• TABLE OF CONTENTS
HIDE
 Title Page
 Dedication
 Acknowledgement
 Table of Contents
 Key to Symbols
 Abstract
 Introduction
 Fluctuation theory for strong electrolyte...
 A Model for direct correlation...
 Application of the model to aqueous...
 Conclusions and recommendation...
 Hard sphere direct correlation...
 Relation of McMillan-Mayer theory...
 Relation of density expansion of...
 Exponential integrals
 Model parameters
 Reference
 Biographical Sketch
 Copyright














Title: Application of fluctuation solution theory to strong electrolyte solutions
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Permanent Link: http://ufdc.ufl.edu/UF00086022/00001
 Material Information
Title: Application of fluctuation solution theory to strong electrolyte solutions
Physical Description: xii, 193 leaves : ill. ; 28 cm.
Language: English
Creator: Cabezas, Heriberto, 1952-
Publication Date: 1985
 Subjects
Subject: Electrolyte solutions   ( lcsh )
Chemical Engineering thesis Ph. D
Dissertations, Academic -- Chemical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (Ph. D.)--University of Florida, 1985.
Bibliography: Bibliography: leaves 188-192.
Statement of Responsibility: by Heriberto Cabezas, Jr.
General Note: Typescript.
General Note: Vita.
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Bibliographic ID: UF00086022
Volume ID: VID00001
Source Institution: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 000880471
oclc - 14971957
notis - AEH8290

Table of Contents
    Title Page
        Page i
    Dedication
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
    Key to Symbols
        Page vi
        Page vii
        Page viii
        Page ix
        Page x
    Abstract
        Page xi
        Page xii
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
    Fluctuation theory for strong electrolyte solutions
        Page 11
        Page 12
        Page 13
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        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
    A Model for direct correlation function integrals in strong electrolyte solutions
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
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        Page 75
        Page 76
    Application of the model to aqueous strong electrolytes
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
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        Page 142
        Page 143
    Conclusions and recommendations
        Page 144
        Page 145
        Page 146
        Page 147
    Hard sphere direct correlation function integral from various models
        Page 148
        Page 149
        Page 150
        Page 151
    Relation of McMillan-Mayer theory to Kirkwood-Buff theory
        Page 152
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    Relation of density expansion of the direct correlation function to virial equation of state: alternate mixing rules
        Page 172
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    Exponential integrals
        Page 180
        Page 181
        Page 182
        Page 183
    Model parameters
        Page 184
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        Page 186
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    Reference
        Page 188
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        Page 190
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    Biographical Sketch
        Page 193
        Page 194
        Page 195
    Copyright
        Copyright
Full Text












APPLICATION OF FLUCTUATION SOLUTION THEORY TO
STRONG ELECTROLYTE SOLUTIONS










By


HERIBERTO CABEZAS, JR.


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



UNIVERSITY OF FLORIDA


1985





















To Flor Maria














ACKNOWLEDGMENTS


I would like to express my sincere gratitude to

Professor J.P. O'Connell, a man of wisdom and knowledge, for

his guidance and encouragement during the course of this

work.

I also wish to thank Drs. G.B. Westermann-Clark and

C.F. Hooper, Jr. for serving on the supervisory committee

and for making very pertinent suggestions regarding this

work.

It is a pleasure to thank Mrs. Smerage for her

excellent typing and patience and Mrs. Piercey for her help

with the figures.

Finally, I am grateful to the Chemical Engineering

Department of the University of Florida for financial sup-

port and for providing the kind of intellectual environment

in which this work could take place. I am also grateful to

the National Science Foundation for providing the financial

support that made this work possible.














TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS.................................. iii

KEY TO SYMBOLS...................................... vi

ABSTRACT.......................................... xi

CHAPTERS

1 INTRODUCTION......... ...... .................. 1

2 FLUCTUATION THEORY FOR STRONG ELECTROLYTE
SOLUTIONS .................................. 11

Introduction................................ 11
Thermodynamic Property Derivatives and
Direct Correlation Function Integrals... 12
Direct Correlation Function Integrals from
Solution Properties..................... 23
Summary ................ .................. .29

3 A MODEL FOR DIRECT CORRELATION FUNCTION
INTEGRALS IN STRONG ELECTROLYTE SOLUTIONS.. 33

Introduction................................... 33
Philosophy of the Model.................... 33
Statistical Mechanical Basis ............... 37
Expression for Salt-Salt DCFI.............. 51
Expression for Salt-Solvent DCFI............ 63
Expression for Solvent-Solvent DCFI........ 69
Summary...................................... 74

4 APPLICATION OF THE MODEL TO AQUEOUS STRONG
ELECTROLYTES ......... ............... ....... 77

Introduction................................ 77
Calculation of Solution Properties from
the Model. .............................. 78
Model Parameters from Experimental Data.... 90
Comparison of Calculated Properties with
Experimental Properties.................. 104








Discussion. ............................... .105
Conclusions ............................... 113

5 CONCLUSIONS AND RECOMMENDATIONS............ 144

APPENDICES

A HARD SPHERE DIRECT CORRELATION FUNCTION
INTEGRAL FROM VARIOUS MODELS................ 148

B RELATION OF McMILLAN-MAYER THEORY TO
KIRKWOOD-BUFF THEORY...................... 152

C RELATION OF DENSITY EXPANSION OF THE
DIRECT CORRELATION FUNCTION TO VIRIAL
EQUATION OF STATE: ALTERNATE MIXING
RULES. ...................................... 172

D EXPONENTIAL INTEGRALS........................ 180

E MODEL PARAMETERS........................... 184

REFERENCES ........................................ 188

BIOGRAPHICAL SKETCH............................... 193














KEY TO SYMBOLS


ai = hard sphere diameter of species i.

a. = distance of closest approach of species i and j.
13
1/2
B = K/II/2

B. = sum of all bridge diagrams, second virial
13 coefficient.

C = mixture third virial coefficient.

C.. = direct correlation function integral for species
13 i and j; two-body factor in third virial
coefficient.

C =direct correlation function integral for
components a and 3.

C.i = third virial coefficient for i, j, k.
ijk
AC.. = short range direct correlation function
13 integral.

c.. = direct correlation function.
1J
Ac.. = short range direct correlation function.
13
D = dielectric constant of solvent or solvent mixture.

D = pure solvent dielectric constant.

E = exponential integral or order n.

e = electronic charge.

AF = spatial integral of Af...

-u../kT
f. = e -1 = Mayer bond functions.
13








Af. = f.. f.HS fLR = differences of microscopic
1] 13 jj ij
two-body coefficient.

g = pair distribution function.


1 n 2
I Z. p = ionic strength
i=l


K2


k

N

N.

N
o
N
Oct
n


n
o

P
INT
qi

r,r.

r.
-1


vii


2
= -8T I = Debye-Huckel inverse length.
DkT


=Boltzmann's constant.

= total number of moles of all species.

= total number of moles of species i.

= total number of moles of all components.

= total number of moles of component a.

= number of different species, integer greater
than one.

= number of different components.

= pressure.

= internal partition function.

= separation between species i and j.

= position vector of i.


6 1/2
= (2 3k3T3) = Debye-Huckel limiting law
D k T

efficient.

= temperature.

= pair potential.

= total system volume.

= partial molar volume of species i.


S
Y




T

u.
13
V

V.
i








Vo = partial molar volume of component a.

W.. potential of mean force.
13
X. = N/N = mole fraction of species i.

N
X = mole fraction of component a on a
oa N
species basis.

Z = dimensionless parameter in exponential integral.

Z. = valence of ion i.
1
a = Euler's constant, empirical universal constant
for ion-solvent correlations.

i = activity coefficient of species i.

a = activity coefficient of component a.

6.. = Kroniker delta.
13
8 = Eulerian angle between a charge and a dipole.

81i,'li = Eulerian angles of dipole of solvent molecule i.

< = isothermal compressibility.

Ki = isothermal compressibility of pure solvent (1).

A. = ideal gas partition function.

. = chemical potential of species i.

l = dipole moment of solvent.
Via = number of species i in component a.

V = total number of species in component a.

n K
S = -6 p. a. = reduced density.
K 6. 1 1
i=l

S = P., osmotic pressure.
N
p density of all species.

p = vector of species densities.


viii







N.
P = = density of species i.


N
oca
po = density of component a.


.ijk = spatial integral of .ijk"

Aijk = spatial integral of Ai .j
1ik ijk

ijk = microscopic three-body coefficient.
HS
Aijk = ijk ijk = difference of microscopic three-

body coefficients.

= orientation dependence of dipole-dipole
interaction.

Q = f dw. = integral over orientation coordinates.
1
. = angular orientation coordinates of i.
1

1 n 2
= Z v. Z..
Y 2 i= y 1



Superscripts


F = Final.

FLL = Friedman's limiting law.

HNC = hypernetted chain.

HS = hard sphere.

KB = Kirkwood-Buff.

LR = long range or field type correlations or
interactions, Lewis-Randall.

MM = McMillan-Mayer.

P = Pure component.

PY = Percus-Yevick.










R

SAT

TB

0,O0


solvent.

component.


Special Symbol



< > = integration over orientation.
w


= Reference.

= Saturated.

= Three body.

= infinite dilution in salt.




Subscripts


= species.

= components.


jj,






0 =


.

.














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of
Doctor of Philosophy


APPLICATION OF FLUCTUATION SOLUTION THEORY TO
STRONG ELECTROLYTE SOLUTIONS

By

Heriberto Cabezas, Jr.

August, 1985

Chairman: Dr. J.P. O'Connell
Major Department: Chemical Engineering

Fluctuation solution theory relates derivatives of the

thermodynamic properties to spatial integrals of the direct

correlation functions. This formalism has been used as the

basis for a model of aqueous strong electrolyte solutions

which gives both volumetric properties and activities.

The main thrust of the work has been the construction

of a microscopic model for the direct correlation func-

tions. This model contains the correlations due to the hard

core repulsion, long range field interactions, and short

range forces. The hard core correlations are modelled with

a hard sphere expression derived from the Percus-Yevick

theory. The long range field correlations are accounted for

by using asymptotic potentials of mean force and the hyper-

netted chain equation. The short range correlations which







include hydration and hydrogen bonding are modelled with a

density expansion of the direct correlation function. The

model requires six parameters for each ion and two for

water. The ionic parameters are valid for all solutions

and those for water are universal.

The model has been used to calculate derivative prop-

erties for six 1:1 electrolytes in water at 25C, 1 ATM. The

calculated properties have been compared to experimentally

determined values in order to confirm the adequacy of the

model.


xii













CHAPTER 1
INTRODUCTION


Aqueous electrolytes are present in many natural and

artificial chemical systems. For example, the chemical

processes of life occur in an aqueous electrolyte medium.

All natural waters contain salts in concentrations ranging

from very low for fresh water to near saturation for geo-

thermal brines. Industrially, electrolytes are used in

azeotropic distillation, electrical storage batteries and

fuel cells, liquid-liquid separations, drilling muds, and

many other processes. Since a quantitative description

of the properties of these systems is required for under-

standing, design, and simulation, the ability to predict

and correlate the solution properties of electrolytes is

both scientifically and technologically important.

In attempting to fill this need, many models of aqueous

salt solutions have been developed. Essentially all describe

only activities of the components but ignore the volumetric

properties. Several extensive reviews of electrolyte solu-

tion models are available in the literature (Pytkowicz,

1979; Mauer, 1983; Renon, 1981). To be concise, the various

models have been classified here into three general cate-

gories and a few examples of each briefly discussed. First,







there are models based on relatively rigorous statistical

mechanical results which can be called "theoretical."

Second, there are those composed of a mixture of rigorous

theory and empirical corrections which can be named "semi-

empirical." Third, there are those models which directly

correlate experimental data and are thus termed "empirical."

Neither this classification nor the following list pretends

to be either unique or all-inclusive.

Among the "theoretical" models, the earliest and still

the most widely accepted is the theory of Debye and Huckel

(1923) which gives the rigorous relation at very low salt

concentration (the limiting law) for salt activity coeffi-

cients but fails at higher salt concentration. This theory

has been amply treated in the literature (Davidson, 1962;

Harned and Owen, 1958). The Debye-Huckel theory considers

an electrolyte solution as a collection of charged hard

spherical ions embedded in a dielectric solvent which is

continuous and devoid of structure. This is the physical

picture generally called the "Primitive Model." The correct

formalism for the application of modern statistical mechani-

cal techniques to the "Primitive Model" is given by the

McMillan-Mayer theory (1945). A major method developed

for this formalism is a resumed hypernetted chain approxima-

tion to the direct correlation function. This, together

with the Ornstein-Zernike equation (1914), forms a solvable

integral equation for the primitive model ion-ion








distribution function which has been used to calculate

the properties of electrolytes up to 1 M salt concentration

(Rasaiah and Friedman, 1968; Friedman and Ramanathan, 1970;

Rasaiah, 1969). This method requires tedious numerical

calculations to obtain the properties. A simpler and more

generalizable approach is the Mean Spherical Approximation

(MSA) which has been applied to both primitive (Blum, 1980;

Triolo, Grigera, and Blum, 1976; Watanasiri, Brule, and

Lee, 1982) and nonprimitive (Vericat and Blum, 1980;

Perez-Hernandez and Blum, 1981; Planche and Renon, 1981)

electrolyte models. The MSA method essentially consists

of solving the Ornstein-Zernike (1914) equation for the

distribution functions subject to the boundary conditions

that the total correlation function is minus one inside

the hard core and that the direct correlation function

equals the pair potential outside the hard core. This

is equivalent to the Percus-Yevick method for rigid nonionic

systems (Lebowitz, 1964). The MSA generally gives good

thermodynamic properties if these are calculated from the

"Energy Equation" (Blum, 1980). It does not yield good

correlation functions and further suffers from the need

to numerically solve complex nonlinear relations for the

value of the shielding parameter at each set of conditions.

This last problem grows progressively worse as the sophisti-

cation of the model increases. Due to their complexity none







of the modern "theoretical" models is widely used in

engineering practice.

The most successful of the semiempirical models is

that due to Pitzer and coworkers (Pitzer, 1973; Pitzer

and Mayorga, 1973; Pitzer and Mayorga, 1974; Pitzer, 1974;

Pitzer and Silvester, 1976). Model parameters for activity

coefficients have been evaluated for a large number of

aqueous salt solutions, but volumetric properties and multi-

solvent systems have not been treated. To construct the

model, Pitzer adopted the "Primitive Model" and inserted

the Debye-Huckel radial distribution function for ions

into the osmotic virial expansion from the McMillan-Mayer

formalism. This latter is analogous to using the "Pressure

Equation" of statistical mechanics (Pitzer, 1977). The

resulting expression contains the correct limiting law.

He then added empirical second and third virial coefficients

which are salt and solvent specific. Although Pitzer's

model correlates aqueous activity coefficients superbly,

it does not add to the fundamental understanding of these

solutions; further, its extension to multisolvent systems

would pose some serious problems associated with the mixture

dielectric constant as has been recently pointed out (Sander,

Fredenslund, and Rasumussen, 1984). Another semiempirical

approach uses the NRTL model for solutions of nonelectrolytes

(Renon and Prausnitz, 1968) adapted for short range ion

and solvent interactions (Cruz and Renon, 1978; Chen, Britt,








Boston, and Evans, 1979) in nonprimitive models of electro-

lyte solutions. Cruz and Renon separate the Gibbs energy

into three additive terms: an elecrostatic term from the

Debye-Huckel theory, a Debye-McAulay contribution to correct

for the change in solvent dielectric constant due to the

ions, and an NRTL term for all the short range intermolecular

forces. Chen et al. adopted a Debye-Huckel contribution

and an NRTL term for the Gibbs energy but no Debye-McAulay

term. More recently, the UNIQUAC model for nonelectrolytes

has been modified for short range intermolecular forces

in electrolyte solutions (Sander, Fredenslund, and Rasmussen,

1984). The resulting UNIQUAC expression has been added

to an empirically modified Pitzer-Debye-Huckel type electro-

static term to form the complete Gibbs energy model.

Although the two NRTL and the UNIQUAC models correlate

activity coefficient data reasonably well even in multi-

solvent systems, they have to be regarded as mainly

empirical. First, their resolution of the Gibbs energy

into additive contributions from each different kind of

interaction is not rigorous. Second, the problems associated

with the mixture dielectric constant are resolved in an

empirical and somewhat arbitrary fashion. As a result,

such models add little to our understanding of these systems

and may not be reliable for extension and extrapolation.

Of the various empirical methods developed, two have

been chosen to be discussed here because they represent







distinct approaches. First, there is the method of Meissner

(1980) which is a correlation for the salt activity coeffi-

cient in terms of a family of curves that are functions

of the ionic strength and a single parameter which can

be selected from a single data point. This method has

been extended to multicomponent electrolyte solutions and

is useful over a wide range of salt concentration (0.1-20

MOLAL), though it is not very accurate. Second, there

is the method of Hala (1969) which is more conventional

in that it consists of a purely empirical model for the

Gibbs energy of the solution. This method is an excellent

correlational tool, but it is not predictive. It has four

parameters per salt-solvent pair.

The existence of so many models to correlate and predict

the thermodynamic behavior of electrolyte solutions is

indicative of the complexity of these systems and, perhaps,

the relatively poor state of the art.

As examples of the physical complexity of electrolyte

solutions, the composition behavior of the salt activity

coefficient (Figure 1) and of the species (ions and solvents)

density (Figure 2) is presented. Figure 1 shows the large

deviation from ideal solution behavior (y= 1) even at

very low salt concentration for all salts. Second, it

indicates that salts of the same charge type show similar

behavior at low salt concentration but are widely different

at higher salt concentration. In Figure 2, the difference







in the salt composition behavior of the species density

is obvious even for relatively similar salts, i.e., the

solution seems to expand for KBr while it seems to contract

for all other salts. The activity coefficient data were

taken from the compilation by Hamer and Wu (1972). For

NaCl and NaBr the density data of Gibson and Loeffler (1948)

were used. For LiCI, LiBr, and KBr the density data were

taken from the International Critical Tables. For KC1

the density data of Romankiw and Chou (1983) were used.

In the hope of improving the situation for obtaining

properties of solutions, a new model of strong aqueous

electrolyte solutions is presented here. This model has

been carefully constructed so that it overcomes a number

of the deficiencies of previous methods. For example,

this model is simple enough for economical engineering

calculations, yet sufficiently sophisticated to rigorously

include all the different interactions (ion-ion, ion-solvent,

solvent-solvent) and the principal physical effects (electro-

static, hard core repulsion, hydration, etc.) that contribute

to each interaction. The model is also extendable to multi-

salt and multisolvent systems in a straightforward fashion.

Finally, it addresses both activity and volumeric

properties.

In the chapters that follow, a detailed development

of the new model is presented. Chapter 2 has the general

relations between solution properties and correlation






8

functions. Chapter 3 has the full development of the new

model. Chapter 4 shows the application of the model to

solutions of aqueous strong electrolytes and the calculation

of solution properties. Chapter 5 has suggestions for further

work and conclusions.
















1.4



1.2




1.0




0.8



0.6



0.4
0.0


Figure 1.


0.5 1.0 1.5 2.0 2.5
1
(Molality) 2


Salt Activity Coefficient in Water at
250C, 1 ATM. Data of Hamer and Wu
(1972).














0.06



0.59


0.58



0.57



0.56



0.55


0.54


0.00 0.02 0.04 0.06

X05


Figure 2.


0.08 0.10 0.12


Species Density in Aqueous Electrolytes
at 250C, 1 ATM. For data sources see
text.













CHAPTER 2
FLUCTUATION THEORY FOR STRONG ELECTROLYTE SOLUTIONS


Introduction


There are three general relations among the thermodynamic

properties of a solution and statistical mechanical correlation

functions. The first two are the so-called "Energy Equation"

and "Pressure Equation" which are obtained from the canonical

ensemble with the assumption of pairwise additivity of inter-

molecular forces. These equations relate the configurational

internal energy and the pressure respectively to spatial

integrals involving the intermolecular pair potential and

the radial distribution function (Reed and Gubbins, 1973;

McQuarrie, 1976). The third relation is the so-called "Com-

pressibility Equation" which is derived in the grand canonical

ensemble without the need to assume pairwise additivity of

intermolecular forces. This equation relates concentration

derivatives of the chemical potential to spatial integrals

of the total correlation function (Kirkwood and Buff, 1951)

and to spatial integrals of the direct correlation function

(O'Connell, 1971; O'Connell, 1981). This last method is

generally known as Fluctuation Solution Theory.

Fluctuation solution theory has been applied to the

case of a general reacting system (Perry, 1980; Perry and







O'Connell, 1984), and the formalism has also been adapted

to treat strong electrolyte solutions which are considered

as systems where the reaction has gone to completion (Perry,

Cabezas, and O'Connell, 1985). The main body of this chapter

consists of a derivation of the general fluctuation solution

theory. Although the final results are identical to those

previously obtained by Perry (1980), the development is

more intuitive and mathematically simpler, though less

general. The remainder of the chapter illustrates the

calculation of direct correlation function integrals (DCFI)

from solution properties and sets theoretically rigorous

infinite dilution limits on the DCFI's.


Thermodynamic Property Derivatives and Direct
Correlation Function Integrals


A general multicomponent electrolyte solution, contain-

ing n species (ions and solvents) formed from no components

(salts and solvents) by the dissociation of the salts into

ions, is not composed of truly independent species due

to the stoichiometric relations among ions originating

from the same salt. It is, therefore, not possible to

change the number of ions of one kind independently of

all the other ions. However, the independence of ions

has been assumed traditionally for theoretical derivations,

and it will lead us to the correct results by a relatively

simple mathematical route. Thus, with the assumption that

any two species i and j are independent of all other species,







Fluctuation Solution Theory gives the following well known

result (O'Connell, 1971; O'Connell, 1981):

1 ^i 6.. c.
1 (2-1)
RT 9N. N. N
T,V,Nkj

where = the chemical potential per mole of
species i.

N. = the number of moles of species i.

N = the total number of moles of all species.

6.. = the Kroniker delta.
13

2
C. = 4Tp J r dr = spatial integral of
13 1=3
the direct correlation function.
N
p = = molecular density of all species.
V


The microscopic direct correlation function
13 W
is an angle averaged direct correlation function defined

by



ij> c1 dw. dw. (2-2)
'J j2 ij 1 J

Q
where Q2 = dw. d .



In order to arrive at the first and simplest of the

desired relations, we define the activity coefficient for

species i on the mole fraction scale as








(T,P) = MP(T) + RT In X.Y.(T,P) (2-3)



where = the reference chemical potential.
1 Ni
Xi = N = mole fraction of species i.

Yi = the activity coefficient of species i.

P = the vector of species mole densities.



By differentiating equation (2-3) with respect to

the number of moles of species j, we obtain


1 i
RT 3N.
T,V,Nk

ny+ i1 1 (2-4)
3N. N. N
3 T,V,Nk j

which upon insertion in equation (2-1) gives


1ny. 1 C..
1 = N1i (2-5)
3N. N
3 T,V,Nkj

and when multiplied by the system volume on both sides

of the equation,


81ny. 1 C..
1 1- (2-6)
p Tp p
J TPkfj

N-r
where p = n-= molar density of species i.
J V







By performing a sum over all species i and j on equation

(2-6)



S n n lny.
v v- v iaej B Pj
a B i=l j=l j T
iT'Pk~j


n n 1-C..
S1 D1 (2-7)
a i=l j=l


where vi = number of species i in component a.

Sa = total number of species in component a.



By noting the definition of the mean activity

coefficient of a component a,


1 n
Iny, = V. Iinyi (2-8)
a v a i-i i
a i=l


and also assuming that species j is formed from an arbitrary

component 8 so that



Pj = VjB Po (2-9)



one then arrives at the first relation


1 alny
1 aOB
T,Po


(2-10)
1 n n 1-C .
i=1 j=l a6








which upon identification of


n n 1-C..
1 C = (2-11)
i=l j=l a3S

assumes the simpler form


alny
PV p= v (1-C ) (2-12)
pv T



Equation (2-12) relates the DCFI to the derivative

of the activity coefficient of any component a (salt or

solvent) with respect to the molar density of any component

3 (salt or solvent) at constant volume, temperature, and

mole number of all components other than 8.

Because most experiments and many practical calculations

are performed at constant pressure rather than constant

volume, it is of interest to derive a relation between

the activity coefficient derivatives at constant pressure

and direct correlation function integrals. First, a change

of variables is executed.

i i
3N. N.
3 T,V,Nkj 3 T,P,Nk/j


yi N .P (2-13)
aN.
T,N T,V,Nk^j

and the following identifications are made,









i |
p = V (2-14)
T,N


aP
av v.
P T,N = r (2-15)
aN. @N. VK
3 TV'Nkj V TPNkj
T,PNkj


Then equations (2-1), (2-14), and (2-15) are inserted into

equation (2-13) to obtain


lli 6. C.. V.V.
1 = -1 13 1 3 (2-16)
RT 3N. N. N VKTRT
3 T,P,Nk 1 T

where = partial molar volume
1 8N.
1 T,P,Nkfi
volume of species i.

V = the system volume.
-1 av
KT V P = isothermal compressibility.
T,N


To develop a relation in terms of activity coefficients,

the chemical potential is written as in equation (2-3)

and the proper constant pressure derivative is taken.


1 i lny
RT aN. = Nj +
3 TP,Nk.j TP,Nkj

6.
13 1 (2-17)
N N


From equations (2-16) and (2-17),





18

81ny. 1 -e. V.V.
1 = L- 1 3
aN N VK RT (2-18)
T,P,Nk#.


By multiplying equation (2-18) by the system volume

and rearranging, one finds


N alny. 1 C V. V.
1 1
pKTRT aN pKTRT TRT K TRT (2-19)
T,P,Nkj

upon which a double summation over species i and j is

performed to obtain


N 1 n n alny
PK RT v v L IiaVji 1N.
T Ta i=l j=l 1
T,P,Nkfj

1 1 n n
1 v. (1-C..) -

KT a 8 i=l j=1

n n
12 1. v V V. (2-20)
(K RT)2 V v i=l j=l a j8 i j
T a B

Equation (2-20) must be simplified so that all the properties

appear as component rather than species quantities. This

is done with the aid of equations (2-8), (2-11), and the

assumption that species j is formed from an arbitrary

component 8 so


Nj = vjNo (2-21)








Additionally, the partial molar volume of a component a

or B is expressed as a sum of the species partial molar

volumes.


-V 1 aV
V- N- N (2-22)
DN NiB No
1 iS Of3
T,P,Nk5i T,P,N



V n
Vo8 N =i i Vi (2-23)
S T,P,N
oY$S
Equation (2-20) is now transformed to

Nva alny
PK RT 8N
T o5
T,P,N


v V (1-C ) V V (2-24)
a s aS oa oS (2-24)
PKTRT K RT KTRT

To make further progress, the relationship of the
Vo
bulk modulus of the solution (p TRT) and the group, -T,

to the direct correlation function integrals must be found.

First, the compressibility equation is derived from the

basic fluctuation theory result of equation (2-1) starting

with the Gibbs-Duhem equation for an isothermal but

nonisobaric process.

n
N N.dP. = VdP (2-25)
i=l 1 1

Upon differentiation of equation (2-25) with respect

to the mole number of an arbitrary component j,










SN. = V P
i=1 i T'V j aN
T,V,N dj T,V,N kj

and by insertion of the equation (2-1),


n
RT Z (6 X.C..) = V
i=l ij 13 N.
ST,V,Nkj


n
1 P 1- 1 x. C.
RT ap T Fk 1 13
RT j i=
T'Pk$j


(2-26)







(2-27)





(2-28)


Equation (2-28) is the general multicomponent compres-

sibility equation expressed in terms of species quantities.

This relation is now transformed to one in terms of com-

ponents by performing a summation over species j and use

of equations (2-9) and (2-29).


n
o
X = 1 Vi X o
S B=lI


1 n ap
RT X Vj
T,PR~i


(2-29)


n n
= 1 X.v C.
i=1 j= 1


(2-30)


1 DP
RT 3p
oa
T,p

n
o n n
1 XI v. Xi v .' C..
@=1 i=1 j=1


(2-31)








which by use of equation (2-11) becomes

n
1 @P 1 = v Xo Ca (2-32)
RT Dp oa 0 0
T,p


Equation (2-32) is the multicomponent compressibility

equation expressed in terms of components. The density

derivative of pressure is related to the partial molar

volume as



-V P aV o (2-33)
ap V 8N
o T,N oa T
T,p o T,P,N

which when inserted in equation (2-32) gives one of the

desired relations.


n
V o
oa v v X (1-C ) (2-34)
KTRT a O~ X


In order to relate the bulk modulus to direct correla-

tion function integrals, the total volume is related to

the partial molar volumes.


n0 n
V= ON = V No (2-35)
VN oa oa oa
a=l oa T,P,N a=1
T,P,N

Dividing equation (2-35) by the mole number of species

(N) yields


1 V n
S= X (2-36)
p N oa oca
a=l








and when applied to equation (2-34)


aP/RTI
D IT,N


n n
o o
RT= 1 VX X (1-Ca)
OK RT l a 6 oa O (
T a=1 8=1


which is the second necessary relation.

Substitution of equation (2-34) and equation (2-37)

into equation (2-24) gives


Nva 1nya
pKTRT 8No
T,P,N
YO


= v
a R


n n
o o
SV vX X [(l-C )(l-C ) (1-C )(1-C )]
y=l 6=1 Y 6X y 06 y6 ay a5

(2-38)

which is further transformed by substituting for the bulk

modulus


Nv ny a
a N
T,P,Noy3


n n
o o
v v I vI v v X Y6[(1-C Y)(1-C a)-(1-C Y)(1-C6 )]
y=l 6=1 o 06
n n
o o
S V X oyX 06(1-C y6
y=l 6=1 (2-39)

Equation (2-39) relates changes in the activity coefficient

of any component a with changes in the mole number of any

component 8 where the process occurs at constant pressure


(2-37)








and temperature to sums of direct correlation function

integrals for components.

In summary, it should be noted that of the various

relations developed in this section,only a few are of prac-

tical importance in relation to this work. These are listed

at the beginning of the next section.


Direct Correlation Function Integrals from
Solution Properties


The previous section consists of a relatively simple

but lengthy derivation of several basic relations between

solution properties and direct correlation function

integrals. The relations that are of most importance to

this work are listed below.


1ainy
PV D v v (1-C ) (2-12)






V n
oco
oa (2-34)
TRT- V V X (1-C ) (2-34)
KTRT a =1 o aB




n n
SP/RT 1 o o
~ P/RT 1 v0 vX X (1-C- 1 (2-37)
P T,N TRT a=l =iTxaB=1









Nva 1nya
PKTRT 8No0
T,P,N,
Y 8


n n
o o

a 8 6=1 Y 6 oy 0
Y=l 6=1


[(l-Cy )(1-Ca ) (1-Cay)(l-C68)]






N N v. v. (1-C..)
S-C = ia jB 1
i=l j=l a Y


(2-38)







(2-11)


Useful bounds on the value of the direct correlation

function integrals as the system approaches infinite

dilution in all components except one (usually the solvent)

can be deduced from the preceding relations. Thus, by

taking the limit of pure solvent (component 1), one obtains


---0M
V
oa = (1-C )0
v K RT al
a1




S (1-C11 )
P K RT
ol 1


(2-40)





(2-41)


N 31ny
ol alnya
V 3N
1 No IT,P,N
Y

N 31ny
ol a
V 3N 0
3 ~o T,P,N
Y


-(1 V K 1KRT



(1-Cal) (L-C )
(1-C(1-C )
(l- 11
#B


2-42a)


(2-42b)








m LIM
where (1-C ) = (1-C )



and where equations (2-42) represent a constant pressure

limit on the DCFI. A corresponding constant volume limit

can be obtained from equation (2-12).

For a binary system consisting of one solvent (1)

and one salt (2), the fluctuation relations become


91ny y
n2 22 (2-43)
ap02 T 01
Po2
T'Pol


o2 = (1-C ) + V2X (1-C ) (2-44)
SRT = 2Xol 12 2 o2 22
T



aP/RT 1 2
p pKRT ol (1-C )+
3P T,N PKTRT 01 11

2 2
2olXo2(-12) + (-22 (2-45)




Nv2 31ny2
pK RT 8No2
T,P,No



2X [(1-C 1 )(1-C22) (1-C12)2] (2-46)
2 ol 11 22 12



v+2 (1-C 1+) + V-2 (1-C 1-)
1 -C12 = (2-47)
12 v







1 C22


2 2
v (1-C )+2+2v (1-C )+ (1-C__ )
+2 ++ +2-2 +- -2


(2-48)


and the respective infinite dilution limits are


V
\K2RT 1-C21
V 2Kl1RT = (C21


1
P p K RT
ol 1


N1 lny2
"2 a2N
o2IT


(l-Cll)


(2-49)


(2-50)


(2-51a)


p 2
Po (V )
22 V2 K RT
2 1


Nol
V2


a1ny2
aN o l
02
T,P,N
o1


-( 1-C122
(1-C ) (1-C12
22 (1-C11)


(2-51b)


The significance of equations (2-51) can be further

understood by realizing that any correct model for the


,P,N ol







activity coefficient of an electrolyte must approach the

Debye-Huckel Limiting Law at very low salt concentration.

Thus, the mean activity coefficient of a salt on the mole

fraction scale is given by this law as


1 n 2 1/2
lny = S ( Z) (2-52)
ny2 i=l 2



6 1/2
where S =( 332T )
S Dk T


Debye-Huckel limiting law coefficient.

e = the electronic charge.

D1 = pure solvent dielectric constant.

k = Boltzmann's constant.

T = temperature.

Z. = valence of ion i.


1 2
I = ZiP = ionic strength.
i=l


and when the proper derivative is taken,

N P -1/2
0ol 2 y ol
2 3N2 4v2
T,P,N 2


n n 2 2
v1 1 i2 Ziz2 (2-53)
i=l j=l







Insertion of equation (2-53) into equation (2-51a)

gives

P -1/2
Sp I n n
Y ol- 2 +
2 i2j22 1
4v2 i=1 j=1


P 2
Po (V O)
21 02 = (1-C2) (2-54)

v2 1RT


which approaches negative infinity as the salt concentration

approaches zero.

In order to construct a model capable of correlating

and predicting the solution properties of electrolytes,

it is helpful to calculate the experimental behavior of

the DCFI's from solution properties. To that purpose,

equations (2-43), (2-44), and (2-45) have been inverted

so that the three DCFI's can be calculated from


1 2
1 C11 = [1 X V p] +
11 2 OKRTo2 o2
XolP TRT


2
X 2 iny2
ol V N 2 (2-55)
2 2 DN
ol T,P,No


Vo2 2
1-C =1[-X V p1-
12 V2XolKTRT -o2 o2



Xo2 N n 2 (2-56)
X N
0ol o2 T
T,P,Nol










2
p(Vo ) N in 2
1 C = -- (2-57)
22 2 vK aN
2 T T,P,Nol



Figures 3-5 show the results of equations (2-55) and

(2-57) for six different salts at 1 ATM and 25C. The

compressibility data used were those of Gibson and Loeffler

(1948) for NaCL and NaBR. For LiCL, LiBR, KCL and KBR

the compressibilities of Allam (1963) were used. The

activity coefficient data were taken from the compilation by

Hamer and Wu (1972). The density data of Gibson and Loeffler

(1948) were again used for NaCL and NaBR. For LiCL, LiBR,

and KBR the density data were taken from the International

Critical Tables. The newer density data of Romankiw and

Chou (1983) were used for KCL. The pure water data were

those of Fine and Millero (1973). The infinite dilution

partial molar volumes were also from Millero (1972).


Summary


The present chapter has introduced the basic relations

of interest, has shown how they have been used to calculate

the experimental behavior of the DCFI's, and has given

some bounds on the values of the DCFI's. The next chapter

introduces a model for correlating the observed experimental

behavior of the DCFI's.














30


25


20



15
1 -C22

10



5


0


0.00 05 02
0.00 0.02


0.04 0.06 0.08 0.10


X02


Figure 3. Salt (2)-Salt (2) DCFI in Aqueous
Electrolyte Solutions at 250C, 1 ATM.
For data sources see text.


0.12
















24



20



1-C2 16


12


8


0.00 0.02


0.04 0.06 0.08 0.10 0.12


X05


Figure 4. Salt (2)-Water (1) DCFI in Aqueous
Electrolyte Solutions at 250C, 1 ATM.
For data sources see text.















28


26



24
1-C 12

22



20


18


0.00 0.02 0.04 0.06 0.08 0.10 0.12

X05



Figure 5. Water (1)-Water (1) DCFI in Aqueous
Electrolyte Solutions at 250C, 1 ATM.
For data sources see text.














CHAPTER 3
A MODEL FOR DIRECT CORRELATION FUNCTION INTEGRALS
IN STRONG ELECTROLYTE SOLUTIONS


Introduction


In order for the formalism introduced in the previous

chapter to be of practical value, a model to express direct

correlation function integrals in terms of measurable

quantities (p, T, x) must be constructed. The present

chapter describes such a model. First, a general physical

picture of electrolyte solutions and its relation to micro-

scopic direct correlation functions is discussed. Second,

a rigorous statistical mechanical basis is laid for the

microscopic direct correlation functions and their spatial

integrals. Third, equations are given for each type of

pair correlations in the system (ion-ioin, ion-solvent,

solvent-solvent). Lastly, a summary is presented of the

model parameters and estimated sensitivity of results to

their values.


Philosophy of the Model


The complex thermodynamic behavior of liquid electro-

lytes is the observable result of the very complex interac-

tions between the species in solution, i.e., the ions and





34

solvent molecules. In the absence of a complete understand-

ing of all these forces, models use simpler or, at least

tractable, interactions which may have the essential charac-

teristics of the real forces. In addition, some semiempiri-

cal terms are used to account for those interactions that

cannot be simply approximated.

Thus the interactions between the ions at long distances

are modeled as those of charges in a dielectric medium

containing a diffuse atmosphere of charges. At very short

range, however, the dominant interaction becomes a hard

sphere-like repulsion. There exist rigorous statistical

mechanical methods to treat these two types of interactions,

but these two are not adequate to correlate and predict

the solution behavior with sufficient accuracy. Interactions

that are important at intermediate ion-ion ranges must

be incorporated. Unfortunately, these intermediate range

forces cannot be simplistically approximated because they

involve strong many-body effects such as dielectric satura-

tion, ion-pairing, polarization, etc., which are not well

understood. In the present model the ionic and hard sphere

interactions are treated theoretically while the rest are

included in a semiempirical fashion.

The interactions between ions and solvent molecules

at large separation can be treated as those of charges and

multipoles in a dielectric medium containing an ionic

atmosphere. In general, quadrupoles and higher order







multipoles are not included, because their contribution

is expected to be numerically insignificant in an aqueous

system. The short range interactions are treated as hard

sphere repulsion. Intermediate range forces for the ion-

solvent case are very important because they include solva-

tion which makes a larger contribution than the long range

charge-multipole forces. Solvation of the ion by the solvent

is intimately related to the partial molar volume of the

salt and must be incorporated if there is to be any hope

of correlating and predicting the volumetric behavior of

the solution. As for ions, the long and short range intera-

tions are treated theoretically while the intermediate

range forces are incorporated semiempirically.

The forces between solvent molecules at long range

can be considered to be those of dipoles in a dielectric

medium which has an ionic atmosphere. Higher order multi-

poles may again be neglected because their contributions

are less important and can be covered in other ways. The

short range forces are again treated as hard sphere repul-

sions. The intermediate range interactions for the solvent-

solvent case are dominated by association type forces such

as hydrogen bonding which make a larger contribution than

the long range dipole-dipole term. As above, the long

and short range interactions are treated theoretically

while the effects of the intermediate range forces are

included semiempirically.







In summary, there are three distinct classes of inter-

action: ion-ion, ion-solvent, solvent-solvent. Each class

has unique contributions from long-range, field-type forces,

short-range, repulsive forces, and intermediate range forces.

Traditionally, models have been written for the excess

Gibbs or Helmholtz energy of the system by adding contribu-

tions from some of the above forces in an ad hoc and, gener-

ally, nonrigorous fashion. The fact that free energy

contributions do not naturally separate into the types

of forces and that experimental values for each cannot

be separately determined has caused many of these models

to be complex and/or inconsistent. Further, they do not

yield volumetric properties along with the activities.

Within the framework of Fluctuation Solution Theory,

the contributions of the pair correlations to the thermo-

dynamic properties can be rigorously added. Thus, there

are terms from the salt-salt, salt-solvent, and solvent-

solvent DCFI's, as shown in Chapter 2. Further, the experi-

mental behavior of each of the three DCFI types can be

separately calculated from solution data as seen in the

previous chapter. It is then possible to construct separate

and accurate models for each one of the DCFI's. These

models can later be manipulated to yield thermodynamic

properties.

As may be inferred from the above discussion, each

of the three types of DCFI's contains long range, short







range, and intermediate range interactions. These can

be theoretically separated into a simple additive form

as will be shown in the next section of this chapter.

It is important to note that the separation is first

developed at the level of microscopic direct correlation

functions which are later integrated to obtain the DCFI's.

Although our particular additive separation of the micro-

scopic direct correlation function is not fully rigorous,

we believe it is more reasonable than a similar resolution

of the radial distribution function into an additive form

(Planche and Renon, 1981). In fact, the radial distribution

function can naturally be resolved only into a multiplicative

rather than an additive form. The intermolecular potential

and, consequently, the potential of mean force can be

approximately decomposed into additive contributions from

interactions of different characteristic range, but this

potential appears in an exponential in the radial distribu-

tion function. Thus, resolving the radial distribution

function into additive contributions is quite inappropriate.


Statistical Mechanical Basis


The above philosophy is a qualitatiave concept which

must be expressed in quantitative terms. To this end,

we now establish a rigorous statistical mechanical basis

for a model of microscopic direct correlation functions.

First, consider the diagrammatic expansion of the direct







correlation function (Reichl, 1980; Croxton, 1975) for

species i and j,


u..
c..(T,p,r ,r ., .) = g. 1 Zn g + B.. (3-1)
13 -j -1- n g ij kT 13

-W.ij ./kT
where gij = e = radial distribution function.

W = potential of mean force.

ui. = pair potential.

B.. = sum of all bridge diagrams also known
as elementary clusters.



Although equation (3-1) is an exact expression for

the direct correlation function, it is of little practical

value because the bridge diagrams cannot be summed analyti-

cally. This series is


Sn n
B.. f.f f .f.f. dr drd d
Bl 2'! Okp fikfk fZjfifkj dkd wkd
k=l =1


+ (3-2)



for a system consisting of n species.


-u. ij./kT
where f.. = e 1 = Mayer bond function.



To obtain the hypernetted chain (HNC) approximation

(Rowlinson, 1965) all of the bridge diagrams are neglected

(B.. = 0). This introduces an error which is second order
1j







in density and ignores some four body contributions. It

is, therefore, exact up to the order of a third virial

coefficient. Thus, the HNC direct correlation function

is


cHNC
ij


= gi
ij


- 1 ng. -
1ij kT


From the definition of the radial distribution

function,


W..
Z.n g..= 11
1] kT


(3-3)


(3-4)


W.. W .
giJ -1 1 kT ) 2
13kT 72T kT


3
1 W.
3! kT


which on insertion into equation (3-3) gives,


2 3
HNC u W.. W..
Cij T 1 3! 1 ) +
ij IT 2! kT 3! kT "


To apply equation (3-6) requires at least approximate

expressions for the potential of mean force in terms of

measurable variables. Such expressions, valid in the limit


(3-5)


(3-6)







of zero salt concentration and large separation between

the two interacting species, are available for ion-ion

interactions from the Debye-Huckel theory and for ion-dipole

and dipole-dipole interactions from more recent work (H'ye

and Stell, 1978; Chan, Mitchell, and Ninham, 1979) which

yields results identical to those of Debye and Huckel for

ionic activities. Thus, the long range direct correlation

function is based on these potentials of mean force, WLR.

Then, our HNC approximation is



LIM
HNC LR
I-*o c.. c.. (3-7)
r ij _0_00 3 13
12


LR 2 LR 3
LR 1 W1 W
cL. + 1 -+ (3-8)
13 kT 2! kT 3! kT



The potentials of mean force, however, are unphysical

inside the hard core of the molecules and must be set equal

to zero.


LR
W R = 0 r.. < a..
i] 1] ji
(3-9)
LR LR
W.. = W.. r.. > a..
ij ij] 3 13


1
where aij = (aii + a..) = distance of closest approach
of species i and j







At contact and inside the core of the molecules, the

direct correlation function is dominated by a very strong

repulsion which is modelled as a hard sphere interaction.

To obtain the appropriate expressions for the hard sphere

direct correlation functions, the Percus-Yevick theory

(1958) was used since it has been shown to give a compres-

sibility equation of state which is in good agreement with

simulation results for hard spheres (Reed and Gubbins,

1973). The Percus-Yevick (PY) microscopic direct correlation

function for hard spheres is zero outside the core. Thus,


HS PY-HS
c.. = c.. (3-10)

HS
uS/kT
PY-HS HS (1 ijkT
cij = gij (1 e ) (3-11)



u. = 0 r.. > a..
13 13
where
HS
u.. = r.. < a..
1] 1] 13


Although the PY microscopic direct correlation function

is formally used in the development that follows, it was

not actually employed in obtaining the final expressions

for the DCFI's. Rather, the expression for the hard sphere

chemical potential as derived from Percus-Yevick theory

through the compressibility equation was used together

with equation (2-1) to obtain the desired relation (see







Appendix A). Although the more exact Carnahan-Starling

(Carnahan and Starling, 1969; Mansoori, Carnahan, Starling,

and Leland, 1971) expression could have been used, it is

somewhat more complex and relatively little improvement

in accuracy would be expected.

At this point, we have established a viable, albeit

traditional, theory for the behavior of the direct correla-

tion function as r.. m and at r.. < a... However, many
1] I] J1
interactions which are important in aqueous electrolyte

systems such as hydration of ions by water, hydrogen bonding

between water molecules, and ion pairing are strongest

at r.. just outside the core. Further, this is that kind

of interaction for which liquid state theory is not well

developed. Therefore, we attempt here to develop a method

for interpolation of the direct correlation function between

long and short range. Because generally available theory

offers little guidance, the method can at best be semiempiri-

cal. For this purpose, the Rusbrooke-Scoins expansion

of direct correlation function (Reichl, 1980; Croxton,

1975) for species i and j in a system of n kinds of species

is now introduced.



cij(T,p,r,rj ,wi'O.) = fij(T) +
n
+ Z pk ijk(T) + (3-12)
k=l


where (T) = J f. fi fj dr
ijk ij ik jk d-k k







Since equation (3-12) represents the entire direct
LR HS
correlation function including cij and cij, these two must

be subtracted to obtain the interpolating function. There-

fore, the complete model for the microscopic direct correla-

tion function for species i and j in a system of n species

is



c.. = c.H + c. + Ac.. (3-13)
] i i] 1]


HS LR
where cij is defined by equation (3-11), c by equation

(3-8), and


HS LR
Ac.. = c c.. c. (3-14)
13 1] 13 3


which is approximated by the Rushbrooke-Scoins expansion

as


HS- LR n
Ac.. = (fi.. f f ) + I (p
] i] ij k= k ijk


HS o LR
P H P o L (3-15)
k ijk k ijk (3-15)

LIM
o LR LIM
where = r.. P
k ijk 13 k ijk
I o

The series in equation (3-12) is truncated at the

first order term in density to be consistent with the HNC







theory and because inclusion of the more complex higher

order terms was empirically unnecessary.

For the sake of simplicity in notation equation (3-15)

is expressed as



Ac = Af.. + n po LR
Skl k ijk ijk3-16)


where Af.. = f.. fHS fLR
13 i] ij ij


HS
A

No attempt was made in this work to analytically calcu-

late the coefficients in equation (3-16); rather, their

spatial integrals were fitted to data. The importance

of equation (3-16), however, is in providing a theoretical

framework for describing the properties for a class of molecu-

lar interactions which are not well understood. Thus,

the first term represents the contribution of pairing or

repulsion in the case of ion pairs, solvation in the case

of ion-solvent pairs, and hydrogen bonding in the case

of solvent pairs. The second term represents the effect

of a third body (k) on the direct correlation between species

i and j. If one or two of the three are solvent and the

rest ions, then this term is dominated by hydration. If

all three species are ions, then this term is dominated







by ion association or repulsion. The physical significance

of these terms will be discussed further below.

As pointed out in Chapter 2, solution properties are

related to spatial integrals of the direct correlation

function. In order to relate this model to thermodynamic

properties, equation (3-13) is integrated over angles first

and separated later. Thus



i c. d dW. (3-17)



C (T,p) = 47'p f r2. dr.. (3-18)
S o 0 ij 13


Cj(T,P) = CHS + CLR + C. (3-19)
ij 13 1j 13


HS
where C is obtained directly from the chemical potential
i3
as shown in Appendix A. Thus, CLR is defined by
ij


LR 47p LR 2
C1 = f r.. dr..
Ci kT o i] ) ij iJ


2 p LR 2 LR2 2dp LR 3 2
+2p> <(w ) 2> r2 dr.-. 2 f <(w ) > r dr.
kT o j 13ij 3kT 0o ij J


+ (3-20)



Lastly, AC.j is defined by formally integrating equation

(3-16).








n LR
ACij = pAFij(T) + p (kAO(T) pk(T)) (3-21)
k=l ijk ijk


where AF(T) = 47 J r.2 dr.
ij o ] W 3 13




A(T) = 47 f r.2 dr..
ijk o ijk 13



LR 2
LR > 2
(T) = 47 r. dr.ij
ijk o


Equations (3-19), (3-20), (3-21), and the expression

for C. from Appendix A are the general forms of the model

for species direct correlation function integrals. To

obtain practical expressions one needs merely to introduce

the appropriate pair potential and potential of mean force

into equation (3-20) and perform the indicated integration

as illustrated in the sections that follow.

Since the coefficients in equation (3-21) are fitted

to data rather than evaluated analytically, it is of

importance to develop mixing rules to reduce the amount

of data necessary to model multicomponent systems. The

aim here is to predict all the coefficients from quantities

associated with no more than two different species so that

only binary or common-ion solution data would be required.

For aqueous electrolytes, the situation can be improved







due to the relative simplicity of ion-ion interactions

which can be generally scaled with the ionic charge (Kusalik

and Patey, 1983). Thus, two and three ion coefficients

are expressed from quantities related to a single ion.

If i, j, and k are ions, then



AF (T) 1 (AFi + AF ) (3-22)
ij 2 ii ji


AQ(T) (A + Aj + AD ) (3-23)
3 iii jjj kkk
ijk
LR
L(T) (L + .. + LR ) (3-24)
ijk 3 11i 33 kkk


If one or two of the species i, j, and k are solvents while

the remainder are ions, then the mixing rule must be

expressed from quantities involving each of the species

and water. The reason for this is that ion-solvent inter-

actions cannot possibly be predicted from solvent-solvent

and ion-ion interactions separately. Therefore, if i is

an ion and j a solvent, then



AF..(T) = AF.. (3-25)



If i and j are ions while k is a solvent, then



AO(T) = (A ii + A k) (3-26)
ijk 2








LR
(T) ( + LR (3-27)
i 2 iik jjk
ijk


If i is an ion and j and k are solvents, then



AM(T) = A-jk (3-28)
ijk

LR LR
$(T) = D(T) (3-29)
ijk ijk


Lastly, if i, j, and k are all solvents, then



AF(T) = AF. (3-30)
ij 13


AI(T) = Ai. (3-31)
ijk ijk

LR
LR
((T) = ijk (3-32)
ijk


It should be noted that these additive mixing rules

are not the only possible ones. In fact, theory would

suggest that geometric mean type mixing rules might be

more appropriate. Geometric mean rules, however, only

work for positive quantities which turned out not to be

the case with our empirically fitted coefficients. This

situation is further discussed in Appendix C.

The last point that needs to be addressed here is

the extension of the model to multisolvent systems. First,







the extension of the expression for c. is well known.
LR
Second, the extension of equation (3-20) for cLR requires

potentials of mean force applicable to the system. Assuming

all solvents are dipolar requires only knowing the dipole

moment of each of the solvent molecules and the dielectric

constant of the solvent mixture. Neither of these are

expected to present a problem in general. Third, the exten-

sion of equation (3-21) for Ac.. involves a few more coeffi-

cients and slightly different mixing rules for some three

body terms. Thus, while equations (3-22) to (3-27) would

remain the same for all solvents, equations (3-28) and

(3-29) where i is an ion and j, k solvents would be altered

to




ijk




D(T) = (LR. + LR (3-34)
ijk 2 ii3 ikk
ijk


which reduce to the previous result only when j and k are

equal. Here, any nonadditive interaction between j and

k has been tacitly ignored because the difference in the

interactions between different solvents is likely to be

less important to direct correlation function integrals

than that from the much stronger ion-solvent interactions.

This assumption is based on previous investigation of







solvent-solvent interactions which are dominated by angle

independent forces (Brelvi, 1973; Mathias, 1978; Telotte,

1985; Campanella, 1983; Gubbins and O'Connell, 1974; Brelvi

and O'Connell, 1975). Finally, equations (3-30), (3-31),

and (3-31) where i, j, and k are solvents would become



AF (T) (AF + AF ) (3-35)
ij 2 ii Fj




1
AM (T) = (At. + A + ) (3-36)
ijk 111 331 kkk



LR
LR 1 LR LR LR
f(T) = i ii + + k) (3-37)
ijk 3 111i 333 kkk
ijk


The above mixing rules for an aqueous system (single

solvent) have been tested against data for a number of

salts and may be regarded as established. The rules for

a multisolvent system, however, have not been tested.

They can only be seen as physically reasonable in the light

of previous experience but still tentative.

The next two sections deal with the application of

the theory developed here to specific interaction in order

to construct practical expressions.







Expression for Salt-Salt DCFI


The salt-salt direct correlation function integral

(C a) can be expressed as a stoichiometric sum of ion-ion

DCFI's (c. ) given by equation (2-11).


n n v. v (1-C..)
1 C = 1 la 18 (2-11)
i=l j=l ae


It is, thus, only necessary to develop general and practical

expressions for the ion-ion DCFI's and insert these into

equation (2-11) to obtain a general expression for the

salt-salt DCFI. The basic model for ion-ion DCFI's is

represented by equation (3-19). The expression for CHS
ij
has been developed in Appendix A and that for AC.. is given

by equation (3-21). This section is then chiefly concerned

with performing the integration in equation (3-20) to
LR
obtain an expression for CLR

The pair potential between two ions is given by


Z.Z .e2
LR z _
u. L3 (3-38)
13 r


Here the potential of mean force is approximated by a gener-

alized form given by the Debye-Huckel theory.


Z.e K(a. .-r. ij)
"LR = i e r > a.. (3-39a)
ij kTr.. D(1+Ka..) l]


LR
W. = 0 r < a.. (3-39b)
ij 13









where


K2 4ne2 n 2
K DkT zi Pi = Debye-Huckel
i=l


inverse length.

D = the dielectric constant of the solvent
or mixture of solvents.



Insertion of equations (3-38) and (3-39) into equation

(3-20) gives


2
LR 4TpZ Z e 2
C f r.i d r..
ij kT i


2 2 4
2rpZ Z .e 2Ka..
+ 1 1 e 13 J
(DkT) (1+Ka..) a


336 3Ka..
2JrpZ.Z .e 3Ka
i e ] f
3 3
3(DkT) (1+Ka..) a..
1J 1J


-2Kr..
e 13 dr.. -



-3Kr.
17
e i rj
e dr.. +
r.. 13
i]


The first term of equation (3-40) contains a divergent

integral. However, when it is introduced into equation

(2-11) which relates it to thermodynamic properties, electro-

neutrality makes the coefficients of the integrals sums

to exactly zero.


(3-40)








2 n n
SkTi 1 Z vjZ f r.. dr.. = 0 (3-41)
v V kT i=la i j=1 1



n
where v. Z. = o
i=l ic 1


The second term of equation (3-40) is integrable and

contains the implications for DCFI's of the Debye-Huckel

limiting and extended laws (see equation 2-54). Then,


27rpZ2 2e 2Ka.. m -2Kr..
i e 13 e dr..
(DkT)2(1+Ka..)2 a. 1J


2 2 -1/2
Z.Z.S pI
= 1 'Y (3-42)
1/2 2
(l+a..IB 1/2)



where 2e6 1/2
where S = ( 2 )
Y D3k3T3


K = B I1/2
Y

e2 1/2
B =K I /2 DkT


1 2
1 2 Zi Pi
i=l


The third term of equation (3-40) is also integrable

but more complex. The integral is the first order member







of a class of functions known as the exponential integrals.

These cannot be evaluated explicitly but a number of

asymptotic expansions and numerical approximations are

available (see Appendix D). It is convenient to express

the integral in dimensionless form.



Letting X = r/ai. then


-3Kr.
e 13
a idr..
r.. 13


-(3Kai)x
= e- dx = E (3Ka..)
1 x 1 l]


(3-43)


where E1(3Ka ij) = the first exponential integral



The third term in equation (3-40) becomes


i ]
27rpZ.Z .e

3(DkT) (l+Ka..)


3Ka..


-3Kr..
e 13
r..- dr..
a.. 13
a.3


3a..B II/2
3 3 2 3a. y12
Z.Z Spe Pe
S(l+a. .B E (3a..B 11/2)
3 1/2 13
(l+a..iB I
1] Y


(3-44)


which contains the implications for DCFI's to a higher

order limiting law for unsymmetric electrolytes (Friedman,

1962). Because of electroneutrality, this term, when







inserted into equation (2-11), is always very small for

symmetric electrolytes, and it approaches zero as the con-

centration of salt decreases. For unsymmetric electrolytes,

however, the sum over the ionic charges is not small and

this term actually diverges logarithmically as the salt

concentration decreases. To further explore the relation

of (3-44) to Friedman's limiting law and to elucidate the

low salt concentration behavior, the exponential integral

(E1) can be expanded for low values of the ionic strength

(I 0).



E (3ai.B I1/2) = n(3aiB I1/2) a + O(I1/2) (3-45)



where a = 0.5772 = Euler's constant.



This expansion is valid only at extremely low ionic

strength. Equation (3-44) then becomes


ji S2p PE (3aB 11/2-
3 Y olEl(3ai ) Y
z33
Z 1
3 ol 2 n + + n 3a



+ 0(11/2) (3-46)



where InI diverges as I 0 while a + In 3a..B are all
13 Y
constant.







The contribution of Friedman's limiting law to the

activity coefficient of a salt (a) is



1 n .3 2
V x ia i
FLL 1 _i=l 2
FLLn y1 i= Z S2 IknI (3-47)
a 3 n 2 Y
i=l La 1


and by taking the first derivative with respect to the

mole number of a salt 8 at I o,



FTL 2 P
Sn y Sypo n n 3
2-1L 01 3 ..zz
N 23v v X 3 3
oNo ITPN 3vaB i=l j=la Z
T,P,N



1 1
(- An I + -) (3-48)
Z 2


Rearrangement of equation (2-24) gives




N any pV V0
N -C oa oB (3-49)
v @N 0CaB$ v K TRT
S oBS a c aB
T,P,N 4



If equations (3-48) and (3-49) are compared, it is

clear that the contribution of Friedman's form of the limit-

ing law to the salt-salt direct correlation function integral

is







2 P
cFLL S ol n n1
CF LLX z z (- Zn I + ) (3-50)
U 3vv ia j i j ]2 2
S 3vaeB i=1 j=1


Comparing equations (2-11) and (3-50) gives the ion-ion

DCFI.

3 3
CFLL S2 P 1 + 1) (3-51)
ij 3 y ol 2 2-


Substitution of equation (3-46) in equation (3-40)

gives the expression for the limiting contribution of the

third term in equation (3-40) to the ion-ion DCFI.

3 3
SS2 P ( nI + + In 3a.. B) (3-52)
3 y o 2 13


Equations (3-51) and (3-52) have essentially the same

behavior as I 0 since they differ only by a small constant

which is negligible compared to ZnI as I 0. Therefore,

equation (3-44) contains the higher order limiting law.
LR
The general expression for C is


-1/2 2 2
S pl12 n n n i. v Z.Z.
cLR 4v v 1/2 2
4t B i=l j=l (l+a..B I )
13 Y

1/2
S3a. .B I /
Sp n n i. Z3Ze E (3ai..B
3v V 1 1/2 3
3a i=l j=l (1 + a..B I2)
11 Y


(3-53)







The expression for CHS is
aB


HS 1 n n HS
CS 1 V V. V. C (3-54)
a ji=l j=l


Lastly, the expression for ACB is


n n
AcB = v I Via. AF. +
a B i=l j=l aB



n n n
+ v (p A po ) (3-55)
Va i=l j=l k=l a j k ijk kjk




Equations (3-53), (3-54), and (3-55) form the complete

model for the salt-salt DCFI.



S = CHS + CLR + AC (3-56)
CaB CaB aB


Since the limits of DCFI's as salt concentration

approaches zero are well defined, it is advantageous to

use equations (3-53) to (3-55) to model the deviations

from this limit. To this purpose, the infinite dilution

limit of the salt-salt DCFI is now explored. From equation

(2-42a)




N n p V
N1 N y = (1-Co) s
S(1T-C ) ola (2-42a)
S N a v vT,P, K RT
S oS T~~Noy a6







it is seen that the constant temperature and pressure limit

has divergent terms associated with the activity coefficient,

a first constant related to the partial molar volume, and

a second constant associated with the activity coefficient

and which is not so well defined. This second constant

is loosely related to a term linear in salt density which

often appears in empirical expressions for the salt activity

coefficient (Guggenheim and Turgen, 1955; Guggenheim and

Stokes, 1969). In the present model the divergent terms

are contained in equation (3-53). The first constant can

be calculated directly from infinite dilution partial molar

volumes and solvent quantities. The second constant must

be fitted to data using terms from equation (3-55) which

have only ion-ion and long range ion-water correlations.

This reflects the fact that triple ion direct correlations

are zero at infinite dilution and any contributing short

range ion-solvent correlations would generally be contained

in the first constant. Thus,


0 LIM LR TB
C XL1 (C aB- A ) (3-57)
a X +91 aB -ca aB




where TB n n n
where VAC = v. v jB ,ijk
SB i=l j=l k=l







p- o_ o
p V V n n
1-c oa oa B p (F
(1-C) KRT v v v (AF
Sa i=l j=1



P LR
+ P 1l ) (3-58)
01 ijl




Finally, the general expression for the salt-salt

DCFI model including the infinite dilution limit is


m LR HS HSo'
1-C = (-C ) (C -HS C
aB cB aaB ca6 )


TB TBm
(A AC ) (3-59)


where cHS = LIM HS
where C C
aB Xol l aB


TBC LIM TB
AC AC
aB X oll aB



P
Pol n n
= 01 y y v PA
V v I I itX jp o1 ijl
Va i=l j=l


Although equation (3-56) can be used in place of

equation (3-59), it was felt that the latter was more

appropriate for calculations at constant temperature and

pressure. Therefore, equation (3-59) was used in the com-

parisons and correlations in this work. In calculations







where pressure varies, equation (3-56) would be more

convenient since it would eliminate the need to obtain

partial molar volumes as a function of pressure.

For illustrative purposes, equation (3-59) will now

be written for a binary system consisting of a solvent

(1) and a salt (2) which dissociates to formvy cations

and v anions.



n = 1 + v +

V = V +

v2 = + -


(1-c22)LR HS
22 = 22 22 (C22


SHSc
22


TB TB-
- (ACTB ACT )
22 22


(3-60)


where


TB TB- C P
(AC22 AC ) -2
2


+ 2v+v _(plAl+-


2
[v (p AD + p AD + p AD ) +
+ 1 1++ + +++ -++


+ p+A++-


+ p-A(_+ ) +


2
+ V (PlAl--_ + p+AI+__ + pA___ )] +

P 2
(P p ) 2 9
01 [v 2A + 2v v A+ + v2A1 ]
2 + ++ + 1+- 1--
V2


(3-61)








R S -1/2 2 4
cLR= ++ +
22 2 1/2 2
4v (l+a BI )
2 ++ y


22 24
2v _Z2Z2 v2 Z
+ ( BI1/22 + (1/2)2
(l+a+ BI ) (l+a BI )
+- -- y


S3a++B I1/2
2 6 + y
+ +


(1+a++B I/)
++ y


2v 3a B 1/2
3 3 3a+- B
2+ Z_ -+


E (3a+ B 1/2)
1 /7


E (3a B 1 /2)
1 --


2 6 3a B II/
2V -- y
v_Z_e


(l+a B 11/2)3


2
P 2
SPol(Vo2) p 2
1 -C22 [V (aF
SC22 =2 -2 + ++
v2 1RT 2


+ 2v+v_ (AF_







(CHS HSm) 1
22 22 2
2


+ 2V v_ (CHS
+ +-


P LR
- P l++) +
o1 1++


P LR 2 P LR
- Po (l ) + v (AF Po l )
S1+- -- 01 1--







[v2 (CHS ) +
+ ++ ++


CHS) + v2 (HS HSm)
+- + __


2
S3
--v
3v2


(3-62)


(3-63)


(3-64)


(1 B1/2 3
(l+a BI )
+- Y


E (3a__ By1/2 )







Expression for Salt-Solvent DCFI


The development in this section parallels that of

the previous one. Thus, a general expression for the

solvent-ion DCFI is derived and then inserted in equation

(2-11) to yield the salt-solvent DCFI relation.

Although any type of interaction can, in principle,

be included, it was assumed here that ion-solvent interac-

tions are dominated by dipole-charge forces at large separa-

tion, and no other interactions were included. The pair

potential for an ion (i) and a dipolar solvent (1) is


LR l e
u = cos 6 (3-65)
ril


where pl = the dipole moment of solvent 1
in Debyes.

S= the Eulerian angle between dipole
and charge.


The potential of mean force is approximated by a func-

tional form inspired by some recent applications of the

mean spherical approximation (Chan, Mitchell, and Ninham,

1979) and of perturbation theory (Hoye and Stell, 1978)

to nonprimitive electrolyte models.

wL lK(ail-ril
LR i 1 e il
il 2 (cos 0) e r > ail (3-66a)
kTri

LR
Wi = 0 r ail (3-66b)
iiil -1







where a is a universal constant that we have set equal

to 4.4 empirically.

Since equations (3-65) and (3-66) are functions of

orientation, it is necessary to first perform the integration

over angles as indicated in equation (3-20).



= u LR dw dwl (3-67)
il 2 i1 i 1


where dwo = sin e.di.d4.
1 1 1 1


S= f dw. = f sin 6.de. f2 7di = 47
1 0 1 1 0 i


When the integral in equation (3-67) is evaluated, it is

found that


LR
= 0 (3-68)
il W


The second term in equation (3-20) has


LR 2 1 I (W)LR2
<(W ) > (W) dw. dw (3-69)
il Q 2 il 1


After the integral in equation (3-69) is evaluated, it

gives


L2 2 2K(ail-ril)
2 Z ile )e
<(W i) > = T 4 (3-70)
ilDkT 4
r.l







The third term in equation (3-20) contains


LR 3 1 )3
<(W ) > ( dw d
ii 2 il 1


which also equals zero.


LR 3
<(W ) > = 0



Therefore, for ion-dipole pairs there is only one term

in equation (3-20).


i 2 Tr c 2 2Ka -2Kril
LR 2I 1I ) 2 il e e
il 3 P DkT ) e f -
aij il
1] 1


dril


(3-71)


(3-72)


(3-73)


The integral in equation (3-73) is also an exponential

integral (E2) which is expressed in dimensionless form

as before.


C -2Kr il -(2Ka )X
S1 il E (2Kail )
2 e dr = 1 f e dX = ail
2 i1 a.i 1 a
ail ril


Equation (3-73) then becomes

2Ka
2 i1
iR -2T lZ el e E2(2Kail
11 3 DkT ail


(3-74)


(3-75)







Then, the general expression for the salt (a) and

solvent (1) DCFI is



S2a iB I1/2
2 11
LR 27p ea 2 2 n E. 1/2
Cl = ( ) E (2a il )
al 3v DkT i=l ail2 il y
a 1=1 11


(3-76)


The expression for CHS is given by


n
HS 1 HS
Cl v Via Cil
a i=l


and the relation for ACal is


n n n
AC v= -P- v. AF. + p vi
al a il a il V a il k la
a l=c a i=l k=l


(Pk Ailk


(3-77)


(3-78)


o LR
- p ik)
k ilk


Again, equations (3-76), (3-77), (3-78), and (3-79)

form the complete general model for the salt-solvent DCFI.


SCHS LR
C = CHS + C + AC
al al al al


(3-79)


As previously discussed, it is convenient, particularly

for isobaric calculations, to use the model only for





67

deviations from infinite dilution. (For nonisobaric calcu-

lations, equation (3-79) would be more appropriate.) The

infinite dilution limit of Cal is given by equation (2-40)
LR
and that of CLR is (see Appendix D)
acl

P 2
P2eep 2 n y Z2
LRO LIM LR 2ol ( ea 2n i (3-80)
1C C = ) P 1 a1 (3-80)
al X ol+1 l 3v DkT 1 ail
ol a i=l a

HS
while the infinite dilution limit of C H is formally


HSm LIM HS (3-81)
C1 =X1 Cl (3-81)



Lastly, the infinite dilution limit of AC is

P
A LIM ol n P LR
al X o1 al i (iFil Pol ill
ol0 i=a


P
Pol n
+ 1 o' pnolA l (3-82)
iV Vi P1 ill
a i=l


The complete general relation for the salt-solvent

DCFI including the infinite dilution limit is

00
V
oc HS HS o LR LRm
1- IT- RT Cl- Cal) (C Cal



(ACal ACal) (3-83)







Finally, equation (3-83) will now be written for a

binary system consisting of solvent (1) and salt (2) with

v+ cations and v_ anions.


V o 02 (HS
21 v K2 RT 21


HS LR
- C2 ) (C2
21 21


(AC21 AC21)


where


HS _HS- 1 HS
21 21 v2 +(C+1
"2


-HS + (HS HSm)
+1 ) -1 -1 )


2
CLR CLR 2- r ( ) e )
21 21 3v DkT
2


2
2 + +
1 [
a+1


2a +B I1/21/
(pe Y E2 (2a+1B
2 1y


\) z2
-p +T
ol a_l


AC C = [v (AF+
1 C21 + +1
2


2a 1B BI/2
(pe E2(2a lBI2) p)]

(3-86)


P LR P LR
- i ) + _(AF Pol )]
01 +11 -1 01 -1


(p p ) +



+ ol + p+ p +
V2 + +1 1+1 + P+A++ + -+
V2


SLR-
21


(3-84)


(3-85)







+ v_(plA11_ + P+A+ + P_A __ )]

P 2
(Pol
[v+ &1+1 + v _l_ (3-87)



Expression for Solvent-Solvent DCFI


The solvent-solvent direct correlation function integral

has the simplest relation since the solvent does not

dissociate so the species and component integrals are the

same.

As previously noted, any type of interaction can gen-

erally be included in this theory, but it was assumed that

solvent-solvent interactions at large separation are domi-

nated only by dipole-dipole forces. The solvent (1)-solvent

(1) pair potential is


LR 1 1
u 3 1 (3-88)
rl
r11

where ( = 2 cos 611 cos e12

sin 011 sin 012 cos (11 012

611' 11 = Eulerian angles of solvent molecule
number 1.

612' 12 = Eulerian angles of solvent molecule
number 2.



The potential of mean force is approximated by a

function inspired by previously mentioned work (Chan,

Mitchell, and Ninham, 1979; Hzye and Stell, 1978).








K(a -r)
LR 1 1 e 1
W = kr D
W11 3 D
kTr11
11


LR
W I 01
11


r > al



r < al
11


(3-89a)



(3-89b)


Again, equations (3-88) and (3-89) are inserted into

equation (3-20) and the required integration over angles

performed.


SLR = L l d11
11 Q 2 11 d11 12


dli= sin 1i deli d li



S= f d6l = f sin 6li dli
o


2Tr
f di = 47
0


The integration of equation (3-90) gives


LR
0
11 W


The second term in equation (3-20) has


LR 2 1 LR 2
<(W ) > f (W ) dw dw
<11 ( 2 (W11 d11 d12


which yields upon evaluation,


LR 2 1 i l11 2
<(W ) > (D -)
11 w 3 DkT


2K(al-rl)
2K(11-11
e
6
11


(3-93)


where


(3-90)


(3-91)


(3-92)







The third term in equation (3-20) has


LR 3 1 LR 3
(W ) > (W ) d
<11 2 11 1 12


which becomes upon integration


LR
<(W )> = o
11


(3-94)


(3-95)


Thus, for dipolar solvents only one term of equation

(3-20) remains after the angle integration.


R = 2 e2Kal -2Krl
1 3 (DkT e r 4
all rll


dr11


(3-96)


The integral in equation (3-96) is also an exponential

integral (E4) which can be expressed in dimensionless form.


-2Krll -(2Kall)X
S11 -(2Ka11 )X E (2Ka )
e 1 e 4 11
d 4 dr l = -- -4 dX = 3
al r11 all 1 X all


(3-97)


Equation (3-96) is then transformed


2alB Il/2
LR 4 ( 1 (1 1)1
C11 3 DkT 3
all


E (2a B 11/2)
4 11


(3-98)







which is the general expression for the solvent-solvent

DCFI.

Since the solvent does not dissociate, there is no
HS
summation over species in C11. However, AC11 does have

a sum over third bodies.


n
CFo R ( 9LR
AC11 = pA1 + (Pkllk llk (3-99)
k=1


Equations (3-98), (3-99), and (3-100) form the complete

general model for the solvent-solvent DCFI.



C = CHS + + AC1 (3-100)
11 11 11 11


Again, the infinite dilution limit of C11 is introduced

so that for isobaric calculations the model need only account

for deviations from the infinite dilution value. Also,

equation (3-100) would be more practical for nonisobaric

cases. The infinite dilution limit of C11 is the bulk

modulus of the pure solvent given by equation (2-41).

The infinite dilution limit of CLR is given by
11

P
LRm LIM cLR 4pol r T D 2
CLR- XLIM CII l ( (3-101)
11 X 01 11 3 fDkT )-0
11

and that for C S is formally
11


HS- LIM HS
C =1 Xol1 CI (3-102)
11 X -1 11 (3-102)
ol







The infinite dilution limit for AC11 is given by



AC LIM ACP (AF P LR P 2 A
S11= Xol 11 01ol 11 ol 111 ll
(3-103)



Finally, the complete general expression for the

solvent-solvent DCFI including the infinite dilution limit

is


1 HS HS" LR LR"
1 C (Cl l ) (C C )
polKRT
--11 11 11 11

(ACll ll) (3-104)



Again, the application of equation (3-104) to a binary

system consisting of solvent (1) and a salt (2) with v
+
cations and V_ anions is shown. However, for the solvent-

solvent DCFI all of the terms except AC11 appear similar

to the general case since they have no summations over

species. Thus, only AC11 is illustrated below.


oo P P LR
AC ACI = (P p )(AF Po +

11 11 01 11 P ll

+ P(P1 111 + P+ A)11 + pA-11) -



(p P)2 A (3-105)
01o 111







Summary


A general statistical mechanical model of the direct

correlation function has been presented. In principle

it is applicable to any system, but it has been specialized

here to treat strong electrolyte solutions. The next chapter

shows the application of this model to six aqueous strong

electrolyte binary solutions. As a preview to the calcula-

tions, the relative magnitude of the three contributions
HS LR
to the DCFI (C C R AC ) will now be discussed, the

model parameters will be listed, and the sensitivity of

solution properties to parameter value considered.

The salt-salt DCFI is dominated at very low salt con-
LR
centration by C n which contains the long ranged electro-
LR
static interactions. However, the magnitude of C 8 decreases

very fast as the salt concentration increases so that above

2M or so in salt density the dominant term becomes C HS

This reflects the increasing shielding of electrostatic

forces by more ions that more frequently repel each other.

AC makes a contribution that is generally not dominant

in either regime but is always numerically significant

above 0.5M.
HS
The salt-solvent DCFI is always dominated by C H with
LR
Cl making a small but not negligible contribution. Due

to the relative strength of the short ranged hydration

interactions, ACal makes the largest contribution after
CHS
Cl*
"al"







The solvent-solvent DCFI is also dominated by CHS

over the entire range of salt concentration up to about
LR
6M. C11 makes a negligible contribution reflecting the

relative weakness of long range dipole-dipole interactions.
HS
Again, the largest term after CI is AC which contains

the short ranged hydrogen bonding between solvent molecules

and the hydration related effect of an ion on two solvent

molecules at short range.

The parameters of the model are species specific and

universal. It is, therefore, necessary to build only a

relatively small set of parameter values to predict the

behavior of a large number of systems. Thus, a hard sphere

diameter (a. ) for each species is required for CHS and
11 a
LR
CaB (where a, B can be salts or solvents). To avoid con-

fusion, the parameters for ACa will be those of a system

with one solvent (1), one salt (2), and many ions (i, j).

Then, AC involves AF PLR which is ion independent,
11 11 ol 111
A11 which is usually neglected, and Al for each ion.
111 ill
AC has AF POaLR i l and A. AC22 includes
1A2 i 01o lil 1il' 1i22
AF. p A and AoP.... This totals to two solvent
11 ol iin, ii, 111
specific parameters if AO11 is neglected and six parameters

for each ion (note that A lii = A.il and AD. = Al )
lil ill 111 ii
three of which involve solvent-ion pairs.

Properties predicted with the model are most sensitive
HS
to the value of the hard sphere diameters because the C

is a very strong function of the diameters. But it is





76

not as sensitive as is the case with other models. This

is due to the fact that the two body coefficients AF..
13
are fitted to infinite dilution quantities that include
HSm
C so there is a degree of compensation for changes in

the diameters. The sensitivity of the results to the value

of the coefficients in ACa is generally small since they

make a small contribution to the DCFI's.













CHAPTER 4
APPLICATION OF THE MODEL TO AQUEOUS STRONG ELECTROLYTES


Introduction


In Chapter 2, the formal relations between DCFI's and

thermodynamic properties were introduced. In Chapter 3, a

model expressing the DCFI's in terms of measurable variables

was constructed. In the present chapter we illustrate the

use of the formal relations and the model in the calcula-

tion of thermodynamic properties. We also explore the

scheme used to fit model parameters; further we compare

calculated values to experimental ones for the salt-salt,

salt-solvent, and solvent-solvent DCFI's and for the bulk

modulus, partial molar volume, and salt activity coeffi-

cient. Finally, a discussion of the above results and a few

conclusions are presented.

The use of Fluctuation Theory in general fluid phase

equilibria problems has been treated in detail by O'Connell

(1981). The specific case of liquids containing super-

critical components has been addressed by Mathias and

O'Connell (1981) and Mathias (1978). The present treatment

generally follows these developments, but there are

important differences for the present case of electrolytes.







Calculation of Solution Properties
from the Model


The formal relations between solution properties and

DCFI's are given by equations (2-12), (2-34), (2-37), and

(2-38) for a system consisting of no components, salts (a,B)

and one solvent (1).


a2enyc
pv
ac ap~ ~
ooY


= V v (1-C )


V n
V o
OT = Ea Vx (1-Ca
ST =1C





n n
aP/RT o o
PT = E Z v VX X (1-C
Tp a=l =l a oa oB aB
a=1 B-l
T,X


NVa aQtny

PK TRT a@i
oT TIPN0


n n
o o
v v V Y v vvX X 0
a y=1 6=1 6 oy 06


[(1-C )(1-C a) (1-Cy )(1-C )]
*Y

(2-12)







(2-34)







(2-37)


(2-38)









ae P/RT 1
where = -
ap I PK RT
ST,N T


In order to evaluate the change in solution density

with pressure while the composition and temperature are

constant, one needs to integrate equation (2-37) from a

known reference density (pR) at the reference pressure (pR)

at the temperature and composition (mole fraction) of the
F
system up to the desired density (p ) at the system pressure

(P).

n n F
R o o P (T,P,X)
v V X X (1-C )dp (4-1)
RT E =i =l oa o R R T,N (
p R(T,R ,X)


Equation (4-1) represents an implicit equation for the
F
unknown density (p ) which can only be solved numerically

with realistic models.

It should be appreciated that equation (4-1) cannot be

applied to an isobaric change because that would imply that

pressure, as well as temperature and composition, were
F R
held constant. Then p would be the same as p so the state

of the system would not vary at all.

To evaluate the change in solution density isothermally

with varying composition, a different approach is required.

To develop the necessary relations we start by considering

that in Fluctuation Theory the pressure is treated as the





80

dependent variable, a function of temperature, density, and

mole fraction.



P = P (T,p,X) (4-2)



Taking the total differential of pressure gives


aP aP
dP dp+ dT
ap T,TN Tp,
T,N p,N


n
o
z aX
a=2 oa


dX
oa
T,pX
oyfa


If the change is isothermal and if we divide by RT,


1 dP
T


1 aP
1- 3- dp
RT 3p T
T,N


1
+ -
RT
C=


o p
S ax
:2 oc


dX
TP oa
T,pX
oyfa


By making some identifications we obtain


n
o
B=I


DP
ax
oY
T,V,X /
oy#a




aN

ax a
Ny4 a


VKT
VK T


DN 0
o3
oa N
oy/8


N
-v X


Inserting equations (4-5) and (4-6) into equation (4-4)

gives


dP 1 aP dp +
RTT RT ap T
T a TN


n
0

ct-2


n -

SK RT
8=1 T


dX
6 -vX
B oa
6eS Boa


(4-7)


(4-3)


(4-4)


(4-5)


(4-6)







We next insert equations (2-34) and (2-37) into

equation (4-7).



n n
1 dP = [= V V X (1-C ) dp +
RT T c=l 8=1 a oaoB 0



n n n
+ 0 V X (1-C pdXo (4-8)
a=2 8=1 y=l Y 6 -vX
c G B-uoc


Equation (4-8) permits us to evaluate the change in

solution density with both pressure and composition along an

isotherm. This equation is also applicable to an isobaric

and isothermal process where the solution density changes as

a function of composition only.

To obtain the density (p ) of a given solution at a

known temperature, pressure, and composition (X ), we
-o
isothermally integrate equation (4-8) from the known

reference density (p ) at a system temperature and a conven-

iently chosen reference pressure (P R) and composition (X )
-o
F F
up to the desired density and composition (p and X ). It
-o
is suggested that for aqueous electrolytes the reference

density be chosen to be that of pure saturated water at the

system temperature.





82



F F
SAT n n P (T,P,X )
P-P o 0 -o
R = I v B SAT X X (1-Ca )) dp +
RT SAT oc oaS
Ra=l =1 Poi (T) T


F
n n n X
o o o oa pdX
+ I I oy(-C) (4-9)
a=2 8=1 y=l Y oy a6 BXa T


In evaluating the integrals of equation (4-9) each

integral involves variables appearing in other integrals.

To explicitly find pF requires further manipulations

discussed below.

The activity coefficient on the mole fraction scale for

any component (a) can be obtained by integration of equation

(3-12) from the reference molar density (P R) to the molar

density of each component (PF ) at constant temperature.

F
Y n p 1-C
n v R dP oS (4-10)
R R R p a T
7 8=1 8 Po T
a P05

where Po5 = Xo8P



Equation (4-10) is applicable to any isothermal change,

isobaric or not, and the reference state composition where

YaR = 1 need not be that chosen. However, for aqueous

electrolytes it is natural to choose pure saturated liquid

water at the system temperature as was done for equation

(4-9).









F F
P (T,P,X)I n P (T,P,X)
ZnYc = al dol + Z f 1-Ca dp
pSATT T 8=2 o p OT
ol
(4-11)

Equation (4-11) can be used for either isobaric or

nonisobaric changes.

In equations (4-9) and (4-11) one can use the DCFI

model represented by equations (3-59), (3-83), and (3-104)

for isobaric integration. But, for nonisobaric integra-

tions with equations (4-1), (4-9), and (4-11) the DCFI model

of equations (3-56), (3-79), and (3-100) will be more

applicable because the pressure behavior of the DCFI

infinite dilution limits, some of which involve salt partial

molar volumes, is not generally available.

The composition behavior of component activity coeffi-

cients on the mole fraction scale at constant temperature

and pressure could also be obtained from equation (2-38)

with composition expressed as mole fractions. Thus, we

express the differential of the activity coefficient of a

component (a) as




no 8nya
dnya = z X dX (4-12)
ST,P B=2 a TPX
TIIoyX$











,nya n
eny o 8 ny aN
= 3o o N(4-13)
06 T,P, Xo l T,P,Noa 0 N
OY 8 oy/c OY7c



oo N
= N (4-14)
ax 6 -v x
N OY4
oyf -

Inserting equations (4-14) and (2-38) into (4-13) and

then putting the resulting expression into equation (4-12)

gives


n n n n
o o PK RT o o
diny = I dX I Z v TV
T,P 6=2 =1 B O 0 Y=l 6=1



XoyXo (1-C 6)(-Ca) (-Cy) (1-C6)] (4-15)


To obtain the activity coefficient, equation (4-15) is

isothermally and isobarically integrated from the reference

to the desired state.


F
n n n n X
o o o o o pK RT
S= I R do 6 v IX iV 6
a=2 B=l y=l C=1 X a 6 oX oB Y
OB T,P


Xo Xo[(1-C 6)(1-C ) (1-C )(1-C6) ]


(4-16)








n n
e o o
where = v v X Xo (1-C )
SRT = 8=1 B a o


6 = Kroniker Delta



Equation (4-16) can only be used for isothermal,

isobaric changes and thus either the DCFI model of equations

(3-59), (3-83), and (3-104) or that of equations (3-56),

(3-79), and (3-100) may be used.

Equations (4-1), (4-9), (4-11), and (4-16) express

integration of the DCFI model formally. However, these

cannot be explicitly evaluated because of the multiple

variables involved in the integrals. To actually evaluate

these integrals requires a change of variables as discussed

by Mathias (1979) and O'Connell (1981). Rather than give

their formal equations, we now give the above relations with

explicit expressions for the present DCFI model. Those

parts that are analytically integrable have been evaluated

while simplified integrals are given for the others. The

DCFI model used is that of equations (3-56), (3-79), and

(3-100) which does not contain the DCFI infinite dilution

limits. This form of the model yields simpler expressions

which can be applied to both isobaric and nonisobaric

changes. We start by rewriting equation (4-1) as









_pR pPY-HS(pX) F PY-HSpRX)
RT RT RT


F R
- (p -p )


n n n
n o 0 1 LR
S 0 0 v v X X X J C L (t)dt -
= oa 0 oy a
U=l =l y=1 o


F F RR n
p p -p p
2 .
i=l



F F FR R R
p P p -p p p
3


n

j=1



n

i=l


FF P,F RR P,R
P P P -pp p
ol ol


n
where p(t) = I(pR +
y=l1


X.X. AF.. -
1 3 13




n n
SI X.X X Aj +
j=1 k=l 1 3 k


n n
S z x.x. ,LR
i=1 j=1 ijl


(4-17)


n
F R o
(P F-P R)t)= I v p (t)
oy oy Y Y


Equation (4-9) can also be changed to

SAT PY-HS F F PY-HS R R
P-P TT) P ( ,X ) P -(p ,X)
1 -o 0
RT RT RT



n n n LR
0 FF RR a C t)
v v (X p -X p ) fp (t)p (t) p dt -
= 8 = oY -oy o o o p(t)p(t)
a=l 8=1 y=1 o







n n n n 1
oo o o 1 1-C (t)
2 ll yl =1 Y o P (t) (t) F (t) dt -
a=2 6=1 Y=I 6=1 0 oy poa(t) a
6 -v p(
rB 8 p(t)


n
F FFF R RRR
S(X. X p p -X X. pp ) AF. -
j=l 1 3 1 3

n n
F F FFFF R R RRRR
S(X. /X. X pp-X. X. A4. +
j=1 k= i j P P -Xi j X P p p ) A ijk
j=l k=l




n
n F F F F P,F R R R R P,R LR
(X. X. p p p -X Xj p p ) p
j= p ol P ijl
j=1


n
1 n
2
2 i=w

n
1
3 i=




n
1
3.
1+






where


R P,R SAT
S=P = T)

R
ol ol
R
X = 1

X R =0 a 1
oU

n
o
X. = v. X
1i L )i Xot
a=l



RR FF R R
p (t) = X p + (X p X p )t
oa oa oa oa


x FF R R
X p -X p
F (t) op(t)
a p(t)


n
0 FF R R
p o (t) (X p -XR p )
oa p oB O
t= )
2
p(t)


(4-18)





88

The expression for PY-HS is given by equation (A-l) and
LR
that for C by equations (3-53), (3-76), and (3-98).

In a similar fashion we transform equation (4-11) to


1
nya =
a


n
i V.
i=l a


F
Pi
i
- n ) (p
R
p
i


n
o

6=2


F


n
Ij=
j=1


n
1 1
a 1=1


PY-HS TpF PY-HS pR
i T, ) pi TT, )
(1 -
RT RT


LR
1 C LR(t)
F R) cl d
ol 1o p(t)




1 CLRt) F
- ) a dt + n -
Sp(t)
o p


SF
ia j


R
- pj)


AF.


n

j=l



n

j=l


n F F
I .. (PijP
k=l la 3 k




F P,F
S. (p .p
i cc ol0


n
o
Pi = p v iaX
a=1


R R
h j ijk




R P,R LR
j- ol )ijl


n
o
= v. P
l ia oa
a=1


and the other quantities have been defined above.


n

a i=



1 n
2v





+ n
2v i
a i=1


where


(4-19)




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