0.45
1.78
0.78
1.04
0. 07
0.9 5
0.62
2.18
0.90
0. 91
0. 18
2. 1 1
2.00
 1.28
0.30
0,72
0.26
0.6 9
2. 25
1.32
0.61
0. 48
0.71
1.31
2. 57
0.54
0.19
0.06
2.23
2.2 2
1.85
1.06
0.47
0.11
2. 22
0.70
1.72
1.25
0. 32
0.57
0.97
0.04
1.85
1.05
0.03
0. 4 1
1. 32
0.3 2
0.27
 1.13
0.72
0.79
1.95
0.56
0.12
0.51
0.67
0. 84
2.27
0.0 3
0. 55
0.38
1.05
1.08
2.75
0.38
0.44
1.11
1.24
1. 14
0.49
0.85
0.52
96
y = O y = L
I
Fig. 5lc. Distributed Potential Along Spherocylinder Axis from y = 0
to y = L Interacting with Molecular Centers in 0 < y < L
and 0 < z < 00
Fig. 5ld. Distributed Potential Along Spherocylinder Axis from y = 0
to y = L Interacting with Molecular Centers in y < 0 and
y > L.
98
Upon substitution of z' equation (57) becomes
..A
M.
x
8tt Â£ p
ws w
r L r
dy
0
d2' 2 8^<2')
w
12
a
ws
ws
(y2+z2)6
(y2+z2)3
(58)
Analytically integrating over the variable y, equation (58)
yields
M^ = 8tt e p L
i ws w
,00
126a12
*
dz' Sws(z,)
ws
1
9
z
512(L2+z2)
+
2, 2n2
+
+
960(L2+z2)3 1120(L2+z2)4 1260(L+z)' 512zL
z8 ,1 1L
rT5+^77tan 7
3a
ws
1 + z
8(L2+z2) 12(l/"+z~)
. 1 1 L
~+ tan
2. z z
8zL
(59)
Upon numerical integration of equation (59) using a 10 point
A
Gauss Quadrature formula, was fit to a function of L with temperature
dependent coefficients.
Ma .cd
X ws
2 3 4
C L + C_L + C0L + C,L
1 / 3 4
L < 3.6 A
Anf / e 1 = C_+C,/L4T? /L^+C /L^+C./L** L > 3.6 A. (510)
v i ws' 5 6 7 8 9
The details of the analytical integration of equation (58) and
expressions for the coefficients in equation (510) can be found in
Appendix B.
16
c) by contrast, Tanford (1973) quotes the results of McAuliffe (1966)
for each carbon group changing the alkane solubility, i,n x^, in water at
25G by 1.49( 0.02). It is not surprising that the nonionics should
show some difference of carbon number effect with head groups and/or
perhaps mean aggregation number. However, it is quite surprising that
these effects do not appear for the ionics.
One explanation which can be advanced is that, except for small
differences in potential energy and in conformational entropy of the hydro
carbon tails in bulk alkanes compared to micelles of amphiphiles (and
even these should probably vary proportionally to the carbon member), the
carbon number dependence of micelles and alkane solubility should be the
same since it is caused solely by the "hydrophobic" effect on the monomers
The variation to be described is
A Â£n x
An
A(AG/N(1 + a)RT)
m
An
(251)
c c
where a = 0 for alkanes and nonionics and a = 1 (?) for ionics and
x is x+ for micelles and x^ for alkane solubility. As noted above, the
value for ionics (0.69) is slightly less than onehalf that for alkane
solubility (1.49). While it is unclear why the nonionics do not have
a value equal to twice that for the ionics, it is possible that the volume
excluded to water by nonionic micelles varies with the nature of the com
pound due to differences of penetration of water around the head groups
which causes differences in the "hydrophobic" effect.
The above observation has been considered by Tanford (1972,1974ab)
in which a theory of Tartar (1955) is expanded for micelle formation and
size distribution. Tanford separates AC^/NRT into a portion linear in the
TABLE OF CONTENTS (Continued)
APPENDIX Page
A PROGRAM FOR CORRELATION OF SPHERICAL GAS
SOLUBILITY PROPERTIES 172
B HELMHOLTZ FREE ENERGY OF INTERACTION BETWEEN
A SPHEROCYLINDRICAL SOLUTE AND SPHERICAL SOLVENT .... 184
Integrations of the Components of the Helmholtz
Free Energy of Interaction 194
Correlation of the Helmholtz Free Energy of
Interaction with O ,L and Temperature 201
s
C PROGRAMS FOR GAS AND LIQUID HYDROCARBON
SOLUBILITY PROPERTIES 209
D PROGRAM FOR CALCULATION OF THERMODYNAMIC
PROPERTIES OF MICELLIZATION 258
BIBLIOGRAPHY 272
BIOGRAPHICAL SKETCH ......... . 280
v
91
From a comparison of Tables 41 and 45 the results of the model
are considerably better than that of Pierotti. A portion of this improve
ment can be attributed to inclusion of temperature dependent hard sphere
diameters, an accurate model for the mixture radial distribution function
and more reasonable values for solutewater interaction energy parameters.
However, use of argon as a reference solute and forcing the inter
action energy parameters to provide an accurate fit of the Henry's con
stant at 298.15K gives the model a strong correlative rather than predic
tive nature. Though somewhat more predictive, Pierotti's model also has
a strong correlative element in the graphical determination of and
Several aspects of this model provide potentially fruitful future
research topics. (1) A valid microscopicmacroscopic match may be pos
sible through the use of a connecting function (possibly a spline func
tion) between Stillinger G(r) in the microscopic region and equation (339).
(2) Extension of the perturbation theory beyond the first term may lead
to improvement of the interaction calculations. Since the series tends
to alternate in sign several terms may be necessary to obtain improvement.
(3) Exploration of other functional forms for 6 (for example, density
rather than temperature dependent).may lead to improved overall temper
ature dependence and in particular better accuracy for heat capacity model
ing. (4) A smaller project would be determination of the requirements to
improve the fit of temperature derivative properties (enthalpy, entropy,
heat capacity) for methane and xenon.
159
between temperature changes to help insure that the entire cell had
reached a uniform temperature.
Temperature measurements utilized a copperconstantan thermocouple
calibrated with boiling points of lower molecular weight alcohols, water
and anisolewater mixtures. As expected a fairly linear relationship was
obtained between the temperature T and thermocouple mf, EMF:
T(C) = 0.680 + 25.0855 EMF 0.4328 EMF2 (71)
where the EMF is in mV. The thermocouple emf was measured within
3
2 x 10 mV with a Leeds and Northrup 7556 guarded potentiometer and
9834 1 null detector. Measurement of a quantity of gas introduced into
the cell was accomplished using a section of larger inside diameter high
pressure tubing of known volume (7.86 0.07 cc) between HP9 and HP10.
Pressure was measured with a 300 psi Heise temperature compensated gauge.
The carbon dioxide was Matheson commercial grade (99.5% min. purity) and
the methane was Matheson C.P. grade (99.0% min. purity).
Auxiliary equipment not shown in Figure 71 included a Gaertner
cathetometer used to magnify the view of the optical cell and determine
relative volumes for multiphase systems. A simple photoelectric cell
detection system was constructed to investigate changes in birefringence
of the sample. Polarized plastic was mounted to fit over the cell win
dows while being protected from the heating tape. A photoelectric cell
was mounted behind one of the polarizers. A Hewlett Packard 3439 A
digital voltmeter was used to measure the amplified photoelectric cell
output voltage. However, since the experimental results are mainly
qualitative this apparatus provided little improvement over direct human
observation.
APPENDIX D
PROGRAM FOR CALCULATION OF THERMODYNAMIC
PROPERTIES OF MICELLIZATION
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor Philosophy
A MOLECULAR THERMODYNAMIC MODEL FOR AQUEOUS SOLUTIONS
OF NONPOLAR COMPOUNDS AND MICELLE FORMATION
By
Robert J. Brugman
June 1979
Chairman: John P. O'Connell
Major Department: Chemical Engineering
A thermodynamic process for micellization has been developed
which provides a basis for better understanding of molecular mechanisms
important in the formation of micelles as well as other processes of
aqueous solution. The foundation of a model of the micelle formation
process is a unified molecular thermodynamic theory of aqueous solutions.
For spherical gases, a modification of scaled particle theory
has been used for the excluded volume contribution while a mean field
theory approach has been used for intermolecular interactions. Very
good correlation of experimental thermodynamic properties was achieved
over a considerable temperature range (277K 358K).
Extension of the model to aliphatic hydrocarbon solutes required
development of an expression for the total interaction energy between
a spherocylindrical solute and a spherical solvent. The radial distri
bution function was considered to be a function of distance from the
spherocylinder surface and the intermolecular potential was distributed
along the spherocylinder axis. The thermodynamic properties of solution
for gaseous hydrocarbons (C^ C^) are well correlated and predicted
xiv
7
All1
m
, 9AG/RT
1 m
NR
N
31/T
AG
m 9Â£nN
P,n NRT 81/T
P.n
(211)
AH 9Â£n x+
m o
9Â£nT
1 J9Â£nN
NRT
AS AH AG
m m m
NR
NRT
_ \9Â£nT
P,n N v.
i 9Â£n x
+
Â£n x + 3 2Â£nN
o
9Â£nT
P,n.
(212)
(213)
AS
9Â£n x
+
NR
; Â£n x 
o
3Â£n T
P.n
N
"9[T Â£n x+]
o
9T
P.n
 2
9[T Â£n N]
9T
+ 1 Â£n
+ 9Â£nN
P.n
o 9Â£nT
P,n.
(214)
The last term in equation (212) is small when the righthand side
of equation (210) is, so the standard state enthalpy change is normally
close to the temperature derivative of the CMC. As determined by Desnoyers
et al. (1978) from direct calorimetry, at lower temperatures (T < 25C)
the micellization process is entropy driven (TAS<0, AH>0), whereas at
m m
higher temperatures it is enthalpy driven (TAS >0, AH<0). This con
m m
siderable variation of AH with temperature is reflected in a large nega
m
tive heat capacity change (ACp) upon micellization. These trends agree
m
qualitatively with those determined using equations (212) and (214) by
Moroi et al. (1975) for sodium alkyl sulfates. Thus theories which
attempt to correlate the data must exhibit considerable flexibility in
their temperature dependence.
The above analysis was developed by Hall and Pethica (1970) for
nonionic species. The small systems analysis has not been applied to
ionic systems where the effect of counterions on the thermodynamics must
be included. This may be due to the extreme complexity of such an effort.
68
An alternative method of calculating discussed by Pierotti
is the KirkwoodMuller formula.
r 2 aia2
dis m c (a1/x1) + (a2/X2>
(417)
where m is the mass of an electron, c is the velocity of light and x^ and
X2 are the molecular susceptibilities of the solvent and solute.
In order to calculate G^,, Pierotti assumed the mean field
approach: the solvent is infinite in extent and uniformly distributed
according to its number density p around the solute molecule. The
number of molecules contained in a spherical shell a distance r from the
2
center of the solute molecule is then equal to 4lTp r dr where dr is the
shell thickness. Combining this with equation (414) and replacing the
summation by an integration gives
r 00
1
kT
4ttp
kT
dis + ^ind
, f4
C,. ex. 0
dxs 12
.,10
dr1
(418)
r v.v r j r
where R is the distance from the center of the solute molecule to the
center of the nearest solvent molecule.
Pierotti obtained an estimate of the distance parameter for
water, afrom a graphical extrapolation of K vs and vs to
obtain values of K and at = 0. Since, from equations (416) (418),
is proportional to
Un K = G^/RT + nRT/V^ at 02 = 0.
(419)
Thus, a can be calculated given values of K and G. Pierotti's value of
w Z
O= 2.75 A was essentially independent of temperature.
110
Results of the Model for Aqueous Solubility
of Aliphatic Hydrocarbons
With the curvature dependence parameter <$ of equation (54)
previously determined from modeling spherical gas solubility as dis
cussed in Chapter 4, parameters remaining in the hydrocarbon solubility
model were the interaction energy parameter e^s for each solute and the
spherocylinder length L which can be broken down into a CH^ group length.
Given a value of L, the energy parameter e was determined by
fitting the experimental standard Gibbs free energy of solution A G
for the desired solute at 298.15K. The remaining gas hydrocarbon solu
bility data were fitted to a segment Lnu as a quadratic function of
Cn.2
inverse temperature
L = A + B/T + C/T2. (541)
t*ri2
Table 53 presents the model results for the gaseous hydrocarbons
using the value of the curvature parameter 6 given in Chapter 4. To illus
trate the lack of sensitivity to parameter values, Appendix C presents a
parallel set of results using the <$ values of Appendix A.
In both cases the model accuracy is very good with the exceptions
of the results at 358.15K. It should however be remembered that the
least accurate experimental data occur at 358.15K and that the correla
tions for the interaction terms in the model are least accurate at this
temperature.
The energy parameter c was for the liquid hydrocarbons shorter
then undecane was determined by fitting A G at 298.15K. With the use
exp
of equation (541) (determined from the smaller hydrocarbons) for L,
Table 55b (Continued)
Solvent
T(K)
AH
c
AH?
l
AH
r,v
cal
ACp
c
ACp?
i
ACp
r ,v
A^cal
RT
RT
RT
RT
R
R
R
R
C8H18
277.15
54. 87
78. 96
0. 00
24.09
146.13
60.33
19.40
105.21
298.15
60.88
77.77
1 .20
15.69
134.35
63.31
15.43
85.97
323.15
66.03
76 .82
2.24
8.55
120.45
66. 75
1 4. 1 0
67.80
358.15
70.46
75. 54
3.38
2. 1 0
102.87
68.61
13 .88
48.14
C9H20
277.15
61 .87
38.73
0.00
26 .9 1
161.38
67.75
22. 63
116.26
298.15
68.41
87.44
1.40
 17.63
148.15
71.56
18.00
94.59
323.15
73.97
86.34
2.61
9.76
132.39
74.72
16.45
74.12
358.15
78 .67
85.32
3.94
2.71
l 12.30
76.49
16. 19
52. 0 0
C10H22
277.15
68 .86
97. 95
0. 00
29. 09
1 76.63
74.65
25.29
127.27
298.15
75.94
96.46
1.56
18.96
161.96
78.74
20.11
103.33
323.15
8 1 .9 1
95.23
2.92
 10.40
144. 33
32. 05
1 8.37
80.65
358.15
86 .83
94. 05
4.40
2.77
121.73
83.68
13.09
56.14
126
108
Analysis of Hydrocarbon. Solubility Data
Henry's constants for the gaseous hydrocarbons (methane, ethane,
propane and butane) wee obtained from the correlation of Wilhelm et al.
(1977) which was based on the experimental data of several workers.
These data were correlated with two temperature functions.
Â£n (1/K) = AQ + a^/T + AÂ£/T2 (537)
Â£n (1/K) = A + A Â£n T + A2 (Â£n T)2 (538),
which gave comparable accuracy.
As shown in Chapter 4,
A G = RT Â£n K (K in atm)
exp
or A G = RT Â£n (K/P ) (539)
exp atm
if Kis in units other than atmospheric. P m is atmosphereic pressure
in corresponding units. The other thermodynamic properties of solutions
can be obtained from appropriate temperature derivatives of equation (539)
Extreme values of the properties calculated with equations (537)
and (538) supplied a bound on the "true" experimental value. These
extreme values included using one standard deviation of the least squares
fit of equations (537) and (538). The "true" experimental values are
shown along with error limits in Table 53.
Determination of the thermodynamic properties of solution for
liquid hydrocarbons requires conversion of mole fraction solubilized
data into Henry's constant data. This can be accomplished by noting
that the liquid hydrocarbon is in equilibrium with its vapor at its
vapor pressure. At equilibrium
OR TR
0057
0058
0059
0060
006 1
0062
0063
0064
0065
0066
0 06 7
0068
0069
0070
0071
0072
0073
0074
0075
0076
0077
0078
0079
0 08 C
008 1
0082
0033
LEVEL 21 MAIN DATE = 79109 22/43/30 P
108 CONT INU ~
WPI TE (6,100 >
DO 110 1=1,3
DC 111 K=1*4
WRITE(6 ,103 ) TT(K} SC ( I K ) ,S I ( I K ) S RV ( I K ) CSS( I,K),EXS( I,K)
111 CONTINUE
110 CONTINUE
WRITE (6.100)
DD 112 14,4
DO 113 K=l,4
WRITS(6,103) TT (K).SC(I,K).SI(I.K).SRV{I,K), CSS( I,K) ,EXS(I,K)
1 13 C CNTINUE
112 CONTINUE
WRITE (6.100)
DO 1141=1,3
DO 115 K=1,4
WRIT= (6.1 03) TT(K ) CP C (I.K),CPI(I,K ), CPRV ( I K ) CCPS( I.K ) ,EXCP{ I ,K
2)
115 CONTINUE
114 CONTINUE
WRITE (6,100)
DO 1 IS 1=4,4
DC 117 K=1 ,4
WRITE (6,103) TT(K) ,CPC(I ,K) ,CPI(I ,K) ,CPRV(I ,K) ,CCPS(I ,K) ,EXCP(I.K
2 )
1 17 CONTINUE
116 CONTINUE
STO
Â£ ND
ho
I1
47
Table 32
Radial Distribution Function for Liquid Water
r(A) g (r)
m
Temperature (C)
4
20
25
50
75
100
0. 10
0.06
0.09
0.30
C.30
0.2 1
0. 1 5
0. 20
0.02
0.03
0.03
0.07
0.06
0.00
0. 30
0.02
0.06
0. 11
0. 16
0.11
0.04
0.40
0.03
0.00
0. 02
0.04
0.0 2
0.04
0.50
0.06
'<1.03
0. 06
0.02
0.0 5
0.09
0. 60
0.03
0.03
0. 06
0.16
0.03
0.0 6
0.70
0.01
0.09
0. 11
0. 22
0.06
0.04
0.80
0.05
0.05
0.02
0.16
0.01
0. 12
0.90
0.05
0.02
0. 01
0. 18
0.0 3
0. 1 8
1. 00
0.04
0.01
0.10
0. 30
0.07
0.12
1. 1C
C. 14
0.06
0.20
0. 36
0.17
0.03
1. 20
0. 14
0.06
0.16
0.29
0.19
0. 10
1.30
0.04
0.03
0.07
0.23
0 17
0.08
1. 40
 0.02
0.10
0. 07
0. 23
0.16
0.05
1.50
0.02
0.07
0. 12
0. 20
0. 17
0. 1 1
1.60
0.09
0.01
0.12
0.14
0.16
0. 19
1.70
0.14
0.04
0. 10
0. 12
0. 15
0.21
1. 80
0. 1 5
0.07
C. 12
0. 10
0. 1 1
0.15
1.90
0.15
0.09
0. 14
0.02
0.0 3
0.0 8
2. 00
0.14
0.08
0.12
0. 16
0.0 1
0.08
2. 10
0.15
0.09
0. 12
0. 15
0.06
0.14
2. 20
0. 17
0.14
0.14
0. 07
0.11
0.15
2. 30
0.13
0. 13
0. 05
0. 14
0.01
0.09
2. 40
0.04
0.03
0. 1 1
0. 29
0.12
0.06
2. 50
0.15
C. 08
0.03
0.09
0.05
0.26
2. 60
0.72
0.63
0.57
0.69
0.66
0.76
2.70
1.62
1.59
1. 51
1.66
1.4 4
1.3 7
2. 80
2.29
2.34
2.22
2.22
1.96
1.78
2.90
2.29
2.41
2. 29
2. 22
2. 07
1 .86
3. 00
1.76
1 .92
1.88
1.90
1.85
1.70
3. 10
1.22
1. 40
1.4 1
1.54
1. 52
1.49
3.20
0.97
1.10
1.13
1.22
1. 23
1. 35
3. 30
0.90
0.93
0.99
1.02
1.09
1.2 3
3.40
n.83
0.80
0. 89
0.97
1.07
1.11
3. 50
0.80
0.78
0.85
1.02
1.07
1.02
Table 45a (Continued)
Solute T(K)
AG
c
RT
AG?
i
RT
AGcal
RT
AG
exp
RT
Kr 277.15
22. 13
12.71
9. 42
9.457
+
298.15
21.66
l l .66
10.00
10 .004
+
323.15
20.97
10.55
10.42
10.410
+
358.15
19. 85
9.22
10.63
10.657
+
Xe 277.15
26.54
17.85
8.69
3.775
+
298.15
25.85
16.40
9.45
9.448
+
323. 1 5
24. 87
l 4.86
10.01
9.953
+
358.15
23.3 4
13.03
10.31
10 .272
+
CH. 277.15
4
23. 03
12.99
10. C4
1 0.1 04
+
298. 15
22.51
11.92
10.59
10 .596
+
323.15
21.76
10.79
10.98
10.946
+ .
358.15
20. 56
9.43
1 1 13
11.133
+ ,
AS
c
R
AS?
i
R
i5Li
R
AS
Â§xÂ£
R
16.90
1.71
13.61
18.18
+.10
14.19
2.45
16.64
 16* 29
+.04
11.31
2. 93
14.23
14.25
+.18
7 .67
3. 19
10.86
11.69
.79
18.39
2. 26
20.65
19.47
+ .12
15.03
3.16
18.19
17.22
+.04
1 1 .47
3. 74
15.21
 14.76
+.24
7. 0 1
4. 06
ll.07
 11.72
+.99
17.16
1.75
18.91
18.06
+.30
 14.34
2. 47
 16.8 l
16.14
+ .04
11.33
2. 94
14.27
14.14
+ .07
7 .54
3. 19
10.73
l1 .66
+ .29
001
001
004
048
001
001
006
062
005
004
003
008
162
Pertinent Calculations for Dissolved Gas Experiments
The potassium oleate microemulsion system studied had the follow
ing formulation.
1.75 g
8.8 cc
40.0 cc
16.0 cc
40.0 cc
potassium hydroxide
oleic acid
hexadecane
hexanol
water
A reaction between potassium hydroxide and oleic acid in water produced
a potassium oleate solution to which the other components were added.
On a mole fraction basis the sample composition was
0.011 potassium oleate
0.054 hexadecane
0.051 hexanol
0.884 Water
Since the pressure at which the gas sample was introduced into
the measuring volume (7.86 cc) was quite low (74.7 psia) the ideal gas
law was sufficient to determine the amount of gas added to the sample
PV 3
n = ^ = 1.63 x 10 moles.
Ri.
This corresponded to a mole fraction of added gas of about 0.0072.
The ratios of added gas to other key components was
gas/potassium oleate = 0.65
gas/hexanol = 0.14
gas/hexadecane = 0.13 .
Table 55a
Contributions to Free Energy and Entropy of Solution of Liquid Hydrocarbons
Solute
T(K)
AG
c
RT
AG
i
RT
AG
LiZ
RT
cal
RT
AS
c
R
AS?
i
R
AS
LiZ
R
cal
R
C5H12
277.15
53 .04
46.99
4.04
10.08
19.15
4. 57
4. 04
27.76
298.15
50. 40
43. 26
4. 01
1 1 15
12.09
7.49
3 .27
22.36
323. 1 5
47.15
33.20
3.93
l 1 .87
4.94
10.91
2.55
18.40
353.15
42 .61
34.08
3.75
12.23
3. 23
 15. 52
1.67
13.95
C6H14
277.15
60.54
55.01
4.80
10.33
1 9.66
5.49
4.80
29.95
298.15
57 .37
50.63
4.76
11.50
11.54
8.96
3. 89
24.38
323.15
53 .49
45.86
4. 66
12.30
3.35
 12.99
3.03
19.36
353.15
4a. 12
39.84
4.45
12.73
5 .93
18.38
1.98
14.43
C7H16
277.15
68 .04
63.46
5.73
10.32
20.17
 6. 49
5. 73
32.39
298.15
64.34
53.39
5. 69
1 1.64
10.98
10.52
4.64
26.14
323.15
5 9.84
52.88
5.57
12.54
1 .76
15.19
3.62
20.56
353.15
53 .64
45.92
5.32
13.04
8.62
2 1. 40
2.37
15.15
121
182
Two sets of watersolute interaction energy parameters ewere
investigated. The first data list which follows corresponds to the
results of Table 45, while the second corresponds to Table Al.
The two resulting functions for the curvature parameter 6 are:
Table 45:
6 = 8.3194896D+00 + 2.6052103EH03/T18.993069D+04/T2. (Al)
Table Al:
= 6.22467929EH00 + 2.1484507D+03/T 14.40244IH04/T2. (A2)
149
Discussion and Suggestion for Future Research
Table 62 presents the model results for the various contributions
to the thermodynamic process for micelle formation. Probably the best
approach to a discussion of these results is to compare the entropy and
enthalpy change for each step as hypothesized in Chapter 2 with the calcu
lations of this chapter.
Step 1 involves removal of surfactant monomers and counterions
from their cavities in aqueous solution. The present model makes the
uncertain assumption that the counterions undergo no change in interac
tion with water upon micellization mainly because no method is readily
available to model such a change. Thus we assume the ion hydration
effects of steps lc and 5c cancel.
Thus, step 1 was modeled by calculating properties associated
with elimination of monomerwater interaction. No model was hypothesized
for changes in waterhead group interactions upon micellization because,
again little knowledge seems to exist concerning such changes. (Host
investigators including the author tend to exclude the head group from
the micelle proper, but some local water structure rearrangement around
the headgroup may occur.) The sign of the calculated enthalpy change is
negative, while the calculated entropy change is positive. Originally
O'/Connell and Brugman (1977) had predicted that the entropy effect of the
monomer cavity was the driving force for micellization. However, the
calculations show that, as with paraffin solubility, the major entropy
effect arises from the interactions between monomers and water. Here,
as with hydrocarbons, the entropy contribution from cavity formation
became less negative as chain length increased. This is partly from the
FORTRAN IV
oo i a
0019
0020
0021
0 022
0023
002 4
0025
0026
0 02 7
0028
0 02 9
0030
003 1
0032
0033
0034
003 5
0036
0037
0038
0039
0040
004 1
G LEVEL 21 MAIN CATE =79108 01/01/23 PA
A 13=0.76631363
A 14=0 .809657804
A 15=0.24062863
C VW= 46 4
CTW=438.7
C READ EXPERIMENTAL MI CELL IZATIQN PROPERTY VALUES
READ (5.1) ((EXMS(I).EXMH(I).EXMG(I))1=1.3)
1 FORMAT (8F10.4)
C READ MICELLE PARTIAL MOLAR VOLUME AND XCMC AT 298K
READ (5,2) ((PMVM( I ).XCMC( I )) .1=1.3)
2 FORMAT (6E12.4)
C READ PARAMETERS FOR ROTATIONAL AND VIBRATIONAL EFFECTS
READ (5,3) ( (VSTAR(I), C( I )), 1=1,3)
3 FORMAT (6F1C.4)
C READ PARAMETERS FOR MONOMERMONOMER INTERACTION IN MICELLE
READ (5,7) (TSTAR(I),1=1,3)
7 FORMAT (3F1C.4)
DO 10 N=1,2
READ (5,9) (ANM(N.M),M=1,5)
9 FORMAT (5F10.5)
10 CONTINUE
C READ HEAD GROUP RADIUS.COUNTERI ON RADIUS AND FRACTION OF 80UND COUNTERIONS
READ (5,8) RHG RCI FBC
8 FORM AT (3F1 0 .4 )
C READ CARBON NUMBER OF SURFACTANT AND AGGREGATION NUMBER OF MICELLE
READ(5.6) ( (NC( I),NAGN( I)),1=1,3)
6 FORMAT (614)
C READ METHANE AND ARGON CHARACTERISTIC VOLUMES AND TEMPERATURES
READ (5,14) CVCH4,CTCH4.CVAR,CTAR
14 FORMAT (4F10.4)
C READ ARGON ENERGY PARAMETER AND COEFFICIENTS FOR ARGON SOLUBILITY
READ (5,15) EPSIAR.EXCl.EXC2.EXC3
260
163
A calculation of the approximate bubble point pressure was
required to insure that sufficient system pressure was maintained to
keep all the added gas in solution.
Empirically it is well established that for a sparingly soluble
gas the solubility is proportional to its vaporphase fugacity (partial
pressure)
f2 = p2 = K2,l x2 (7~ }
provided the gas pressure is not too high. Since we could, at best,
estimate the Henry's constant K for carbon dioxide or methane in this
^ 9 L
microemulsion system using the value for water, this expression was
probably sufficient for a firstorder estimate. Furthermore, since the
Henry's constant for carbon dioxide or methane in the larger alkanes is
less than that for water (de Ligny and van der Veen, 1972), this calcula
tion should provide an upper bound on the pressure requirement. The
results are shown below.
K 1 (Wilhelm et al., 1976) P
(psi) (psi)
C02 24,055 174
CH4 586,354 4230
Both these pressures are easily within the capability of the experi
mental apparatus.
Table C2b
Contributions to Enthalpy and Heat Capacity of Solution of Liquid Hydrocarbons
Solute
T(K)
AH
c
AH?
X
AH
r,v
AH
cal
ACp
c
ACp?
ACp
r, v
A^cal
RT
RT
RT
RT
R
R
R
R
C5H12
277.15
28 .23
46.03
0.00
17.80
98. 06
34.89
11.95
75.11
298.15
32.79
45.32
0.74
11.79
88.29
36.95
9.50
60.84
323.15
36 .70
44.75
1.38
6.67
78.67
38.97
8. 68
48. 38
35e .15
40 .26
44.30
2. 08
 1.95
68. 1 5
41.05
3.55
35.68
C6H14
277.15
33.27
53.08
0.00
19.81
112.06
40.67
14.19
85.53
298.15
38 .40
52.29
0.88
13.01
100. 58
42. 98
11.28
68. 89
323.15
42.75
51.65
1.64
7.26
89.00
45.05
10.31
54 .26
358.15
46.61
51.10
2 .47
2.02
76. 12
46.76
10.15
39.5 1
C7H16
277.15
38.31
60.62
0.00
22.31
126.06
46. 6 1
16.96
96.41
298.15
44.02
59. 73
1.05
14.67
112.87
49.25
13.48
77.10
323.15
48 .80
59.01
l .96
8.25
99.34
51.56
12.32
60. i C
358.15
52 .96
58.38
2. 95
2.47
84. 04
53.23
12.13
42.94
252
61
of a combination of two distinct correlation functions to model the
results of Figure 32.
The most remarkable fact about these results is that even though
the integrands do not scale, the integrals do over wide ranges of
conditions.
FORTRAN
000 I
0002
0003
0004
0005
0006
0007
oooa
0009
00 1 o
00 l 1
00 l 2
00 L 3
001 4
0015
00 1 6
00 1 7
00 1 8
00 19
0020
002 1
0022
0023
0024
0025
IV G LEVEL 21 CAL FUN DATE = 79 109 22/43/30
SUBROUTINE C ALFUN ( M N F ,X )
IMPLICIT FE AL*8( AH *0Z )
DIMZNSION C(4) ,T(7), TT(4),DCH4(4),SRV(7,4 >,HRV( 7,4),GRV(7,4) ,UL(7)
2 *'AC (7 *4) S AC(7,4) H AC (7,4)X(3)GI(74)SI(74)HI(74) ,UUL (4) ,
3HHSDW 4),HH(7),HI I{ 7> SSI (4,4) HH I ( 4,4) ,GC(4 ,4) .HC(4,4} SC(4 4) ,
4GGI(4.4).CPC(4.4)*CPI(4,4),CCPS(4,4),CGS(4,4).CHS(4,4),F(18),HARV(
54,4) ,GAFV( 4,4) SAR V( 4 ,4 ) CP AR V(4 ,4) DHHH ( 7 ) HHH ( 7) ,TDCH4 (4 ) OHH( 7 )
6,DHII(7 ),CSS(4 4),EXG(4,4),EXS( 4,4) ,EXH(4,4) ,EXCP(4.4) ,XTC(99) ,
75PSI (4)
C CM MON/A/EXS.EXH,E XG,EXCP
COMMON/B/GC,GGI,HC.HHI,SC.SSI,CPC.CP I.HHSDW.DCH4.UUL
C CMMCN/C/CSS.CGS.CHS,CCPS,TT
C CMMON/0/GAFV.HAPV,SARV,CPARV
CCMMON/E/XTC.EPS I,C,CVW,CTW.EXC1,EXC2,EXC3,CVCH4,Â£PSIA9,CTCH4
CC.MMCN/F/C VAP ,CTAF
P 1=3 I 4 l 59265D0
XK = l ,38066
P K= 1,93 7
C PARAMETER IN EXPRESSIONS FOR HARD SPHERE DIAMETER
A 1 = 0.54006832
A 2= 1 .2669302
A3=0 .05132355
A4=2.9107424
A5= 2 .5167259
A6=2.1595955
A 7=0 .64269552
AS=0.17565885
A 9=0.1 3874624
A 10= 17.952388
Aii=0.48197123
A12 = 0. 76696099
A 13=0.76631363
212
277
Prigogine, I., The Molecular Theory of Solutions, North Holland Publish
ing Co., Amsterdam (1957).
Reed, R. M. and K. E. Gubbins, Applied Statistical Mechanics, McGraw
Hill, New York, N. Y. (1973).
Reid, R. C., J. M. Prausnitz and T. K. Sherwood, The Properties of
Gases and Liquids, McGrawHill, New York, N. Y. (1977).
Reiss, H., "Scaled Particle Methods in the Statistical Thermodynamics
of Fluids," Advan. Chem. Phys., j?, 1 (1965).
Reiss, H., "Scaled Particle Theory of Hard Sphere Fluids to 1976," in
Statistical Mechanics and Statistical Methods in Theory and
Application, Ed., U. Landman, Plenum Press, New York, N. Y. (1977).
Reiss, H. and R. V. Casberg, "Radial Distribution Function for Hard
Spheres from Scaled Particle Theory, and an Improved Equation of
State," J. Chem. Phys., 61, 1107 (1974).
Reiss, H., H. L. Frisch and J. L. Lebowitz, "Statistical Mechanics of
Rigid Spheres," J. Chem. Phys., .31, 369 (1959).
Rigby, N., J. P. O'Connell and J. M. Prausnitz, "Intermolecular Forces
in Aqueous Vapor Mixtures," I & EC Fund., j3, 460 (1969).
13
Roberts, R. T. and C. Chachaty, C Relaxation Measurements of Molec
ular Motion in Micellar Solutions," Chem. Phys. Lett., 22, 348
(1973).
Sexsmith, F. H. and H. J. White, "The Absorption of Cationic Surfactants
by Cellulosic Materials. I. The Uptake of Cation and Anion by
a Variety of Substrates," J. Colloid Scil, JL4, 598 (1959a).
Sexsmith, F. H. and H. J. White, "The Absorption of Cationic Surfactants
by Cellulosic Materials. III. A Theoretical Model for the Absorp
tion Process and a Discussion of Maxima in Absorption Isotherms for
Surfactants," J. Colloid Sci., 14, 630 (1959b).
Shinoda, K., "The Effect of Chain Length, Salts and Alcohols on the
Critical Micelle Concentration," Bull, Chem. Soc. Japan, 26,
101 (1953).
Shinoda, K. and E. Hutchinson, "PseudoPhase Separation Model for Thermo
dynamic Calculations on Micellar Solutions," J. Phys. Chem.,66,
577 (1962).
Shinoda, K. and T. Soda, "Partial Molal Volumes of Surface Active Agents
in Micellar, Singly Dispersed, and Hydrated Solid States," J. Phys.
Chem., 67, 2072 (1963).
Shoor, S. K., R. D. Walker and K. E. Gubbins, "Salting Out of Nonpolar
Gases in Aqueous Potassium Hydroxide Solutions," J. Phys. Chem.,
73, 312 (1969).
FORTRAN IV G LEVEL 21
CALFUN
DATE
79 I 09
22/43/30
0 193
0 194
0195
0196
01 97
0198
0199
0200
020 1
0202
0203
0204
0205
0206
0207
0208
0209
0210
021 1
0212
021 3
021 4
021 5
0216
0217
0218
021 9
0220
0 22 1
0222
15 CC'NtinuE
H=C.5000
MN=7
CALL DPET5{ H ,HH ,DHH ,MN, IER)
CALL DDET5( 6,HI I.DHII.MN, IER)
CALL DDET51H.HHH.DHHH.MN,IER)
C SECTION FOR CALCULATING PROPERTIES AT DESIRED TEMPERATURES
SSI { I K ) =S 1(4* I )
HHICI,K)=HI(4,I)
GC{T,K)=WAC(4,I)
HCCI ,K)=HAC(4,I)
SC(I ,K) = SAC<4.I}
GGI ( I, K ) = GI (4, I 1
SAPV(IK)SRV(4,I)
GARV{I,K)=GRV(4,1)
HARV ( I ,K)=HRV(4, I )
CPC(I.K)=DHH(4)
CPI { I,K ) = DH II( 4 )
C PAR VC I *K) = DHHH(4)
C SECTION FOR CALCULA TI NG TOTAL PROPERTIES
CCPS
CGS(I ,K)=GC {I K)+GGI( I,K)+GARV( I,K)
C HS( I K)=HC(I,K1+HHI(I.K)+HARV I,K)
CSS I I,K)=SC(I,K)+SS I ( I,K)+SARV( I.K )
11 CONTINUE
4 CONTINUE
DO 12 1=2,4
DO 3 K=1,3
J=6 ( 12 ) +K
F(J)=(E XG(I K)CGS(I.K))/(EXG{I,K))*100.00
F (J +3)= (EXS ( I ,K )CSS( I. K) )/( EXS< I K) >* 100. 00
3 C ONT I NU E
FORTRAN
0059
0060
006 1
0062
0063
0064
0065
0066
0067
0068
0069
0070
007 1
0072
0073
0074
0075
0076
0077
0 078
IV G LEVEL 21 CALFUN DATE = 79108 00/49/39
HIAR = GIAR+S IAR
C SECTION TO CALCULATE INTERACTION PROPERTIES FOR OTHER SOLUTES
G K J,I )=EPSI{I )*< 6.720 24.954D + 03/C T J))+6.548D + 05/(T{J)**2)0.760
2D+00 *HSDG(I )0.7925D + 00*(HSDG1 I >**2) )/{T{J ) }
SI {J ,1 )=EPSI (I)*(4.954D+03/(T{J)**2)l3.096D+05/(T(J)**3)0.76D+0
20*DHSDG(I)1 535D + 0 0 THSDG(I)*DHSDG(I))
H I ( J I ) = G I ( J I ) +S I ( J I ).
C SECTION TO CALCULATE EXPERIMENTAL PROPERTIES FOR ARGON
GEX AR {EX Cl+EXC2*DL0G(T{J ) )+EXC3*(DLOGtT{J) )**2) )
HEXA R=E XC2+2.0*EXC3*DL0G(T(J))
SEXAR=(HEXARGEXAR)
C CAVITY PROPERTIES OBTAINED BY DIFFERENCE
WAC ( J.3)=GE XARGIA P
HAC(Ji3) = HÂ£XAR HIAR
SAC( J.3 > =SEXARSI AR
I F( I .EQ.3) GO TO 12
C SECTION TO CALCULATE CAVITY PROPERTIES FOR OTHER SOLUTES
DEL=X Cl ) + l .OD+03*X(2)/T(J)X{3)*l.00+04/{T(J )T*2 )
DDEL= l .0D+03*X(2)/(T( J)**2) +2. 0*1.00+ 04*X (3 ) /( T ( J ) **3 >
W AC( J. I ) = (PI*RK/XK J *{STW* ( ( DCWG( I ) * 2 ) ( DC WG {3)* *2) ) 4. 0* ( S TV*DEL*
2(DCWG(I)DCWG(3))))/(RK*T
S AC(J I )=(PI/XK)*{STW*{2.0+DCWGl I)*ODCWG{ I)2. 0*0CWG< 3)*D0CWG<3 > )
2 + DST fc* ( DCWG I ) **2 ) { DCVi G (3 ) **2 ) )4.0*ST W *DEL *( DDCXG( I )DDCWG( 3) )
34.0*DSTW*DEL*{0CWG( I )DCWG( 3) )4.0*ST**DDEL*(OCwG(I )DCWG{3) ))
4 + SAC(J,3)
HAC{J,I)=WAC{J,I)+SAC(J,I)
C SECTION TO OBTAIN HARD SPHERE DIAMETERS AND CAVITY AREAS AT INPUT
C DATA TEMPERATURES
12 IF(J.EQ.4) GO TO 35
GO TO 10
35 HHSDM K)=HSDW
HHSDG I K)=HSDG(I )
Table Cld
Contributions to Heat Capacity of Solution of Gaseous Hydrocarbons
Solute
T(K)
ACp
c
ACp?
ACp
r,v
ASpcal
ACp
exp
R
R
R
R
R
CH4
277.15
42.05
 13 .98
0.0
28. 07
26.709
+5.7
298. 15
39.14
8.78
0.0
30 .36
25 .55 1
2.5
323.15
37 .32
4.69
0.0
32.63
24.484
+1.5
358.15
36.47
1.21
0. 0
35.26
23.845
+5.1
C2H6
277. 15
56.05
13.18
3.77
41.64
38 .777
+8.3
293.15
51 .43
 19.50
2.99
34. 92
37.091
+3.6
323.15
47.66
20.73
2.74
29.61
35 .496
+2 .i
35a. 1 5
44 .40
22 .07
2.69
25.03
34.580
+7.4
C3H8
277.15
70 .05
26.25
6.50
50.30
47. 70 0 +1Q.2
298. 15
53.71
27.43
5.17
4 1.46
45.591
+4.4
323.15
58 .00
28.77
4.72
33.95
43.598
+2.6
358.15
52.33
30.58
4.65
26.41
42.416
+9.0
232
FORTRAN IV G LEVEL 21
MAIN
DATE
79i oa
00/49/39
0064
WRITS (6,24)
0065
24
FORMAT (i'.///////////)
0066
DO 25 L=7 9
0067
DO 26 K=1,4
0068
WRITE (6.20) TT(K),HC ,CPI(
2L.K) .CCPS(L.K) .EXCP(L.K)
0069
26
CCNTINUS
0070
25
CONTINUE
007 1
STOP
0072
END
175
100
Since d cos 9 = sin 9 d8, equation (515) can be written
MB2
i
= P.
w
d cos 9
'o
a
w
, o 2 D hs,
dw' 8tt w e g (w )
ws ws
12
7.
ws
(L2+w2+ 2Lw cos 9)^
ws
(L2+w2+2Lw cos 9)3
(516)
Analytically integrating over the variable cos 9, equation (516)
becomes
D
m 8n p Â£
mB2 it_ws
i L
ws
J0
w dw' gBs(w')
ws
w
,12
ws
12
7
WS
10(L2+w2)5
+
ws
lOL+w)10 4(Lfv)4
2 2 2
4(L +w )
(517)
where w = w + a a
w ws
B2
Upon numerical integration of equation (517), was fit to
a function of L with temperature dependent coefficients.
,B2 ,._D ^ 3
Â£n( M. /Â£ 1 = (T.+C, Â£n L + C1c(Â£n L) + (V (Â£n L)'
v i ws' 13 14 15 16
+ C1?(2n L)4 + Clg(Â£n L)5.
(518)
The details of the analytical integration of equation (516)
and expressions for the coefficients in equation (518) can be found in
Appendix B.
FORTRAN Â£ V G LEVEL
21
MAIN
79 1 oa
00/49/39
DATE =
0009
00 1 O
00 1 1
0012
00 1 3
00 1 4
00 1 Â£
00 l 6
001 7
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
003 1
0032
0032
0034
003 5
0036
NN=9
READ (5,1) ((EPSI (I ).CV(I) ,CT{ I ) ), I = t,9)
1 FORMAT (3F10.4)
READ (5,2) CVW.CTW, EXCl ,EXC2, EXC3
2 FORMAT (5E14.6)
00 3 K=1,4
READ (5 ,4 ) TT(K)
4 FORMAT (FIO.4)
DC 5 1=1,NN
READ (5,6) EXS(I K) ,EXH(I K),EX G(IK)* EX CP(I K)
FORMAT (4F10.4)
5 CCNT I NU c
3 CONTINUE
C INITIAL ESTIMATES OF PARAMETERS TO BE DETERMINED
X(l )=8.06
X ( 2) =2.53
X(3) = 19.16
C VA05AD IS A NONLINEAR FITTING ROUTINE FROM HARWELL
CALL VA05AD(M,N,F,X,1.0D03,l0.0,1.00,100,10,W)
CALL CALFUN(M,N,F,X)
WRITE (6,10)
10 FORMAT Cl*,///////////)
DC 7 L=i ,3
DO 3 K=l 4
WRITE (6.9) TT(K),GC(L,K),GGI(L,K) ,CGS(L,K),EXG(L,K) ,SC(L ,K) SI
2K),CSSL,K) ,EXS(L.K)
9 FORMAT (/,10X,F6.2.3X.F5.2.2X,F6.2,2X,F5.2,2XF6 .3,8X,F6.2,2X,F5.2
2.22X.F6.2))
8 CONTINUE
7 CONTINUE
WRITE (6,11)
11 FORMAT Cl',///////////)
FORTRAN IV G LEVEL 21
MAIN
DATE
79 l OS
01/01/23
0140
0 14 1
0 142
0143
0 144
0 145
0146
0147
0148
0149
0150
0151
0152
0 153
0 154
0 155
0 156
0 157
0 158
0159
0160
0 16 1
0162
20 *GC 3*D LL/( UL J ) **3 ) + SC4/ ( LL( J) **3 ) 3.0*GC4* CUL/ (UL(J)**4) +SC5/(UL
3< J)**4)4.0*GC5*DUL/( UL ( J )**5) )
C CCNTINUCUS DISTRIBUTION
C OUTSIDE REGION Y LT O Y GT.L
GCl=XT C{67)+XT C(68)+T(J)+XTC(69)*T(J)**2
GC2=XTC(70) + XTC (71 ) *T( J)+XTC(72 )*T{ J )**2
GC3=XTC(73)+XTC(74)*T{J)+XTC(75)*T( J)**2
GC4=XTC(76)+XTC(77)*T(J)+XTC(78}*T(J)**2
GCGLLI = ( GCi/UL J)+GC2/(UL(J)**2)+GC3/ (UL{ J)* ) +GC4/
SCl=XTC(68>+2.O*XTCC69)*T(J)
SC2 = XTC(71 ) +2.0*XTC(72) *T { J)
SC3= XTC(74)+2O+XTC(75)*T(J)
SC4=XTC( 77 1+2.0 XTC(78) +T(J )
SCGLLI = (SCI /UL ( J)GC1 *DUL/(UL (J )**2.) +S C2/ (UL ( J ) **2 >2 O*GC2*DUL/( U
2L(J)**3)+SC3/(UL(J)**3)3.0*DUL*GC3/{UL(J)+*4) + SC4/(UL(J)**4)4.O *
2DUL*GC4/(UL(J)**5) )
C DISCRETE DISTRIBUTION
C OUTSIDE REGION Y LT.O Y GT.L
GC1=XTC(82)+XTC(83)*T(J)+XTC(84)*T{J )**2
GC2= XTC ( 85)+XTC(36) *T( J)+XTC ( 37) *T ( J )T*2
GC3=XTC (88) +XTC (89 ) *T ( J >+XTC( 90 )*T( J )**2
GC4=XTC(91)+XTC(92)*T(J )+XTC(93)*T(J)**2
GC5= XT C(94)+XTC(95)*T(J) + XTC(96)*T(J>**2
GC6=XTC (97 ) +XTC (98 ) *T ( J )+XTC(99 )*T ( J )**2
G EE=DE XP(GC1+GC2TDLQG(UL(J) )+GC3*DLCG(UL(J) )**2+ GC4*DLOG(UL(J) >**
23 +GC5*DL0G(UL(J))**4 + GC6 +DLOG(UL(J ) )**S)
GDE=GEE+0.50*(XTC(79)+XTC(Q0)*T(J)+XTC(81) *T{J)**2)
SC1=XTC(83)+2.0+XTC(84)*T(J)
S C2 =XT C(86) +2.0 *XT C(37)*T(J)
SC3=XTC(89)+2.OTXTC(90)*T(J)
SC4=XTC(92)+2.0 4XTC( 93)+T( J)
SC5=XTC(95)+20+XTC(96)*T(J)
Table 53b
Contributions to Enthalpy of Solution of Gaseous Hydrocarbons
Solute
T(K)
AH
c
RT
AH
. X
RT
AH
rv
RT
AH
cal
RT
AH
exp
RT
CH4
277.15
5 .91
14.74
0.0
8.84
7.954
+.30
298.15
8.20
 14.39
0. 0
6. 19
5.549
+ .05
323.15
1 0 .44
13.72
0 .0
3.28
3.188
+ .07
358.15
13.02
 12.62
0.0
0. 40
0.526
+.28
C2H6
277. 15
12.90
24.03
0.00
11.13
11.441
+ .43
298.15
15.72
23.65
0 .23
7.69
7.952
+.07
323.15
13.38
23.36
0.43
4. 55
4.53 5
+.10
3 5 8. 1 5
21 .22
23.17
0.66
1 .29
0 .683
+ .41
C3H8
277.15
19.90
32.81
0.00
12.91
 13.234
+ .53
298.15
23.25
32.55
0. 40
8.90
9.052
+ .08
323.15
26.32
32.41
0.75
5.33
4 .909
+ .13
358.15
29.43
32.42
1.13
 1. e6
0.248
+.50
113
18 APRIL 1979
DATA LIST FOR SPHERICAL GAS SOLUBILITY
c
14.1369
3. 1880
10.9457
24.4841
c
14.8364
2.0323
12.8013
45.4404
c
l5.3800
2.6157
12.7637
61 .3689
c
17.6095
 5. 49 89
1 2. 1 1 42
63.4675
c
358.1500
c
9. 61 75
2.0736
1 1 .6912
13 .3166
c
10.010l
1.6784
11.6891
17. 1263
c
11.1676
0.1375
11 .0307
22.5264
c
1 l. 6910
1.03 1 0
10.6574
24.6553
c
11.7212
 1.4444
10.2723
29.4162
c
11.6608
0.5255
11.1334
23.8450
c
10.2063
2. 5628
12.7693
44.68 04
c
9.1344
3.5724
12.7081
60.2365
c
1 l. 1 877
1.1515
12.3357
62 .3251
<781
39
Table 31
Surface Tension and Curvature Parameter Calculated
for Liquid Water at Its Saturated Vapor Pressure
Using the Pierotti Approximation
T
^v(expt.)
(K)
(dyne/cm)
277.15
75.07
298.15
72.01
323.15
67.93
348.15
63.49
373.15
58.78
473.15
37.81
573.15
14.39
YjEq.(324)]
(dyne/cm)
6[Eq.(32
0
A
51.44
0.5026
54.97
0.5022
58.35
0.5010
60.96
0.4992
62.86
0.4970
63.82
0.4845
52.18
0.4648
Fig. 71. High Pressure Experimental Apparatus
158
165
Table 72
Pressure Dependence of Two Phase Region
Temperature = 67.5C
Pressure (psia)
440
1440
2000, 3500, 7000
Anisotropic Fraction
of Visible Sample
0.33
0.30
0.30
Table 73
k
Effect of Dissolved Methane
P = 7000 psi
Temperature Anisotropic Fraction
of Visible Sample
75C
68C
64C
61C
Q.25
0.30
0.50
0.90
Mole fraction of added methane was 7.2 x 10 as noted
in previous section.
FORTRAN IV
0116
01 1 7
o i a
o i g
O 120
0121
0 122
0 122
0124
0125
0126
0127
0128
0129
0 130
0131
0132
0133
0134
0135
0136
0137
O 138
0139
G LEVEL 21 MAIN OATE = 79108 01/01/23 P
C CONTINUOUS DISTRIBUTION SECTION
C O
GC1= XTC(1>+ XTC(2)*T(J)+XTC<3)*T(J)**2
GC2=XTCC4J+XTCC5)*T
G C3=XT C (7 )+ XT C( 8 ) *T(J)+XTC(9)*T(J) * 2
GC4=XTC( 10) +XTC (11 )*T
GC5=XTC(13)+XTC{ 14)*T(J)+XTC( 15)*T( J)**2
GCOLLI=DEXF(GC1+GC2/UL{J)*GC3/(UL {J)**2)+GC4/(UL(J)**3)+GC5/< UL(J
2)*4 ) )
SC1 = XTC(2)+2.0*XTC( 3>*T( J )
SC2=XTC(5)+2.0*XTC(6)*T{J)
SC3=XTC( 8)*2.0*XTC( 9)*T< J)
S C4 =XT C { 11 )+2 O *XT C ( i 2 ) *T ( J )
SC5=XTCC14)+2,0+XTC(15)*T(J)
SC0LLI=GCOLLI*(SC1+SC2/UL{J)GC2*DUL/(UL(J)**2)+SC3/{UL(J)**2)2.
2 0*GC3*DLL/(UL(J)**3)+SC4/(UL(J)**3)3.0 + GC4*DUL/(UL{J ) **4 )+ SC5/(UL
3(J )**4)4.0*GC5*DUL/(UL(J)**5))
C DISCRETE DISTRIBUTION SECTION
C 0< Y
GC1= XTC(16)+XTC (17)*T(J)+XTC( 13)*T(J)* *2
GC2 =XT C(19)*XTC(20)*T C J)FXTC(21 )*T(J > **2
GC3= XTC(22)+XTC(23)*T< J)FXTC(24)*T(J)**2
GC4= XT C{25)+XTC(26) *T{ J)+XTC(27)*T(J)**2
GC5=XTC(28)+XTC(29)*T{J)+XTC30)*T{J)**2
GD OLLI = DE XP(G C1+GC2/UL(J)+GC3/UL( J)**2)+ GC4/(UL(J)**3)+GC5/(ULC J
2)**4) )
SCI=XTC(17)+2.0*XTC(18)*TJ)
SC2= XTC(20)+2.0 *XTC(21)*T< J)
SC3=XTC(23)+2.0*XTC(24)*T(j)
SC4= XTC(26)+2 O *XTC(27)*T{J)
S C5= XT C{29)+2O *XTC{30)*T(J)
SDOLL I=GDOLLI*(SC1+SC2/UL(J)GC2*DUL/(UL(J)**2 ) +SC3/(UL(J))2.
265
Table Ala (Continued)
Solute
T(K)
AG
c
RT
AG
i
RT
AG
cal
RT
AG
exp
RT
AS
c
R
AS
i
R
Alcal
R
AS
exp
R
CF4
277.15
30 .33
18.43
11.91
11.891
+ 003
19.25
2. 22
21.57
21.99
+ .29
298.15
29. 4 1
16.93
12.48
12 .478
+ .002.
15 .35
3.13
18.48
18.52
+ .06
323.15
28.16
15.36
12.81
12.80 I
+.005
11 .32
3.64
14.96
14.84
+ .30
358.15
26 .27
13.49
12. 79
12.769
+ .083
6.42
3. 90
10.32
10.21
+1.38
SF6
277. 15
35. 94
24.36
11.38
11.548
+. 008
2 l .3 0
2.89
24.19
25.08
+ .30
298.15
34.73
22.40
12.33
12.330
+.010
16 .49
3.78
20.2 7
20.38
+ .08
323.15
33.12
20 .35
12. 77
12.764
+ .005
11.51
4.34
15.85
15.38
+ .36
358. 15
30. 71
17.92
12.79
12.708
+. 101
5.53
4.62
10.15
9.13
+1.74
nC,.
277.15
44 .66
34.25
10.41
10.400
+.087
23.68
3. 92
2 7. 6 0
27.66
+1.94
298.15
42. 96
31.53
11.44
l1.441
+.010
17.56
4.92
22.48
22.75
+ .58
323.15
40.7 5
28 .68
12.07
12.114
+ .015
1 1 .26
5. 33
16.78
 17.61
+; .22
353.15
37.49
25 .32
12. 17
12.336
+.045
3.7 1
5. 84
9.55
11.19
+1.30
FORTRAN
0179
0 180
0181
0182
0183
0 184
0185
0186
0187
O 188
0189
O 190
0 19 1
0192
0192
01 94
0 195
O 196
019 7
0198
0199
0200
0201
0202
0203
IV G LEVEL 21 MAIN OATE = 79108 01/01/23 PA
DO 18 N=l,2
GSUMGSUM+M*ANM{N M)/ { { TB AR **N } *( VB ARM* *M ) }
SSUM=SSUM+M*ANM(N,M)/( ((TBAR**{N + 1 ) ) ( V B ARM *M ) )*TSTAR( I } )+ANN< N.M
2)*M*DVBARM/{(TBAR** N)* C VBARM**(M + l ) } )
18 CONTINUE
17 CONTINUE
WMMI ( I, J) = C { I )*GSUM
SMMI(I,J)=WMMI(I,J)+T{J)*C(I)*SSUM
HMMI{I.J)=WMMI{I,J)+SMMI{I,J)
C SECTION TO CALCULATE COUNTER1QN ADSORPTION PROPERTIES
RH0CM=2.00*C,75/( (185.0NAGN(I ) )*2.718 )
RHOCS= < i .OFBC) *RHOS( I )
WCA
SCA ( I J ) = WCA ( I J )
HCA( I, J )=O.OOD+O0
C SECTION TO CALCULATE TOTAL PROPERTIES
WT{ I J )=WMON{ I J) + WMI C( I J) + WCI (J) tWRV I J)+WCA{ I J) +
2WMW( I, Jl+WMVK I,J )
HT( I J) =HMCM I J) +HMI C I J )+HC I (J ) 4HRV ( I J )+FCA { I, J ) +
2HMW( I,J l+HMMI (I ,J)
STC I J) =SMCM I J) +SMIC I. J J+SCI (J ) +SRV { I. J )+SCA ( I J ) +
2SMW(I,JJ+SMMI{I.J)
11 CONTINUE
4 CONTINUE
WRITE (6,100
100 FORMAT { 1 ',///////////>
DO 1 01 1=1 ,3
DO 102 J=1,4
IF(J.EQ.2) GO TO 117
WRITE (6,104) T(J) ,WMW(I,J) ,WMCCI,J) ,WMIC(I,J),WRV ( I J ) ,WMMI (I,J )
2,WCA(I,J),WT(I,J)
104 FORMAT {/,1 OX,F7.2,3X ,F6.2.2X,F7.2,2X5(F6.2,2X) )
268
xml record header identifier oai:www.uflib.ufl.edu.ufdc:UF0008953900001datestamp 20090209setSpec [UFDC_OAI_SET]metadata oai_dc:dc xmlns:oai_dc http:www.openarchives.orgOAI2.0oai_dc xmlns:dc http:purl.orgdcelements1.1 xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.openarchives.orgOAI2.0oai_dc.xsd dc:title A molecular thermodynamic model for aqueous solutions of nonpolar compounds and micelle formationdc:creator Brugman, Robert Jamesdc:publisher Robert James Brugmandc:date 1979dc:type Bookdc:identifier http://www.uflib.ufl.edu/ufdc/?b=UF00089539&v=0000105581592 (oclc)000087919 (alephbibnum)dc:source University of Floridadc:language English
Table C3
Energy Parameter Values and Length Function
Hydrocarbon
e /k
ws
Methane
266.2
Ethane
403.4
Propane
543.2
Butane
686.2
Pentane
824.2
Hexane
960.3
Heptane
1105.9
Octane
1242.8
Nonane
1395.1
Decane
1535.8
Undecane
1693.3
Dodecane
1835.2
Tridecane
1998.3
Tetradecane
2141.7
Lou = 0.49403381 + 3.07754241H02/T
 2.3198407D+04/T with T in K.
32
where the initial condition P^iO) = 1 has been applied (a cavity of zero
radius is always empty).
An important relationship can be derived between PQ(r) or G(r)
and the work of cavity formation W(r). This relation shall be derived
in the canonical (T,V,N) ensemble. The probability density of finding
N
a specific configuration R = R^,...,R^ is given by Hill (1956) and
BenNaim (1974) as
P(RN)
exp f g u(RN)]
 exp [ 3 U(RN)]dRN
(36)
1 N
where 3 = (kT) and U(R ) is the interaction energy among the N particles
N
at.the configuration R Thus, the probability of finding an empty spher
ical region of radius r, centered at Rq may be obtained from equation (36)
by integrating over all the region Vv(r) where v(r) denotes the spher
ical region of radius r.
Po(r) =
Vv(r)
P(RN)dRN
(37)
The following relation exists between the Helmholtz free energy,
of a system and the corresponding partition function in the canonical
ensemble
exp [BU(RN)]dRN (38)
3
where A is the momentum partition function, and no internal degrees of
freedom are ascribed to the particles.
Similarly the free energy of a system with a cavity of radius r
at R is given by
o
exp [(3A(T, V,N) ] =
. 3N
exp H3A(T,V,N;r)] =
N!A
3N
Vv(r)
exp [6U(RN)]dRN. (39)
55
Table 34 (Continued)
r(A) c (r)
m
Temperature (C)
4
20
25
50
75
100
4. 60
0.36
0.62
0.33
0 .25
0.16
0.13
4,63
0.36
0.63
0. 34
0.2 7
o. i a
0.14
4.66
0.37
0 .64
0.34
0.29
0.19
0.14
4. 69
0.37
0.64
0. 34
0.30
0.20
0.15
4.72
0.37
0.64
0.34
0.3!
0. 2 1
0. 1 5
4.75
0.36
0.63
0.33
0.31
0 .21
0.15
4.78
0.35
0.62
0. 33
0. 30
0. 20
0.15
4.81
0.34
0.60
0.32
0.29
0.19
0. 15
4. 84
0.33
0.58
0.31
0.28
0.18
0.15
4.87
0.3 1
0.54
0.30
0. 25
0. 1 7
0.14
4.90
0.3 0
0.5 1
0.29
0.23
0.16
0.14
4.93
0.28
0. 47
0. 27
0.21
0.15
0.13
4.96
0 .27
0.44
0. 26
0. 1 8
0. 1 4
0. 1 3
4. 99
0.25
0.4 1
0.24
0.16
0.14
0.13
5.02
0.23
0.38
0. 23
0.14
0.14
0. 1 2
5. 05
0.22
0.36
0.21
0.12
0.13
0.12
5. 08
0.20
0.34
0. 19
0.10
0.13
0.11
5.11
0.18
0.32
0. 17
CO
o
o
0. 1 2
0. 1 1
5. 14
0.16
0.30
0.15
0.07
0.1 l
0.11
5. 17
0. 1 4
0. 28
0. 13
0. 05
0.09
0.10
5.20
0.12
0 .26
0.11
0.03
0.08
0. 1 0
5. 23
0.10
0.24
0.09
0.0 l
0.06
0.09
5. 26
0.07
0.21
0. 07
0. 0 1
0. 04
0.08
5.29
0.05
0.18
0.05
0.04
0.03
0.07
5.32
0. 03
0. 15
0. 04
0. 06
0.02
0.06
5.35
0.0 1
0.12
0. 02
0.07
0.0 1
0.05
5.38
0.01
0.09
0.00
0.09
0.00
0 .04
5.41
0.02
0. 07
0. 01
0. 1 0
0. 01
0.03
5.44
0.03
0.05
0.03
0.11
0.0 1
0. 02
5. 47
0.04
0.04
0.04
0.12
0.0 1
0.01
17
carbon number, nc, a portion which depends upon the area of the hydro
carbon core in the micelle, A^, plus a portion dependent only on the area
per head group A Tanford's empirical expression for an ideal solution is
1 i ag
 Â£n x + Un N + Â£n x1 =
N m N 1 NRT
 [krk2nc + k3V + i/A <252>
where the constants are positive, the <5_^ are constants and there may be
as many as three different terms in the 6^. sum. The first group
of terms on the righthand side is the same as (p/N ]J) in equation (18)
ml
g1 gI
while the summation is apparently (p /N p, ). No distinction is made
ml
by Tanford between enthalpy and entropy contributions. However, the second
term is likely to be enthalpy dominated, whereas the first is mixed imply
ing certain temperature variations of the k^ and 6^. Tanford identified
^RM t^iat f an ellipsoid whose minor axis is that of the flexible
hydrocarbon chain length plus 3A. (An extensive discussion of the role of
geometric constraints in micellization can be found in Israelachvili,
Mitchell and Ninham (1976)..) Tanford's values for k^ and k^ are apparently
derived empirically for micelles since they are not the same as those for
alkane solubility.
A preferable approach is the concept of BenNaim (1971) Tenne and
BenNaim (1977) and Pratt and Chandler (1977), who indicate that the
"hydrophobic" effect arises from aggregation of the volume (cavity) occu
pied by the hydrocarbon which is excluded from occupancy by the water.
As calculated using scaled particle theory (the application of scaled
particle theory and subsequent modifications to aqueous solutions is dis
cussed in considerable detail in Chapter 3), this effect is essentially
FORTRAN IV G LEVEL 21
MAIN
DATE
79 109
22/43/30
N3
O
VO
000 l
0002
C OBJECTIVE TO MODEL SOLUBILITY PROPERTIES OF GASEOUS HYDROCARBONS
C EXPLANATION OF INPUT DATA
C FDF EXPLANATION CONSULT LIQUID HYDROCARBONS PROGRAM LIST
C
IMPLICIT REAL*a(AH0Z)
DIMENSION S XS( 4,4) ,EXH(4,4) ,EXG(4,4) ,EXCP(4,4),GC(4,4),GGI(4,4),
2HC(4 ,4 ) HI(4.4),SC(44).SI(44) CP C ( 4,4),CPI (4,4) ,HSD W{4) ,
3AREAC(4,4) CSS(4,4) ,CGS(4,4) .CHS(4,4 ), CCP3 (4 4),TT(4 ) .F{ 13), X(3),
4W(500),H SD G(4,4),CPR V(4,4) ,GRV( 4,4) ,HRV(4,4) ,SRV(4,4) ,HSDCH4(4),
5UL4 ) T C { 9 9 ) ,EPS 1(4),C(4)
0003
C OMMON/A/EXS,EXH,E XG,EXCP
000 4
COMMON/ B/GC GGI HC HI, SC, SI .CPC ,C=>I HSDW HSDCH4 UL
0005
CCMMON/C/CSS,CGS,CHS,CCPS,TT
0006
C CMM OM/D/GR V HRV.SPV.CPRV
0007
C CMMON/E/TC.EPSI,C,CVW,CTW,EXCl,EXC2.EXC3,CVCH4,EPSIAP ,CTCH4
0008
CCMMON/F/CVAR,CTAP
0009
READ (5,1) (T C(K) ,K= 1,99)
00 l 0
1
FORMAT (6E13.6)
00 1 1
READ (5,2) ICPSI(L),L = l ,4)
00 12
2
FORMAT (4F10.4)
0013
READ (5,3) { C{ I ) ,1=1 ,4)
0014
3
FORMAT (4F10.4)
001 5
READ (5,4) CVW,CTW, CVCH4, EPS IAR,CTCH4,CVAR,CTAR
0016
4
FORMAT (7F1 0.4)
00 17
READ (5,7) EXC1,EXC2,EXC3
00 18
7
FORMAT (3E15.6)
0019
DO 17 K=l,4
002 0
READ (5,6) TT(K)
002 l
6
FORMAT (FI 0.4)
0022
DO 9 1= 1,4
0023
READ (5.10) EXS(I,K),EX H(IK),EXG(I,K).EXCP( I,K)
0024
10
FORMAT (4F10.4)
Table 45a
Contributions to Free Energy and Entropy of Solution
Solute
T(K)
AG
c
RT
AG?
i
RT
AGcal
RT
AG
exp
RT
AS
c
R
AS?
i
R
AS ,
cal
R
AS
exp
R
He
277. 15
12.65
0.36
11.79
1 1.802
+.001
13.17
0. 13
13.30
 13.20
4*.
298.15
12.64
0 .78
11.86
11 .863
+.001
11.37
0. 2 1
12.08
12.15
+ .
323.15
12. 52
0.71
11.82
11.841
+.002
10.51
0.26
10.7/
11 .02
+ .
358.15
12.24
0.61
1 1 .63
1 1 .691
+.005
3.70
0.29
8. 99
9.62
+ .
Ne
277.15
14.70
3.15
11.55
11.543
+.001
 14.02
0. 48
14.50
14.59
298.15
14. 59
2. 88
11.71
11.710
+.001
12.42
0.72
13.15
13 25
+ .
323.15
14.36
2 .60
11 .76
l 1 .773
+.002
10.73
0. 89
11.61
11.81
+ .
358.15
13.89
2.25
11.64
11.689
+.035
3.51
^0. 98
9.49
10.01
+.
Ar
277.15
19.34
9. 13
10.16
l0.159
+ .003
15.78
1.3 1
 17.05
17.13
4,
298.15
19.0 1
8.42
10.59
10.588
+.002
 13.52
1.89
15.41
 15.42
4,
323.15
18.49
7.60
10. 88
10.883
+.009
1 1.09
2.27
13.37
13.53
4*.
358. 15
17.61
6 .63
10.99
1 1 .031
+.068
8.0 0
2.49
10.49
11.17
4.
08
02
09
41
11
03
13
57
08
30
.04
.96
FORTRAN IV G LEVEL 21
CALFUN
DATE
79109
22/43/30
0 100
0101
0102
0 103
0104
0105
0106
0 10 7
0108
0109
0 110
01 1 1
01 l 2
0 113
0 114
0 115
0116
0 117
one
0119
0120
012 1
0 122
GC1= XTC{ 16)+XTC(17)?T{J)+XTC{ 13)*T(J ) *2
GC2 =XT C(19)+XTC(20)*T(J)+XTC(21 )*T( J )**2
GC3=XTC( 22)+XTC(23)*T( J ) <XTC ( 24 ) T ( J)**2
GC4=XTC(25)+XTC(26)*T(J)FXTC(27)*T{J)**2
GC5=XTC(23)+XTC(29)*T( J)+XTC(30 )*T(J )**2
GD0LLI=DEXP(GC 1+GC2/UL ( J ) +GC 3/ ( L ( J )**2 ) 4GC4/( UL ( J)**3 ) +GC5/ (ULl J
2)**4>)
SCI =XTC( 17 M2.0 *XTC ( 18 ) *T ( J)
SC2=XTC( 20)+2.0*XTC(21)*T( J)
S C3XTC ( 23 ) +2 0 *XTC ( 24 ) *T ( J )
SC4 = XTC(26)+2 0*XTC(27)*T(J)
SC5=XTC(29)+2.0*XTC(30)*T{ J)
SDOLLI=GDOLL I (SCI +SC2/UL ( J )GC2+DUL/ ( UL ( J ) **2) +SC3/ UL ( J ) **2)2.
20*GC 3*D UL/( UL( J )*3) + sC 4/( UL( J) **3) 3.0*GC4*DLL/(UL( J 1**4 > *S C5/ (UL
2 ( J ) *4 ) 4.0 *GC5*DUL/ ( UL ( J ) *'*5 ) )
C CONTINUOUS DISTRIBUTION
C OUTSIDF REGION Y LT. O Y GT. UL
GCl=X''C (67 ) +XTC (68) *T ( J MX TC ( 69 ) *T ( J ) *2
GC2=XTC(70)+XTC71 )*T( J)+XTC(72)*T{ j)k*2
GC3=XTC(73)+XTC(74)*T(J)+XTC(75)*(J)**2
GC4=XTC(76)+XTC(77)*T(J)+XTC(76)*T(J )**2
GCGL L I = ( GC 1 /UL { J) +GC2/( UL( J)**2) <CC3/ (UL(J)'**3) +GC4/ (UL ( J ) **4 ) )
SC1=XTC (68 ) +2.0 *XTC{ 69) *T J )
SC 2 = XTC(71 )+20 *X TC{72)(J)
SC3=XTC(74)+2.0*XTC(75)T(J)
SC4=XTC (77 > +2.0*XTC{78 ) t ( J )
SCGLLI=(SCI/UL(J)GC1*OUL/(UL(J)**2)+SC2/(UL(J)**2)2.0YGC2*DUL/(U
2L(J)**3)+SC3/(UL(J ) **3)3.0*DUL *GC 3/(UL(J)**4)+SC4/(UL(J)**4>4.0*
3DUL* GC4 / { UL ( J > **5 ) )
C DISCRETE DISTRIBUTION
C OUTSIDE REGION
GC1 =XTC (82)+XTC(33)*T(J)+XTC(84)*T (J ) *2
217
160
Operating Procedures
Introduction of mercury into the pressure intensifier was
accomplished by forcing mercury from the reservoir with compressed air
until the level in the reservoir sightglass remained constant. The inten
sifier was then used to force some mercury back toward the reservoir,
sweeping any trapped air from the intensifier region. The portion of the
system bounded by HP4, HP6, HP8 and HP10 was evacuated to about 0.4 mm Hg
pressure through HP5 to remove any air which might be inadvertently intro
duced into the sample. Mercury was then displaced from the intensifier
into this evacuated region.
The optical cell was disconnected from the system at HP10.
A small quantity of mercury (~ 12 cc) was injected into the cell prior
to the sample to provide both a dense material to facilitate mixing as
well as a material, other than the sample, which could be displaced from
the cell. The sample was introduced with a syringe in sufficient quan
tity to insure that very little air space could exist in the cell.
The cell was reattached at HP10 and pressurized with the
intensifier. Pressure relaxation was minor after a pressure change was
accomplished, indicating that very little air was trapped in the cell.
After desired temperature and pressure studies were made on the sample,
the cell was sealed from the system by closing HP10.
To begin the procedure for introducing a gas sample into the
cell, the pressure was released with the intensifier and mercury to the
right of HP4 drained into either the mercury reservoir or into a container
which could be attached at LP2 in place of the vacuum line. Considerable
104
Upon numerical integration of euqation (526), was fit to
a function of L with temperature dependent coefficients.
M1 el(C28 + C29 L + C30 + Si L3)
L < 3.60 A
= eÂ£g(C /L + C33/L2 + C34/L3 + C35/L4) L > 3.60 A. (527)
The details of the analytical integration of equation (525) and
expressions for the coefficients in equation (527) can be found in
Appendix B.
Changes in Rotational and Vibrational Degrees of Freedom
of Aliphatic Hydrocarbons Upon Solution
The canonical partition function Q is related to the Helmholtz
free energy A by the simple relation
A = k T in Q. (528)
For a pure fluid, the generalized van der Waals partition func
tion is
Q =
r iN
V '
N!
A'
( )N ( )N ( )N
4repulsiony '^attraction ',"r,v'/
(529)
where
A =
h
(2tt m kT)
and
Van der Waals suggested that
= Zf
^repulsion V
q = exp
attraction
zi.
2kT
(530)
where the free volume is the volume available to the center of mass of
a single molecule as it moves about the system holding the position of all
18 APRIL 1979
DATA LIST FOR MICELLE PROGRAM
C
C
c
c
c
c
c
c
c
c
c
c
13.QQQQ
Q.OOQQ
10.6QQ 0.05QQ 10.6500
15.3700
1.8470D+02 2.3600D03 2.1920D+02 5.9900D04 2.5310D+02
13.QQQQ
0.0853 2.3022 0. 1023
350.5600 365.5400 385.8700
7.04677 7.22636 3.16538
3.56999 11.35209 10.85375
2.5200 1.0100 0.7500
8 27 10 41 12 64
96.0300 190.6000 74.9000
0l89016D+02 0.488330D+03
277. 1500 298. 1500 323. I 500
2.6969
14.34352
3. 61310
0.1237
1 .26227
7. 34334
2 .9004
150.7000
0.1703060 + 03
358.1500
0 14521 8D + 02
C2893.0066 3624.0368 4371.0368
15.50QQ 0.13QQ
1.4600004
ro
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
// / P
o y.
jhn P. O'Connell, Chairman
Professor of Chemical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Robert D. Walker
Professor of Chemical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Jopd 6. Biery
Professor of Chemical/fengineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of .
Doctor of Philosophy.
orge x. O'
George x. Ohoda
Professor of Materials Science
and Engineering
138
C. CarnahanStarling Equation of State for Mixtures
of ArbitraryShaped Rigid Bodies
The CarnahanStarling equation is
cs p AB
BP
(1"Y) + (1Y)2
2 2 12
 B C g B C
+ r + 1 ;
(68)
(1Y) (1Y)
Using equations (62) and (63) the corresponding thermodynam
ically consistent relation for the chemical potential of a component of
the mixture is
By^s = In
p h
(2Trmi kT)
3/2
V. p V. AB B2C V. ~ B2C V. a. R.B
+ .24 + _i^+9 + i1 1
UV (1Y)2
(1Y)'
(1Y)
(1Y)
2 1 2 122212 2 2 3
b. RTA 4 b. R7BC 4 af R7B 4 a7 RTB 4 Y)
 in (1Y) + 4=4r + 1 + 9 1 1
(1Y)
(1Y)
(1Y)
(1Y)'
4. (1YA 9 bj RiBC r(1 2 Y) Â£n (1Y)
Y J Y 1 (1Y)2 Y
i V. b2c
9 i
_ + (12Y) 2 Jn (1Y)
2 %3 Y
(69)
'(1Y) (1Y)'
Equations (65), (67) and (69) were obtained by trial and error
on several expressions given the corresponding expressions for hard sphere
mixtures (Reed and Gubbins, 1973), the equation of state and the constraints
of equations (62) and (63). Equation (65) is the same as that derived
by Boublik (1975).
A rigid body equation of state such as those discussed above can
be used to calculate the entropy change upon compression of the dispersed
monomers to the micelle density. This accounts only for the effect of
increased density and does not include intermolecular forces between the
chains in the micelle interior nor chain conformation change. With the
assumption that no enthalpy change is involved in this process
a
261
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u.
6W**2.0) *3.00
0055 TRCH4=T{JJ/CTCH4
Table 35 (Continued)
. *1/3 2 .
4tt p V r c (r)
m
Temperature (C)
4 
20
25
50
75
100
1.80
4.02
0.27
1.85
2.72
1.82
4.88
1.85
4.64
2.18
3.13
0.56
2.76
3.26
1.90
4.16
1.79
1.68
0.69
2.94
3.65
1.95
3.31
4.33
1.73
0.20
3.40
4.37
2.00
2.56
4.65
1.18
0.24
3.42
4.60
2.05
1.50
5.81
0.25
0.20
2.80
3.80
2.10
1.25
' 7.11
0.60
0.35
2.54
1.69
2.15
0.77
6.25
0.96
1.66
0.60
0.59
2.20
2.10
3.63
0.53
3.29
1.65
0.23
2.25
1.88
2.84
2.78
3.78
2.10
1.67
2.30
1.99
2.65
3.06
5.48
2.11
1.12
2.35
2.53
0.18
3.40
6.91
2.98
0.69
2.40
1.41
2.29
3.04
7.77
2.98
0.54
2.45
1.94
0.31
2.72
8.07
1.89
1.25
2.50
0.15
0.00
2.30
6.74
1.87
0.57
2.55
0.10
0.56
2.03
6.51
1.26
0.34
2.60
1.37
0.76
2.05
5.29
0.22
0.53
2.65
1.66
0.13
1.10
3.84
0.33
3.62
2.70
1.77
1.64
1.66
2.94
0.91
2.45
2.75
2.93
1.86
1.32
3.75
0.63
2.62
2.80
1.00
2.63
1.54
2.94
1.14
. 1.88
2.85
1.87
2.49
1.55
1.17
0.95
1.53
2.90
1.96
2.41
2.36
1.10
0.95
2.22
2.95
1.73
1.32
1.38
0.55
1.28
CO
iH
1
3.00
2.43
2.74
1.62
0.08
1.27
2.73
3.05
2.25
1.52
0.55
1.40
0.88
0.74
3.10
1.11
1.08
0.85
2.37
0.53
0.01
3.15
0.85
1.29
1.56
2.83
1.35
0.65
3.20
0.82
1.26
1.40
2.99
1.70
0.07
3.25
1.68
1.12
1.20
2.91
2.09
0.10
3.30
2.05
0.02
0.26
3.68
1.04
0.06
3.35
0.63
0.12
0.28
1.63
0.77
0.49
3.40
] .12
0.62
1.77
1.36
0.48
2.64
3.45
0.18
0.65
1.75
3.16
0.24
1.48
3.50
1.48
0.58
2.83
1.61
0.45
1.33
gas constant
separation between molecules
entropy
temperature
characteristic temperature
interaction energy
volume
characteristic volume
work of cavity formation
mole fraction
reduced solvent density
Greek Letters
a = fraction of counterions bound to micelle
= solvent coefficient of thermal expansion
(*2 = solute polarizability
3 = 1/KT
y = surface tension
CO
Y = planar surface tension
T. = relative adsorption of i to water
i,w
S = curvature dependence parameter for surface tension
A = denotes a property change
Â£ = interaction energy parameter
y = chemical potential
p = number density
a = potential distance parameter, hard sphere diameter
xi
79
We have chosen to approximate 4> (r) by a LennardJones form
ws
with a.. = R.
1J
(r) = 4e '
ws ws
R
lrJ
12
V6
(434)
where Â£^g includes both dispersion and induction interaction. An approx
imate expression for Â£^g from Ribgy et al. (1969)
Â£' = e
ws ws
1 +
2
a M
s w
4e a3 (a + 2a )3I
WS ws ws
(435)
utilizes StockmayerKihara potential parameters E and 0 along with
ws ws
solute polarizabilities and the dipole moment of water u .
w
Table 44 lists the values of E^g calculated from equation (435)
compared to those required to obtain an exact fit of the standard free
energy of solution AG at 298.15K. The two values are in close agree
ment in most cases.
In order to evaluate the integrals in equation (433), an
approximation must be formulated for the radial distribution function of
a hard sphere mixture. We have chosen the formulation of Mathias (1978).
hs
He chose to approximate g_^.(r) in terms of the value at contact
and an equivalent pure hard sphere distribution function. An average hard
sphere diameter can be defined as
n ., n
. ,a+l. r .a
d = h x, d, / E x. d. .
av i=l 1 1 i=l 1 1
(436)
We used a = 3; equivalent to a volume fraction average.
hs.
If g..(d,.) is the contact value of the radial distribution function of
ij ij
hs P
the mixture and g (d ) is the radial distribution function of a pure
av
Table C2b (Continued)
Solute
T(K)
AH
c
AH
i
AH
r, v
i5cal
ACp
c
ACp
i
ACp
r ,v
RT
RT
RT
RT
R
R
R
R
C8H18
277.15
43.35
67.69
0.00
24.34
140.06
52. 03
19.40
107.44
298.15
49.63
66. 69
1.20
15.87
125.16
54.95
15.43
85 .64
323.15
54.86
65.89
2 .24
8.79
109.68
57.46
14.10
66.32
3 58.15
59 .31
65.16
3. 38
2.47
91.97
59. 1 1
13.88
46.74
C9H20
277.15
48.39
75. 58
0. 00
27.18
154.06
58.01
22.63
1 18.69
298.15
55.24
74.46
1.40
17.32
137.45
61.22
1 8.00
94. 23
323.15
60 .91
73. 54
2. 6 1
 10.02
120.01
63.91
16.45
72.55
358.15
65.66
72.70
3.94
3.10
99.90
65.53
16.19
50.56
C10H22
277.15
53 .44
32.83
0.00
29.40
168.07
63.48
25.29
129.87
298.15
60.85
81.60
1.56
 19. 18
149.73
66.91
20.11
102.93
323. 1 5
66.96
80.58
2.92
10.70
130.35
69.74
18.37
78.9 8
358.15
72 .0 1
79.62
4.40
3.21
107.83
71. 27
16.09
54.64
BIBLIOGRAPHY
Aniansson, E. A. G., S. N. Wall, M. Almgren, H. Hoffmann, I. Keilmann,
W. Ulbricht, R. Zana, J. Lang, and C. Tondre, "Theory of the Kinetics
of Micellar Equilibria and Quantitative Interpretation of Chemical
Relaxation Studies of Micellar Solutions of Ionic Surfactants,"
J. Phys. Chem., 80, 905 (1976).
Ashton, J. T., R. A. Dawe, K. W. Miller, E. B. Smith, and B. J. Stickings,
"The Solubility of Certain Gaseous Fluorine Compounds in Water,"
J. Chem. Soc. (A), 1793 (1968).
Baker, E. G., "Origin and Migration of Oil," Science, 129, 871 (1959).
BenNaim, A., "Statistical Mechanical Study of Hydrophobic Interaction.
I. Interaction Between Two Identical Nonpolar Solute Particles,"
J. Chem. Phys., 54, 1387 (1971).
BenNaim, A., Water and Aqueous Solutions. Introduction to a Molecular
Theory, Plenum Press, New York, N.Y. (1974).
Benson, B. B. and D. Krause, "Empirical Laws for Dilute Aqueous Solutions
of Nonpolar Gases," J. Chem. Phys., 64, 689 (1976).
Beret, S. and J. M. Prausnitz, "Perturbed HardChain Theory: An Equation
of State for Fluids Containing Small or Large Molecules," AICHE J.,
21, 1123 (1975).
Bienkowski, P. R. and K. C. Chao, "Molecular Hard Cores of Normal Fluids,"
J. Chem. Phys., 63, 4217 (1975).
Boublik, T., "Statistical Thermodynamics of Convex Molecule Fluids,"
Mol. Phys., 27, 1415 (1974).
Boublik, T., "Hard Convex Body Equation of State," J. Chem. Phys., 63,
4084 (1975).
Buff, F. P., "The Spherical Interface. I. Thermodynamics," J. Chem. Phys.,
19, 1591 (1951).
Carnahan, N. F. and N. E. Starling, "Equation of State for Nonattracting
Rigid Spheres," J. Chem. Phys., _51, 635 (1969).
Carnahan, N. F. and K. E. Starling, "Intermolecular Repulsions and the
Equation of State for Fluids," AICHE J., IB, 1184 (1972).
Chung, H. W. and I. J. Heilwell, "A Statistical Treatment of Micellar Solu
tions," J. Phys. Chem., ]_h, 488 (1970).
272
Table 53c (Continued)
Solute
T(K)
As
c
R
AS?
i
R
AS
r,v
R
AS
cal
R
AS
exp
R
C4H10
277. 15
18.65
3. 56
3.05
25.26
24.635
+.53
298.15
12.65
5 .98
2.47
21.10
21.183
+.08
32315
6.54
8.88
 1.92
17.34
 17.559
+.13
358. 15
0.54
15.21
1 .26
15.93
13.100
+ .52
116
18 APRIL 1979
DATA LIST FDR SPHERICAL GAS SOLUBILITY
143.2744
50.0000
39.0000
c
175. 0504
6 C. 4800
45.2000
c
245.2719
74 .9000
150.8000
c
273.6779
88.4600
209.4000
c
300.4175
114.4600
289.7000
c
266.2096
96.0300
190.6000
c
261.7464
147. 0000
227.6000
c
282.5186
203.9400
318.7000
c
305.62 08
312.0900
433 .8000
c
0. 464000D+02 0.438700D + 03 i
c
277.1500
c
13.2008
1.3993
11.8020
c
14.5949
3. 0504
ll.5432
c
17.1263
6.96 88
10.1588
c
18.1832
8. 72 74
9.4568
c
19.4716
 10. 6963
8.7749
c
18.0574
7.9536
10.1038
c
2 1. 9930
1 0. 1 033
11.8910
c
25.0780
13.5313
11.5480
c
27.6598
17.2758
10.3998
c
298.1500
c
12.1490
0.2898
11 .3631
c
13. 2511
1.53 91
1l.7099
c
15.4152
4. 8253
10.5884
c
16.2859
6.2829
10.0036
c
17.2169
7.76 92
9.4476
c
16.1449
5.5487
10.5956
c
18.5304
6.05 17
12.4784
c
20.3774
e. 0395
12.33 02
c
22.7479
11.3061
11.4413
c
323. 1 500
c
11.0216
0.8177
11.8412
c
11.8067
0.0367
11.7726
c
13. 5330
2.6438
10.8829
c
14.2476
3.8401
10.4096
c
14.7811
4.8290
9.9530
488330D+03 0.1 7 03 06D+03 0.1452180+02
14.6855
18.3223
23.9104
26.4419
31.4142
26.7086
48.4399
65.80 78
69.7333
14.1772
18.1278
23.4826
25.5964
30.5637
25.5511
46.5476
63.0146
65.5813
13.8450
17.6950
23.1102
25.0780
29.9799
185
67
where 6 is the isothermal compressibility of the solvent.
Pierotti calculated the partial molar Gibbs free energy of
creating a cavity in a fluid using the scaled particle theory approach
of Reiss et al. (1959) (Equations 326 and 327). Appropriate temper
ature derivatives yield H^, Cp and S^, the molar entropy of cavity
formation.
The interaction energy of a nonpolar solute with a polar solvent
can be described in terms of dispersion, induction and repulsive inter
actions. Pierotti approximated the dispersion and repulsive interactions
by a LennardJonespairwise additive potential while the inductive inter
action was described by an inverse sixth power law. The total interaction
energy per solute molecule was given by
G. = C,. (Z (r "6 cr r~12) } C. L r"6 (414)
x dis V p p 12 p v md p p v '
where r is the distance from the center of the solute to the center of
P
the Pj.^ solvent molecule and is the distance at which the dispersion
and repulsive energies are equal.
Cdis 4 e12 12= 4
(415)
where and are the energy parameters for the solvent and solute,
respectively, and a and o^ are the corresponding distance parameters in
the LennardrJones potential.
Cind l 2
(416)
where is the solvent dipole moment and is the solute polarizability.
LIST OF TABLES (Continued)
Table Page
54 Energy Parameter Values and Length Function 119
55a Contributions to Free Energy and Entropy of
Solution of Liquid Hydrocarbons . . 121
55b Contributions to Enthalpy and Heat Capacity of
Solution of Liquid Hydrocarbons 125
56 Infinite Dilution Heat Capacity of Surfactants
in Water at 298.15K 130
61 Comparison of Properties of Hard Spheres with Those
of Some NonSpherical Particles 137
62a Contributions to Gibbs Free Energy of Micellization . 141
62b Contributions to Enthalpy of Micellization . 142
62c Contributions to Entropy of Micellization 143
63 Parameter Values for Micellization Model . 145
71 Temperature Dependence of Two Phase Region . 164
72 Pressure Dependence of Two Phase Region 165
73 Effect of Dissolved Methane 165
Ala Contributions to Free Energy and Entropy of Solution . 187
Alb Contributions to Enthalpy and Heat Capacity
of Solution 190
Bla Parameters for Temperature Dependence of 0
Interaction Correlation Coefficients (a^ = 3.40 A) . 202
Blb Parameters for Temperature Dependence of 0
Interaction Correlation Coefficients (a = 3.60 A) . 204
s
Blc Parameters for Temperature Dependence of 0
Interaction Correlation Coefficients (a = 3.80 A) . 206
Cla Contributions to Free Energy of Solution of
Gaseous Hydrocarbons 226
Clb Contributions to Enthalpy of Solution of
Gaseous Hydrocarbons .... 228
vii
Table C2b (Continued)
AH
ah
o
ffll
o
Solute
T)K)
c
X
r,v
RT
RT
RT
C14H30
277.15
73 .60
114.05
0.00
298.15
83 .30
 112.32
O
.
Cl
323.15
91.17
110.84
3.87
358.15
97 .4 1
 109.35
5.84
i5cal
>
Ol
o o
ACp?
i
ACp
r,v
RT
R
R
R
R
40.44
224.07
87. 16
3 3.53
170.43
26.94
198.89
91.46
26.66
134.09
15.80
171 .69
94.72
2 4.36
101.34
6. 10
139.54
95.62
23.98
67.90
94
where R ^ is the reference sphere radius, Rg is the solute spherocylinder
radius and L is the length of the cylindrical portion of the spherocylinder
Combining equations (51) and (53) yields
00 2 2
y [47r(RgRref) + 2it Rg L]
6(8ttR + 2ttL 8ttR c)
s ref
4tt(R2R2 J + 2ttR L
s ref s
(54)
Since the process being considered here is performed at constant
pressure and constant overall volume, GG c = A A where G G 
s ref s ref s ref
is the change in Gibbs free energy upon creation of the solute cavity from
the reference cavity. Other thermodynamic property changes due to forma
tion of the spherocylindrical cavity are calculated from the appropriate
temperature derivatives of equation (54).
Free Energy of Interaction Between a Spherocylindrical
Solute and Spherical Solvent
Expressions for the Helmholtz free energy of interaction between
a spherocylindrical solute and a spherical solvent are necessary to model
aqueous solubility of aliphatic hydrocarbons. We chose to divide the
intermolecular potential into discrete contributions fixed at Y = 0 and
Y = L (Figure 5la) and a continuous distribution of potential along the
axis of the spherocylinder. Four distinct contributions to the total
potential arise from this model and will be discussed individually.
A. Consider the case of a potential fixed at y = 0 interacting
with molecular centers in the region 0y'L and 0
obtain the Helmholtz free energy of interaction M we must integrate the
product of the intermolecular potential g(r), the solvent density
199
dx
r126x 12
ws
+
+
C1. 8, 2, 2. 6, 2 2.2 4. 2^ 2.3' 110ri 2. 2, 2.4
512z (x +z ) 768z (x +z ) 960z (x +z ) 1120z (x +z )
1260(x2+z2)5 512z"x
1 X
9~ tan T
126a
12
ws
1536z6(L2+z2)
+
1536z8 3840z4(L2+z2)2 3840z8 6720z2(L2+z2)3 6720z8
+
10,080(L2+z2)4 10,080z8 512z
L 1 L
9 tan T
(B17)
Combining equations (B8), (B14), (B15), (B06) and (B17)
yields equation (521).
Consider finally the case of a differential potential, d<> (r)
continuously distributed along the spherocylinder axis interacting with
molecular centers in y < 0 and y > L. The expression for the Helmholtz
free energy of interaction (equation 525) is
a12
ws
c
fL
r 1
i
_ 167T p e
,.D w ws
dx
d cos 0
* 2 hs *
dw w g (w ^
a/i. 
x L
o ^
o J
aw
(x2+w2+2xw cos @)8
ws
(x2+w2+2xw cos 0)3
(B18)
Again a double integral must be evaluated analytically,
evaluate the integral
First
f 1
d cos 0
12
7
WS
o
ws
(x2+w2+2xw cos 0)8 (x2+w2+2xw cos 0)3
Fig. 21. Contributions of Species to Property Changes of Micellization
hO
ro
CALFUN
79 109
22/46/19
FORTRAN IV G LEVEL 21
DATE =
0 124
0 125
0 126
0127
0128
0129
0 130
0 13 1
0 132
0 133
0134
0 135
0136
0137
013 8
0139
014 0
0 141
0 142
0 143
0144
0 145
0 146
0147
SC3=XTC(89>+2.0*XTCC90)*T(J)
SC4=XTC(92 ) +2.0 "XTCt 93 ) *T ( J )
SC 5 = XTC{95)+20*XTC(96)*T(J)
SC6=XTC{98)+2.0*XTC(99)*T(J)
SDE=GEE*(S C1+SC2*DL0G(UL{J) )+GC2*DUL/UL(J ) +SC3*DL0G(UL(J))**2+2.*
2G C3*tDLQG ( UL ( J ) ) *D UL/UL ( J ) + SC4 *DLOG ( UL { J) )**3 +3.0 GC4 CLO G (UL { J ) ) *
3 2* DU L/UL(J)+SC5 *DL OG(UL(J) )**4+4.0*GC5*DL0G(UL( J) )**3*DUL/UL{J) + SC
4 6*DLCG(UL(J))**5F50*GC6DLOG(UL{J) )**4 *DUL/UL{J >)+0.50*(XTC(80)
52 .0*XTC { 81 ) *T( J ) )
GKJ.I ) ={DF*EPS I ( I ) ( GDOLL I + GDE ) +CF *EP SI ( I ) *(GCOLLI+GCGLLI ))/CT(J)
2)
SUJ.I ) = DF *EP S I (I ) *( SDOLL I + SDE )+CF *EPSI ( I ) *( SCOLL HSCGLLI lDDF+EPS
1 I { I > (GDOLLI+GDE)DCF*EPSI(I)*( GCOLLI + GCGLLI )
HI ( J I)=GI( J I )+SI(J I)
C SECTION FOR CALCULATING HEAT CAPACITIES
10 CONTINUE
DO 15 J=l,7
HH(J}=HAC(J,I)*T(J)
H I I ( J) =Ht ( J .1 ) *T( J)
HHH{ J ) = HP. V { J I ) *T ( J )
15 CONTINUE
H = 0. 5000
MN=7
CALL DDET5{H,HH,DHH,MN,IER)
CALL D0ET5 H.HI I ,DHI I ,MN, IER)
CALL DDET5(H,HHH.DHHH.MN,IER)
C SECTION FOR CALCULATING PROPERTIES AT DESIRED TEMPERATURES
SS I { I K ) = SI (4,1)
HHI(I.K)=HI(4.I)
GC(I .K) =WAC(4,I)
HC(I,K)=HAC(4,I)
SC(I .K) =SAC(4,1)
244
TABLE OF CONTENTS (Continued)
CHAPTER Page
4 (Continued)
Contributions to the Thermodynamic Properties
of Solution from Intermolecular Forces 78
Analysis of Spreading Pressure of a Solute Occupying
a Cavity 81
Discussion of Results and Suggestions for Future
Research 84
5 AQUEOUS SOLUBILITY OF ALIPHATIC HYDROCARBONS ...... 92
Introduction 92
Calculation of Thermodynamic Properties of Cavity
Formation forAliphatic Hydrocarbons 93
Free Energy of Interaction Between a Spherocylindrical
Solute and Spherical Solvent 94
Changes in Rotational and Vibrational Degrees of Freedom
of Aliphatic Hydrocarbons Upon Solution 104
Results of the Model for Aqueous Solubility of
Aliphatic Hydrocarbons 110
Comparison with Infinite Dilution Properties
of Surfactants 129
Suggestions for Future Work 129
6 MODELING OF THE THERMODYNAMIC PROPERTIES OF MICELLIZATION 132
Introduction and Review of Thermodynamic Process
for Micelle Formation 132
Derivation and Application of an ArbitraryShape
Hard Body Equation of State 134
Contributions to a Model for the Thermodynamics
of Micellization 139
Discussion and Suggestions for Future Research 149
7 EXPERIMENTAL INVESTIGATION OF PHASE BEHAVIOR AND
TRANSITIONS FOR CONCENTRATED SURFACTANT SOLUTIONS .... 156
Introduction 156
Experimental Objectives ..... 156
Description of Experimental Apparatus 156
Operating Procedures 160
Pertinent Calculations for Dissolved Gas Experiments . 162
Results and Suggestions for Future Work 166
8 SUMMARY AND CONCLUSIONS 168
iv
95
Fig. 5la. Fixed Potential at y=0 Interacting with Molecular Centers
in 0
Fig. 5lb. Fixed Potential at y=0 Interacting with Molecular Centers
in y < 0 and y > L
CHAPTER 8
SUMMARY AND CONCLUSIONS
A thermodynamic process for micellization has been developed
which should provide a basis for a better understanding of molecular
mechanisms important in the formation of micelles as well as other pro
cesses of aqueous solution. Analysis of this process supports the
hypothesis that micellization at normal temperatures is primarily driven
by large positive entropy changes of the water when the monomers are
aggregated.
2. A model for the aqueous solution properties of spherical
gases has been developed using a modified scaled particle theory (cavity
j
creation) for the excluded volume effect and a mean field theory for the
intermolecular interaction contribution. Correlation of the experimental
data is quite good. The results are highly sensitive to the temperature
j
dependence of the curvature effect on the water surface tension. The
I
j
results show that the enthalpy, entropy and heat capacity are more sensi
I
I
tive to the temperature dependence of the curvature effect than to that
I 00
of the planar surface tension y For example, a nonzero second deriva
!
I
five of the curvature dependence is essential to obtain a reasonable
I
I
correlation of the heat capacity. Also, significant changes in the inter
!.
action energy parameter can easily be compensated by a reasonable
change in 6.
168
LIST OF TABLES
Table Page
31 Surface Tension and Curvature Parameter Calculated
for Liquid Water at Its Saturated Vapor Pressure
Using the Pierotti Approximation 39
32 Radial Distribution Function for Liquid Water 47
33 Direct Correlation Function for Liquid Water ...... 50
34 Direct Correlation Function for Liquid Water 53
35 Reduced Direct Correlation Function for Liquid Water . 58
4la Solution Properties from Pierottis Model 70
4lb Enthalpy and Heat Capacity Contributions 71
42 Universal Correlation for the Reduced Hard
Sphere Diameter 76
43 Characteristic Parameters 77
44 Intermolecular Potential Energy Parameter 80
45a Contributions to Free Energy and Entropy of Solution . 85
45b Contributions to Enthalpy and Heat Capacity of Solution 88
51 c Parameter Values for Aliphatic Hydrocarbons 107
52 Properties Required to Analyze Liquid Hydrocarbon
Solubility ..... .......... 109
53a Contributions to Free Energy of Solution of
Gaseous Hydrocarbons Ill
53b Contributions to Enthalpy of Solution of
Gaseous Hydrocarbons 113
53c Contributions to Entropy of Solution of
Gaseous Hydrocarbons .... 115
53d Contributions to Heat Capacity of Solution
of Gaseous Hydrocarbons 117
vi
FORTRAN IV G LEVEL 21
CALFUN
DATS
79 109
22/46/19
0078
0 079
0080
008 I
0082
0083
0084
0085
0086
0087
0088
0089
0090
0091
0092
0093
0094
0095
0096
0097
0098
0099
0100
DDF= 0 OD +00
DCF =0.0 0 D +0 0
C LARGE UL VALUE SECTION
C CONTINUOUS DISTRIBUTION SECTION
C 0< Y
GC t= XTC ( 1 ) + XTC { 2) *T( J) + XTC (3 ) T ( J) **2
GC2= XT C(4)+XTC(5)*T(J )+XTC(6)T(J)**2
GC3=XTC(7)+XTC(8)*T(J)+XTC(9)*T(J)**2
GC4=XTC(10)+XTC(11)*T(J)+XTC(12)*T(J)**2
GC5=XTC(13)+XTC(14)*T
GCOLLI=OEXP(GC1+GC2/UL(J}+GC3/(UL(J)**2)+GC4/(UL(J ) **3)+GC5/(UL(J
2)**4 ) )
SC1=XTC(2)+2.0*XTC(3)*T(J)
SC2= XTC ( 5) + 2. 0* XTC ( 6 > T ( J )
SC3=XTC(8)+2.0*XTC(9)*T (j)
SC4 = XTC ( 1 1 ) +2.0 *XTC( 1 2 ) *T ( J )
SC5=XTC( 14)+2.0*X7C(15)*T(J)
S CO L LI=GCO LLI *{S C1+S C2/UL(J)GC2*DUL/CUL { J) **2)+SC3/C UL(J)** 2}2.
20*GC 3TDUL/ UL(J)**3)*SC4/(UL(J)**3)3.0*GC4*DUL/ (UL(J ) **4)+SC5/(UL
31J)~T4)4.0*GC5*0UL/{UL(J)**5))
C DISCRETE DISTRIBUTION SECTION
C 0
GC1=XTC(16)+XTC(17)*T{J)+XTC{18)*T(J)*T2
GC2=XTC(19)+XTC(20)*T(J)+XTC(21) *T(J>**2
GC 3= XTC ( 22) +XTC (23) *T(J)+XTC(24)*T(J)**2
GC4=XTC(25)+XTC{26J *T(J)+XTC(27)*T( J )**2
GC5=XTC(28)+XTC(29)*T(J)+XTC(30)*T(J)**2
GDOL LI = DEXPGC HGC2/ULJ) +GC3/(UL(J)**2)FGC4/(UL(J)**3)+GC5/(U L(J
2)**4))
SC 1=XTC( 17) +2. 0*XTC ( l 8) *T< J)
SC2=XTC (20 ) +2.0 *XTC( 2 1 ) *T< J )
SC3 = XTC(23)+20 *X T C(24 )+T ( J )
37
So far all relations are derived for a hard sphere fluid. The
application of this theory to real fluids proceeded along three lines.
One was by Yosim and Owens (1963) which involved using hard sphere
diameters for real fluids determined from experiment on one property
such as surface tension, isothermal compressibility or thermal expan
sivity in expressions for the other properties. The results were
reasonably good for nonpolar species but poor for polar substances, par
ticularly water. Another procedure was to predict entropy of phase
change by a clever thermodynamic cycle. The same experience was encoun
tered here as with the other procedure. Finally, Pierotti (1963) explored
the results of the theory for gas solubility in liquids. The process he
used consisted of creation of a cavity in the liquid, with the free energy
change calculated from the above expressions. The second step was to fill
the cavity with the solute, calculating the free energy effects as the
sum of contributions of pairwise intermolecular forces.
Scaled Particle Theory and Aqueous Solutions
Pierotti (1965) applied his scaled particle theory for solubility
to aqueous solutions of nonpolar gases.. Somewhat surprisingly, he found
it possible to predict enthalpies, entropies and heat capacities of solu
tion with the physical assumption that water molecules arrange themselves
spatially in the pure liquid as would hard spheres of an appropriate size
and several mathematical approximations. Considering the complexity of
water structure and interactions, this success seems quite fortuitous, since
the only explicit information required about the molecular structure of
O
water is a, the distance of closest approach which he set at 2.75 A.
106
Therefore, the partition function for a polyatomic pure fluid is
rNc
Q =
N!
N
V
A3
\ J
~4>
T exp 2kT
f(T) .
(533)
Consider the change in Helmholtz free energy due to changes in
rotational and vibrational motions when a polyatomic hydrocarbon is taken
from gas phase density to that of pure water. We set : = 0, assuming
that this contribution has been accounted for in the interaction Helmholtz
free energy.
From equation (528) it is apparent that
AA
= H KTIn ^rV'0lutl"
r> V ^r,v ideal gas
(534)
AA = RT Â£n
r, v
If
V
i c1
AG = AA = RT(cl) Â£n (V^/V).
r, v r, v f
(535)
The other thermodynamic property changes are determined from the
appropriate temperature derivatives of equation (535).
Beret and Prausnitz chose the free volume expression of Carnahan
and Starling (1972)
r \ 2
T
*4=
 4
i4
v J
here x = = 0.7405 and v =
(536)
In this case v is the water molar
v
* 2
volume and v = 1.2227 x 10 liter/mole (Gmehling et al., 1979).
Table 51 lists values of c for aliphatic hydrocarbons from Gmehling et al.,
1979) obtained by fitting PVT data.
136
Using equations (62) and (63) the corresponding thermodynamically
consistent relation for the chemical potential of a component of the
mixture is
eu!r = An
P h~
(2ir m kT)
3/2
V. p V. AB B2C V.
+ + ^ + 3 _L
(1Y)
b.
(1Y)'
(1Y)'
2 1 2 12 2 2
a. R.B b. R;A b. R.BC j; a? RTB
, ii i i 3 i i 6 r i
+ Jen (1Y) H ^ + z H .
(1Y)
(65)
(1Y)
(1Y)'
(1Y)'
Note the only difference between equations (65) and (61) lies in
splitting the last term into two similar terms.
B. PercusYevick Pressure Equations of State for
Mixtures of ArbitraryShaped Rigid Bodies
The pressure equation is
BP
P P_
1 2
, 3 B AB
(1Y) (1Y)2
(1Y)
2
(66)
Using equations (62) and (63) the corresponding thermodynam
ically consistent relation for the chemical potential of a component of
the mixture is
By.P = An
P h
(27rmi kT)
3/2
V. p V. AB B2C V. a. R.B
+ __ + A + __l + Â£n (1Y)
(1Y) (1Y)
(1Y)
1 2 12221222 3
+ + 3 bi R1 BC + 6*1 *1 + I *f RI B 2 Y)
(1Y> (1Y)2
(1Y)'
(1Y)'
+
in qYn I fhill + in qY)
(1Y)'
i vi 2c r i
(12Y) 2 An (1Y)
1Y)2 (1Y)3 Y
+
(67)
Table Clb
Contributions to Enthalpy
AH
AH
Solute
T(K)
c
RT
c
RT
CH4
277.15
8.06
16 .
298.15
10 .34
 16.
323.15
12. 49
15.
358.15
14.86
14 .
C2H6
277.15
13.10
24.
298. 1 5
15.95
23 .
323.15
1 8.54
23.
358. 15
2 1.21
23.
C3H8
277. 1 5
18.14
31 .
298.15
21.57
30 .
323. 15
24.59
30.
358.15
27.56
30 .
Solution of Gaseous Hydrocarbons
AH
r,v
RT
cal
RT
AH
exp
RT
0.0
8.82
7.954
+ .30
0.0
6. 14
5.549
+ .05
0.0
3 .22
3.188
+.07
0.0
0.41
0.526
+ .28
0. 0 0
l 0.99
11.441
+ .43
0.23
7.53
7.952
+ .07
0.43
4.47
4.53S
+ .10
0.66
1.33
0.683
.4!
0 .00
13.01
13.234
+ .53
0.40
8. 88
9.052
+.08
0.75
5 .29
4 .909
+.13
1.13
1.85
0.248
+.50
of
3.8
48
71
45
09
72
45
25
15
0 5
64
54
18 APRIL 1979
DATA LIST FOR 3PHER ICAL GAS SOLUBILITY
C 14.1369
3. iaeo
1 0. 9457
24.4841
C 14.8364
2.0323
12.3013
45.4404
C 15.3800
2.6157
12.7637
61 .3689
C 17.6095
5. 4989
12.1142
63.4675
C 358.1500
C 9.6175
2.0736
1 1 .6912
13.3166
C 10.0101
1. 6784
1 t.6891
17.1263
C 1 1.1676
0.1375
1 1.0307
22.5264
C 1 1.6910
1.0310
10.6574
24.6553
C 11.7212
 1. 4444
1 0.2 723
29.4162
C 11.6608
0.5255
1 1.1334
23.8450
C 10.2063
2.5628
12.7693
44.6804
C 9.1344
3.5724
12.7081
60.2365
C 11.1877
1.15 15
12.3357
62.3251
Table 53c
Contributions to the Entropy of Solution of Gaseous Hydrocarbons
AS
AS?
Solute
T(K)
c
R
X
R
CH,
4
277. 1 5
17. 13
1.75
298.15
14.32
2.4 7
323.15
11.32
2. 94
358.15
7.54
3.19
C2H6
277. 15
17 .63
1 .85
293. 15
13.76
3.20
323.15
9.73
4 .8 1
353.15
4.84
7.0 1
C3H8
277. 15
18.14
2.3 0
298.15
13 .21
4 .43
323.15
8.13
6.90
353. 15
2.15
10.24
o
tV)
<
AS
AS
r,v
cal
exp
R
R
R
0.0
13.38
 18 .057
+ .30
0.0
 16.79
16.145
+.04
0.0
14.26
14.137
+ .07
0.0
10.73
11 .66 1
+.29
1.27
20. 76
21.042
+.42
1.03
13.00
1 8 .259
+.06
0 .30
15.34
 15.340
+.11
0.53
12.30
 1 l 751
+ .4 2
2.20
22.64
22.995
+.52
 1 .78
19.42
 19.577
+ .08
1.39
16.42
15.939
+.13
0.91
13 .30
11 .59 0
+.51
115
Table 4la
Solution Properties from Pierotti's Model
Solute
Temperature
AG
c
AG?
X
AG
exp
AS
c
AS0
exp
RT
RT
RT
RT
_R
R:_
Helium
298.15
12.20
0.67
11.53
11.86
12.35
12.15
323.15
12.18
0.61
11.57
11.84
11.62
11.02
Neon
298.15
12.63
1.65
10.98
11.71
12.72
13.25
323.15
12.60
1.52
11.08
11.77
11.90
11.81
argon
298.15
14.66
4.35
10.31
10.59
14.66
15.42
323.15
14.59
3.99
10.60
10.88
13.25
13.53
Krypton
298.15
15.33
5.80
9.53
10.00
14.97
16.29
323.15
15.29
5.24
10.05
10.41
13.25
14.25
Xenon
298.15
17.26
8.06
9.20
9.45
16.58
17.22
323.15
17.18
7.36
9.82
9.95
15.00
14.78
Methane
298.15
16.15
5.66
10.49
10.60
15.66
16.14
323.15
16.10
5.19
10.91
10.95
14.27
14.14
21
micellization. The breakdown is into overall energetic effects in various
regions of the system, entropic contributions due to changes in molecular
conformation and excluded volume effects. The last is chosen as a sepa
rate category (which has enthalpy and entropy contributions) because it
has been modeled in several theories as the "hydrophobic" effect. Electro
static effects are indicated which would be restricted to ionic amphi
philes. Finally, a distinction is made between those contributions which
are associated with aspects internal to a micelle, including conforma
tional constraints and those which occur across the interface between the
micelle and the aqueous environment.
Effects arising from the properties of salt species are restricted
to ionic amphiphiles. Important enthalpic effects are possible changes
in ion hydration (uncertain at present) upon binding to the micelle sur
face as well as increased electrostatic interaction of the ions with the
much greater surface charge density of the micelle relative to the amphi
phile monomer. The latter effect has been extensively modeled; the most
recent and detailed being that of Stigter (1975ab). Unfortunately, due
to the great complexity of detail Stigter's model is not readily adapt
able to a general theory of micellization.
Two significant entropy effects are attributable to the salt
species, both resulting from binding of oppositely charged ions to the
micelle surface. The volume excluded to the solvent by the ions is
aggregated at the micelle surface resulting in an entropy increase as mod
eled by the scaled particle theory. Secondly, the ion configurational
entropy is considerably decreased as modeled by Stigter.
166
Results and Suggestions for Future Work
Considerable difficulty maintaining a pressure seal and
interpretation of early results limited the scope and depth of this
investigation.
As apparent from Table 71, the fraction of the visible sample
which is anisotropic decreases with increasing temperature reasonably
smoothly, indicating a firstorder phase transition. The results are
presented as a range of values since considerable variation may occur,
probably due mainly to temperature control problems with sample aging as
a possibility. An independent experiment at atmospheric pressure in
a constant temperature oven confirmed these results. Since only about
35% of.the sample volume was visible between the windows of the pressure
cell, the temperature range for the twophase region is only about one
third that of the atmospheric pressure experiment in which the entire
sample was visible.
Table 72 presents results concerning the effect of pressure on
the relative phase volumes. The temperatureanisotropic fraction rela
tion differs from that of Table 71 since this is a different sample of
the potassium oleate system. The results of Table 72 clearly indicate
no effect of pressure on the relative volumes. Some earlier results
appeared to suggest enhancement of the anisotropic phase as a function
of increasing pressure. However, given early experimental difficulties
and improved technique, the latter results are probably more accurate.
Table 73 concerns the influence of dissolved methane on the
phase behavior. No significant enhancement of the isotropic or anisotropic
APPENDIX B
HELMHOLTZ FREE ENERGY OF INTERACTION BETWEEN
A SPHEROCYLINDRICAL SOLUTE AND SPHERICAL SOLVENT
97
hs
p and the hard sphere solventsolute radial distribution function g
w 6ws
over the appropriate volume. In our approach the unlike radial distri
tis
bution function is modelled as a function of z, the perpendicular
distance to the axes of the spherocylinder. The intermolecular poten
tial (r) is modelled as a LennardJones form,
ws
From Figure 5la the volume element dV = 2ir z dy dz and
r = /y2 + z2.
= p dV
i wj ws ws
(55)
Substituting a LennardJones potential form
= p
i w
dV 4e
D
ws
12
7
WS
ws
hs t f *
8a(z >
(56)
where the LennardJones size parameter awg is effectively the sphero
hs
cylinder cavity radius and is the point at which g becomes nonzero.
The discrete portion of the distributed potential is denoted as
4e (r) where
ws ws
12 6
a o
T / \ WS ws
*vs(r) 12
Substituting for dV and r yields
L
. .A
M. = p
i Kw
dy
dz 2tt z 4e
D
ws
r12
a
ws
ws
(y2+z2)6 (y2+^2)3
hs / *
(57)
Since equation (437) provides an expression for g as a
function of z' = z + a a equation (57) would be better rearranged
av ws
as a function of zr. (av a volume fraction average essentially equal
to the pure water a for dilute solutions.)
Table C2b (Continued)
Solute
T(K)
AH
c
AH
X
AH
r ,v
ARal
>
m
O O
ACp?
i
ACp
r,v
iECi
RT
RT
RT
RT
R
R
R
R
C11H24
277.15
58 .43
90.97
0.00
32.49
182.07
69. 63
26. 80
139.24
298.15
66.47
39.60
1.66
21.43
162.02
73.31
21.31
110.03
323.15
73.01
36.47
3.09
12.36
140.68
76. 28
1 9. 48
83.89
358.15
78 .36
87.37
4.67
 4.34
115.76
77. 69
19.17
57.24
C12H26
277.15
63.52
98. 28
0. 00
34.75
196.07
75.15
28.32
149.23
298.15
72 .08
96.79
1.75
22.96
174.31
79.03
22. 52
117.80
323.15
79 .0 7
95.55
3. 27
 13.21
151.02
82. 09
20.58
89.51
358.15
34.71
94.32
4.93
4.68
123.68
83.35
20.25
60.59
C13H28
277.15
68 .56
106.68
15.63
22.49
210.07
8 1.55
9.55
138.06
298.15
77 .69
 105.06
15. 18
12.19
186.60
85.66
9.06
1 10 .00
323.15
85.12
103.70
14.69
3.89
161.36
88.83
3.59
81.12
358.15
9 1 .06
 102.33
14.07
2.30
131.6 1
89. 93
8. 1 0
49. 78
254
74
Since dG = cL4 + d(PV), and our cavity formation process is at
ct 8
constant pressure and constant overall volume (dV = dV ), G G
s ref
= A A
s ref
The free energy of creating the argon reference cavity is
obtained by difference between the experimental free energy of solution
and the interaction contribution discussed in the next section. The
other cavity contributions to the thermodynamic properties of solution
are obtained through the appropriate temperature derivative of equa
tion (426).
Calculation of the thermodynamic properties of cavity formation
requires a model for the hard sphere diameters of the solute and solvent
as a function of temperature. We chose the model of Mathias (1978).
From considerations involving the direct correlation function he
postulated that the reduced hard sphere diameter should be some universal
function of reduced temperature and reduced density. The functional
form was obtained by fitting the experimental reduced isothermal compres
sibility to that obtained for a hard sphere fluid from the Carnahan
Starling equation (Carnahan and Starling, 1969). The rather complicated
function is shown in Table 42. Table 43 shows the values of the char
acteristic parameters for water and solutes of interest in this chapter.
Slight adjustments in these parameters compared to those of Mathias were
sufficient (with helium and neon as notable exceptions) to provide a con
sistent fit of the solubility data. Mathias chose the particular form
because it has the following features:
a) At high reduced temperatures the hard sphere diameter is
a function of reduced temperature only.
APPENDIX C
PROGRAMS FOR GAS AND LIQUID HYDROCARBON
SOLUBILITY PROPERTIES
206
Table Blc
Parameters for Temperature Dependence of
Interaction Correlation Coefficients
Coefficient
0
s
A
n .
O
= 3.80 A
B
n
D
n
C1
0.849988D+00
0.722813D02
0.13Q440D04
C2
0.104030D+01
0.401482D02
0.543200D05
C3
0.533194D+00
0.223701D02
0.327500D05
C4
0.726711D01
0.362763D03
0.557000D06
C5
0.165575D+01
0.294656D02
0.614200D05
C6
0.590079D+00
0.515182D02
0.838100D05
C7
0.113939D+02
0.825450D01
0.131290D03
C8
0.525518Df02
0.487378D+00
0.775848D03
C9
0.110752D+03
0.953354D+00
0.151580D02
C10
0.577680D+01
0.000 D+00
0.000 IH00
C11
0.214413D01
0.000 D+00
0.000 D+00
C12
0.422360D04
0.000 D+00
0.000 D+00
C13
0.133347D+01
0.409142D02
0.798500D05
C14
0.392375D+00
0.886390D04
0.9100Q0D07
C,c
0.288964D+00
0.266120D03
0.433000D06
12
A possible resolution of this conflict between the two approaches
to the thermodynamics of ionic amphiphiles lies in the new specific ion
surfactant electrode technique (Evans, 1978). When combined with ion
specific electrodes for the counterion, measurements on long chain sur
factants permit the fractional charge on the micelle to be determined.
Present estimates yield values of a = 0.8, whereas inferred results gave
values closer to 0.5.
Effect of Added Salt on Micellization
An extremely important aspect of the thermodynamics of micellization
is concerned with the effect of added salt on the CMC of ionic amphiphiles.
Examination of the better data reported by Mukerjee and Mysels (1971) for
systems such as alkyl ammonium chlorides and bromines, sodium alkyl sulfates
and alkyl sulfonates with added salts such as the sodium halides, potassium
nitrate and potassium bromide up to 1M, confirm the relation first given by
Corrin and Harkins (1947), Hobbs (1951) and by Shinoda (1953) and described
in detail by Mijnlieff (1970) and Lin and Somasundaran (1971). The data can
be reproduced to within the estimated experimental error with
&n X^/x+ = Kr Ln
~(x+ + x2)'
+o
(236)
fo "f*
where x^ is the CMC without added salt, x^ is the value with added salt
of mole fraction x^ and Kr is a constant independent of the salt whose
value is 0.66 .03 for anionic amphiphiles and 0.58 .03 for
cationics. The data are insufficient and probably of inadequate accuracy
to properly test this correlation for multiply charged salts. Mijnlieff
FORTRAN IV G LEVEL 21
MAI N
DATS
79108
01/01/23
0073
0074
0075
0 07 6
0077
007e
0079
0080
008 1
0082
0083
0064
0085
0086
0087
0088
0089
0090
0091
0092
0093
GEXAR=(EXCi+EXC2*DL0G(T(J))+EXC3*(OLOGT J ) )**2))
HEX A REX C2+2. O *EXC 3*DL0G T(J))
SEXAR=HEXARGE XAR
C SECTION TO CALCULATE ARGON CAVITY PROPERTIES BY DIFFERENCE
WACAR=G2XARGIAR
HACA R=HEXARHI A R
SACA R=SEXARS TAR
C CALCULATE PARTIAL MOLAR VOLUME IN MICELLE AS A FUNCTION OF TEMPERATURE
PPMVMI >=PMVM ( I )*( 1 .0+( T{ J >293.15)/700. 00)
DPMV M=FMVM C I )/700
C CALCULATE SURFACTANT DENSITY IN AQUEOUS SOLUTION AND IN MICELLE
RHOM(I} = 0 6 023/PPMVM( I )
RFOS
C CALCULATE SURFACE TENSION CURVATURE PARAMETER AND SEGMENTAL LENGTH
STW=1.162D+021 .477D01*T(J)
D STW=1. 477D01
DEL=3.3241071D+00+2.6065614D+03/T(J)19.001953D+04/(T(J)**2)
DDEL=2.6065614D+03/T(J)**2)+2.0*19.001953D + 04/ T(J )**3 )
XL( J >=0.527 78918+3. 19 46 78D+ 02/T ( J ) 2.871 56 1 9 D +0 4 / ( T { J )**2>
DXL{J)=3.1946780+02/(T{J)**2)+2.0*2.87l5619D+04/{T(J)**3)
C SECTION TO CALCULATE MONOMER CAVITY PROPERTIES
XMONL=XL(J>*{NC(I>1)
DXMONL=CXL (J)*(NC(I)1 )
W MO N(I J)={PI/(XK* T J) ) >*(XM0NL*STW*DCWCH4+STW*( (CCWCH4**2)(DCWA
2R **2 ) ) 4 0 C EL *STW ( DC WCH 4DC WAR) ST W*DEL* 2. 0*XMONL)WACAR
SMON ( I J ) = ( PI/XK ) ( DX MCNL*ST W *DCW CH4 + XM0NL DST W +DCWCH4+XMQNL AST V*
2DDCWG4+DSTW+(OCWCH4**2)(DCWAR**2) )+3TW*(2.0*DCwCH4*CGCWG42.0 *DC
3 W AR* DDCW GA )4.0 +DOEL *ST W* ( DCW CH 4DC WAR )4 .0*0 EL *D S TV* ( DC WCH4DCWR
4) 4.0*DEL*STW+ ( DDCWG4 DDCWG A ) DSTW*DEL *2 .0 *X MONLST rf*DDEL *2 O +XMO N
5LST W *DEL *2.0*D XMONL)SAC AR
H MON (I J)=WMCN{ ItJ) +SMON{ I J )
C SECTION TO CALCULATE COUNTERION CAVITY PROPERTIES
PA
263
65
Several equations for correlating An K as a function of temper
ature are available in the literature (Benson and Krause, 1976). Two
expressions are considered here.
An (1/K) = aQ + a1 An T + a2(Â£n T)2 (48)
An (1/K) = bQ + b /T + b2/T2. (49)
A standard leastsquares routine was utilized to determine the
parameter values and their standard deviations in equations (48) and
(49). Extreme values of,the thermodynamic properties calculated at
one standard deviation of the parameters were used to provide an estimate
of the bounds on the true value. For purposes of subsequent modeling the
average value of the two extremes was used as the "true" experimental
property value. The "true" experimental values and error limits are
listed in Table 45.
27
The next step in the process involves collapsing the amphiphile
and counterion cavities. Scaled particle theory calculations yield
significant entropy and enthalpy contributions, whose relative magnitude
is temperature dependent.
The third step involves compression of the dispersed monomers to
micellar density, with restriction of the monomer head group to the
micelle surface. An appreciable entropy decrease will occur due to both
the compression and the restriction placed on head group location. The
enthalpy will increase since the electrostatic repulsion between the
head groups at the micelle surface will more than counteract attractive
Van der Waals interactions between the compressed hydrocarbon chains.
Step 4 is essentially the reverse of step,2 with creation of
a micelle cavity. This has small entropy and enthalpy changes because
of the surface area and curvature dependence of the excluded volume
effect. On a monomer basis the magnitude of the changes in step 4 is
much less than those of Step 2.
The final step of this thermodynamic process for micelle
formation involves placement of the compressed monomers in the micelle
cavity and binding of the dispersed counterions to the micelle surface.
The enthalpy decrease accompanying this step results from reduction of
the head group repulsion at the micelle surface because of the counterion
binding and creation of attractive interaction between the head groups
and water.
Entropy changes are mixed with an entropy decrease due to binding
counterions to the micelle surface and a possible entropy increase due to
water structure rearrangement around the occupied cavity.
84
Discussion of Results and Suggestions
for Future Research
With accurate values for the characteristic volume and temper
ature and the interaction energy parameter as previously discussed, the
final fit to the experimental data involves fitting 6 to a temperature
dependence of the form
6 = A + B/T + C/T2 (T in K) (450)
A minimum sum of squares fit resulted in A 8.3194896
B = 2,605.2103
C = 189,930.69
O
Temperature (K)
6(A)
277.1.5
1.39
298.15 
1.72
323.15
2.08
358.15
2.53
Experience has shown that a fit of similar accuracy to Table 45
can be obtained with a different set of interaction energy parameters
Â£ and consequently 6 values. An example considerably different from that
of Table 45 can be found in Appendix A along with the computer program
which determines the coefficients in the 6 function (equation (450).
The surprisingly large magnitude of 6 relative to the radius of
the cavities involved may arise from several sources: (1) inadequacy in
using the surface tension of pure water for the calculations involving an
evacuated cavity, (2) indicative of strong structural changes in water
caused by cavity formation, and (3) covers other inadequacies in the model.
25
remembered that the species whose entropy is increased is the solvent
water not the amphiphile. When the excluded volume is coalesced many
more configurations are available for the water molecules than when
it is dispersed.
A Thermodynamic Process for Micelle Formation
Development of a meaningful theory of micellization requires
integration of the various effects discussed in the previous section
into a thermodynamic process for micelle formation. Such a process is
illustrated in Figure 22. The calculation of the change in a thermody
namic property between two equilibrium states is independent of the
path followed between these states. Thus the process in Figure 22 can
be developed for conceptual expedience rather than physical reality.
Note that the process involves three parallel paths for the amphiphile,
counterions and solvent.
The initial step of the process involves removal of the
amphiphiles and an appropriate fraction of the counterions from their
cavities in solution at constant density. Since the density is main
tained constant, there is essentially no change in the entropy of the
monomers and counterions. However, enthalpy and entropy changes in the
solvent will occur due to removal of these species from their excluded
volume cavities. The enthalpy change will be positive due to elimination
of the intermolecular interactions between the amphiphile and water.
An entropy change (probably negative) will arise due to rearrangemnt of
local water structure around the cavities. Frank and Wen (1957) have
examined this effect for ions.
28
As previously noted, experimentally the overall process of
micelle formation is entropy driven at lower temperatures and become
enthalpy driven with increasing temperature. Since step 2 is the only
one with an appreciable entropy increase at lower temperatures (T 35C)
the driving force (at least at lower temperatures) for micellization
must be aggregation of the volume excluded to the solvent by the amphi
phile monomers. Since there are likely to be only weak temperature
variations in the other steps of the process, step 2 must also reflect
the change to an enthalpy driving force at elevated temperature. Since
there is a large heat capacity effect in it, agreement with this trend
is expected and observed. Further discussion and calculations involving
the more significant stages of this process are included in Chapter 6.
Table Alb
Contributions to Enthalpy and Heat Capacity of Solution
AH
AH?
Solute
T(K)
c
1
RT
RT
He
277.15
4 .52
6 .03
298.15
5.62
 6 C3
322.15
6.66
5.85
358.15
7.89
5.45
Ne
277. 15
5. 1 1
8. 06
298.15
6.46
8.0 1
323.15
7.74
7.74
358. 15
9.22
7.18
Ar
277.15
6.69
13.61
298.15
8.56
13 .38
323.15
1 0.33
12.82
358.15
12.33
11.83
AH
AH
ACp
cal
exp
Q
RT
RT
R
1 .50
1.399
.152
21.21
*4
<*
o
1
0.290
.020
19.35
0 .80
0 .8 18
.091
18.87
2 .44
2.074
.387
19. 96
2.94
3.050
.111
25.42
l .55
l .539
.026
23.40
1
o
o
o
0.037
.133
22.65
2.0 4
1 .678
.533
23.35
6.92
6.969
.077
34.62
4.81
4.825
.034
32.22
2 .48
2 .644
.288
30.91
0 .50
0.138
.893
30. 74
ACp
R
o
4Cpcal
R
ACp
exp
R
7.53
13.68
14.69
2.1
4.37
14 .48
14.18
0.6
2.72
16.14
13.85
2.2
0.86
19.09
13.32
3.5
9.12
16 .30
18.82
2.9
5.88
17.52
18.13
0.9
3.26
19.39
17.70
3.0
0 .99
22.35
17.13
4.6
 12.75
21.87
23. 9 1
3.0
3.1 1
24.11
23.48
2.0
l
.
P
O
26 .50
23.11
4.8
 1.22
29.52
22.53
7.0
278
Smelser, S., An Xray Diffraction Study of the Structure of Argon in the
Dense Liquid Region, Ph.D. Dissertation, California Institute of
Technology (1969).
Stigter, D., "On the Adsoprtion of Counterions at the Surface of Deter
gent Micelles," J. Phys. Chem., 68, 3603 (1964).
Stigter, D., "Micelle Formation by Ionic Surfactants. I. Two Phase
Model, GouyChapman Model, Hydrophobic Interactions," J. Colloid
Interface Sci. 47, 473 (1974a).
Stigter, D., "Micelle Formation by Ionic Surfactants, II. Specificity
of Head Groups, Micelle Structure," J. Phys. Chem., J78, 2480
(1974b).
Stigter, D., "Micelle Formation by Ionic Surfactants, III. Model of
Stern Layer, Ion Distribution, and Potential Fluctuations,"
J. Phys. Chem., 79, 1008 (1975a).
Stigter, D., "Micelle Formation by Ionic Surfactants. IV. Electro
static and Hydrophobic Free Energy from SternGouy Ionic Double
Layer," J. Phys. Chem., 79, 1015 (1975b).
Stillinger, F. H., "Structure in Aqueous Solutions of Nonpolar Solutes
from the Standpoint of Scaled Particle Theory," J. Solution Chem.,
2, 141 (1973).
Stillinger, F. H. and M. A. Cotter, "Free Energy in the Presence of
Constraint Surfaces," J. Chem. Phys., J55, 3449 (1971).
Sutton, C. and J. A. Calder, "Solubility of HigherMolecularWeight
nParaffins in Distilled Water and Seawater," Env. Sci. Tech.,
8, 654 (1974).
Tanaka, M., S. Kaneshina, K. Shinno, T. Okajima and T. Tomida,
"Partial Molal Volumes of Surfactant and its Homologous Salts
Under High Pressure," J. Colloid & Interface Sci., 4J5, 132 (1974).
Tanford, C., "Micelle Shape and Size," J. Phys. Chem., 76, 3020 (1972).
Tanford, C., The Hydrophobic Effect: Formation of Micelles and Bio
logical Membranes, John Wiley & Sons, New York, N. Y.. (1973).
Tanford, C., "Thermodynamics of Micelle Formation: Prediction of
Micelle Size and Size Distribution," Proc. Natl. Acad. Sci. USA,
71, 1811 (1974a).
Tanford, C., "Theory of Micelle Formation in Aqueous Solutions,"
J. Phys. Chem., ^8, 2469 (1974b).
196
1
6
a
WS
1 1
?
O
1
*
4Lw
4 2 2 2
(L+w; (L +w )
Combining equations (B7) and (B5) yields equation (517).
Consider now the case of a differential potential dd> (r)
ws
.continuously distributed along the spherocylinder axis from y = 0 to
(B7)
y = L interacting with molecular centers in 0 < y < L and 0 < z < 00.
The expression for the Helmholtz free energy of interaction (equation
520) is
r 8ir e p
MC = ws_w
x L
ws
[L
f L
. 00
dx
0
dy
dz z
a
0
w
12
ws
(z +(yx) )
21 6
(z2+(yx)2)3
We must now evaluate analytically the double integral
CL
L
12
6
dx
^0
dy
0
ws
ws
(z2+(yx)2)6
(z2+(yx)2)3
Substituting q = y x, equation (B9) becomes
L
Lx
dx
dq
*0 ^
x
,12
ws
ws
, 2, 2,6 2 2,3
(z +q ) (z +q )
The integral
Lx
dq
12
ws
ws
x
, 2. 2,6 2 2,3
(z +q ) (z +q )
is of the form
(B8)
(B9)
(B10)
dx
(a+bx2)"^1
which can be evaluated from standard tables. The result is
15
Finally, after some rearrangement
o o _
yMQ yM0 Q o
 ~ y2
N N N
RT
+
= (1+K)
2 + (1K)x^/x2
(1K')
Â£n
+ (l + (lK'Jx^/x^ Â£n (1 + x^/x2)
 Â£n x
+o
(249)
Thus, theories for the standard state Gibbs' free energy change should be
of the above form. When x2/x^ 1 or high salt concentration equation
(249) becomes
o o
^MQ _y_H0 + (1 + K,)y2
N N
RT
 (1 + Kf) Â£n x+ = constant .
(250)
Some Theories for Free Energy Changes
Upon Micellization
Before proceeding to describe the theories for calculating Ag,
consideration should be given to a significant phenomenological observa
tion previously discussed by Lin and Somasundaran (1971). From tables
of Mukerjee and Mysels (1971), the critical micelle concentration for
amphiphiles with paraffinic tails varies with the number of carbons in the
following way:
a) for all ionics such as sulfates, sulfonates and alkanoates, each
+
additional carbon changes fn x^. within experimental error by a value of
0.69( 0,02) with negligible effect of temperature and added salt concen
tration (Lin and Somasundaran (1971) cited values essentially the same);
b) for nonionics such as oxyethylene 3 alcohols, 6 alcohols, and n and
cbetaines the variation is from 1.09 to 1.28 although it is constant
for each compound;
46
In practice the accessible range of scattering angles is limited
to finite values of the. variable s < s
max
Fourier transformation of
the structure function,
Hm(s) H [I(s) ]/2
(343)
yields a correlation function
gm(r) = 1 + (2rr2pr) 1
r s
max
s H (s) sin (sr)ds
m
(344)
which becomes exactly equal to the function g(r) only if s^ > , Also
since the Xray scattering center of a water molecule is so close to the
oxygen atom the i?m(r) determined is essentially the oxygenoxygen atom
correlation function in liquid water.
A direct correlation function, as proposed by Ornstein and
Zernike (Reed and Gubbins, 1973), may be defined by the following equation
(r12) = (g(r19) l) 
12
c(r ) (g(r ) l)dr.
(345)
13' 23 J 3
where c(r) is the direct correlation function.
The direct correlation function can be obtained from the structure
function H (s) as follows (Fisher, 1964)
m
c (r) = (27TZpr) ^ f s H (s) l + H (s)l X sin (sr)ds.
m m v m '
(346)
Tables 32 and 33 contain g (r) and c (r) calculated from the
m
m
structure function H (s) data of Narten and Levy (1971) at several temper
m
atures. Table 34 contains an expansion of Table 33 in the region of prin
cipal structural features. Note the unusual local extremum in c (r)
m
O
at r s; 2.9 A.
Gubbins and O'Connell (1974) present a remarkable correspondence
between the reduced isothermal compresibility for several molecules
152
Step 5 was modeled considering only the entropy change upon
counterion adsorption onto the micelle, yielding the expected negative
entropy change. This effect is denoted by subscript ca in Table 62.
A comparison of results from the total model (denoted by
subscript cal in Table 62) with experiment shows that the model predicts
a positive entropy change partially balanced by a positive enthalpy
change leading to a negative free energy of micellization (however not
as negative as experiment). Which significant effects have been omitted
from the model?
Changes in interaction between the counterions and water upon
micellization are likely to result in a positive enthalpy change and
positive entropy change assuming loss of counterion hydration upon
micellization. Restriction of the monomer headgroups to the micelle
surface is likely to lead to an entropy decrease which is dependent
significantly on chain length since the shorter chain is likely to have
a greater fraction of its rotational and vibrational freedom eliminated.
This should help improve the chain length variation of the total entropy
which is not strong enough in the present model. In addition, an excess
positive enthalpy change will occur due to the electrostatic repulsions
between head groups constrained to the micelle surface. The adsorbed
counterions will moderate this repulsion to some significant degree;
AH should not be zero,
ca
Appreciable negative enthalpy and entropy changes would arise
from inclusion of watermicelle interactions in the model; however, since
only a modest fraction of each monomer chain is close enough to the
micelle surface to interact with water, the enthalpy change may be
139
TAS = AA = 
V
pdV =
p, 2
M1 p
dp
(610)
where 1 denotes dispersed surfactants and 2 denotes surfactants in the
micellar state. To facilitate integration we let A' = A/p, B' = B/p,
C' = C/p and Y' = Y/p.
Using equation (68) we obtain
TAS
1
(1Y'p)
A 'B'
(1Y'p)2
 b'2cp
(1Y'p)2
2 2 
4 B' C'p
+ ^ T
(1Y'p>
(611)
TAS = RT
Jin
(1Y'p)
, A'B'
Pl Y'(1Y'p)
h2 2
P1 9Y2
iaY'p) +7iW
+
2B'2C'
9 Y
,2
1
) iP.
(1Y'p) 2(1_yf
P)
K
Upon rearrangement
equation (612) yields
AS P n
(1Y2)/p2
A2B2 Vt B2C2
Y2
Jin
(lY1)/p1
Y2P2(1_Y2) Y1P1(1_Y1) 9Y2p2
_dY2)2
+ in (1Y )
B1C1
9Y2P,
!k1 L(lYi)
j + in (1Y )
(612)
(613)
Contributions to a Model for the
Thermodynamics of Micellization
This section considers the contributions to a model for the
thermodynamics of micellization for a series of sodium surfactants, those
of octyl, decyl and dodecyl sulfate.
A. Cavity Aggregation
Steps 2 and 4 of Figure 61 involve monomer, counterion and
micelle cavities. First, the thermodynamic properties associated with
120
the other thermodynamic properties were predicted for the liquid hydro
carbons. Due to severe uncertainty in the experimental solubility
(Baker,. 1959; Franks, 1966; Sutton and Calder, 1974) for hydrocarbons
larger than decane, Â£ was determined by extrapolation of values for
the shorter hydrocarbons. Unfortunately, such extrapolation is also
quite uncertain due to fluctuations in e as a function of carbon number.
ws
Table 55 presents results of predictions for larger hydrocarbons
for which an extremum occurs in AGa^ at decane. Appendix C again con
tains a parallel calculation using the S values of Appendix A.
As demonstrated by the results of this chapter and Appendix C,
a lack of uniqueness problem exists in the model parameters since both
E and Lare unknown, or at least uncertain, and model results are not
ws
highly sensitive to either. A modest variation in one parameter can
easily be compensated for by a modest variation in the other.
Thomas and Meath (1979) present an excellent discussion concern
ing the variation of dispersion energy coefficients (= ea^) when dif
6
ferent thermodynamic properties are fitted. Thus we would not expect to
be able to find e values obtained from fitting a thermodynamic property
to model gas solubility. We have done this because no alternative approach
is readily available.
The temperature dependence of L obtained in the modeling is much
larger than that of the hard sphere diameter for the spherical solutes.
While the complexity of the model for the interaction terms prevented using
a temperature dependent spherocylinder radius and thus, the large temper
ature dependence of L may be partially compensating for this, we expect
that the major reason is coiling of the chain, with resultant shortening,
as the temperature increases.
103
rL
rTT/2
rco
f c 1
. .D2
M = p
i w
dxf
o J
\
o
'O
o
dwf 8tt w'2
0
sin 0 r
Â£
WS
L
,
x Â£<*,
12
O
ws
ws
(x,2+w'^+2xrw' eos 0f)8 (x,2+w,2+2x,w' eos 0')8
(524)
where w = wf + O O
w ws
Clearly = M^2 and since sin 0 d0 = d eos 0
M? = MD1 + M02
i i
C
L
C1
e
ws
dx
d cos 0
J
0 J
0 +
"k 2. lis ^
dw w Â§Wg(w )
,12
O
ws
ws
oo 6 2 2 3
(x +w +2x w eos 0) (x +w +2x w eos 0)
(525)
Upon analytical integration over x and cos 0, equation (525)
becomes
M =
i
C
r12
f Q Q
P e
w ws
* hs *
d gus(w )
w
ws
(L+w) w
L
9
lOw
9 (L+w)9
8 8
,7 7
+
+
8(L+w)8 7(L+w)7
+ (L+w) + (L+w)8w~* + (L+w^w^ t (L4w) 1
6(L+v)8 5(L+w)^ 4(L+w)^ 3(L+w)8 2
2w+L
w+L
+ 2
2 /t 22\ 4 2 2.2 6 2 2.3 8 ,2 2,4
+ w ~(L +w ) w (L +w ) w (L +w ) w (L +w )
9,T2.2, ,,.,2,2,2 .^2^ 2x3 q 2 2.4
2 (L +w ) 4(L +w ) 6 (L +w ) 8 (L +w )
L+W
/_ 22.%
i(L +w ) 2J
a
ws
8w~
w
(L2+w2) (
2w+L
w+L
+ 3+ + in
J (L+u)
((L+w)2
2 2
UL +w )J
. (526)
LIST OF FIGURES
Figure Page
21 Contributions of Species to Property Changes
of Micellization 22
22 A Thermodynamic Process for Micelle Formation 26
31 Contact Correlation Function; Comparison of
Different Models . 42
32 Reduced Direct Correlation Functions .... 57
5la Fixed Potential at y = 0 Interacting with Molecular
Centers in 0 < y < L and 0 < z < 00 95
5lb Fixed Potential at y = 0 Interacting with Molecular
Centers in y < 0 and y > L 95
5lc Distributed Potential Along Spherocylinder Axis
from y = 0 to y = L Interactint with Molecular
Centers in 0 < y < L and 0 < z < 00 . 96
5ld Distributed Potential Along Spherocylinder Axis
from y = 0 to y = L Interacting with Molecular
Centers in y < 0 and y > L 96
61 A Thermodynamic Process for Micelle Formation ...... 133
71 High Pressure Experimental Apparatus 158
ix
Table 53a (Continued)
Solute
T(K)
AG
c
RT
AG?
X
RT
AG
r>v
RT
C4H10
277,15
45.54
38.80
3.05
29Â£. 15
43. 43
35.73
3.03
323.15
40 .80
32.39
2.97
355.15
37. 1 0
28.20
2. 83
AG
cal
 RT
AG
exp
RT
9.79
9.809
+ .011
10.73
10.729
+ .006
1 1.38
11.395
+.005
11.73
11.775
+. Oil
112
45
From equation (334 )
9 G(r)
3r
2.
+  