• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 Introduction
 Thermodynamics of micellizatio...
 Scaled particle theory modified...
 Modeling of spherical gas...
 Aqueous solubility of apliphatic...
 Modeling of the thermodynamic properties...
 Experimental investigation of phase...
 Summary and conclusions
 Appendix
 Bibliography
 Biographical sketch
 Copyright














Group Title: molecular thermodynamic model for aqueous solutions of nonpolar compounds and micelle formation
Title: A molecular thermodynamic model for aqueous solutions of nonpolar compounds and micelle formation
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 Material Information
Title: A molecular thermodynamic model for aqueous solutions of nonpolar compounds and micelle formation
Series Title: A molecular thermodynamic model for aqueous solutions of nonpolar compounds and micelle formation
Physical Description: Book
Language: English
Creator: Brugman, Robert James
Publisher: Robert James Brugman
Publication Date: 1979
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Bibliographic ID: UF00089539
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: oclc - 05581592
alephbibnum - 000087919

Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
        Page iv
        Page v
    List of Tables
        Page vi
        Page vii
        Page viii
    List of Figures
        Page ix
        Page x
        Page xi
        Page xii
        Page xiii
    Abstract
        Page xiv
        Page xv
    Introduction
        Page 1
        Page 2
        Page 3
    Thermodynamics of micellization
        Page 4
        Page 5
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    Scaled particle theory modified for aqueous solutions
        Page 29
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    Modeling of spherical gas solubility
        Page 62
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    Aqueous solubility of apliphatic hydrocarbons
        Page 92
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    Modeling of the thermodynamic properties of micellization
        Page 132
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    Experimental investigation of phase behavior and transitions for concentrated surfactant solutions
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    Summary and conclusions
        Page 168
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    Appendix
        Page 171
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    Bibliography
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    Biographical sketch
        Page 280
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    Copyright
        Copyright
Full Text












A MOLECULAR THERMODYNAMIC MODEL FOR AQUEOUS SOLUTIONS
OF NONPOLAR COMPOUNDS AND MICELLE FORMATION










By

ROBERT J. BRUGMAN


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY








UNIVERSITY OF FLORIDA


1979















ACKNOWLEDGMENTS


I wish to express my deepest appreciation to Dr. John O'Connell

for his interest and enthusiastic guidance throughout my graduate studies.

Working with him has been a truly valuable and enjoyable experience.

I also wish to thank Drs. J. C. Biery, G. Y. Onoda and F. A.

Vilallonga' and. Prof. R. D. Walker for serving on my supervisory committee.

It is a pleasure to thank the faculty and students of the Depart-

ment of Chemical Engineering for providing an enjoyable, pressure-free

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 X-Ray Diffraction.. . . . . . . . 45

4 MODELING OF SPHERICAL GAS SOLUBILITY . . . . . 62

Introduction . . . . . . . .. . .. . 62
Thermodynamic Properties of Solution from
Experimental Data . . . . . . . . . 63
Application of Scaled-Particle 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 for-Aliphatic 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 Arbitrary-Shape
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

3-1 Surface Tension and Curvature Parameter Calculated
for Liquid Water at Its Saturated Vapor Pressure
Using the Pierotti Approximation . . . . . 39

3-2 Radial Distribution Function for Liquid Water . . . 47

3-3 Direct Correlation Function for Liquid Water . . 50

3-4 Direct Correlation Function for Liquid Water . . . 53

3-5 Reduced Direct Correlation Function for Liquid Water . 58

4-la Solution Properties from Pierotti's Model . . . . 70

4-lb Enthalpy and Heat Capacity Contributions . . . . 71

4-2 Universal Correlation for the Reduced Hard
Sphere Diameter . . . . . . . . . 76

4-3 Characteristic Parameters . . . . . . . . 77

4-4 Intermolecular Potential Energy Parameter . . . . 80

4-5a Contributions to Free Energy and Entropy of Solution . 85

4-5b Contributions to Enthalpy and Heat Capacity of Solution 88

5-1 c Parameter Values for Aliphatic Hydrocarbons . . . 107

5-2 Properties Required to Analyze Liquid Hydrocarbon
Solubility . . . . . . . . . . . 109

5-3a Contributions to Free Energy of Solution of
Gaseous Hydrocarbons . . . . . . . . 111

5-3b Contributions to Enthalpy of Solution of
Gaseous Hydrocarbons . . . . . . . . 113

5-3c Contributions to Entropy of Solution of
Gaseous Hydrocarbons . . . . . . . . 115

5-3d Contributions to Heat Capacity of Solution
of Gaseous Hydrocarbons . . . . . . . 117

vi












LIST OF TABLES (Continued)


Table Page
5-4 Energy Parameter Values and Length Function . . . 119

5-5a Contributions to Free Energy and Entropy of
Solution of Liquid Hydrocarbons . . . . . 121

5-5b Contributions to Enthalpy and Heat Capacity of
Solution of Liquid Hydrocarbons . . . . . 125

5-6 Infinite Dilution Heat Capacity of Surfactants
in Water at 298.150K . . . . . . . . 130

6-1 Comparison of Properties of Hard Spheres with Those
of Some Non-Spherical Particles . . . . . 137

6-2a Contributions to Gibbs Free Energy of Micellization . 141

6-2b Contributions to Enthalpy of Micellization . . . 142

6-2c Contributions to Entropy of Micellization . . . . 143

6-3 Parameter Values for Micellization Model . . . . 145

7-1 Temperature Dependence of Two Phase Region . . . 164

7-2 Pressure Dependence of Two Phase Region . . . . 165

7-3 Effect of Dissolved Methane . . . . . . . 165

A-la Contributions to Free Energy and Entropy of Solution . 187

A-lb Contributions to Enthalpy and Heat Capacity
of Solution . . . . . . . . . . 190

B-la Parameters for Temperature Dependence of o
Interaction Correlation Coefficients (a = 3.40 A) . 202
s
B-lb Parameters for Temperature Dependence of o
Interaction Correlation Coefficients (as = 3.60 A) . 204

B-1c Parameters for Temperature Dependence of o
Interaction Correlation Coefficients (as = 3.80 A) . 206

C-la Contributions to Free Energy of Solution of
Gaseous Hydrocarbons . . . . . . . . 226

C-lb Contributions to Enthalpy of Solution of
Gaseous Hydrocarbons . . . . . . . . 228


vii











LIST OF TABLES (Continued)


Table Page
C-lc Contributions to Entropy of Solution of
Gaseous Hydrocarbons . . . . . . . . 230

C-ld Contributions to Heat Capacity of Solution of
Gaseous Hydrocarbons . . . . . . . . 232

C-2a Contributions to Free Energy and Entropy of
Solution of Liquid Hydrocarbons . . . . . 248

C-2b Contributions to Enthalpy and Heat Capacity of
Solution of Liquid Hydrocarbons . . . . . 252

C-3 Energy Parameter Values and Length Function . . . 256


viii
















LIST OF FIGURES


Figure Page

2-1 Contributions of Species to Property Changes
of Micellization . . . . . . . . . . 22

2-2 A Thermodynamic Process for Micelle Formation . . . 26

3-1 Contact Correlation Function; Comparison of
Different Models . . . . ... . . . . 42

3-2 Reduced Direct Correlation Functions . . . ... . 57

5-la Fixed Potential at y = 0 Interacting with Molecular
Centers in 0 < y 5 L and 0 < z < .. . . . . . 95

5-lb Fixed Potential at y = 0 Interacting with Molecular
Centers in y < 0 and y > L . . . . . . . 95

5-ic Distributed Potential Along Spherocylinder Axis
from y = 0 to y = L Interactint with Molecular
Centers in 0 S y S L and 0 z
5-ld Distributed Potential Along Spherocylinder Axis
from y = 0 to y = L Interacting with Molecular
Centers in y < 0 and y > L . . . . . .. 96

6-1 A Thermodynamic Process for Micelle Formation . . . 133

7-1 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 = monomer-monomer interaction property

0 = absence of added salt

ref = refers to reference solute

s = solvent property

w = water

ws = water-solute 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 isotropic-anisotropic 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 water-hydrocarbon 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 center-to-center

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

water-micelle 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 isotropic-anisotropic phase transition in lyotropic

liquid crystals. A study of a single system containing potassium oleate

showed a two-phase 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 (2-1)
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 (2-2)
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 (2-3)
\ + @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 (2-4)
+ xo
x +x
o o T,P

which is essentially equivalent to equation (2-3) for sharp CMC points.

This expression can be used to obtain (Hall and Pethica (1970))


x = (N2 2N)x (2-5)
1 m












From the definition of x in equation (2-2)


x = x /(N N) (2-6)
m o

+ + 2 2 -
xl = xo(N 2N)/(N N) (2-7)

which then yields

AG + 1 + 2
-- -n x = n x+ n N + n(l N/N
NRT o N o


+ Yn [(1-2N/N 2)/(1-N/N2) (2-8)

To assume that polydispersity is unimportant the right-hand

side of equation (2-8) 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 (2-9)

equation (2-8) becomes, in first order approximation

AG n x + 2%n N 1
---n x = (2-10)
NRT N

The right-hand side of equation (2-10) 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 (2-11)
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 (2-12)
NRT nT P,n N 0Pn

AS0 AH0 AG
m m m (2-13)
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 + (2-14)
T P,n P,nJ

The last term in equation (2-12) is small when the right-hand side

of equation (2-10) 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 (2-12) and (2-14) 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 (2-15)
1 m

and the thermodynamic formulation is

o el
0i = Pi + P + RT Zn a (2-16)

o el
m = p + p + RT kn a (2-17)
m m m m

+ + o o el el + -- +
P Np = 0 = (m Npo) + (,p Np ) + RT(Zn a N n a ) (2-18)
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 three-fourths 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 Debye-Huckel theory leading to

iel = i + RT An yl (2-19)

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) (2-20)
C C JC
el
o 1m + RT n a (m) (2-21)
p (solution) = V micellee) = p (m) + -- c
c c c N
or el
el
( o m P = RT Zn(a (s)/a (m)) (2-22)
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 (2-23)
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 (2-1) where all the species are uncharged gives

AG
m n x_ x+ (2-24)
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(1-a)
NM + NCG+ M-(- (2-26)
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 ) ] (2-27)
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 (2-1) gives
m

AG
m0 + + ,
= n x1(x) (2-28)
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) (2-29)
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 (2-28) yields

AG0 +
m +
= (l+a) kn x (2-30)
NRT










In order for equation (2-30) to yield equation (2-25), 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 semi-empirical

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 (2-2) and (2-27) with mole

fractions for activities at all concentrations plus

xc = x C(x xl) (2-31)

yields the relation
1/N
x =1 (2-32)
X J K[x -t(x -x )]


where K E exp [-AGo/NRT]. (2-33)
m
At small values of x, x1 = x but at larger values of x x> two

limiting cases appear


x 1/x= K(l-c)o a < 1 (2-34)
1 o

1/2N 1/2
x = (x /N) /K a = 1. (2-35)
1 o

Equation (2-34) is chosen by Sexsmith and White (1959b) which indicates

a rapidly decreasing monomeric concentration with total amphiphile while

equation (2-35) 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 (2-36)
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 (2-37)

and the mass action relation for amphiphile (1) and salt (2) as


N P' + Q 2 Q MQ (2-38)

Now for an ideal solution where the added salt has a common ion with the

amphiphilic salt

PMQ = Q (2-39)
MQ MQ
o
u1 = + RT in XlXc (2-40)


P2 = 02 + RT Zn x2xc (2-41.)

where xc = xl + x2 is the mole fraction of counterion in the system from

both a 1-1 amphiphilic salt and a 1-1 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) (2-42)
NRT NRT L )

In the limit x2 = 0, Q = 0

o --o
MO 1_ +0

WNRT 2 n (2-43)

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)]. (2-44)
N

For the correlation of equation (2-36) 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 (2-45)

Mijnlieff shows that the reciprocity relation


(- 2-46)
Dn2 n1
T,P,n1 T,P,n2

leads to
Q (1 + K') < 0. (2-47)
N 2 + (I-K')x /x

In the limit x /x 1
2 1

(1 + K')x
K)x2 (2-48)
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 + (l-K')x_/x2 2


+ ( + (I-K')x /x2) n (1 + xn/x2 Pn x (2-49)

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

(2-49) becomes
S0O
Q 10 + (1 + K')p2
-2
SN +o(2-50)
RT (1 + K') Zn xl = constant. (2-50)




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

c-betaines 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 (2-51)
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 one-half 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

= [-kl-k2nc + k3AM] + Z6i/ARM (2-52)


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 right-hand 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 Ben-Naim (1971), Tenne and

Ben-Naim (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 hydrocarbon-chain 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 (l-y)2 (1-y) (l-y)2
l y (1y) (-y)


+ (__1 18y )2 (2-53)



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 (2-54)
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-- (2-55)
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 (2-53) is

the most important.) Thus
2
A sphere = = 2r = Acylinder (2-56)
sphere 1 cylinder
with

k = c + c nc (2-57)

50 2
so = c' + c' n (2-58)
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- (2-59)
-- = k' k' A + k' A /N
NRT RT 1 2 1 3 m

= [a' + b'n ] + c' O2/N (2-60)
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 rigid-body 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 2-1 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. 2-1. 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 hard-body 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 micelle-solvent











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 2-1, 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 Ben-Naim (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 2-2. 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 2-2 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. 2-2. 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 X-ray 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 two-dimensional 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 (3-1)

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)[1-4rr pG(r)dr]. (3-2)
o o
Expanding P (r+dr) to first order in dr yields

P
P (r+dr) = P (r) + Dro dr + ... (3-3)

Combining equations (3-2) and (3-3) yields

Sn P
D = 4Tr 2pG(r). (3-4)
Dr

Upon integration
(r
P (r) = exp [- 4Wr pG(r')dr'] (3-5)
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

Ben-Naim (1974) as

N exp [- g U(RN]
P(R) = exp (R N (3-6)
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 (3-6)
o
by integrating over all the region V-v(r) where v(r) denotes the spher-

ical region of radius r.

P (r) = ...... P(RN)dRN (3-7)


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 (3-8)
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 (3-9)
N:A f V-v(r) f










Thus the ratio of (3-8) and (3-9) gives

exp {-[A(T,V,N;r) A(T,V,N)]} =

SV-V...... exp [-BU(RN)]dRN
J V-V(r) (3-10)

f ... exp [-U(RN)]dRN
V
= P (r).

Equation (3-10) 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'. (3-11)
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 (3-12.)

(a+b)/2 2
W(r) = kTp 4Tr' G(r')dr'. (3-13)
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. (3-14)
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 (3-15)
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). (3-16)
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 (3-15) represents the first three terms in this series.

When 0 5 r 5 a/2 all terms in equation (3-16)'beyond n=l vanish. In this

range equation (3-14) applies.

As r begins to exceed a/2, two solvent centers can fit into the

cavity, so the n = 2 term in series (3-16) 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 cavity-solvent inter-

face. Thus










W(r) = (47Tp/3)r3 + (4iry)r 2- (161Ty6)r + 0(1)* (3-17)

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]. (3-18)

The integral relation (equation.3-11) between W(r) and G(r) results

in the following large-r behavior for G(r)

2y 4y 6
G(r) = pkT p+ kTr 2 + (3-19)
PGT ) +pkTr2

Subsequent efforts (Tully-Smith 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 (3-4) and (3-14) yield

G(r) = (1 47r3p/3)- for r a/2 (3-20)

and for W(r) from equation

W(r) = kT kn(l 4pr3p/3) for r a/2. (3-21)

For very large cavities, r -- c, from equation 0-19)

P
G(r) = pkT (3-22)

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 (3-19) 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. (3-23)

The coefficient A in equation (3-23) was determined from

equation (3-22). Expressions for B and C were determined by matching

values and derivatives of equations (3-19) and (3-20) when r equalled

a/2. If P is the experimental value the expressions lead to

S 3y kT 1 3 (3-24)
Y.= 7a2 V-y 2 ( 2 pkT2 (3-4


6 a + 3y (3-25)
8 2+y-2(1-y)2(P/pkT)

where y = rpa 3/6.

The lower solid curve in Figure 3-1 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 (3-11) with this expression for G(r)

yields

W(r) = Ko + K r + K2r2 + K3r3 (3-26)

where the coefficients are

K = kT[- kn(l-y) + 4.5z2] 1 r Pa3 (3-27a)
o 6

K = (kT/a)(6z + 18z2) + T Pa2 (3-27b)


K = (kT/a 2)(12z + 18z 2) 2a Pa (3-27c)


K3 = 4ir P/3 (3-27d)

where z = y/(l-y).

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 (3-24)

and (3-25) at several points along the saturation curve for water. He

also includes measured liquid-vapor interfacial tension for comparison.

Table 3-1 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 liquid-vapor interfacial tension, y and the radial distribu-

tion function, g(r), as input data.

The most accurate determination to date of the oxygen-oxygen

pair correlation function g (2)(r) in liquid water can be determined

from Fourier transformation of the structure function data as determined

from X-ray 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 (3-16) will be correct for G(r) in the range 0
For larger r, at least the pair term in P(r), equation (3-15), 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 3-1

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.(3-24)]
(dyne/cm)


51.44

54.97

58.35

60.96

62.86

63.82

52.18


6[Eq. (3-25)]
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

(3-4) and (3-14):
3]-1 0
G(r) = [1 (47/3)pr ] (0 r 5 1.20 A) (3-28)

while from equations (3-4) and (3-15)

1 + {2r dt g (t)t (t-2r)
r
G(r) = 2r-- (3-29)
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 3-19) truncated after the fourth term

2y G G
P + v 2 4 o
G(r) pkT + Pr + -- (1.95 A < r < o). (3-30)
r r










-3
The r term is missing in equation (3-30) 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 (3-29) and the macro-

scopic series (3-30) can be used to fix their values. Series (3-10) 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 3-1. 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 hydrogen-bonding

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

hydrogen-bond 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. 3-1.










Curvature Dependence of Surface Tension


The expression for the curvature dependence of the surface

tension (equation 3-18) 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 one-component 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 (3-31) reduces to equation (3-18).

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 (3-31) can be rearranged to yield

Z Zn y 2[(1+ 6/r) 1] (3-32)
31 + 2[(l + 6/r)]

Equations (3-18) and (3-19) imply that G(r) in the macroscopic

region should be written as

G(r) = p + 2 (3-33)
pkT pkTr

Since the first term is negligible at atmospheric pressure

G(r) 2 (3-34)
pkTr

This leads to

D r G(r) 2 D (335)
Dr pkT Dr

2_ D n y (3-36)
pkTr 8D n r

Substituting equation (3-32)

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 (3-38)
(rG)o [1 + 2(1 + 6/r')3] r'

where (rG)o = 2 yo/pkT.

Substituting X = 6/r', equation (3-38) can be rewritten in

dimensionless form with finite limits
6/r
nn rG = 2[(1 + X) 1] dX (339)
-n = dX. (3-39)
(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 (3-19).










From equation (3-34)

SG(r) = 2y + 2 Y (3-40)
@r pkTr2 pkTr Br


= -6y 3 (3-41)
pkTr I + 2(1 + 6/r)

From the form of equation (3-41) 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 3-1 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 X-Ray Diffraction

The total scattered intensity in electron units per molecule,

I(s), obtainable from X-ray 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 (3-42)
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 Field-Molecular 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 (3-43)

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 X-ray scattering center of a water molecule is so close to the

oxygen atom the gm(r) determined is essentially the oxygen-oxygen 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


(3-45)




structure




(3-46)


Tables 3-2 and 3-3 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 3-4 contains an expansion of Table 3-3 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


(3-44)















Table 3-2

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 3-2 (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 3-2 (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 3-3

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 3-3 (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 3-3 (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 3-4


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 3-4 (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 3-4 (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. (3-47)
T IT

Figure 3-2 and Table 3-5 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

(3-47) 4pV*1/3 'r2c(r) as a function of reduced distance r* = r/V at

several temperatures is presented in Table 3-5. Figure 3-2 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 3-2 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. 3-2. Reduced Direct Correlation Functions
-20 \
Ar(1080K)




-40_


Ar(85K)


-60
0 0.5 r 1.0 1.5


Fig. 3-2. Reduced Direct Correlation Functions











Table 3-5

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 3-5 (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 3-5 (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 3-2.

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 solute-sized

cavity in the solvent and introduction of an interacting solute molecule

into the cavity. Using scaled-particle theory for the first step and

a mean-field theory using the Lennard-Jones 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. (4-1)

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 (4-2)






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 (4-1) becomes
diss odissdiss RT (4-3)
2 (4-3)

For equilibrium
Gdiss = G2 (4-4)
2 2
and

AG = o 'diss G'g = RT(kn x Sn f ) (4-5)
or
AG = RT in K (4-6)

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) (4-7)
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 (4-8)


Sn (1/K) = b + bl/T + b2 /T2 (4-9)

A standard least-squares routine was utilized to determine the

parameter values and their standard deviations in equations (4-8) and

(4-9). 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 4-5.










Application of Scaled-Particle 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 ) (4-10)

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 (4-11)


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 + RT2-I (4-12)
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 (4-13)
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 3-26 and 3-27). 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 Lennard-Jones 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 r-6 (4-14)
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 (4-15)

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 Lennard-Jones potential.

Cnd = 2 2 (4-16)
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 Kirkwood-Muller formula.


2 ait2
Cdis = 6 m c (/X) + (a2/X2) (4-17)

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 (4-14) and replacing the

summation by an integration gives


Gi 47rp Cdis +Cind Cdis 12 dr' (4-18)
k-T -L kT ,4 --0 drl (4-18)
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 (4-16) (4-18),

G is proportional to o2

kn K = G /RT + Zn(RT/V ) at c2 = 0. (4-19)

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 (4-10),

(4-14) (4-18) is
8 Cind Tp
n K + 3 G /RT in (RT/V) =
6 kT 012

13
(11.17p/T)(C /k) (e /k) a3 (4-20)
w 2 12

Pierotti determined E /k from the best linear fit of the left-
w
hand side of equation (4-20) 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 4-1 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 (4-10)

have been lumped with the cavity terms in Table 4-1. Considering that

no fitting of solute parameters was done, the results are quite good

except for the heat capacities.










Table 4-la


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 4-lb


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 3-30)

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 (3-39) should allow a more rigorous calculation of G(r)

than the series result of Stillinger. However, equation (3-41) 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 (3-29) for G(r) in the region where

r contains two water molecular centers can pass through an extremum and

be used to match equation (3-39). 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 (4-21)

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. (4-22)

Since our cavity creation process is constant pressure with

dVO = -dVB

dA = yda + j adJ (4-23)

Using the one term macroscopic approximation

y = Y (1-6 J) d = YO. (4-24)

The work of changing the cavity from that of the reference

solute (argon) is then

As A ref = y (a-aref 6[(aJ)s (aJ) ref]

6(a J a J )
y (a -a ) 1 s s ref ref (4-25)
as-a ref (a a re)

Since for a sphere

a = 4rR2 and J = 2/R

A Aref = 47 y R2 R2 1 f[]26 (4-26)
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 (4-26).

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 4-2. Table 4-3 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 4-2

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 (4-27)

T < 0.73: fs = a14 exp [-a15 T] (4-28)


-r N d
3 aV 2
3V* = fs + a2/exp [a4(p+ aT) 2] -


a3/exp [a5(p-+alT a6)2] + a9/exp [al0{(T-a13)2 +


all(p al2)2}] (4-29)



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 4-3

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 + ... (4-30)
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 first-order perturbation theory.

A basic relation of statistical mechanics is

S-kT n (4-31)
SiT,V,Ny .


With the hard sphere as the reference state, equations (4-30)

and (4-31) yield

y Phs 2 (4-32)
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 water-solute inter-

molecular potential and gs (r) is the water-solute hard sphere radial

distribution function. Since G = P


R hs (r) r2dr. (4-33)
Gi = 4Tpw ws(r) gws (r) r dr. 4-33)











We have chosen to approximate ws(r) by a Lennard-Jones form
ws
with 0.. E R.
13


(w(r) = 4c' (4-34)
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 (4-35)
ws 4 3 (a + 2a )3
ws ws ws ws

utilizes Stockmayer-Kihara potential parameters and 0 along with

solute polarizabilities and the dipole moment of water V .
w
Table 4-4 lists the values of c' calculated from equation (4-35)
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 (4-33), 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 (4-36)
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 4-4

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 (4-35) whereas other

values result from solubility data fit at 298.150K.

Pure component parameter values for equation (4-35) 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] (4-37)
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 Percus-Yevick equation.

We evaluated equation (4-33) for several R values over the

temperature range considered in this work and obtained the following

accurate correlation.


G. = ' (6.7202-4.954 x10 3/T + 6.548 x10 5/T 1.52R-3.17R 2). (4-38)
1 ws

The other interaction contributions to the thermodynamic

properties H., S., Cp. can be obtained from the appropriate temperature

derivatives of equation (4-38).



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 (4-39)
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 (4-40)
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" (4-41)
1,w i w i 1 w
z z
Also for any change in the dividing surface from z1 to z2,


F r. = (z2-zl) (C-C") (4-42)
1 z2 2 1 2 1 1
and
Cf
r. = r. (z-z)C + (4-43)
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 (4-44)
2,w 1
and

(y-Y ) = (Z-ZI) C R T (4-45)
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 r-cule by Reiss and Tully-Smith, 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. (4-46)
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. (4-47)

Since there is only one r-cule or solute molecule in the volume

47T 3
3 Ri'


C' = C' = (4-48)
r s 4 N R3
3 o i

Subtracting equation (4-48) from (4-47) we obtain

[(6-6') + (61 61)]
Y Yrw =N R3 R T. (4-49)
r,w rw g
3 o 1

Since (6-6') and (61-6') 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 (6-6') = (61-6{) = 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) (4-50)

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 4-5

can be obtained with a different set of interaction energy parameters

C and consequently 6 values. An example considerably different from that

of Table 4-5 can be found in Appendix A along with the computer program

which determines the coefficients in the 6 function (equation (4-50).

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 4-5a

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




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