A LIQUID EQUATION OF STATE

FOR AQUEOUS STRONG ELECTROLYTES

By

KENRIC A. MARSHALL

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

1989

To my beloved family, and especially my sister, Clare,

whose mid-week phone calls greatly encouraged me.

ACKNOWLEDGEMENTS

I would like to thank Professor J. P. O'Connell for his

support and encouragement throughout the course of my

studies at the University of Florida. Special thanks go out

for including me in his sabbatical trip, which was not only

a tremendous educational experience, but was culturally

broadening as well.

I wish to thank Aa. Fredenslund, P. Rasmussen, and J.

M0llerup of the Instituttet for Kemiteknik, Danmarks

Tekniske HOjskole for their hospitality during my stay in

Denmark.

I am grateful to the members of my supervisory

committee, Professor G. B. Westermann-Clark, and Professor

Michael C. Zerner.

I would especially like to thank Mrs. Nancy Mishoe for

her outstanding typing. I would also like to thank Mr.

Roderick Hagen, Mr. Timothy Vaught, Ms. Dede Rumble, and Mr.

Dirk Anderson for their help with preparing the figures.

Finally, I would like to thank the University of

Florida, the Department of Energy, and N.A.T.O. for

financial support during my studies.

iii

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS. . . . . . . .

ABSTRACT . . . . . . . . .

CHAPTERS

1 INTRODUCTION. . . .

The Debye-Hickel Theory

Activity Coefficients .

Volumetric Properties .

Thermal Properties . .

A New Approach. . . .

Experimental. . . .

2 FLUCTUATION PROPERTIES--THEORY AND

EXPERIMENT . . . . . .

Theory. . . . . . . .

Relation of DCFI to Experimental

Measurements .. . . . .

General Application . . . .

3 MODELING FLUCTUATION PROPERTIES .

The Cabezas Model . . . .

Consequences of the Cabezas Model

Ionic Additivity. . . . .

Short-Range Contributions--A New

Formulation. . . . . .

Long-Range Contributions--McMillan-Mayer

to Kirkwood-Buff Transformation. . .

Model Equations . . . . . . .

4 CORRELATION OF SOLUTION PROPERTIES. . .

DCFI Testing . . . . .....

Extension to Densities and Activity

Coefficients . . . . . . .

iii

. . . . . 1

S . . 16

S . . 22

S . . 30

Page

Fitting to Density and Activity Coefficient

Data . . . . . . . . . .

5 DISCUSSION AND EXTENSIONS OF THE DCFI

MODEL. . . . . . . . .

Property Calculation Review . . .

Ionic Additivity. . . . . .

Activity Coefficients . . . .

Densities . . . . . . .

Isothermal Compressibilities. . .

Discussion. . . . . . . .

6 EXPERIMENT . . . . . .

Sample Containment and Measurement. .

Pressure Generation and Measurement .

Temperature Control and Measurement .

Accomplishments, Shortcomings, and

Recommendations . . . . .

7 CONCLUSIONS AND RECOMMENDATIONS . .

APPENDICES

A PERCUS-YEVICK HARD SPHERE EQUATIONS . .

B McMILLAN-MAYER TO KIRKWOOD-BUFF SYSTEM

TRANSFORMATION--MULTI-COMPONENT CASE .

Multi-Salt, Single Solvent . . . .

Multi-Salt, Multi-Solvent . . . .

C MODEL PARAMETERS . . . . . .

D PHYSICAL PROPERTY DATA FROM MODEL . .

E ACTIVITY COEFFICIENT MODEL CONTRIBUTIONS.

REFERENCES . . . . . . . . . .

BIOGRAPHICAL SKETCH . . . . . . . .

. 101

. 101

. 102

S. 103

S. 104

S. 106

S. 106

. 136

. 137

S. 140

S. 142

. 143

. 150

S. 154

. 157

. 157

. 159

. 162

S. 168

S. 186

S. 204

S. 210

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

A LIQUID EQUATION OF STATE

FOR AQUEOUS STRONG ELECTROLYTES

By

Kenric A. Marshall

May 1989

Chairman: Dr. J. P. O'Connell

Major Department: Chemical Engineering

The formalism of Fluctuation Solution Theory relates

concentration derivatives of thermodynamic properties to

spatial integrals of statistical mechanical direct

correlation functions (DCFI). This is an alternative to the

traditional Gibbs energy and partition function methods to

obtain physical properties. Theoretical development and

analysis of experiment have been done to obtain an accurate

DCFI model for densities and activity coeffficients of

aqueous strong electrolyte solutions.

A model for the DCFI for the species of ions and

solvents has been established. The long-range field

interactions are based on an ionically additive form of the

extended Debye-HUckel theory. The connection between the

McMillan-Mayer system, in which the Debye-Huckel result is

derived, and the Kirkwood-Buff system, which is more

convenient for modelling, has been investigated. Short-

range interactions have been modelled using a density-

concentration form inspired by the Rushbrooke-Scoins virial

expansion.

The proposed model was used within the framework of

Fluctuation Solution Theory to construct equations for the

activity coefficients and solution densities. Parameters

were determined for seventeen salts of various charge types.

These included quantities for solvent-solvent interactions,

ion-solvent interactions, and ion-ion interactions at 250C

and 1 atmosphere. The predictions from the proposed

activity coefficient model compare quite favorably with

existing correlations which do not yield densities. Ambient

pressure densities predicted by the model are excellent with

good accuracy up to 1000 bars.

vii

CHAPTER 1

INTRODUCTION

Solutions of aqueous electrolytes are important in many

chemical systems. All natural waters contain electrolytes

in varying amounts. Freshwater streams and rivers tend to

be relatively pure, while the oceans are more concentrated

in electrolytes, and hypersaline solutions may be found in

enclosed seas or percolating through geological formations.

Natural waters may be highly complex with regard to both the

number and variety of salts present. They are subject to

wide ranges of temperature and may be subject to

extraordinarily high pressures as well (to 20 kbar).

Electrolyte solutions are also important in chemical

processing. Salts are used in applications such as

azeotropic distillation and industrial waste treatment

processes. Electrolyte solutions in industrial processes

are not subject to as wide a variation in temperature or

pressure as is found in geothermal solutions. They are,

however, often used in mixtures containing non-aqueous

organic compounds, and in solutions where rapid chemical

reactions occur.

2

Electrolytes are commonly found in complex systems both

in nature and in industry. Unfortunately, physical property

information for electrolyte solutions is generally limited

to binary or ternary systems at ambient temperature and

atmospheric pressure. The available data are correlated

with empirical expressions, but these expressions cannot be

extended to more complex conditions in a predictive manner.

Theoretical expressions have been advanced to model a single

property, such as the solute activity coefficient or partial

molar volume, or solution isothermal compressibility. While

some attempts have been made to develop a coherent formalism

for electrolyte physical property correlations, a

satisfactory treatment has yet to be advanced.

Debye-Huckel Theory

Any discussion of physical property correlations for

solutions of aqueous electrolytes must begin with an

examination of the pioneering work of Debye and HUckel

(1923). This work laid the foundation for all subsequent

theoretical developments. Many excellent reviews of Debye-

Hickel theory and later developments are available in the

literature (Mayo and Mou, 1979; Friedman, 1981; Mauer, 1983;

Renon, 1981). It is not the purpose of this work to repeat

their findings. However, it is appropriate to discuss the

salient points of currently available models and formalisms

in order to contrast them with this work.

Debye and Huckel considered the case of charged hard

spheres embedded in a continuous dielectric medium. They

solved the linearized Poisson-Boltzmann equation to obtain

the electrostatic potential for this system. The activity

coefficient corresponding to this potential is determined by

integrating the potential through a changing process.

GUntelberg (1926) offered an alternative charging process to

that of Debye and Huckel. Pitzer (1977) has taken the

electrostatic potential of Debye and Huckel and used it to

determine the radial distribution function. By application

of the pressure equation of statistical mechanics, a result

similar to that of Debye and Huckel is recovered, and higher

order terms were elucidated.

The result of Debye and HUckel serves as the basis for

the description of the effect of long-range forces on the

physical properties of electrolyte solutions. The basic

result which accounts for the finite size of the ions is

known as the extended Debye-HUckel equation. At very low

concentrations, the effect of the size of the individual

ions may be neglected. This result is known as the Debye-

HUckel limiting law. Many critiques of the basic

assumptions of Debye-Huckel theory have been advanced and

are recognized (Onsager, 1933; Frank and Thompson, 1959;

Fowler and Guggenheim, 1965; Guggenheim and Stokes, 1969).

There are several areas of controversy in the field of

thermodynamics of electrolytes. One area is the role of the

dielectric constant in the modelling equations. While many

investigators use the pure solvent dielectric constant,

others have used a form of the dielectric constant that is

dependent on the salt concentration of the solution

(Prausnitz and Liu, in press). A second point is the

representation of solute concentrations through the choice

of concentration scales. Models commonly used in

engineering applications use molality or mole fraction as

concentration variables. Theoretical studies generally

apply the molar concentration scale, with only the solute

concentrations accounted for explicitly (Pailthorpe,

Mitchell, and Ninham, 1984; Martin, G6mez-Estevez, and

Canales, 1987). Finally, the development of models in which

the parameters depend only on the ions present, rather than

the salts from which they are formed, has been sought

(Glueckhauf, 1955; Guggenheim and Turgeon, 1955).

Activity Coefficients

Many models for activity coefficients of aqueous

electrolytes have been developed in recent years. For

engineering applications, simplicity of form (a minimum

number of parameters) while retaining a high degree of

accuracy is of paramount consideration. Perhaps the

simplest correlation in use is the Meissner correlation

(Meissner and Tester, 1972; Meissner and Kusik, 1973(a);

Meissner and Kusik, 1978; Meissner, 1980). This correlation

was developed empirically as a graphical method for

predicting activity coefficients. A single parameter

correlating equation was developed from graphs of the

activity coefficient reduced by the charge as a function of

molal ionic strength. The method has also been extended to

multi-component solutions (Meissner and Kusik, 1972;

Meissner and Kusik, 1973(b); Meissner, Kusik and Field,

1979).

The most successful of the current activity coefficient

models, both in terms of the number of salts for which

parameters have been determined, and in the number of

investigators using it, is the Pitzer model (Pitzer, 1973;

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

and Kim, 1974; Pitzer, 1975; Pitzer and Silvester, 1976;

Pitzer, Peterson, and Silvester, 1978; Pitzer and Silvester,

1978(a); Pitzer and Silvester, 1978(b); Pitzer and Bradley,

1979; Zemaitis et al., 1986). In this model, the long-range

interactions are modelled by the extended Debye-HUckel

equation in which the inter-ionic separation distance takes

a universal value for all salts. The short-range

interactions are modelled as quadratic in salt molality.

This expression contains three salt-specific parameters.

For the case of 2-2 salts, a fourth salt-specific parameter

is added.

The activity coefficient data for over 275 salts have

been correlated using the Pitzer formalism. The temperature

dependence of these parameters has been extensively

investigated for NaCl from 25 to 3000C. The method has been

extended to multi-component solutions. The multi-component

case requires the addition of extra parameters to account

for double- and triple-ion interactions.

A third illustration of current trends in activity

coefficient modelling for electrolytes is the method due to

Chen and co-workers (Chen et al., 1979; Chen et al., 1980;

Chen et al., 1982). Unlike the methods of Meissner and

Pitzer, Chen's method uses mole fractions as the composition

variable. The long-range forces are modelled using the

expression developed by Pitzer, except normalized to the

mole fraction scale. Short-range effects are described by

the Non-Random Two Liquid (NRTL) model of Renon and

Prausnitz (1968). No method has been proposed to account

for temperature dependence of the short-range contribution,

although Chen has noted that for NaCl, FeC12, KCl, and KBr,

these variations appear to be small. The method of Chen has

been applied to over 130 systems.

While the activity coefficient models discussed thus

far are important, there are many other models that are

available for engineering applications (e.g., Sander,

Fredenslund, and Rasmussen, 1984; Corvalan-Quiroz, 1986;

Bromley, 1972, 1973, 1974). These models were chosen for

discussion because they illustrate different streams of

development in current modelling practice. The Meissner

correlation is an example of a purely empirical modelling

approach. The Pitzer model was developed by adopting a form

suggested by analysis of the pressure equation of

statistical mechanics. The Chen model represents the use of

a short-range expression developed for non-electrolytes and

adapted to electrolytes by the addition of the Debye-HUckel

expression for long-range interactions.

Volumetric Properties

In addition to activity coefficients, the volumetric

properties of aqueous electrolyte solutions are very

important in chemical process design and engineering. The

volumetric properties include the solution density,

component partial molar volumes, isothermal

compressibilities, and isobaric expansivities. These

properties may be used to determine the volume of the system

at a given set of conditions, and predict the variations in

the system volume as the state is changed.

Historically, the volumetric properties of electrolyte

solutions have been correlated empirically. Rather than

developing an equation of state to predict the solution

density, most efforts have been focused on modelling

component salt partial molar volumes, or even more commonly,

the salt apparent molal volume. The apparent molal volume

approach treats the solvent as having a partial molar volume

that is constant and equal to its pure component value

(ideal solution). All composition-related volume changes

are attributed to the solute.

The Masson equation (Masson, 1929) is one of the

earliest attempts to correlate apparent molal volumes. It

models the apparent molal volume as a linear function of the

square root of the molar concentration. The intercept of

this equation is the partial molar volume at infinite

dilution. The slope is experimentally determined and is

salt-specific. The Masson equation is an empirical

equation. Because of its empirical nature, this equation

does not always yield accurate values of the partial molar

volume of infinite dilution when used as an extrapolating

equation. Parameters for the Masson equation are available

for many salts at a wide range of temperatures (Millero,

1972).

Redlich and Rosenfeld (1931) developed the first

theoretically inspired model for apparent molal volumes.

They determined the Debye-HUckel limiting slope for the

apparent molal volume. Redlich and Meyer (1964) proposed an

extrapolating equation which added a term linear in molarity

to the Masson equation. They replaced the empirical slope

of Masson equation with the Debye-HUckel limiting slope and

chose the coefficient of the new term to be empirical. Owen

and Brinkley (1949) proposed a similar equation based on a

theoretical limiting slope derived from the extended Debye-

Hickel equation, rather than the limiting law expression.

Pitzer has also proposed a model for apparent molal

volumes (Pitzer, 1979). This equation was derived by taking

the derivative with respect to pressure of the Pitzer

activity coefficient expression. The Pitzer model for

apparent molal volumes has a Debye-HUckel slope that differs

from that derived by Redlich and Meyer by a factor of 3/2.

The Pitzer apparent molal volume equation requires three

salt-specific parameters. These parameters are identified

as the pressure derivatives of the three activity

coefficient model parameters.

The isothermal compressibilities and isobaric

expansivities can be represented by the related quantities,

the partial molal compressibility and the partial molal

expansibilities. Millero (1979) reports correlating

expressions for these quantities based on the Redlich-Meyer

expression for apparent molal volumes. They are derived by

taking derivatives of the apparent molal volume with respect

to pressure and temperature. A Pitzer formalism for partial

molal compressibilities has been advanced and tested for

multi-component solutions (Kumar, Atkinson, and Howell,

1982). This approach requires the addition of three new

salt-specific parameters.

Thermal Properties

The effect of temperature on the activity coefficient

of a salt is taken into account through the partial molar

enthalpy. While correlating equations for the apparent

molal volume are developed from the derivative with respect

to pressure of the activity coefficient, expressions for the

apparent molal enthalpy are derived by taking the derivative

of the activity coefficient with respect to temperature.

Analogs of the Owen-Brinkley equation have been used by many

investigators to successfully correlate the enthalpy data of

many electrolyte solutions (Millero, Hansen, and Hoff, 1973;

Leung and Millero, 1975; Leung and Millero, 1976; Duer et

al., 1976). While reasonable results for apparent molal

volumes may be obtained from an equation derived from the

Debye-HUckel limiting law, the extended Debye-Huckel

equation must be used to correlate the enthalpy data. The

Pitzer equation has been extended to enthalpies and, as in

the case of apparent molal volumes, three new salt-specific

parameters are added.

The effect of temperature on the enthalpy is accounted

for by the constant pressure heat capacity. For electrolyte

solutions, the heat capacity data is generally represented

in terms of the apparent molal heat capacity. An extended

Debye-HUckel expression is used as the correlating equation.

Millero (1979) gives a comprehensive review of available

data sources for the apparent molal heat capacities of many

electrolyte solutions.

A New Approach

From a review of the available physical property

correlations for aqueous electrolyte solutions, it is

evident that a unified, theoretically based formalism is

needed. While the Pitzer formalism is an attempt to achieve

this end, it has drawbacks. Foremost among these is the

need to introduce new parameters for each additional

property that is modelled.

In the discussion that follows, the formalism of

Fluctuation Solution Theory will be reviewed. A model for

the physical properties of aqueous electrolytes will be

developed and applied within the Fluctuation Solution Theory

framework. Salt activity coefficient, solution density,

component partial molar volume, and isothermal

compressibilities are the focus of this effort. The

advantage of the Fluctuation Solution Theory formalism in

terms of fewer number of parameters to successfully model

these properties will be exploited.

Another issue which will be addressed is the ionic

additivity of model parameters. A proper theory should

treat the contributions of the ions as additive species

since they are constrained only by overall charge

neutrality. In addition, such a treatment has the major

advantage of reducing the size of the parameter set as the

number of salts is much larger than the number of

constituent ions.

A final issue concerns the conversion from McMillan-

Mayer to Kirkwood-Buff variables. The true Debye-Huckel

expression is derived in the McMillan-Mayer system, in which

the natural variables are temperature, volume, solute mole

number, and solvent chemical potential. The Fluctuation

Solution Theory formalism has the Kirkwood-Buff canonical

variables of temperature, volume, solute mole number, and

solvent mole number. Other models use variables of

temperature, pressure, and component mole numbers (Lewis-

Randall variables). The proper transformation of the Debye-

HUckel expression from McMillan-Mayer to Kirkwood-Buff and

Lewis-Randall variables will be derived.

Experimental

Finally, an effort was undertaken to measure the

volumes of aqueous electrolytes as a function of

temperature, pressure, and salt concentration. This effort

was not successful. The apparatus which was used will be

described in detail, with special attention paid to the

13

extensive modifications which were made. Suggestions for

further modifications will be advanced.

CHAPTER 2

FLUCTUATION PROPERTIES--THEORY AND EXPERIMENT

The aim of molecular thermodynamics has been to apply

principles of molecular physics and statistical mechanics to

predict physical properties of chemicals and their mixtures

(Prausnitz, 1969). One of the results of this approach is

the identification of which physical properties, or collec-

tions of properties, are most amenable to modelling.

Statistical mechanics provides three avenues for connecting

models for microscopic behavior to macroscopic properties.

One method is to model the partition function, which can be

used to calculate the Helmholtz free energy. The inter-

molecular pair potential and the radial distribution

function, when spatially integrated using the proper

formulations, lead to expressions for the system pressure

("Pressure Equation") or the configurational internal energy

("Energy Equation") (Reed and Gubbins, 1973; Mohling, 1982).

Thermodynamic manipulations relate the results of these

equations yield the desired properties, usually the free

energies and fugacities. One drawback to the use of these

general equations is that both assume pairwise additivity of

intermolecular potentials. A second and more serious

objection is that the models derived from these equations

must be differentiated to obtain activity coefficients. The

differentiation process highlights shortcomings of models.

Statistical mechanics provides a third method of

relating microscopic interactions to macroscopic

thermodynamic properties. This formalism, known as

Fluctuation Solution Theory, relates spatial integrals of

the total correlation function (TCFI) to concentration

derivatives of the chemical potential and the pressure

(Kirkwood and Buff, 1951). For pure components, this is

known as the "Compressibility Equation." The equations are

derived without assuming pairwise additivity. A variation

of this equation relates TCFI to spatial integrals of the

direct correlation function (DCFI) via the Ornstein-Zernike

equation to concentration derivatives of the chemical

potential (O'Connell, 1971; O'Connell, 1981). The direct

correlation function is generally shorter ranged than the

radial distribution function so that accurate modelling may

be simpler. The great attraction in the use of Fluctuation

Solution Theory to the modelling of thermodynamic properties

is that models derived from this approach are integrated to

yield activity coefficients. The integration process tends

to minimize the effect of model shortcomings. In addition,

changes in density can be obtained from the same quantities.

A formalism exists for applying Fluctuation Solution

Theory to the case of a system with multiple partial

chemical reactions (Perry, 1980; Perry and O'Connell, 1984).

A treatment for strong electrolytes has also been developed

(Perry, Cabezas, and O'Connell, 1988) as a special case in

which all of the reactions have gone to completion (complete

dissociation). This work develops models for this case.

Theory

In order to properly consider the application of

Fluctuation Solution Theory to electrolytes, certain basic

facts concerning solutions of strong aqueous electrolytes

need to be recognized. First, thermodynamics can only give

information concerning the components present in solution.

Components refer to all solvents and undissociated salts.

Components are the substances which are used to prepare

electrolyte solutions. For the case of strong electrolytes,

all salts completely dissociate into their constituent ions.

These ions, and all solvents, are referred to as species.

(Solvent molecules are considered to be both components and

species.) It should be noted that the amount of a given

ionic species in solution is not independent of the other

ionic species due to the constraints of electroneutrality

and stoichiometry.

In his derivation of the Fluctuation Solution Theory

equations for general multi-component electrolyte solutions,

17

Cabezas (1985) begins with the fundamental equation obtained

by Perry and O'Connell (1984)

1 i 1- i (2-1)

RT Nj 'T,V,N j Ni N

where

ui = the chemical potential of species i,

Ni = the number of moles of species i,

N = the total number of moles of all species,

6ij = the Kroniker delta,

R = the gas constant,

T = absolute temperature.

The DCFI for spherically symmetric potentials is defined by

C = 4n p J c.ir2dr (2-2)

with

Cij = the molecular centers direct correlation

function,

p = N/V = the molecular density of all species,

V = the system volume.

At this point, the stoichiometric relationships among ions

are introduced to develop the component equations from the

species equations.

The number of moles of a given species, j, is related

to the number of moles of components, 0, by

n

N = E v Np (2-3)

P =1 j

where

Np = the number of moles of component p,

vjp = the stoichiometric coefficient of species j in

component 3,

no = all components.

Analogous expressions relating species and component

densities and species and component mole fractions may also

be defined.

n

PJ = E v Pp (2-4)

Pj = Nj/V = the density of species j,

pp = Np/V = the density of component 3,

and

xj = n x (2-5)

J 13=1 jP P

Xj = Nj/N = the mole fraction of species j,

xp = NP/N = the mole fraction of component P.

It should be noted that the species mole fractions sum to

unity, while the component mole fractions do not.

The derivation of Cabezas, starting with equation (2-1)

will not be reproduced here, but the principle equations for

modelling will be discussed. The first equation of interest

relates DCFI to the derivative of the mean ionic activity

coefficient with respect to the molar density of component 3

akny

p- P TP v (1-C a) (2-6)

where

1 n

ny E v.ianyi (2-7)

V i=l

and

4i-4i

inyi = = natural log of activity coefficient of

RT

species i,

4 = the chemical potential of species i at its

reference state,

a = the sum of the stoichiometric coefficients of

component a,

n = all species.

and

1 n n

1 C = E via v (1-C. .) (2-8)

X)Cp i=l j=l

At this point it would be possible to discontinue the

analysis. A model for either the direct correlation

function, or the direct correlation function integral, could

be formulated, and this model could be integrated in

equation (2-6) to yield the activity coefficient. Because

this equation is written in terms of component density, the

solution density would be necessary to find the

activity coefficient. Alternately, the model equation for

the activity coefficient could be converted to a molality or

a mole fraction basis. While no models currently used in

engineering practice are density based, conversion from a

theoretically derived density based equation to one of the

other bases is very common in the modelling of electrolyte

solutions (Pitzer, 1977; Bromley, 1973).

It is possible to obtain a relation between pressure,

density, and composition using the isothermal, non-isobaric

Gibbs-Duhem equation.

n

SNidui = VdP (2-9)

i=1

where

P = the system pressure.

Combining equations (2-1) and (2-9) relates the derivative

of the system pressure with respect to component density to

the DCFI.

1 IP n

--E- = V0 x x(1-C ) (2-10)

RT Dp T VlPyta =1

This equation is the key to calculating solution densities

for use in equation (2-6). A more complete discussion of

this point will be made in the Application section of this

chapter.

Finally, the thermodynamic identification

@P V

--= (2-11)

p T,py~ T

where

Va = the partial molar volume of component a

KT = the isothermal compressibility of the solution

can be made. Substitution equation (2-11) into (2-10),

multiplying by the mole fraction of component a, and summing

over all components a yields

aP/RT 1 n0 n

1 = v v xx x(1-C a) (2-12)

Sp T,N pcTRT a= = =1

Relation of DCFI to Experimental Measurements

Given a model for the DCFI, integration of equations

(2-10) and (2-6) yield the solution density and component

activity coefficients. It is possible to develop a model

for the DCFI from the microscopic direct correlation

function. To gain modelling insight at this level, one must

generally resort to examining the results of molecular

simulation. These techniques are very valuable for

exploring the impact of different effects on molecular

interactions. However, they offer little additional insight

into the more complex cases, such as aqueous electrolytes,

where not all of the effects present are well understood.

An alternative approach is to construct a model from

examining the relationships between the DCFI and

experimental quantities. While equations (2-6) and (2-10)

are useful for finding densities and activity coefficients

from a DCFI model, they are not useful for developing a DCFI

model. The first reason is that the derivative in equation

(2-6) is taken with respect to constant temperature, volume,

and mole number. Experimental measurements on solutions are

generally made at fixed pressure rather than constant

volume. This point will be discussed more thoroughly in the

next chapter. The second reason is that the summations on

the right hand side of equations (2-6) and (2-10) do not

allow unlike component DCFI to be separated from like

component DCFI for analysis.

For the purpose of developing a DCFI model for

electrolyte solutions, it is necessary only to examine the

case of a single electrolyte in a single solvent. The

summations in equations (2-6) and (2-10) give the proper

extension to the multi-solvent, multi-salt case once a model

for each type of DCFI is known. For the one-solvent (1)-

one-salt (2) case, the principle equations of the previous

section become

and

2 I 22

=ny2 2 1-C22 (2-13)

aP2 T'Pl P

aP/RT 2

P/T 2x (1-C12 + x2(1-C22) (2-14)

Sp2 'T,1 KTRT

aP/RT I 1

p TN PTR x2(1-C11) + 2v2x1x2(1-C12)

ap 'T,N pKTRT

+ V22x(1-C22)

(2-15)

where, because the solvent does not dissociate, v1= 1.

Inversion of equations (2-13)-(2-15) yields separate

equations for DCFI for solvent-solvent (C11), solvent-salt

(C12), and salt-salt (C22) interactions in terms of

derivatives of experimental quantities at constant volume.

Conversion from a constant volume to a constant pressure

system yields the following equations:

1

1 C = 2 (-x2V2P)

X1PKTRT

2

x2 kny

+ V2 N (2-16)

x 2N2 T,P,N1

V2 x2 ny2 I

1 C (l-x2V2) + N (2-17)

2X1KTRT x1 -N2 T,P,N1

and

-2

PV2 N atny2

1 C22 2 + (2-18)

2 KTRT v2 aN2 T,P,N1

Figures 2-1 through 2-3 show typical results for the salts

NaC1, NaBr, LiCl, and LiBr at 250C and 1 atm pressure. For

all salts, the compressibility data of Allam (1963) and

activity coefficient data of Hamer and Wu (1972) were used.

For the density of NaCl, the data of Romankiw and Chou

(1983) were used, while the International Critical Tables

provided the density data for the remaining salts.

Equation (2-18) presents a particular difficulty for

use in modelling the properties of aqueous electrolytes.

The activity coefficient for electrolytes in solution is

generally represented as a Debye-Hickel term plus an

expansion in powers of molality. A typical equation of this

type is the equation used by Hamer and Wu (1972) in their

data correlation.

IZ Z_ | AmIm 2

log0 = + Im + CI + *** (2-19)

1 + B*

where

Zi = the valence of ion i,

B*, ,C ... = empirical constants

1 n

I E Z m = ionic strength, molality basis

2 i=l1

m = 1000-N2-N-1 *M1 = solution molality

1/2 3/2

(2rNA)/2 e 1/2

m nl0 4ee 0kT)

= the Debye-HUckel constant, molality basis

= 0.5108 kg1/2 mol-1/2 at 25C

NA = Avogadro constant = 6.02252 x 1023 mol-1

e = elementary charge = 1.60210 x 10-19 C

eo = permittivity of free space = 8.85417 x 10-12C2-j-l1m-1

e = the pure solvent dielectric constant permittivityy)

= 78.4472 for water at 250C

k = Boltzman constant = 1.38054 x 10-23 J K-1

M1 = solvent molecular weight.

Differentiation of equation (2-19) according to the

prescription of equation (2-18) yields as its leading

N ~9ny2 |Iz+Zl Am(100l nl0)

S= (2-20)

2 DN2 T,P,N1 2 v2x1M14m(1+B m )2

It is seen that this term diverges as I1/2 in the limit of

infinite dilution of the salt (Im 0). This difficulty may

be removed by subtracting out the Debye-Huckel limiting law

term from both sides of equation (2-20), defining a

convergent short-range activity coefficient as

c LL

N any I N DRny N Dny 2

ST,P,N I (2-21)

2 DN2 T,P,N1 V2 2 T'PN1 2 N2 T,P,N1

where

N ~.ny2L Z Z_ Am(1000n 10)

--- = I (2-22)

22 2N2 T'P'N 2 1MJIm

= the Debye-HUckel limiting law expression for the

activity coefficient derivative with respect to

salt mole number.

Combining equations (2-20), (2-21) and (2-22) yields

N Many2 A B*(1000tnl0) (2+B*JIm)

S- (2-23)

M aN T,P,N 2

2 2 1 2v2X1M1 (1+B*,JI)

Using this expression, a convergent short-range salt-salt

DCFI can be calculated

-2 c

pV2 N atny

1 C2 + (2-24)

S2KTRT v2 8N2 T,P,N1

In the limit of infinite dilution, the activity coefficient

derivative term of equation (2-24) using the general model

form of (2-19) becomes

N ny2 1000knl0

= (AmB + 3) + 1 (2-25)

S2 aN2 T,P,N1 v2M1

where the factor of 1 added to equation (2-25) converts

equation (2-19) from a molality basis to the mole fraction

basis required by equation (2-18).

In deciding how many terms to include in their

correlating equation for a given salt, Hamer and Wu chose

the number of terms which minimized the standard deviation

of the data from the predicted value of the correlation. In

practice, this means that equations comprised of fewer terms

than the number of terms selected by Hamer and Wu may be

equally capable of representing the experimental data. For

each of these equations for a given salt, the values of B*

and 3 may vary by as much as 15 to 20%. For sodium chloride

at 250C and 1 atm pressure, the activity coefficient

derivative contribution to the infinite dilution convergent

salt-salt DCFI is about 90% of the total infinite dilution

value. The choice of the number of terms to use in equation

(2-19) will significantly affect the DCFI in the low-

concentration region, as will using other forms. Equation

(2-25) shows that parameter values for a given salt-salt

DCFI model will also be influenced by the form of

the correlation. This difficulty will be addressed more

fully in Chapter 4.

A final equation of interest for relating DCFI to

experimental quantities is derived by converting equation

(2-13) from a constant volume to a constant

pressure derivative. The result is divided by the reduced

bulk modulus (pKTRT). Equations (2-14) and (2-15) are

substituted into the resulting equation, yielding

Nva Znya n0 n0

TRT NP 'T,P,N y=l 6=1

S[(1-Cy)(1-Ca, ) (1-C y)(1-C8 )] (2-26)

While this equation was not used in this work, it is very

important for determining the general applicability of

Fluctuation Solution Theory to a class of solutions. For a

binary mixture, such as a one salt-one solvent mixture, this

equation reduces to

Nv2 8Jny2 n 22

2- 2TP = \2x1[(1-C11)(1-C22) (1-C21)(1-C12)]

pKTRT aN2 T,P,N1

(2-27)

The right-hand side of this equation reflects the difference

between like-like and unlike interactions, as measured

by DCFI. In the case of ideal (or nearly ideal) solutions,

the unlike DCFI (C12 and C21) are near the mean of the like

DCFI (C11 and C22). The right-hand side of equation (2-27)

is then a small difference of large numbers, with the

differences being on the order of the experimental error.

The method is clearly not applicable in these

30

near-ideal situations. Examination of the experimental data

for aqueous electrolytes shows that this difference is

usually large, because electrostatic forces make the salt-

salt DCFI much different from the short-range solvent-

solvent and solvent-salt DCFI. This suggests that

Fluctuation Solution Theory can be applied in this case.

General Application

As yet, no mention has been made of a specific model

for the DCFI for electrolyte solutions. While the

introduction of a model is reserved for the next chapter,

the general connection of the DCFI to calculating densities

and activity coefficients is described here. An insight

into modelling the partial molar volume of electrolytes is

obtained from examination of Fluctuation Solution Theory

general equations.

The following discussion follows the general method

outlined by O'Connell (1981). The variation of the total

pressure difference with solution density is given by

1 1 n P

dP =- E - dp (2-28)

RT RT a=1 pa T,py

Substitution of equation (2-10) into (2-28) and integrating

from an initial condition for which temperature, pressure,

and composition are specified to a final state at the same

temperature, and a specified pressure and composition yields

f r pf

Pf pr n n a (1-C )

= ZE v p p P dp, (2-29)

RT a=1 =1 pr p T,p

A model for the various DCFI may be inserted into the right

hand side of this equation and integrated. Given T, pf, pr

and all pr equation (2-29) is solved by trial and error for

the density.

Equation (2-6) may be used to calculate activity

coefficients in a similar manner. The activity coefficient

differences for a solution as the component densities vary

are given by

n 09nny

diny, = aS dp (2-30)

a=1 app TpY#P

Insertion of equation (2-6) into (2-30) and integration

yields

r

n 0 (1-C )

=ny = v Eo dpp (2-31)

a=1 p TPy

The DCFI models are inserted into this expression to yield

an equation for the activity coefficient of component a.

The densities calculated from equation (2-29) are used in

this equation. Thus, given a model for the DCFI, both

densities and activity coefficients of solutions may

be calculated. In contrast to other modelling approaches,

Fluctuation Solution Theory allows the basic model to be

integrated, rather than differentiated, to obtain these

properties. Integration should allow more accurate

predictions of the densities and activity coefficients.

As a final comment on the application of Fluctuation

Solution Theory to modelling thermodynamic properties,

equation (2-14) will be re-examined.

2 2

v2x1(1-C12) + x2(1-C22) (2-14)

KTRT

The first term on the right hand side of this equation

reflects salt-solvent interactions. The second term

reflects salt-salt interactions. Currently, many

investigators focus on the apparent molal volume. The

apparent molal volume is related to the partial molar volume

by

m1/2 (d,

2 v 2 -

V2 = v + --1- ( 2-32)

djm

where

(v = the salt apparent molal volume.

Equation (2-31) suggests that, for the purpose of advancing

a theory based on the molecular interactions present in

33

electrolyte solutions, the quantity (V2/KTRT) is more

appropriate than the apparent molal volume. In subsequent

chapters, a model for the DCFI of electrolytes in solution

will be advanced, and this assertion, as well as the

consequences of equations (2-29) and (2-31), will be

explored in detail.

0 1 2 3 4 5 6

7 8 9 10 11

* NaCI

0 LiCI

* NaBr

SLIBr

Molality (mol/kg)

Figure 2-1. Water (1) Water (1) DCFI in aqueous

electrolyte solutions at 250C, 1 atm.

See text for data sources.

.__________ f

- - _ _

- - - ^ -

~~~/ ^_ ,__ ____

'- imt^^^ !!'-""

. -- - z ; _ i

- - __,,,_ _ _ _

I - ? - - - -

I -I

. . -' -- -_ -- - -.- -."-]- - -

0 1

2 3

9 10

Molality (mol/kg)

Figure 2-2. Water (1) Salt (2) DCFI in aqueous

electrolyte solutions at 250C, 1 atm.

See text for data sources.

* NaCI

0 LICI

A NaBr

A LiBr

* ^, / -I -. -

-F^.^-^E^r m

-/^^ E======

\^f

P-t-T

46

44

42

40

(N

0 (N 38

36

o 34

32

30

28

26

24

22

20

* ~--- -

I -

* - -- .- -

- --- - - -

---------

3 4 5 6 7

Molality(mol/kg)

Figure 2-3. Convergent salt (2) salt (2) DCFI in aqueous

electrolyte solutions at 250C, 1 atm. See

text for data sources.

* NaCI

LiCI

A NaBr

SLiBr

8 9 10

CHAPTER 3

MODELLING FLUCTUATION PROPERTIES

In Chapter 2, a formalism was presented for relating

DCFI to the fluctuation properties. The prescription for

relating DCFI to experimental quantities was developed as

well. This chapter will begin with a discussion of the DCFI

model developed by Cabezas (1985). The strengths and

weaknesses of this model will be examined, and a new DCFI

model will be presented. Finally, the activity coefficient

and total pressure equations will be derived by integrating

the new model in equations (2-29) and (2-31).

The Cabezas Model

Cabezas (1985) proposed a DCFI model in which the

direct correlation function is separated into long-range and

short-range contributions. This form was inspired by the

work of Stell, Patey and H0ye (1981).

C = CS + LR (3-1)

ap ap ap

SR

C = short-range DCFI

LR

C = long-range DCFI.

ap

He further subdivided the short-range expression into a

hard-sphere contribution, to account for repulsive forces

between species at contact, and a term resembling a virial

expansion to account for all other short-range affects.

SR CHS

C = CH + AC (3-2)

where theoretical expressions can be used for C and

correlations developed for ACc. Cabezas chose the Percus-

Yevick compressibility equation as the basis for the hard

sphere expression (Appendix A). The virial expression was

inspired by the Rushbrooke-Scoins expansion of the direct

correlation function for species i and j in an n-species

system (Croxton, 1975; Reichl, 1980).

n

cij(T,P,r ,rj.,iJj) = fij(T) + k Pk ijk(T) + *** (3-3)

where

-u. /kT

f = e 1 = Mayer bond function

= j

9 ijk(T) = -f f.i fik fjk drkd

0 = dwi = integral over orientation coordinates.

39

The hard sphere and long-range direct correlation functions

were subtracted from equation (3-3), and the remainder was

integrated using equation (2-2) to obtain the virial

expression.

n

AC. = p A F i(T) + p E PkA ijk(T) (3-4)

k=l

where

00

2

AF. (T) = 4n < Af.ij> r.j dri

Sf r2

0

Aik (T) = 4T < A > r .dr.

The A in the above expressions signifies that long-range and

hard sphere contributions are subtracted from the total.

Summing overall species yields

1 n n

AC E E v. vj C.

a v p i=l j=1 la

p n n

-- E vi\ i aFi =

V p i=l j=l

p n n n

+ --- E E v v. 0jPk Aijk (3-5)

va v i=l j=1 k=1 l i

In equation (3-5), the species density may be replaced by

the component density. By defining

n n

AF v. iaVjAFij (3-6)

i=1 j=1

and

n n n

ADaoy 5E E E v ia~vjBkyADijk (3-7)

i=1 j=1 k=1 k ijk

the virial contribution to the DCFI becomes

p p n

&Cap aFap + apy

AC =- AF + pAc (3-8)

Traditionally, long-range interactions between ions

have been modelled using the Debye-HUckel treatment, or one

of its variants. Cabezas derived a long-range expression

for the DCFI by making the quadratic hypernetted chain

approximation to the direct correlation function

QHNC + 1 (3

c = - + -- + *** (3-9)

13 kT 2! kT J

where

uij = pair potential

Wij = potential of mean force.

For the case of ion-ion interactions, Cabezas modeled the

pair potential with a Coulomb potential.

2

Z Z e

u = (3-10)

lij

The potential of mean force was approximated with a form

suggested by the extended Debye-HUckel theory.

2 K(a i-ri.)

LR ZiZje2 e 3

j = r i>aij (3-11a)

13 kT r.j ee0(l+

ij 1J

LR

ij = 0 rijS aij (3-11b)

where

2 4e n

2 2

K E ZP = Debye-Huckel inverse length

es kT i=1

1

a = (aii+ajj) = distance of closest approach of

species i and j.

Equations (3-10) (3-11b) were inserted into equation (3-

9). The result was integrated using equation (2-2) and

summed over all species. The expression for the long-range

salt-salt DCFI is

c I-1/2 n 2 2

CLR = ya] ipjpi

a= 2

4 V i=1 j=1 (1+a1iB /2)

2 2 2 3ajB Y 1/21/2

Sp nC n ia. Z.Z.e aj E 3a.ij.ByI1/2

3v v i=1 j=1 (+aB1/2

(3-12)

with

e2 '3/2

S = (2n)1/2 e] (3-12a)

es kT

8ne2 /2

B = o- (3-12b)

See kT

1 n

2

I = E Zi.p = ionic strength, density basis.

2 i=l

and

S-3B I/2r..

< i1/2> r e 13

E 3aijByI 2)= j 3dr ij

1 13 ij

r.

a. 13

= the first exponential integral.

In a similar manner, Cabezas derived expressions for

the long-range interactions between ions and solvent

molecules, and solvent molecules with other solvent

molecules. Solvent molecules were treated as dipoles in the

pair potentials. For the ion-dipole case, the potential of

mean force was approximated with a functional form suggested

by applications of the mean spherical approximation (Chan,

Mitchell, and Ninham, 1979) and perturbation theory (H0ye

and Stell, 1978) to non-primitive electrolyte models. The

functional form for the potential of mean force for dipole-

dipole interactions was inspired by the same source. The

expansion for the salt (a)-solvent (1) long-range DCFI is

2mpf e )2 n v z e Y /2

S 22 il

LR -2 E ZaE 12)

al 3v ekT i=l ai 2 (2aiBYI

(3-13)

where

S= empirical constant, arbitrarily set to 4.4

I1 = dipole moment of solvent

and

-ii 13

E2(2ailBjI /2j= e 2 dr il

aj il

aij i

= the second exponential integral.

The solvent (1)-solvent (1) long-range DCFI expression is

_LR 4 IL 2 2allB /2

C1 ----

11

3 s kT

E4(2a 11B /2

(3-14)

E4(2a11Bj1/2) = f

1/2

e- 2 B I1/2

-2a Bll I

e

4 drill

a r11

11

= the fourth exponential integral.

The solvent dipole moment in equation (3-14) is written as a

product to indicate that for the mixed solvent case, the

dipole moment of each solvent must be accounted for in this

expression.

Cabezas tested his DCFI model using the DCFI data for

six salts: LiCd, LiBr, NaCl, NaBr, KC1, and KBr. From his

where

analysis, he reached two important conclusions. The first

conclusion was that diameters derived from ionic crystal

radii measurements could be used in both the hard sphere and

long-range portions of the model. The second conclusion was

that the virial expression could apparently be decomposed

into additive ionic contributions.

In testing his model, Cabezas was successful with the

solvent-solvent and salt-solvent DCFI. The difficulty was

encountered in modelling deviations of the salt-salt DCFI

from the infinite dilution value as salt concentration

increased. The general form for modelling in this manner is

1 C = -C) HS-C -C -C ) AC -AC

ap ap ap

C LR -C (3-15)

where the superscript (-) refers to the infinite dilution

state. For the salt-salt case, this choice of modelling

approach presents a problem. As alluded to in Chapter 2,

the salt-salt DCFI diverge as I-1/2. Cabezas approached the

problem by rearranging the terms in equation (3-15).

1 C 1-C 00 LR HSC HS-1

1 "Ca ap ap ap p

STB TBTooB

c _-C J (3-16)

where the following special definitions have been made:

0a P VaV p-p

o2

TB TBoo PP1-P PP2

-C C = A + -- (3-18)

ap ap = apl V aPY

pO = reference (pure solvent) density

K = reference (pure solvent) isothermal compressibility.

At low concentrations, equation (3-16) contains divergences

which obscure the other model contributions. This is a

critical problem because the two-body parameter, AFga, is in

essence the zero concentration intercept. This parameter is

important at the activity coefficient level and needs to be

fitted using a more precise method, such as that suggested

in Chapter 2.

Consequences of the Cabezas Model

One of the interesting conclusions reached by Cabezas

was that the crystal (bare ion) radii of the ions could be

used in both the long-range and hard sphere portions of

the model. This conclusion is very different from that

reached by most investigators. For the long-range portion

of the model, the ion separation distance (diameter) has

been treated in one of two ways. In engineering

correlations, it is generally treated as a universal

constant for all salts. Theoretical studies, such as that

of Pailthorpe, Mitchell, and Ninham (1984), and Martin,

Gom6z-Estevez, and Canales (1987) used fitted diameter

values.

The two above-mentioned theoretical studies model

activity coefficients and osmotic coefficients using an

extended Debye-HUckel term for long-range forces and a hard

sphere term for the short-range repulsions. These

investigators fitted a single diameter for each salt for the

two terms of the model. Both groups found best fit

diameters which were not equal to the sum of the bare

ion radii. The fitted values were in fact found to decrease

as bare ion radii increased. Table 3-1 shows a comparison

of Pauling bare ion diameters with the fitted diameters for

one of the models tested by Martin et al. (1987).

This disagreement, as well as unsatisfactory results

for activity coefficient predictions using the Cabezas model

and parameters, led to a re-examination of the salt-salt

DCFI fit. Rather than using equation (3-16), the salt-salt

DCFI were analyzed using an equation similar to equation (2-

20). To do this, we begin with equation (3-12), neglecting

the higher order second term. To derive the convergent

long-range expression, the limiting law is subtracted from

the extended expression.

LR C = LR LR LL

ap aS ap

(3-19)

LR C

CLR = convergent long-range salt-salt DCFI

-1/2

S pl n n

LR LL S yI n 22

C E V. iv Z i Z

a 4Va V 1i=1 j=1 l i

(3-20)

The resulting expression is

-1/2

S pi 1 n

4v v i=1

n v iaV Z2

E (l+aij)

j=1 (1+Ka. )2

n n

- E E v. iv.Z.Z

i=1 j=1 1

-1/2

S pI n

Y E

4v D

i=1 j=1

2

. v. Z. Ka.

i1L j1P I 13

(1+Ka..ij)2

SBp n n a.z .v v 2

SE E ic j

4v: V i=1 j=1 (1+Kaij)2

The convergent salt-salt DCFI are modelled as

with

cLR C

ap -

(3-21)

S C cHS VIR LR C (3-22)

1 a- = S C C (3-22)

with

C

C = convergent salt-salt DCFI.

The expression on the left-hand side of equation (3-22) is

calculated using a form similar to equation (2-24). In

generating the data, a slightly different form of (2-24) was

developed by subtracting equation (3-20) from (2-20).

2 -1/2

pV2 N alny+ S pi/2

1 C22 + -+ Y Vi2j2 2

22 2 KV M4vi2 j2ZiZ j

2 RT 2 N2 T,P,N1 4v ij

(3-23)

This was done to allow a consistent comparison of the data

and model in equation (3-22) by guaranteeing that the same

term was removed.

By rearrangement of equation (3-22), the virial

contribution can be isolated.

HS LRC C

AC 1 c C (1-C ) (3-24)

Equation (3-24) can be further broken down by combining it

with equation (3-8), dividing by the total density, and

multiplying by Va.p*

AF + p = P (-HS CLRc-(l-C )) (3-25)

AF y=l a+y 1 ap ap ap3

For the special case of a one salt-one solvent system,

equation (3-25) becomes

2

V2 HS LRC C

AF2 + p 221 + p2A222 (1-C -C2 -(1-C )) (3-26)

22 + 221 +2 222 22 22 22

This equation should be roughly linear in salt component

density (p2) if the salt-salt DCFI can be successfully

modelled with the Cabezas model, and if the correct

parameter values (diameters) are chosen for the hard sphere

and long-range contributions.

The right-hand side of equation (3-26) is graphed as a

function of salt component density in Figures 3-1 (LiC1,

NaC1, and KC1) and 3-2 (LiBr, NaBr, and KBr). The ionic

crystal radii of Marcus (1983) were used in order to follow

Cabezas as closely as possible. (Pauling and Marcus

diameters are very similar.) It is evident that the result

is non-linear at low concentrations, particularly for the

lithium and sodium salts. The potassium salts show somewhat

linear behavior, but the reason for this is obvious on

examination of Table 1. The fitted diameters for potassium

are close to the Pauling diameters. The conclusion from

this analysis is that ionic crystal radii do not give

acceptable results in modelling salt-salt DCFI.

Ionic Additivity

Another result reported by Cabezas was the possibility

that the virial contribution to the model could be

decomposed into terms which were dependent only on the ions

present in solution. He proposed a set of simple mixing

rules for the two- and three-body terms. These mixing rules

are based on certain assumptions made about the terms in the

summations of equations (3-6) and (3-7). As an example, the

case of the salt-salt two-body parameter will be examined.

The salt-salt two-body parameter is, upon expansion,

given by

AF22 = 2AF + 2v 2AF + 2 AF__ (3-27)

22 +2 ++ +2 -2 +- -2

Cabezas assumed that the anion-cation term could be broken

down into cation-cation and anion-anion terms according to

the mixing rule.

1

AF++ = (AF +++AF) (3-28)

++ ++

2

Using this rule, equation (3-27) becomes

2 2

AF22 = +2 AF + v2 -V2(AF+++AF__) + v 2AF__

22 +2 ++ +2 -2 ++ -2 -

= v2(+2AF+++ v_2AF__)

(3-29)

Similar mixing rules were adopted for the other parameters

which comprise the virial term.

If mixing rules of the type represented by equations

(3-27)-(3-29) are valid, then it is implied that the virial

contributions, as determined by subtracting out the hard

sphere and long-range terms, should display a simple

additivity as well. The overall salt-salt DCFI do not show

such behavior. Therefore, if the virial term is to be

simply additive, then the non-additive effects must be

removed in either the hard sphere or long-range

contributions. Figure 3-3 compares the convergent long-

range contribution for LiC1 calculated using the Marcus

diameter with the same quantity calculated using the

diameter for LiC1 determined by Martin et al. Figure 3-4

makes the same comparison for NaBr. The density data

sources are the same as given in Chapter 1. It is clear

that since the high concentration values of the long-range

contribution do not depend on salt diameter, this term is

not the source of non-additive effects.

Since at higher salt concentrations (>3m) the

convergent long-range contributions tend toward a common

value, this is the appropriate range to consider non-

additive effects from the hard sphere contribution. Hard

sphere salt-salt DCFI calculated using Marcus diameters as a

function of salt density are shown in Figure 3-5.

Figure 2-3 gives the convergent salt-salt DCFI calculated

according to (3-23), also as a function of salt density.

The Percus-Yevick hard sphere DCFI are ordered as

LiCl>NaCl=LiBr>NaBr. The convergent salt-salt DCFI are

ordered as LiBr>NaBr>LiCl>NaCl.

Thus, while the Percus-Yevick hard sphere form yields a

slightly non-additive contribution, this effect is not

large. Furthermore, the hard sphere equations do not

provide the sort of "de-ordering" to justify the simple

mixing rules proposed by Cabezas for the virial parameters.

Unfortunately, the alternative is to search for new mixing

rules, or to accept that some of the parameters defined by

equations (3-6) and (3-7) will be salt-specific.

Short-Range Contributions--A New Formulation

In light of the difficulties with the Cabezas model

uncovered in the previous section, it is evident that the

model needs revision. In particular, the role of the hard

sphere and virial contributions need to be re-examined.

Examination of Figure 3-5 indicates that the hard sphere

DCFI are linear. The virial contribution is effectively

linear as well. It is evident that the hard sphere and

virial contributions, as formulated by Cabezas, can be

replaced by a single virial term which includes both hard

sphere and intermediate range effects. This is advantageous

because it allows the number of model parameters to

54

be reduced. In the Cabezas formulation, the hard sphere and

long-range diameters were assumed to be the same, though it

is now clear that these diameters need to be fitted

separately. Rather than introduce a new set of

parameterized diameters, the hard sphere term can be

removed.

A remaining question concerns the ionic additivity of

virial term parameters. As alluded to in the previous

section, the simple additivity reported by Cabezas does not

seem to exist. By making assumptions of the type

represented by equation (3-28), Cabezas made an a priori

assumption concerning unlike interactions between charged

species at small separations. Rather than taking this

approach, model comparison with data can elucidate mixing

rules for some of the virial parameters.

By combining equations (2-10) and (2-11), we obtain

V n

a x(1-Cap) (3-30)

KTRT P=l

For the case of one salt (2) in one solvent (1) at infinite

dilution, this equation becomes

_c__

V

-- 2(1-C21) (3-31)

KIRT

T

55

At infinite dilution, the long-range contribution, as given

by equation (3-13), is zero. Substituting (3-8) into (3-31)

yields, upon rearrangement,

-m o o2

p- p o2

V2 P P

-= 1R - AF A112 (3-32)

oT 12 112

V2KTRT v2 2

It is well-known that the partial molar volume at infinite

dilution is ionically additive in a simple fashion (Millero,

1979). Therefore, it can be asserted that a simple mixing

rule of the type represented by equation (3-28) is

appropriate for the two-body salt-solvent parameter and the

three-body solvent-solvent-salt parameter. Unfortunately,

there are no other equations which yield similar insight for

the other parameters, AF22, A0122, and A0222. Thus, these

virial parameters will be fitted and the results examined to

see if any mixing rules are suggested.

Long-Range Contributions--McMillan-Mayer

to Kirkwood-Buff Transformation

Now that the short-range model has been revised, the

long-range expressions can be examined. The long-range

DCFI expressions are given by equations (3-12)-(3-14).

Cabezas (1985) reported solvent-solvent long-range

contributions which never exceeded 0.01% of the overall

solvent-solvent DCFI. For this reason, the solvent-solvent

long-range DCFI are neglected in subsequent applications.

A similar conclusion was reached concerning salt-solvent

long-range DCFI. While making a larger contribution than

the solvent-solvent long-range term, the salt-solvent long-

range term is still small, linear, and tends not to vary

greatly among the six salts examined. This term is

therefore absorbed into the virial contribution.

The salt-salt DCFI presents a more difficult problem.

A beginning can be made by recognizing that the second term

of equation (3-12) contains the Friedman limiting law

(Friedman, 1962). The Friedman limiting law is a higher

order contribution, and is small at even high salt

concentrations. Goldberg and Nuttall (1978) have

investigated this term and report that it cannot be isolated

experimentally, and that it does not affect the values of

model parameters. Hence, this term is also neglected.

The long-range salt-salt DCFI now becomes

-1/2 2 2

LR S pl n n vi Z ZZ.

CLR y- a ji j (3-33)

ap 4v v i i j=l +aijB1/2

There is, however, an inconsistency in this equation. By

substitution of a model of the form of equation (3-1) into

equation (2-6), we can identify the long-range salt-salt

DCFI with the appropriate activity coefficient derivative.

cLR n

-CR -p ap Ta (3-34)

SapP Tp

Note that the natural variables in (3-34) are the Kirkwood-

Buff variables of temperature, volume, and component mole

numbers. Equation (3-33) was derived from a McMillan-Mayer

system, where the natural variables are temperature, volume,

solute mole number, and solvent chemical potential. It is

evident that equations (3-12) and (3-33) are not correct

identifications in light of equation (3-34). Equation (3-

33) is re-identified for a two component system, as

EDH -1/2 22

p D.ny S pi n n v v Z Z

2 = -2 E E

V2 P2 T,V,41 4 2 i=l j=1 1+aijB 1/2

(3-35)

where

EDH

kny2 = extended Debye-HUckel activity coefficient.

Equation (3-35) must be transformed to the appropriate set

of independent variables for use in (3-34).

The transformation is begun by introducing the chain

rule for the activity coefficient in the McMillan-Mayer

system.

E EDH

EDH 2

d 2ny2 =

9N2 T,V,UL1

EDH

+any2

+ 2

aT

EDH

any2

dN2 + 2 dul

a1 T,V,N2

EnDH

dT + 2 TN2

V,N2' 1 2TN'1

For constant temperature and volume, this becomes

DH nEDH

N EDH 2

dIny2 =

9N2 T,V,41

EDH

SAny 2 I

dN + I

au1 T,V,N2

Dividing by dN2 and fixing the solvent mole number, N1,

yields

EnDH

amn2 T

aN2 T,V,N1

EDH

= nY2

D N2 T,V,4 1

EDH

9ny2 1_ T

+ -a- I=

+ 1 T,V,N2 MN2 T,V,N

1 2 2

(3-38)

Applying equation (2-6) to (3-38) yields

EDH

aRny2

T,V,N1 aN2

1 2

EnyDH

9any2 1-C1

+ 2-= IT VV2

T,V,ul 91ny1 T,V,N2 N

anyEDH a M-

+ a-y2 as -N1 V 1-12

T,V,+l ae T,V,N29N1 T,V,N2a9ny1 T,V,N2 N

as 'TV~aNT,V,N2 N

(3-36)

(3-37)

EDH

Any2

aN2

EDH

aN2

59

EDH EDH -C

8tny2 aIny2 as 1-C12

= + 2 12

3N2 T,V,41 3e T,V,N 2 aN1 T,V,N 2 1-C11

(3-39)

where the solvent dielectric constant, e, has

been introduced. Now, combining equations (3-34) and (3-39)

yields

L n EDH

.LR p ny2D

-C

-C22

N2 p2 T,V, 1

EnyEDH ( '1

2-iny2 e l-C12

+ p -

S T, Tp2 1-C11

(3-40)

An analogous expression for the solvent-salt long-range term

can be derived by dividing equation (3-37) by dN1 and fixing

solute mole number. The result is

LR n EDH e

-C2 1

2 V2 e IT,p2 ;1 '~ P2

(3-41)

Multi-component expressions for equations (3-40) and (3-41)

are presented in Appendix B.

The extended Debye-HUckel solute activity coefficient

expression is

S S I1/2 n n v. V. Z2Z2

Yny DH= y i z 1 j

Sn 2) i=1 j=l (1+a..B I 2)

i=l z

(3-42)

Noting the definition of Sy and By given in equations (3-

12a) and (3-12b), combination of equations (3-40) and (3-42)

yields

-1/2

-CLR S I-1/2

C22 2 -

4v

2

dAne

ap,

n

i=l

Tn

IT,p2 1-C1 i=l

I,.l-112

n2 2

E i2 j2 i j

j=1 (l+aijBjIl/2

n v. '. Z2Z2(3+2a.B 11/2)

n j2 j2Zi j ijy y

j=1 (1+aijBy 11/2 2

(3-43)

Similarly, the salt-solvent long-range DCFI becomes

LR S pI/ ane n n

-CLR _y 1 E

21 n 2

2 E Vi2Z z 2 1 Tp2 i=1 j=1

i=l

vN j2Z Z 3+2a.B 11/2

Ui2j Z j( aB11/ )

(1+aijB ll/2)2

J( 2 j

(3-44)

61

In summary, the Cabezas model for the DCFI of solutions

of aqueous electrolytes has been reformulated. The hard

sphere contribution has been eliminated in favor of

retaining the virial contribution to account for short-range

interactions. The long-range expressions have been

simplified. They have also been transformed into the

appropriate set of independent variables for use in the

Fluctuation Solution Theory formalism. In the next section,

the new DCFI model will be integrated to derive the model

equations for densities and activity coefficients.

Model Equations

The new DCFI model has the form

C = AC + CLR (3-45)

The virial contribution is given by equation (3-8). The

long-range contribution is given by equation (3-43) for

salt-salt interactions, equation (3-44) for salt-solvent

interactions, and is set equal to zero for solvent-solvent

interactions.

The equation of state relating system pressure and

solution density for this model is given by substituting

equation (3-45) into (2-29)

f f

f r no no P f P A

P -P o o a P a P AC

= E v vs - dp1 dpa

RT a= =l r Pp r p

P P

a p

Pf LR

PP CL dp( L

r p

f

n n P P AC

+quai Po dpn (3-46)

= r p

f

p LR

+1 ~ dp1 (3-46)

r p

pa

Restricting the example to the one solute-one solvent case,

and substituting the appropriate model expressions into

equation (3-46) yields

f r 1 1

p p f r 1 f2 r2) ( f3 r3)

RT 2 1 1l 3 i l 1 1 -P

RT 2

f f f2 f f f2

F2 P1 2 112 l 2 122 P1 P2

1 1 2 S

f2 f3 Y

AF22 P2 - 222 P2 n

2 3 3 2

B E v2 Z

1i=1

n n v. v. ZZ2Z 2a. B I /2+(a.i.B I1

x E Ei2 j2 ij 13 Y 2n13 +aijByi 1/2

x E S 3 _ai_-i- 2ln/1+a. .B I

i=1 j=l a?. 1+a..B 11/2 I y1

13 13 Y

S n n 1 Vij2 2 1/2 3+2aB 1/2

SPl vi2v2 Zi 2I ?3+2a ijBy I

+ nYE E

n Z2 i=l j=l r + 1/22

2 E v. p 1+a.iB I/2

i=1 2 i 1

f

aZnE 2S n n P2

x --- dp + n E S

9P1 1P2 2 E v 2Z2 i= j=l

1i12 1

i=l

2 2 1/2 B 1/2n

Si2 j2Z2Z2p2I /2f3+2a B I1/ 1-C a dnP2

(l+aBiB 11/22 1-C1 p1 T2

(3-47)

The reference state has been taken as the infinite dilution

state for the solute (p5 = 0). The last two terms of

equation (3-47) need to be numerically integrated and will

be discussed more thoroughly in the next chapter.

64

A similar analysis may be made beginning with equations

(2-31) and (3-45). For the solute activity coefficient, the

result is

f r f 1 f2 r2

9ny2 = Pn(p /p) AF12(pp 1 1 A112( -1

2

f f f

- AF22P2 122P 1

1/2

S I/2

Y

12 i 2

n

i=l

n

i=l

v iV 2Z 2 1i/2 3+2a B 1/2

i2 j2 i jP2 ij y

(i+aijBI 1/2)

S n

+ y

n 2

2 2^Z. i=1

i=l Z

1

2

n

E

j=1

n

j=1

f2

222P2

22

i2 j2 i j

l+ka.i

f

1

P1

P1ne

8pl

I dpl

T,p2

f

n 2

j=

^o

v2 2 1 3+2aiI/22 3+2a 1 2 -C1 ns

i2j2 j2Zi j 12id

1+a..B Il/2B2 l- 11 p1 T, P2

(3-48)

As with equation (3-47), the last two terms must be

numerically integrated.

Equations (3-45), (3-47), and (3-48) present the model

equations for an equation of state for aqueous strong

electrolytes. In the next chapter, these equations

will be extended to the correlation of electrolyte solution

properties. Model parameters will be fitted, and the

implications of this model will be discussed.

Table 3-1.

Best Fit Values of aj, Model of Martin et al.

(1987); Selected Salts.

Salt rb aij

NaF 2.31 2.80

LiC1 2.41 4.16

NaCl 2.76 3.54

KC1 3.14 3.37

RbCl 3.29 3.31

CsCl 3.50 3.15

NaBr 2.90 3.81

aDiameter fit over molality

a in units of angstroms.

bSum of bare ion radii from

1927).

range of 0.1-1.0 mol-kg-1

Pauling's radii (Pauling,

0

E

(E

0 1 2 3 4 5 6 7 8 9 10

Molarity (mol/)

Figure 3-1.

Test of the assumptions of the Cabezas

model, Chloride Salts.

68

0

Os -500.

S\ LiBr

E %% -1500

P._ KBr

NO

-200-

0 1 2 3 4 5 6 7 8

Molarity (mol/l)

Figure 3-2. Test of the assumptions of the Cabezas

model, Bromide Salts.

- -- i Marcus DIA

-17

-1 : i i i Martin, et al

-20'

-23 I i i i

-21

:I3 i I

-28 i i i i I i i !-- I i j--- ---

-25

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Molality (mol/kg)

Figure 3-3. Convergent long-range salt (2)-salt (2)

DCFI, LiC1.

-1

-2

-1

-2

-2

-a,

-2

-2

-2

-2

-2

-2

-2

-2

d* Ti I

-I i i j I I .

-aU

30 o.r---i ,

Si i I !

i2- I -4 J 1 -- 1- - ;

^:l __ ' i _i i !

*5

6

7

6

0.0 0.5 1.0

1.5 2.0

Marcus DIA

* Martin, et al

3.0 3.5 4.0 4.5 5.0

Molality (mol/kg)

Figure 3-4. Convergent long-range salt (2)-salt (2)

DCFI, NaBr.

II I

I

rm

i i i i

n ,.'

I I!

-45- LiBr"

-55---

60---

0 1 2 3 4 5 6 7 9 10

Molality (mol/kg)

Figure 3-5. Hard sphere salt (2) salt (2) DCFI

from Percus-Yevick compressibility

equation. Diameters from Marcus (1983).

CHAPTER 4

CORRELATION OF SOLUTION PROPERTIES

In previous chapters the development of Fluctuation

Solution Theory for electrolytes was outlined and a model

for the DCFI was proposed. In this chapter the model is

tested. Comparisons are made with DCFI experimental data to

obtain model parameters. Densities and activity

coefficients are calculated from the parameters. Finally,

determination of model parameters from measured density and

activity coefficient data will be discussed.

The determination of model parameters has been obtained

with a generalized least squares algorithm for maximum

likelihood fitting. Developed by Britt and Luecke (1973),

this method estimates model parameters given data containing

experimental error and subject to constraints. Starting

guesses are generated by the Deming approximate algorithm

(Deming, 1943). All calculations have been performed on a

microcomputer.

DCFI Testing

In this section, parameters for the DCFI model will be

estimated in order to test the model with DCFI calculated

from equations (2-16), (2-17), and (3-23). The model will

72

73

be tested for four salts--LiCd, NaCl, LiBr, and NaBr. This

set was selected because of the availability of experimental

data, especially isothermal compressibilies for each salt.

Some of the drawbacks of this choice of salts will also be

discussed.

Following Cabezas (1985), parameter fitting began at

the solvent-solvent DCFI level. This is the most convenient

starting point because the solvent-solvent DCFI has the

fewest parameters. Since there is no long-range solvent-

solvent term, the model is given by

1 C11 = 1 pAF11 pp A 1- P2A 112 (4-1)

for the case of one salt in one solvent. In equation (4-1)

there are two parameters specific to the solvent (AF11 and

Ac111) and one parameter unique to the salt (A112) which is

additive for the ions.

For parameter estimation of the four test salts by

simultaneous fitting, a total of six parameters are

determined--the two solvent specific parameters plus one

salt parameter for each of the four salts. The constraint

function that was chosen is

f = (1-C11) P (-C 11) (4-2)

where

(l-C11)EXP = solvent-solvent DCFI as calculated from

equation (2-16)

(1-C11)MOD = solvent-solvent DCFI as calculated from

model equation (4-1).

Two sets of calculations were performed on the solvent-

solvent DCFI. In the first case, no assumptions concerning

the ionic additivity of the salt parameters were made. In

the second case, the simple ionic additivity rule suggested

by equation (3-32) was used to eliminate one of the salt

parameters.

Ac11d l= A c A a Allb (4-3)

where

a = NaCl

b = LiC

c = NaBr

d = LiBr

The parameters for the two cases are given in Table 4-1.

The standard error of fitting for each of the two cases is

almost identical, as are the parameters estimated for the

two cases. Thus the assumption of ionic additivity for

solvent-solvent-salt parameters is accurate. Figure 4-1

shows the model predictions for the four salts using the

ionic additivity rule.

Next, the parameters for the salt-solvent DCFI were

calculated. The salt-solvent DCFI model is

SLR

1 C21 = 1 --(pF12+pp 112+pp2122) 21

V2

(4-4)

where

LR

C21

= the long-range salt-solvent DCFI from equation

(3-44).

The constraint function is again given by

f = (1-C21) EXP (-C21)MOD

(4-5)

where

(1-C21)EXP = salt-solvent DCFI as calculated from

equation (2-19)

(1-C21)MOD = salt-solvent DCFI from equation (4-4).

In equation (4-4), the AA112 are parameters fitted to C11.

A difficulty in applying this equation is that values for

the ion diameters are necessary to calculate the long-range

contribution to the model. The salt-solvent long-range

contribution is small in magnitude and could not be used to

accurately determine the ion diameters. The problem was

avoided by first calculating the virial parameters, AF12 and

A0122, using equation (4-5) with the long-range contribution

set to zero. The parameter estimates for the Al112s were

used to fit the salt-salt DCFI. For the salt-salt DCFI, the

long-range contribution is significant and may be used to

estimate the ion diameters. These diameters were then used

in the C12 calculation. This procedure was repeated until

there were no significant changes in the parameter

estimates.

Three cases were studied in the estimation of the salt-

solvent DCFI parameters. In the first case, no additivity

rules were assumed. In the second case, simple additivity

of the form suggested by equation (3-32) was used to

eliminate the AF12 parameter for LiBr.

AFId = AFlb AFla + AF1c (4-6)

In the third case, simple additivity was used to eliminate

the A122 for LiBr as well as the AF12 parameter.

A(Idd = AIbb Alaa + c (4-7)

This assumes that the solvent-ion-ion quantities are

additive, i.e.

AI1+_ = (A1I++ + AI-_)

(4-7a)

The parameter estimates for the three cases are listed in

Table 4-2.

Making the assumption of simple additivity for the AF12

parameters does not significantly affect the quality of the

fit. The same assumption does affect the quality of the fit

when used to eliminate one of the A4122 parameters.

The impact of this assumption on the prediction of activity

coefficients and densities will be explored in the

next section. Figure 4-2 shows the model prediction for the

salt-solvent DCFI for the four test salts using the ionic

additivity rule for AF12 only. Figure 4-3 shows the case of

the assumption of simple additivity for all virial

parameters.

Finally, the salt-salt DCFI model was examined. The

convergent salt-salt DCFI model is

1

C LRC

1 22 = 1 -(A22+p1122+p2A 222) C22 (4-8

2

where

C2 C= the convergent salt-salt DCFI as calculated from

equation (3-43).

The constraint function is

C EXP C MOD

f = (1-C ) (1-C 2) (4-9)

22 22

where

C EXP

(1-C2) = convergent salt-salt DCFI calculated from

equation (3-23)

C MOD

(l-C22) = convergent salt-salt DCFI from

equation (4-8).

In addition to the virial parameters, the ion diameters are

estimated from the convergent salt-salt DCFI data.

Two cases were examined for the estimation of salt-salt

DCFI parameters. The first case was the case of no

additivity assumption. In the second case the simple

additivity rule was applied to all virial parameters. The

parameter estimates for the two cases are listed in Table

4-3.

Comparison of the two cases reveals that, as in the

salt-solvent case, the assumption of simple additivity for

the virial parameters involving salt-salt interactions makes

the fit poorer. Figure 4-4 shows the no additivity

assumption for the salt-salt DCFI. Figure 4-5 shows the

case where simple additivity has been assumed for all virial

parameters. The mixing rule appears to give fair agreement,

79

but the full consequences of this set of assumptions require

more investigation.

The fitted values of the ionic diameters highlight a

drawback of the four salts that were chosen for study. Most

modelling efforts for activity coefficients or osmotic

coefficients replace the summation in the extended long-

range term with a single term so that, at the DCFI level,

22 22

n n vv ZZi2

n n j2 2 i j i=l

Z1 1 22 (4-10)

i=l j=1 l +aijBY1i/22 laBi1/22

where

a = effective "salt" diameter.

The salt diameter, a, of equation (4-10), is either fitted

for each salt or assumed to be a universal constant.

Investigators who have fitted the diameter for each salt

report a trend of decreasing a-sizes as one moves down a

cation group on the periodic table (see Table 3-1). On the

other hand, the values of a increase as one moves down an

anion group. Many investigators have noted this trend. The

cation effect is generally ascribed to the hydration of the

smaller cations, such as lithium, while the larger cations,

such as cesium, show little or no hydration. The anion

effect is attributed to the fact that anions weakly hydrate,

so that the crystal radii of the ions are considered to be a

good measure of the size of the anion in solution (Neilson

and Enderby, 1983; Enderby et al., 1987).

The full summation of equation (4-10) was intended to

separate the sizes of the ions in an additive fashion.

Unfortunately, the behavior of the four salts which were

analyzed is similar enough, as measured by DCFI, that the

physical interpretation of the ion diameters has been lost

(see Table 4-3). In essence, the anion terms play a role

similar to the value of the parameter a on the right-hand

side of equation (4-10). The terms on the left-hand side of

equation (4-10) which contain the "cation" information have

been effectively reduced to perturbations.

While the un-physical nature of the fitted ion

diameters is disconcerting, valuable insights may be gained

in any case. From Table 4-3 it is apparent that the fitted

ion diameters are not strongly affected by the mixing rule

assumed in the virial contribution to the model. This

indicates that the two contributions to the model dominate

at different ranges of salt concentration. Figure 4-6 shows

the convergent salt-salt DCFI for NaCl broken down into the

long-range and virial contributions and confirms this

assertion. For this model, short-range effects (high salt

concentration) are modelled by the virial term and long-

range effects (low salt concentration) are accounted for by

the Debye-HUckel term. This separation of effects is in

81

contrast to other engineering correlations, such as those of

Pitzer and Chen, where a universal diameter is assumed for

all salts. In these models, complex salt concentration

dependence must be built into the short-range terms to

account for low salt concentration behavior that could be

modelled by the long-range term. While the fitting of

separate diameters for each ion adds additional parameters

to the model, it ultimately will allow future model

improvements to be made on a more rational basis.

Extension to Densities and Activity Coefficients

The parameters estimated from the DCFI modelling of the

previous section may be used to calculate densities and

activity coefficients. The density at 25C, 1 atm pressure,

and a given solution composition, specified in molality for

convenience, is found by iterative solution of equation

(3-47). The resulting density is used in equation (3-48) to

calculate activity coefficients. Since most activity

coefficient data are reported on the molal scale, while this

model predicts activity coefficients on the molar scale, the

model predictions must be converted to make comparisons.

The conversions equation is

CYc = domym (4-11)

where

c = 1000 p2 = salt molarity,

do = mass density of the solvent,

m = molality of the solution,

Yc = activity coefficient on the molar scale,

Ym = activity coefficient on the molal scale.

Figure 4-7 compares the densities predicted by the

parameters fitted from DCFI data with the density data of

Romankiw and Chen (1983) for NaCl and density data from the

International Critical Tables for the other salts. Here

ionic additivity is assumed only for the parameters of

equation (3-32) (A112 and AF12). This case is referred to

as the limited additivity case. Figure 4-8 is the case

where ionic additivity has been assumed for all parameters.

The densities predicted through the use of either set of

assumptions are quite reasonable.

Figure 4-9 gives a comparison of activity coefficients

predicted from the limited additivity case with activity

coefficients from the National Bureau of Standards

correlation of Hamer and Wu (1972). Figure 4-10 gives the

same comparison for the full additivity case. Again, the

results of the two cases are comparable. However, neither

assumption yields results for the activity coefficients that

are within experimental error. On this basis, no conclusion

can be reached concerning the validity of the assumption of

ionic additivity for the virial parameters.

Fitting to Density and Activity

Coefficient Data

While model parameters may be fit using DCFI data, as

in the previous section, the parameters may also be fitted

directly to density and activity coefficient data. Rather

than having a constraint for each of the DCFI, this fitting

requires two constraints. The first constraint utilizes

equation (3-47) to fit parameters to density data.

EXP p MOD

f =- (4-12)

^RT; RT,

where

EXP

IAP EXP pressure at which reference

= density data was pressure

IRT taken J

AP MOD

I-A = predicted pressure difference from the right

IRTJ

hand side of equation (3-47).

Experimental densities are used in equation (4-12). All

density data used were atmospheric pressure, which is the

same as the reference pressure. Therefore, the experimental

pressure difference term on the right-hand side of equation

(4-12) was always zero.

84

At this stage, the parameters are passed to the second

constraint.

EXP MOD

f = Y2 2 (4-13)

where

EXP

Y2 = salt activity coefficient from experiment

MOD

Y2 = activity coefficient from equation (3-48).

Note that experimental densities are not used in this

equation. As a result, densities are calculated from the

model after use of the constraint (4-12). These densities

are then used in the constraint equation (4-13).

This more complicated procedure introduces an

additional difficulty. As noted in Chapter 3, the modelling

equations representing the conversion from McMillan-Mayer to

Kirkwood-Buff variables must be numerically integrated (see

equations (3-47) and (3-48)). This means that the fitting

routines become extraordinarily long in duration,

particularly because of the trial and error solution for the

model density required between the two constraint equations.

This problem has been solved in the following way. The

contribution of these correction terms to the DCFI are small

(see Table 4-4 for NaCl). The contributions at the activity

85

coefficient level are expected to be even smaller. This is

because at this level, the salt-solvent term goes as the

difference in solvent density between the final and

reference states while the salt-salt term goes as the salt

density. The same holds true for the density calculation.

Thus, these terms will tend to cancel. Therefore, the

McMillan-Mayer to Kirkwood-Buff terms have been neglected in

the density calculations, but retained in the activity

coefficient calculation. This allows the parameter fitting

to be accomplished in a reasonable amount of time.

Examination of the contribution of this term in the activity

coefficient serves as an indicator of cases where the term

is significant and thus should be retained in the density

model.

Using this algorithm, parameters were fit for seventeen

salts. The initial fitting was carried out on the four

salts NaCl, NaBr, KC1 and KBr. The potassium salts were

used in place of the lithium salts to try to avoid the

problems encountered in the ion diameter fitting. The

solvent specific parameters AF11 and A111 values derived

from the DCFI fitting were retained for convenience. This

assumption will be briefly investigated in the next chapter.

After the parameters for the first four salts were

determined, the parameter table was expanded by adding

either anions or cations one at a time. The simple mixing

86

rule was retained for the AF12 and A112 parameters. A

listing of the parameters that were fitted is found in

Appendix C. An attempt was made to extend the simple

additivity rule to all virial parameters for the four test

salts. The results will be discussed in the next chapter

where the model predictions will be examined more closely.

Table 4-1. Parameters for Solvent-Solvent DCFI.

No Additivity Rule

-965.08169.51

12481.33054.7

-19323.63615.7

-15885.43543.3

-13029.44796.4

-9550.94573.6

0.74

Simple Additivity

-967.27222.28

12520.64004.7

-19278.04744.5

-15825.84569.4

-12958.96244.5

-9506.7b

0.73

aS.E. = standard error of fit.

calculated from equation (4-3).

AF11 has units of (ml/mol).

A112 have units of (ml/mol)2.

S.E. = [- E .

d v k=l k=1 k ,

where

nd = number of data points,

n, = number of variables,

ZOkk = observed data of variable Z,

ZMkN = model generated estimate of variable Z,

rkk = standard deviation of the error in ZOkP.

AF11

h~11a

A4)~lla

A'D~llb

A'Dlld

S.E.a

---

Table 4-2. Parameters for Salt-Solvent DCFI

Additivity for

No Additivity Additivity for AF12 All Parameters

AFla 827.020.47 827.200.46 827.060.75

AFib 603.162.64 603.452.31 602.873.81

AFic 353.133.82 353.752.82 354.564.67

AFld 130.513.47 130.00a 130.37a

ADlaa -75236.2226.5 -75232.6221.6 -75391.0365.1

Ailbb -38085.41054.0 -38085.4716.3 -36139.4

ADicc -70951.41426.7 -71128.61180.0 -67680.61781.4

Acidd -25880.21346.3 -25717.91148.4 -28429.0b

S.E. 1.04 1.02 1.69

calculated from equation (4-6).

calculated from equation (4-7).

AF12 have units of (ml/mol).

A 112 have units of (ml/mol)2.

Table. 4-3. Parameters for Salt-Salt DCFI.

AF22 have units of (ml/mol).

aCalculated using the simple additivity rule.

AQ222 have units of (ml/mol)2.

ai have units of Angstroms.

AFaa

AFbb

AFcc

AFdd

ADaaa

ADbbb

A~ccc

ADddd

aNa+

aLi+

aCl-

aBr-

S.E.

No Additivity

3714.9025.66

1329.5635.17

3046.3254.36

195.8055.07

-218877.13850.7

-157652.07876.5

-255044.28733.0

-162022.99980.6

8.800.60

9.800.85

1.080.32

0.7190.47

0.60

Additivity for All

Virial Parameters

3722.3533.42

1220.0543.21

2856.0669.65

353.76a

-219074.24990.2

-147389.08966.9

-247840.111012.1

-176154.9a

8.830.76

9.871.10

1.060.46

0.7140.61

0.77

Table 4-4. Activity Coefficient Model Contributions NaCl.

Molality In Y2 ln(pf/pr) n A Y2 kn yR MM-KB

0.1 -0.2488 0.002 0.004 -0.253 0

0.2 -0.3086 0.004 0.008 -0.317 0

0.3 -0.3439 0.006 0.013 -0.358 0

0.4 -0.368 0.008 0.018 -0.387 0

0.5 -0.3854 0.009 0.024 -0.41 0

0.6 -0.3984 0.011 0.03 -0.429 0

0.7 -0.4079 0.013 0.036 -0.445 0

0.8 -0.4151 0.015 0.043 -0.459 0

0.9 -0.4203 0.016 0.05 -0.471 0

1.0 -0.4236 0.018 0.058 -0.482 0

1.1 -0.4259 0.02 0.065 -0.492 0

1.2 -0.4268 0.021 0.073 -0.5 0

1.3 -0.4267 0.023 0.082 -0.508 0

1.4 -0.4256 0.024 0.091 -0.516 0

1.5 -0.4238 0.026 0.1 -0.523 0

2.0 -0.4055 0.033 0.148 -0.55 0

3.0 -0.3368 0.046 0.261 -0.586 0

4.0 -0.2431 0.057 0.389 -0.61 -0.001

5.0 -0.1351 0.066 0.528 -0.627 -0.002

6.0 -0.0185 0.074 0.673 -0.641 -0.003

6.1 -0.0065 0.075 0.687 -0.642 -0.003

nAY2 = virial contribution to model

tny R = long-range model contribution

MM-KB = contribution from McMillan-Mayer to Kirkwood-

Buff system.

*- ---- --_-----_--_--_

olll

-- ,-r --------

0 26 7 i

0 1 2 3 4 5 6 7 8 9 10 11 12

* NaCI

LICI

* NaBr

A LIBr

Molality (mol/kg)

Figure 4-1. Water (1) Water (1) DCFI. Ionic additivity

assumption for AO112 in equation (4-1).

* NaCI

o LiCI

A NaBr

LiBr

0 1 2 3 4 5 6 7 8 9 10

Molality (mol/kg)

Figure 4-2.

Water (1) Salt (2) DCFI from equation (4-4).

Ionic additivity for AF12, A$112. No

assumption about form of A 122

* NaCI

SLICl

NaBr

& LiBr

0 1 2 3 4 5 6 7 8 9 10

Molality (mol/kg)

Figure 4-3.

Water (1) Salt (2) DCFI from equation (4-4).

Ionic additivity for all parameters.