A LIQUID EQUATION OF STATE
FOR AQUEOUS STRONG ELECTROLYTES
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
To my beloved family, and especially my sister, Clare,
whose mid-week phone calls greatly encouraged me.
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
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.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS. . . . . . . .
ABSTRACT . . . . . . . . .
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. . . . . .
to Kirkwood-Buff Transformation. . .
Model Equations . . . . . . .
4 CORRELATION OF SOLUTION PROPERTIES. . .
DCFI Testing . . . . .....
Extension to Densities and Activity
Coefficients . . . . . . .
. . . . . 1
S . . 16
S . . 22
S . . 30
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 . .
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 . . . . . . . .
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
Kenric A. Marshall
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
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.
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
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.
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).
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,
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
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.
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,
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
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
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
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.
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
extensive modifications which were made. Suggestions for
further modifications will be advanced.
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.
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
In his derivation of the Fluctuation Solution Theory
equations for general multi-component electrolyte solutions,
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
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)
Cij = the molecular centers direct correlation
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
The number of moles of a given species, j, is related
to the number of moles of components, 0, by
N = E v Np (2-3)
P =1 j
Np = the number of moles of component p,
vjp = the stoichiometric coefficient of species j in
no = all components.
Analogous expressions relating species and component
densities and species and component mole fractions may also
PJ = E v Pp (2-4)
Pj = Nj/V = the density of species j,
pp = Np/V = the density of component 3,
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
p- P TP v (1-C a) (2-6)
ny E v.ianyi (2-7)
inyi = = natural log of activity coefficient of
4 = the chemical potential of species i at its
a = the sum of the stoichiometric coefficients of
n = all species.
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
SNidui = VdP (2-9)
P = the system pressure.
Combining equations (2-1) and (2-9) relates the derivative
of the system pressure with respect to component density to
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
Finally, the thermodynamic identification
p T,py~ T
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
2 I 22
=ny2 2 1-C22 (2-13)
aP2 T'Pl P
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
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 C = 2 (-x2V2P)
+ 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
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
IZ Z_ | AmIm 2
log0 = + Im + CI + *** (2-19)
1 + B*
Zi = the valence of ion i,
B*, ,C ... = empirical constants
I E Z m = ionic strength, molality basis
m = 1000-N2-N-1 *M1 = solution molality
(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)
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
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
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)
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
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
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
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.
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
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
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
diny, = aS dp (2-30)
a=1 app TpY#P
Insertion of equation (2-6) into (2-30) and integration
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.
v2x1(1-C12) + x2(1-C22) (2-14)
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
2 v 2 -
V2 = v + --1- ( 2-32)
(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
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
Figure 2-1. Water (1) Water (1) DCFI in aqueous
electrolyte solutions at 250C, 1 atm.
See text for data sources.
- - _ _
- - - ^ -
~~~/ ^_ ,__ ____
'- imt^^^ !!'-""
. -- - z ; _ i
- - __,,,_ _ _ _
I - ? - - - -
. . -' -- -_ -- - -.- -."-]- - -
Figure 2-2. Water (1) Salt (2) DCFI in aqueous
electrolyte solutions at 250C, 1 atm.
See text for data sources.
* ^, / -I -. -
0 (N 38
* ~--- -
* - -- .- -
- --- - - -
3 4 5 6 7
Figure 2-3. Convergent salt (2) salt (2) DCFI in aqueous
electrolyte solutions at 250C, 1 atm. See
text for data sources.
8 9 10
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
C = short-range DCFI
C = long-range DCFI.
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.
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).
cij(T,P,r ,rj.,iJj) = fij(T) + k Pk ijk(T) + *** (3-3)
f = e 1 = Mayer bond function
9 ijk(T) = -f f.i fik fjk drkd
0 = dwi = integral over orientation coordinates.
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
AC. = p A F i(T) + p E PkA ijk(T) (3-4)
AF. (T) = 4n < Af.ij> r.j dri
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
AF v. iaVjAFij (3-6)
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
uij = pair potential
Wij = potential of mean force.
For the case of ion-ion interactions, Cabezas modeled the
pair potential with a Coulomb potential.
Z Z e
u = (3-10)
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 = 0 rijS aij (3-11b)
2 4e n
K E ZP = Debye-Huckel inverse length
es kT i=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
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
S = (2n)1/2 e] (3-12a)
B = o- (3-12b)
I = E Zi.p = ionic strength, density basis.
< i1/2> r e 13
E 3aijByI 2)= j 3dr ij
1 13 ij
= 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
S= empirical constant, arbitrarily set to 4.4
I1 = dipole moment of solvent
E2(2ailBjI /2j= e 2 dr il
= the second exponential integral.
The solvent (1)-solvent (1) long-range DCFI expression is
_LR 4 IL 2 2allB /2
3 s kT
E4(2a 11B /2
E4(2a11Bj1/2) = f
e- 2 B I1/2
-2a Bll I
= 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
Cabezas tested his DCFI model using the DCFI data for
six salts: LiCd, LiBr, NaCl, NaBr, KC1, and KBr. From his
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
c _-C J (3-16)
where the following special definitions have been made:
0a P VaV p-p
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
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
CLR = convergent long-range salt-salt DCFI
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
The resulting expression is
S pi 1 n
4v v i=1
n v iaV Z2
j=1 (1+Ka. )2
- E E v. iv.Z.Z
i=1 j=1 1
S pI n
. v. Z. Ka.
i1L j1P I 13
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
S C cHS VIR LR C (3-22)
1 a- = S C C (3-22)
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).
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
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
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.
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,
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.
AF++ = (AF +++AF) (3-28)
Using this rule, equation (3-27) becomes
AF22 = +2 AF + v2 -V2(AF+++AF__) + v 2AF__
22 +2 ++ +2 -2 ++ -2 -
= v2(+2AF+++ v_2AF__)
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
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
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
a x(1-Cap) (3-30)
For the case of one salt (2) in one solvent (1) at infinite
dilution, this equation becomes
-- 2(1-C21) (3-31)
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.
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.
-CR -p ap Ta (3-34)
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
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
d 2ny2 =
dN2 + 2 dul
dT + 2 TN2
V,N2' 1 2TN'1
For constant temperature and volume, this becomes
N EDH 2
SAny 2 I
dN + I
Dividing by dN2 and fixing the solvent mole number, N1,
D N2 T,V,4 1
9ny2 1_ T
+ -a- I=
+ 1 T,V,N2 MN2 T,V,N
1 2 2
Applying equation (2-6) to (3-38) yields
+ 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
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
where the solvent dielectric constant, e, has
been introduced. Now, combining equations (3-34) and (3-39)
L n EDH
.LR p ny2D
N2 p2 T,V, 1
EnyEDH ( '1
2-iny2 e l-C12
+ p -
S T, Tp2 1-C11
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
2 V2 e IT,p2 ;1 '~ P2
Multi-component expressions for equations (3-40) and (3-41)
are presented in Appendix B.
The extended Debye-HUckel solute activity coefficient
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)
Noting the definition of Sy and By given in equations (3-
12a) and (3-12b), combination of equations (3-40) and (3-42)
-CLR S I-1/2
C22 2 -
IT,p2 1-C1 i=l
E i2 j2 i j
n v. '. Z2Z2(3+2a.B 11/2)
n j2 j2Zi j ijy y
j=1 (1+aijBy 11/2 2
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
vN j2Z Z 3+2a.B 11/2
Ui2j Z j( aB11/ )
J( 2 j
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.
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
The equation of state relating system pressure and
solution density for this model is given by substituting
equation (3-45) into (2-29)
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
PP CL dp( L
n n P P AC
+quai Po dpn (3-46)
= r p
+1 ~ dp1 (3-46)
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
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
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
aZnE 2S n n P2
x --- dp + n E S
9P1 1P2 2 E v 2Z2 i= j=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
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.
A similar analysis may be made beginning with equations
(2-31) and (3-45). For the solute activity coefficient, the
f r f 1 f2 r2
9ny2 = Pn(p /p) AF12(pp 1 1 A112( -1
f f f
- AF22P2 122P 1
12 i 2
v iV 2Z 2 1i/2 3+2a B 1/2
i2 j2 i jP2 ij y
2 2^Z. i=1
i2 j2 i j
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
As with equation (3-47), the last two terms must be
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.
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
range of 0.1-1.0 mol-kg-1
Pauling's radii (Pauling,
0 1 2 3 4 5 6 7 8 9 10
Test of the assumptions of the Cabezas
model, Chloride Salts.
E %% -1500
0 1 2 3 4 5 6 7 8
Figure 3-2. Test of the assumptions of the Cabezas
model, Bromide Salts.
- -- i Marcus DIA
-1 : i i i Martin, et al
-23 I i i i
:I3 i I
-28 i i i i I i i !-- I i j--- ---
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Figure 3-3. Convergent long-range salt (2)-salt (2)
d* Ti I
-I i i j I I .
30 o.r---i ,
Si i I !
i2- I -4 J 1 -- 1- - ;
^:l __ ' i _i i !
0.0 0.5 1.0
* Martin, et al
3.0 3.5 4.0 4.5 5.0
Figure 3-4. Convergent long-range salt (2)-salt (2)
i i i i
0 1 2 3 4 5 6 7 9 10
Figure 3-5. Hard sphere salt (2) salt (2) DCFI
from Percus-Yevick compressibility
equation. Diameters from Marcus (1983).
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
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
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
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)
(l-C11)EXP = solvent-solvent DCFI as calculated from
(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
Ac11d l= A c A a Allb (4-3)
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
1 C21 = 1 --(pF12+pp 112+pp2122) 21
= the long-range salt-solvent DCFI from equation
The constraint function is again given by
f = (1-C21) EXP (-C21)MOD
(1-C21)EXP = salt-solvent DCFI as calculated from
(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
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
AI1+_ = (A1I++ + AI-_)
The parameter estimates for the three cases are listed in
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
Finally, the salt-salt DCFI model was examined. The
convergent salt-salt DCFI model is
1 22 = 1 -(A22+p1122+p2A 222) C22 (4-8
C2 C= the convergent salt-salt DCFI as calculated from
The constraint function is
C EXP C MOD
f = (1-C ) (1-C 2) (4-9)
(1-C2) = convergent salt-salt DCFI calculated from
(l-C22) = convergent salt-salt DCFI from
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
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,
but the full consequences of this set of assumptions require
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,
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
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
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)
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
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)
IAP EXP pressure at which reference
= density data was pressure
IRT taken J
I-A = predicted pressure difference from the right
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.
At this stage, the parameters are passed to the second
f = Y2 2 (4-13)
Y2 = salt activity coefficient from experiment
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
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
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
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
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 ,
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.
Table 4-2. Parameters for Salt-Solvent DCFI
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.
Additivity for All
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-
*- ---- --_-----_--_--_
-- ,-r --------
0 26 7 i
0 1 2 3 4 5 6 7 8 9 10 11 12
Figure 4-1. Water (1) Water (1) DCFI. Ionic additivity
assumption for AO112 in equation (4-1).
0 1 2 3 4 5 6 7 8 9 10
Water (1) Salt (2) DCFI from equation (4-4).
Ionic additivity for AF12, A$112. No
assumption about form of A 122
0 1 2 3 4 5 6 7 8 9 10
Water (1) Salt (2) DCFI from equation (4-4).
Ionic additivity for all parameters.