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
