
Citation 
 Permanent Link:
 https://ufdc.ufl.edu/UF00086022/00001
Material Information
 Title:
 Application of fluctuation solution theory to strong electrolyte solutions
 Creator:
 Cabezas, Heriberto, 1952 ( Dissertant )
O'Connel, John P. ( Thesis advisor )
Hooper, Charles F. ( Reviewer )
WestermannClark, Gerald B. ( Reviewer )
 Place of Publication:
 Gainesville, Fla.
 Publisher:
 University of Florida
 Publication Date:
 1985
 Copyright Date:
 1985
 Language:
 English
 Physical Description:
 xii, 193 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Chemicals ( jstor )
Correlations ( jstor ) Diameters ( jstor ) Electrolytes ( jstor ) Experimental data ( jstor ) Ions ( jstor ) Parametric models ( jstor ) Solvents ( jstor ) Teeth ( jstor ) Water tables ( jstor ) Chemical Engineering thesis Ph. D Dissertations, Academic  Chemical Engineering  UF Electrolyte solutions ( lcsh )
 Genre:
 bibliography ( marcgt )
nonfiction ( marcgt ) thesis ( marcgt )
Notes
 Abstract:
 Fluctuation solution theory relates derivatives of the thermodynamic properties to spatial integrals of the direct correlation functions. This formalism has been used as the basis for a model of aqueous strong electrolyte solutions which gives both volumetric properties and activities. The main thrust of the work has been the construction of a microscopic model for the direct correlation functions. This model contains the correlations due to the hard core repulsion, long range field interactions, and short range forces. The hard core correlations are modeled with a hard sphere expression derived from the PercusYevick theory. The long range field correlations are accounted for by using asymptotic potentials of mean force and the hypernetted chain equation. The short range correlations which include hydration and hydrogen bonding are modeled with a density expansion of the direct correlation function. The model requires six parameters for each ion and two for water. The ionic parameters are valid for all solution and those for water are universal. The model has been used to calculate derivative properties for six 1:1 electrolytes in water at 25c, 1 ATM, the calculated properties have been compared to experimentally determined values in order to confirm the adequacy of the model.
 Thesis:
 Thesis (Ph. D.)University of Florida, 1985.
 Bibliography:
 Bibliography: leaves 188192.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Heriberto Cabezas, Jr.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 000880471 ( ALEPH )
14971957 ( OCLC ) AEH8290 ( NOTIS )

Downloads 
This item has the following downloads:

Full Text 
APPLICATION OF FLUCTUATION SOLUTION THEORY TO
STRONG ELECTROLYTE SOLUTIONS
By
HERIBERTO CABEZAS, JR.
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE
UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1985
To Flor Maria
ACKNOWLEDGMENTS
I would like to express my sincere gratitude to
Professor J.P. O'Connell, a man of wisdom and knowledge, for
his guidance and encouragement during the course of this
work.
I also wish to thank Drs. G.B. WestermannClark and
C.F. Hooper, Jr. for serving on the supervisory committee
and for making very pertinent suggestions regarding this
work.
It is a pleasure to thank Mrs. Smerage for her
excellent typing and patience and Mrs. Piercey for her help
with the figures.
Finally, I am grateful to the Chemical Engineering
Department of the University of Florida for financial sup
port and for providing the kind of intellectual environment
in which this work could take place. I am also grateful to
the National Science Foundation for providing the financial
support that made this work possible.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS.................................. iii
KEY TO SYMBOLS...................................... vi
ABSTRACT.......................................... xi
CHAPTERS
1 INTRODUCTION......... ...... .................. 1
2 FLUCTUATION THEORY FOR STRONG ELECTROLYTE
SOLUTIONS .................................. 11
Introduction................................ 11
Thermodynamic Property Derivatives and
Direct Correlation Function Integrals... 12
Direct Correlation Function Integrals from
Solution Properties..................... 23
Summary ................ .................. .29
3 A MODEL FOR DIRECT CORRELATION FUNCTION
INTEGRALS IN STRONG ELECTROLYTE SOLUTIONS.. 33
Introduction................................... 33
Philosophy of the Model.................... 33
Statistical Mechanical Basis ............... 37
Expression for SaltSalt DCFI.............. 51
Expression for SaltSolvent DCFI............ 63
Expression for SolventSolvent DCFI........ 69
Summary...................................... 74
4 APPLICATION OF THE MODEL TO AQUEOUS STRONG
ELECTROLYTES ......... ............... ....... 77
Introduction................................ 77
Calculation of Solution Properties from
the Model. .............................. 78
Model Parameters from Experimental Data.... 90
Comparison of Calculated Properties with
Experimental Properties.................. 104
Discussion. ............................... .105
Conclusions ............................... 113
5 CONCLUSIONS AND RECOMMENDATIONS............ 144
APPENDICES
A HARD SPHERE DIRECT CORRELATION FUNCTION
INTEGRAL FROM VARIOUS MODELS................ 148
B RELATION OF McMILLANMAYER THEORY TO
KIRKWOODBUFF THEORY...................... 152
C RELATION OF DENSITY EXPANSION OF THE
DIRECT CORRELATION FUNCTION TO VIRIAL
EQUATION OF STATE: ALTERNATE MIXING
RULES. ...................................... 172
D EXPONENTIAL INTEGRALS........................ 180
E MODEL PARAMETERS........................... 184
REFERENCES ........................................ 188
BIOGRAPHICAL SKETCH............................... 193
KEY TO SYMBOLS
ai = hard sphere diameter of species i.
a. = distance of closest approach of species i and j.
13
1/2
B = K/II/2
B. = sum of all bridge diagrams, second virial
13 coefficient.
C = mixture third virial coefficient.
C.. = direct correlation function integral for species
13 i and j; twobody factor in third virial
coefficient.
C =direct correlation function integral for
components a and 3.
C.i = third virial coefficient for i, j, k.
ijk
AC.. = short range direct correlation function
13 integral.
c.. = direct correlation function.
1J
Ac.. = short range direct correlation function.
13
D = dielectric constant of solvent or solvent mixture.
D = pure solvent dielectric constant.
E = exponential integral or order n.
e = electronic charge.
AF = spatial integral of Af...
u../kT
f. = e 1 = Mayer bond functions.
13
Af. = f.. f.HS fLR = differences of microscopic
1] 13 jj ij
twobody coefficient.
g = pair distribution function.
1 n 2
I Z. p = ionic strength
i=l
K2
k
N
N.
N
o
N
Oct
n
n
o
P
INT
qi
r,r.
r.
1
vii
2
= 8T I = DebyeHuckel inverse length.
DkT
=Boltzmann's constant.
= total number of moles of all species.
= total number of moles of species i.
= total number of moles of all components.
= total number of moles of component a.
= number of different species, integer greater
than one.
= number of different components.
= pressure.
= internal partition function.
= separation between species i and j.
= position vector of i.
6 1/2
= (2 3k3T3) = DebyeHuckel limiting law
D k T
efficient.
= temperature.
= pair potential.
= total system volume.
= partial molar volume of species i.
S
Y
T
u.
13
V
V.
i
Vo = partial molar volume of component a.
W.. potential of mean force.
13
X. = N/N = mole fraction of species i.
N
X = mole fraction of component a on a
oa N
species basis.
Z = dimensionless parameter in exponential integral.
Z. = valence of ion i.
1
a = Euler's constant, empirical universal constant
for ionsolvent correlations.
i = activity coefficient of species i.
a = activity coefficient of component a.
6.. = Kroniker delta.
13
8 = Eulerian angle between a charge and a dipole.
81i,'li = Eulerian angles of dipole of solvent molecule i.
< = isothermal compressibility.
Ki = isothermal compressibility of pure solvent (1).
A. = ideal gas partition function.
. = chemical potential of species i.
l = dipole moment of solvent.
Via = number of species i in component a.
V = total number of species in component a.
n K
S = 6 p. a. = reduced density.
K 6. 1 1
i=l
S = P., osmotic pressure.
N
p density of all species.
p = vector of species densities.
viii
N.
P = = density of species i.
N
oca
po = density of component a.
.ijk = spatial integral of .ijk"
Aijk = spatial integral of Ai .j
1ik ijk
ijk = microscopic threebody coefficient.
HS
Aijk = ijk ijk = difference of microscopic three
body coefficients.
= orientation dependence of dipoledipole
interaction.
Q = f dw. = integral over orientation coordinates.
1
. = angular orientation coordinates of i.
1
1 n 2
= Z v. Z..
Y 2 i= y 1
Superscripts
F = Final.
FLL = Friedman's limiting law.
HNC = hypernetted chain.
HS = hard sphere.
KB = KirkwoodBuff.
LR = long range or field type correlations or
interactions, LewisRandall.
MM = McMillanMayer.
P = Pure component.
PY = PercusYevick.
R
SAT
TB
0,O0
solvent.
component.
Special Symbol
< > = integration over orientation.
w
= Reference.
= Saturated.
= Three body.
= infinite dilution in salt.
Subscripts
= species.
= components.
jj,
0 =
.
.
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of
Doctor of Philosophy
APPLICATION OF FLUCTUATION SOLUTION THEORY TO
STRONG ELECTROLYTE SOLUTIONS
By
Heriberto Cabezas, Jr.
August, 1985
Chairman: Dr. J.P. O'Connell
Major Department: Chemical Engineering
Fluctuation solution theory relates derivatives of the
thermodynamic properties to spatial integrals of the direct
correlation functions. This formalism has been used as the
basis for a model of aqueous strong electrolyte solutions
which gives both volumetric properties and activities.
The main thrust of the work has been the construction
of a microscopic model for the direct correlation func
tions. This model contains the correlations due to the hard
core repulsion, long range field interactions, and short
range forces. The hard core correlations are modelled with
a hard sphere expression derived from the PercusYevick
theory. The long range field correlations are accounted for
by using asymptotic potentials of mean force and the hyper
netted chain equation. The short range correlations which
include hydration and hydrogen bonding are modelled with a
density expansion of the direct correlation function. The
model requires six parameters for each ion and two for
water. The ionic parameters are valid for all solutions
and those for water are universal.
The model has been used to calculate derivative prop
erties for six 1:1 electrolytes in water at 25C, 1 ATM. The
calculated properties have been compared to experimentally
determined values in order to confirm the adequacy of the
model.
xii
CHAPTER 1
INTRODUCTION
Aqueous electrolytes are present in many natural and
artificial chemical systems. For example, the chemical
processes of life occur in an aqueous electrolyte medium.
All natural waters contain salts in concentrations ranging
from very low for fresh water to near saturation for geo
thermal brines. Industrially, electrolytes are used in
azeotropic distillation, electrical storage batteries and
fuel cells, liquidliquid separations, drilling muds, and
many other processes. Since a quantitative description
of the properties of these systems is required for under
standing, design, and simulation, the ability to predict
and correlate the solution properties of electrolytes is
both scientifically and technologically important.
In attempting to fill this need, many models of aqueous
salt solutions have been developed. Essentially all describe
only activities of the components but ignore the volumetric
properties. Several extensive reviews of electrolyte solu
tion models are available in the literature (Pytkowicz,
1979; Mauer, 1983; Renon, 1981). To be concise, the various
models have been classified here into three general cate
gories and a few examples of each briefly discussed. First,
there are models based on relatively rigorous statistical
mechanical results which can be called "theoretical."
Second, there are those composed of a mixture of rigorous
theory and empirical corrections which can be named "semi
empirical." Third, there are those models which directly
correlate experimental data and are thus termed "empirical."
Neither this classification nor the following list pretends
to be either unique or allinclusive.
Among the "theoretical" models, the earliest and still
the most widely accepted is the theory of Debye and Huckel
(1923) which gives the rigorous relation at very low salt
concentration (the limiting law) for salt activity coeffi
cients but fails at higher salt concentration. This theory
has been amply treated in the literature (Davidson, 1962;
Harned and Owen, 1958). The DebyeHuckel theory considers
an electrolyte solution as a collection of charged hard
spherical ions embedded in a dielectric solvent which is
continuous and devoid of structure. This is the physical
picture generally called the "Primitive Model." The correct
formalism for the application of modern statistical mechani
cal techniques to the "Primitive Model" is given by the
McMillanMayer theory (1945). A major method developed
for this formalism is a resumed hypernetted chain approxima
tion to the direct correlation function. This, together
with the OrnsteinZernike equation (1914), forms a solvable
integral equation for the primitive model ionion
distribution function which has been used to calculate
the properties of electrolytes up to 1 M salt concentration
(Rasaiah and Friedman, 1968; Friedman and Ramanathan, 1970;
Rasaiah, 1969). This method requires tedious numerical
calculations to obtain the properties. A simpler and more
generalizable approach is the Mean Spherical Approximation
(MSA) which has been applied to both primitive (Blum, 1980;
Triolo, Grigera, and Blum, 1976; Watanasiri, Brule, and
Lee, 1982) and nonprimitive (Vericat and Blum, 1980;
PerezHernandez and Blum, 1981; Planche and Renon, 1981)
electrolyte models. The MSA method essentially consists
of solving the OrnsteinZernike (1914) equation for the
distribution functions subject to the boundary conditions
that the total correlation function is minus one inside
the hard core and that the direct correlation function
equals the pair potential outside the hard core. This
is equivalent to the PercusYevick method for rigid nonionic
systems (Lebowitz, 1964). The MSA generally gives good
thermodynamic properties if these are calculated from the
"Energy Equation" (Blum, 1980). It does not yield good
correlation functions and further suffers from the need
to numerically solve complex nonlinear relations for the
value of the shielding parameter at each set of conditions.
This last problem grows progressively worse as the sophisti
cation of the model increases. Due to their complexity none
of the modern "theoretical" models is widely used in
engineering practice.
The most successful of the semiempirical models is
that due to Pitzer and coworkers (Pitzer, 1973; Pitzer
and Mayorga, 1973; Pitzer and Mayorga, 1974; Pitzer, 1974;
Pitzer and Silvester, 1976). Model parameters for activity
coefficients have been evaluated for a large number of
aqueous salt solutions, but volumetric properties and multi
solvent systems have not been treated. To construct the
model, Pitzer adopted the "Primitive Model" and inserted
the DebyeHuckel radial distribution function for ions
into the osmotic virial expansion from the McMillanMayer
formalism. This latter is analogous to using the "Pressure
Equation" of statistical mechanics (Pitzer, 1977). The
resulting expression contains the correct limiting law.
He then added empirical second and third virial coefficients
which are salt and solvent specific. Although Pitzer's
model correlates aqueous activity coefficients superbly,
it does not add to the fundamental understanding of these
solutions; further, its extension to multisolvent systems
would pose some serious problems associated with the mixture
dielectric constant as has been recently pointed out (Sander,
Fredenslund, and Rasumussen, 1984). Another semiempirical
approach uses the NRTL model for solutions of nonelectrolytes
(Renon and Prausnitz, 1968) adapted for short range ion
and solvent interactions (Cruz and Renon, 1978; Chen, Britt,
Boston, and Evans, 1979) in nonprimitive models of electro
lyte solutions. Cruz and Renon separate the Gibbs energy
into three additive terms: an elecrostatic term from the
DebyeHuckel theory, a DebyeMcAulay contribution to correct
for the change in solvent dielectric constant due to the
ions, and an NRTL term for all the short range intermolecular
forces. Chen et al. adopted a DebyeHuckel contribution
and an NRTL term for the Gibbs energy but no DebyeMcAulay
term. More recently, the UNIQUAC model for nonelectrolytes
has been modified for short range intermolecular forces
in electrolyte solutions (Sander, Fredenslund, and Rasmussen,
1984). The resulting UNIQUAC expression has been added
to an empirically modified PitzerDebyeHuckel type electro
static term to form the complete Gibbs energy model.
Although the two NRTL and the UNIQUAC models correlate
activity coefficient data reasonably well even in multi
solvent systems, they have to be regarded as mainly
empirical. First, their resolution of the Gibbs energy
into additive contributions from each different kind of
interaction is not rigorous. Second, the problems associated
with the mixture dielectric constant are resolved in an
empirical and somewhat arbitrary fashion. As a result,
such models add little to our understanding of these systems
and may not be reliable for extension and extrapolation.
Of the various empirical methods developed, two have
been chosen to be discussed here because they represent
distinct approaches. First, there is the method of Meissner
(1980) which is a correlation for the salt activity coeffi
cient in terms of a family of curves that are functions
of the ionic strength and a single parameter which can
be selected from a single data point. This method has
been extended to multicomponent electrolyte solutions and
is useful over a wide range of salt concentration (0.120
MOLAL), though it is not very accurate. Second, there
is the method of Hala (1969) which is more conventional
in that it consists of a purely empirical model for the
Gibbs energy of the solution. This method is an excellent
correlational tool, but it is not predictive. It has four
parameters per saltsolvent pair.
The existence of so many models to correlate and predict
the thermodynamic behavior of electrolyte solutions is
indicative of the complexity of these systems and, perhaps,
the relatively poor state of the art.
As examples of the physical complexity of electrolyte
solutions, the composition behavior of the salt activity
coefficient (Figure 1) and of the species (ions and solvents)
density (Figure 2) is presented. Figure 1 shows the large
deviation from ideal solution behavior (y= 1) even at
very low salt concentration for all salts. Second, it
indicates that salts of the same charge type show similar
behavior at low salt concentration but are widely different
at higher salt concentration. In Figure 2, the difference
in the salt composition behavior of the species density
is obvious even for relatively similar salts, i.e., the
solution seems to expand for KBr while it seems to contract
for all other salts. The activity coefficient data were
taken from the compilation by Hamer and Wu (1972). For
NaCl and NaBr the density data of Gibson and Loeffler (1948)
were used. For LiCI, LiBr, and KBr the density data were
taken from the International Critical Tables. For KC1
the density data of Romankiw and Chou (1983) were used.
In the hope of improving the situation for obtaining
properties of solutions, a new model of strong aqueous
electrolyte solutions is presented here. This model has
been carefully constructed so that it overcomes a number
of the deficiencies of previous methods. For example,
this model is simple enough for economical engineering
calculations, yet sufficiently sophisticated to rigorously
include all the different interactions (ionion, ionsolvent,
solventsolvent) and the principal physical effects (electro
static, hard core repulsion, hydration, etc.) that contribute
to each interaction. The model is also extendable to multi
salt and multisolvent systems in a straightforward fashion.
Finally, it addresses both activity and volumeric
properties.
In the chapters that follow, a detailed development
of the new model is presented. Chapter 2 has the general
relations between solution properties and correlation
8
functions. Chapter 3 has the full development of the new
model. Chapter 4 shows the application of the model to
solutions of aqueous strong electrolytes and the calculation
of solution properties. Chapter 5 has suggestions for further
work and conclusions.
1.4
1.2
1.0
0.8
0.6
0.4
0.0
Figure 1.
0.5 1.0 1.5 2.0 2.5
1
(Molality) 2
Salt Activity Coefficient in Water at
250C, 1 ATM. Data of Hamer and Wu
(1972).
0.06
0.59
0.58
0.57
0.56
0.55
0.54
0.00 0.02 0.04 0.06
X05
Figure 2.
0.08 0.10 0.12
Species Density in Aqueous Electrolytes
at 250C, 1 ATM. For data sources see
text.
CHAPTER 2
FLUCTUATION THEORY FOR STRONG ELECTROLYTE SOLUTIONS
Introduction
There are three general relations among the thermodynamic
properties of a solution and statistical mechanical correlation
functions. The first two are the socalled "Energy Equation"
and "Pressure Equation" which are obtained from the canonical
ensemble with the assumption of pairwise additivity of inter
molecular forces. These equations relate the configurational
internal energy and the pressure respectively to spatial
integrals involving the intermolecular pair potential and
the radial distribution function (Reed and Gubbins, 1973;
McQuarrie, 1976). The third relation is the socalled "Com
pressibility Equation" which is derived in the grand canonical
ensemble without the need to assume pairwise additivity of
intermolecular forces. This equation relates concentration
derivatives of the chemical potential to spatial integrals
of the total correlation function (Kirkwood and Buff, 1951)
and to spatial integrals of the direct correlation function
(O'Connell, 1971; O'Connell, 1981). This last method is
generally known as Fluctuation Solution Theory.
Fluctuation solution theory has been applied to the
case of a general reacting system (Perry, 1980; Perry and
O'Connell, 1984), and the formalism has also been adapted
to treat strong electrolyte solutions which are considered
as systems where the reaction has gone to completion (Perry,
Cabezas, and O'Connell, 1985). The main body of this chapter
consists of a derivation of the general fluctuation solution
theory. Although the final results are identical to those
previously obtained by Perry (1980), the development is
more intuitive and mathematically simpler, though less
general. The remainder of the chapter illustrates the
calculation of direct correlation function integrals (DCFI)
from solution properties and sets theoretically rigorous
infinite dilution limits on the DCFI's.
Thermodynamic Property Derivatives and Direct
Correlation Function Integrals
A general multicomponent electrolyte solution, contain
ing n species (ions and solvents) formed from no components
(salts and solvents) by the dissociation of the salts into
ions, is not composed of truly independent species due
to the stoichiometric relations among ions originating
from the same salt. It is, therefore, not possible to
change the number of ions of one kind independently of
all the other ions. However, the independence of ions
has been assumed traditionally for theoretical derivations,
and it will lead us to the correct results by a relatively
simple mathematical route. Thus, with the assumption that
any two species i and j are independent of all other species,
Fluctuation Solution Theory gives the following well known
result (O'Connell, 1971; O'Connell, 1981):
1 ^i 6.. c.
1 (21)
RT 9N. N. N
T,V,Nkj
where = the chemical potential per mole of
species i.
N. = the number of moles of species i.
N = the total number of moles of all species.
6.. = the Kroniker delta.
13
2
C. = 4Tp J r dr = spatial integral of
13 1=3
the direct correlation function.
N
p = = molecular density of all species.
V
The microscopic direct correlation function
13 W
is an angle averaged direct correlation function defined
by
ij> c1 dw. dw. (22)
'J j2 ij 1 J
Q
where Q2 = dw. d .
In order to arrive at the first and simplest of the
desired relations, we define the activity coefficient for
species i on the mole fraction scale as
(T,P) = MP(T) + RT In X.Y.(T,P) (23)
where = the reference chemical potential.
1 Ni
Xi = N = mole fraction of species i.
Yi = the activity coefficient of species i.
P = the vector of species mole densities.
By differentiating equation (23) with respect to
the number of moles of species j, we obtain
1 i
RT 3N.
T,V,Nk
ny+ i1 1 (24)
3N. N. N
3 T,V,Nk j
which upon insertion in equation (21) gives
1ny. 1 C..
1 = N1i (25)
3N. N
3 T,V,Nkj
and when multiplied by the system volume on both sides
of the equation,
81ny. 1 C..
1 1 (26)
p Tp p
J TPkfj
Nr
where p = n= molar density of species i.
J V
By performing a sum over all species i and j on equation
(26)
S n n lny.
v v v iaej B Pj
a B i=l j=l j T
iT'Pk~j
n n 1C..
S1 D1 (27)
a i=l j=l
where vi = number of species i in component a.
Sa = total number of species in component a.
By noting the definition of the mean activity
coefficient of a component a,
1 n
Iny, = V. Iinyi (28)
a v a ii i
a i=l
and also assuming that species j is formed from an arbitrary
component 8 so that
Pj = VjB Po (29)
one then arrives at the first relation
1 alny
1 aOB
T,Po
(210)
1 n n 1C .
i=1 j=l a6
which upon identification of
n n 1C..
1 C = (211)
i=l j=l a3S
assumes the simpler form
alny
PV p= v (1C ) (212)
pv T
Equation (212) relates the DCFI to the derivative
of the activity coefficient of any component a (salt or
solvent) with respect to the molar density of any component
3 (salt or solvent) at constant volume, temperature, and
mole number of all components other than 8.
Because most experiments and many practical calculations
are performed at constant pressure rather than constant
volume, it is of interest to derive a relation between
the activity coefficient derivatives at constant pressure
and direct correlation function integrals. First, a change
of variables is executed.
i i
3N. N.
3 T,V,Nkj 3 T,P,Nk/j
yi N .P (213)
aN.
T,N T,V,Nk^j
and the following identifications are made,
i 
p = V (214)
T,N
aP
av v.
P T,N = r (215)
aN. @N. VK
3 TV'Nkj V TPNkj
T,PNkj
Then equations (21), (214), and (215) are inserted into
equation (213) to obtain
lli 6. C.. V.V.
1 = 1 13 1 3 (216)
RT 3N. N. N VKTRT
3 T,P,Nk 1 T
where = partial molar volume
1 8N.
1 T,P,Nkfi
volume of species i.
V = the system volume.
1 av
KT V P = isothermal compressibility.
T,N
To develop a relation in terms of activity coefficients,
the chemical potential is written as in equation (23)
and the proper constant pressure derivative is taken.
1 i lny
RT aN. = Nj +
3 TP,Nk.j TP,Nkj
6.
13 1 (217)
N N
From equations (216) and (217),
18
81ny. 1 e. V.V.
1 = L 1 3
aN N VK RT (218)
T,P,Nk#.
By multiplying equation (218) by the system volume
and rearranging, one finds
N alny. 1 C V. V.
1 1
pKTRT aN pKTRT TRT K TRT (219)
T,P,Nkj
upon which a double summation over species i and j is
performed to obtain
N 1 n n alny
PK RT v v L IiaVji 1N.
T Ta i=l j=l 1
T,P,Nkfj
1 1 n n
1 v. (1C..) 
KT a 8 i=l j=1
n n
12 1. v V V. (220)
(K RT)2 V v i=l j=l a j8 i j
T a B
Equation (220) must be simplified so that all the properties
appear as component rather than species quantities. This
is done with the aid of equations (28), (211), and the
assumption that species j is formed from an arbitrary
component 8 so
Nj = vjNo (221)
Additionally, the partial molar volume of a component a
or B is expressed as a sum of the species partial molar
volumes.
V 1 aV
V N N (222)
DN NiB No
1 iS Of3
T,P,Nk5i T,P,N
V n
Vo8 N =i i Vi (223)
S T,P,N
oY$S
Equation (220) is now transformed to
Nva alny
PK RT 8N
T o5
T,P,N
v V (1C ) V V (224)
a s aS oa oS (224)
PKTRT K RT KTRT
To make further progress, the relationship of the
Vo
bulk modulus of the solution (p TRT) and the group, T,
to the direct correlation function integrals must be found.
First, the compressibility equation is derived from the
basic fluctuation theory result of equation (21) starting
with the GibbsDuhem equation for an isothermal but
nonisobaric process.
n
N N.dP. = VdP (225)
i=l 1 1
Upon differentiation of equation (225) with respect
to the mole number of an arbitrary component j,
SN. = V P
i=1 i T'V j aN
T,V,N dj T,V,N kj
and by insertion of the equation (21),
n
RT Z (6 X.C..) = V
i=l ij 13 N.
ST,V,Nkj
n
1 P 1 1 x. C.
RT ap T Fk 1 13
RT j i=
T'Pk$j
(226)
(227)
(228)
Equation (228) is the general multicomponent compres
sibility equation expressed in terms of species quantities.
This relation is now transformed to one in terms of com
ponents by performing a summation over species j and use
of equations (29) and (229).
n
o
X = 1 Vi X o
S B=lI
1 n ap
RT X Vj
T,PR~i
(229)
n n
= 1 X.v C.
i=1 j= 1
(230)
1 DP
RT 3p
oa
T,p
n
o n n
1 XI v. Xi v .' C..
@=1 i=1 j=1
(231)
which by use of equation (211) becomes
n
1 @P 1 = v Xo Ca (232)
RT Dp oa 0 0
T,p
Equation (232) is the multicomponent compressibility
equation expressed in terms of components. The density
derivative of pressure is related to the partial molar
volume as
V P aV o (233)
ap V 8N
o T,N oa T
T,p o T,P,N
which when inserted in equation (232) gives one of the
desired relations.
n
V o
oa v v X (1C ) (234)
KTRT a O~ X
In order to relate the bulk modulus to direct correla
tion function integrals, the total volume is related to
the partial molar volumes.
n0 n
V= ON = V No (235)
VN oa oa oa
a=l oa T,P,N a=1
T,P,N
Dividing equation (235) by the mole number of species
(N) yields
1 V n
S= X (236)
p N oa oca
a=l
and when applied to equation (234)
aP/RTI
D IT,N
n n
o o
RT= 1 VX X (1Ca)
OK RT l a 6 oa O (
T a=1 8=1
which is the second necessary relation.
Substitution of equation (234) and equation (237)
into equation (224) gives
Nva 1nya
pKTRT 8No
T,P,N
YO
= v
a R
n n
o o
SV vX X [(lC )(lC ) (1C )(1C )]
y=l 6=1 Y 6X y 06 y6 ay a5
(238)
which is further transformed by substituting for the bulk
modulus
Nv ny a
a N
T,P,Noy3
n n
o o
v v I vI v v X Y6[(1C Y)(1C a)(1C Y)(1C6 )]
y=l 6=1 o 06
n n
o o
S V X oyX 06(1C y6
y=l 6=1 (239)
Equation (239) relates changes in the activity coefficient
of any component a with changes in the mole number of any
component 8 where the process occurs at constant pressure
(237)
and temperature to sums of direct correlation function
integrals for components.
In summary, it should be noted that of the various
relations developed in this section,only a few are of prac
tical importance in relation to this work. These are listed
at the beginning of the next section.
Direct Correlation Function Integrals from
Solution Properties
The previous section consists of a relatively simple
but lengthy derivation of several basic relations between
solution properties and direct correlation function
integrals. The relations that are of most importance to
this work are listed below.
1ainy
PV D v v (1C ) (212)
V n
oco
oa (234)
TRT V V X (1C ) (234)
KTRT a =1 o aB
n n
SP/RT 1 o o
~ P/RT 1 v0 vX X (1C 1 (237)
P T,N TRT a=l =iTxaB=1
Nva 1nya
PKTRT 8No0
T,P,N,
Y 8
n n
o o
a 8 6=1 Y 6 oy 0
Y=l 6=1
[(lCy )(1Ca ) (1Cay)(lC68)]
N N v. v. (1C..)
SC = ia jB 1
i=l j=l a Y
(238)
(211)
Useful bounds on the value of the direct correlation
function integrals as the system approaches infinite
dilution in all components except one (usually the solvent)
can be deduced from the preceding relations. Thus, by
taking the limit of pure solvent (component 1), one obtains
0M
V
oa = (1C )0
v K RT al
a1
S (1C11 )
P K RT
ol 1
(240)
(241)
N 31ny
ol alnya
V 3N
1 No IT,P,N
Y
N 31ny
ol a
V 3N 0
3 ~o T,P,N
Y
(1 V K 1KRT
(1Cal) (LC )
(1C(1C )
(l 11
#B
242a)
(242b)
m LIM
where (1C ) = (1C )
and where equations (242) represent a constant pressure
limit on the DCFI. A corresponding constant volume limit
can be obtained from equation (212).
For a binary system consisting of one solvent (1)
and one salt (2), the fluctuation relations become
91ny y
n2 22 (243)
ap02 T 01
Po2
T'Pol
o2 = (1C ) + V2X (1C ) (244)
SRT = 2Xol 12 2 o2 22
T
aP/RT 1 2
p pKRT ol (1C )+
3P T,N PKTRT 01 11
2 2
2olXo2(12) + (22 (245)
Nv2 31ny2
pK RT 8No2
T,P,No
2X [(1C 1 )(1C22) (1C12)2] (246)
2 ol 11 22 12
v+2 (1C 1+) + V2 (1C 1)
1 C12 = (247)
12 v
1 C22
2 2
v (1C )+2+2v (1C )+ (1C__ )
+2 ++ +22 + 2
(248)
and the respective infinite dilution limits are
V
\K2RT 1C21
V 2Kl1RT = (C21
1
P p K RT
ol 1
N1 lny2
"2 a2N
o2IT
(lCll)
(249)
(250)
(251a)
p 2
Po (V )
22 V2 K RT
2 1
Nol
V2
a1ny2
aN o l
02
T,P,N
o1
( 1C122
(1C ) (1C12
22 (1C11)
(251b)
The significance of equations (251) can be further
understood by realizing that any correct model for the
,P,N ol
activity coefficient of an electrolyte must approach the
DebyeHuckel Limiting Law at very low salt concentration.
Thus, the mean activity coefficient of a salt on the mole
fraction scale is given by this law as
1 n 2 1/2
lny = S ( Z) (252)
ny2 i=l 2
6 1/2
where S =( 332T )
S Dk T
DebyeHuckel limiting law coefficient.
e = the electronic charge.
D1 = pure solvent dielectric constant.
k = Boltzmann's constant.
T = temperature.
Z. = valence of ion i.
1 2
I = ZiP = ionic strength.
i=l
and when the proper derivative is taken,
N P 1/2
0ol 2 y ol
2 3N2 4v2
T,P,N 2
n n 2 2
v1 1 i2 Ziz2 (253)
i=l j=l
Insertion of equation (253) into equation (251a)
gives
P 1/2
Sp I n n
Y ol 2 +
2 i2j22 1
4v2 i=1 j=1
P 2
Po (V O)
21 02 = (1C2) (254)
v2 1RT
which approaches negative infinity as the salt concentration
approaches zero.
In order to construct a model capable of correlating
and predicting the solution properties of electrolytes,
it is helpful to calculate the experimental behavior of
the DCFI's from solution properties. To that purpose,
equations (243), (244), and (245) have been inverted
so that the three DCFI's can be calculated from
1 2
1 C11 = [1 X V p] +
11 2 OKRTo2 o2
XolP TRT
2
X 2 iny2
ol V N 2 (255)
2 2 DN
ol T,P,No
Vo2 2
1C =1[X V p1
12 V2XolKTRT o2 o2
Xo2 N n 2 (256)
X N
0ol o2 T
T,P,Nol
2
p(Vo ) N in 2
1 C =  (257)
22 2 vK aN
2 T T,P,Nol
Figures 35 show the results of equations (255) and
(257) for six different salts at 1 ATM and 25C. The
compressibility data used were those of Gibson and Loeffler
(1948) for NaCL and NaBR. For LiCL, LiBR, KCL and KBR
the compressibilities of Allam (1963) were used. The
activity coefficient data were taken from the compilation by
Hamer and Wu (1972). The density data of Gibson and Loeffler
(1948) were again used for NaCL and NaBR. For LiCL, LiBR,
and KBR the density data were taken from the International
Critical Tables. The newer density data of Romankiw and
Chou (1983) were used for KCL. The pure water data were
those of Fine and Millero (1973). The infinite dilution
partial molar volumes were also from Millero (1972).
Summary
The present chapter has introduced the basic relations
of interest, has shown how they have been used to calculate
the experimental behavior of the DCFI's, and has given
some bounds on the values of the DCFI's. The next chapter
introduces a model for correlating the observed experimental
behavior of the DCFI's.
30
25
20
15
1 C22
10
5
0
0.00 05 02
0.00 0.02
0.04 0.06 0.08 0.10
X02
Figure 3. Salt (2)Salt (2) DCFI in Aqueous
Electrolyte Solutions at 250C, 1 ATM.
For data sources see text.
0.12
24
20
1C2 16
12
8
0.00 0.02
0.04 0.06 0.08 0.10 0.12
X05
Figure 4. Salt (2)Water (1) DCFI in Aqueous
Electrolyte Solutions at 250C, 1 ATM.
For data sources see text.
28
26
24
1C 12
22
20
18
0.00 0.02 0.04 0.06 0.08 0.10 0.12
X05
Figure 5. Water (1)Water (1) DCFI in Aqueous
Electrolyte Solutions at 250C, 1 ATM.
For data sources see text.
CHAPTER 3
A MODEL FOR DIRECT CORRELATION FUNCTION INTEGRALS
IN STRONG ELECTROLYTE SOLUTIONS
Introduction
In order for the formalism introduced in the previous
chapter to be of practical value, a model to express direct
correlation function integrals in terms of measurable
quantities (p, T, x) must be constructed. The present
chapter describes such a model. First, a general physical
picture of electrolyte solutions and its relation to micro
scopic direct correlation functions is discussed. Second,
a rigorous statistical mechanical basis is laid for the
microscopic direct correlation functions and their spatial
integrals. Third, equations are given for each type of
pair correlations in the system (ionioin, ionsolvent,
solventsolvent). Lastly, a summary is presented of the
model parameters and estimated sensitivity of results to
their values.
Philosophy of the Model
The complex thermodynamic behavior of liquid electro
lytes is the observable result of the very complex interac
tions between the species in solution, i.e., the ions and
34
solvent molecules. In the absence of a complete understand
ing of all these forces, models use simpler or, at least
tractable, interactions which may have the essential charac
teristics of the real forces. In addition, some semiempiri
cal terms are used to account for those interactions that
cannot be simply approximated.
Thus the interactions between the ions at long distances
are modeled as those of charges in a dielectric medium
containing a diffuse atmosphere of charges. At very short
range, however, the dominant interaction becomes a hard
spherelike repulsion. There exist rigorous statistical
mechanical methods to treat these two types of interactions,
but these two are not adequate to correlate and predict
the solution behavior with sufficient accuracy. Interactions
that are important at intermediate ionion ranges must
be incorporated. Unfortunately, these intermediate range
forces cannot be simplistically approximated because they
involve strong manybody effects such as dielectric satura
tion, ionpairing, polarization, etc., which are not well
understood. In the present model the ionic and hard sphere
interactions are treated theoretically while the rest are
included in a semiempirical fashion.
The interactions between ions and solvent molecules
at large separation can be treated as those of charges and
multipoles in a dielectric medium containing an ionic
atmosphere. In general, quadrupoles and higher order
multipoles are not included, because their contribution
is expected to be numerically insignificant in an aqueous
system. The short range interactions are treated as hard
sphere repulsion. Intermediate range forces for the ion
solvent case are very important because they include solva
tion which makes a larger contribution than the long range
chargemultipole forces. Solvation of the ion by the solvent
is intimately related to the partial molar volume of the
salt and must be incorporated if there is to be any hope
of correlating and predicting the volumetric behavior of
the solution. As for ions, the long and short range intera
tions are treated theoretically while the intermediate
range forces are incorporated semiempirically.
The forces between solvent molecules at long range
can be considered to be those of dipoles in a dielectric
medium which has an ionic atmosphere. Higher order multi
poles may again be neglected because their contributions
are less important and can be covered in other ways. The
short range forces are again treated as hard sphere repul
sions. The intermediate range interactions for the solvent
solvent case are dominated by association type forces such
as hydrogen bonding which make a larger contribution than
the long range dipoledipole term. As above, the long
and short range interactions are treated theoretically
while the effects of the intermediate range forces are
included semiempirically.
In summary, there are three distinct classes of inter
action: ionion, ionsolvent, solventsolvent. Each class
has unique contributions from longrange, fieldtype forces,
shortrange, repulsive forces, and intermediate range forces.
Traditionally, models have been written for the excess
Gibbs or Helmholtz energy of the system by adding contribu
tions from some of the above forces in an ad hoc and, gener
ally, nonrigorous fashion. The fact that free energy
contributions do not naturally separate into the types
of forces and that experimental values for each cannot
be separately determined has caused many of these models
to be complex and/or inconsistent. Further, they do not
yield volumetric properties along with the activities.
Within the framework of Fluctuation Solution Theory,
the contributions of the pair correlations to the thermo
dynamic properties can be rigorously added. Thus, there
are terms from the saltsalt, saltsolvent, and solvent
solvent DCFI's, as shown in Chapter 2. Further, the experi
mental behavior of each of the three DCFI types can be
separately calculated from solution data as seen in the
previous chapter. It is then possible to construct separate
and accurate models for each one of the DCFI's. These
models can later be manipulated to yield thermodynamic
properties.
As may be inferred from the above discussion, each
of the three types of DCFI's contains long range, short
range, and intermediate range interactions. These can
be theoretically separated into a simple additive form
as will be shown in the next section of this chapter.
It is important to note that the separation is first
developed at the level of microscopic direct correlation
functions which are later integrated to obtain the DCFI's.
Although our particular additive separation of the micro
scopic direct correlation function is not fully rigorous,
we believe it is more reasonable than a similar resolution
of the radial distribution function into an additive form
(Planche and Renon, 1981). In fact, the radial distribution
function can naturally be resolved only into a multiplicative
rather than an additive form. The intermolecular potential
and, consequently, the potential of mean force can be
approximately decomposed into additive contributions from
interactions of different characteristic range, but this
potential appears in an exponential in the radial distribu
tion function. Thus, resolving the radial distribution
function into additive contributions is quite inappropriate.
Statistical Mechanical Basis
The above philosophy is a qualitatiave concept which
must be expressed in quantitative terms. To this end,
we now establish a rigorous statistical mechanical basis
for a model of microscopic direct correlation functions.
First, consider the diagrammatic expansion of the direct
correlation function (Reichl, 1980; Croxton, 1975) for
species i and j,
u..
c..(T,p,r ,r ., .) = g. 1 Zn g + B.. (31)
13 j 1 n g ij kT 13
W.ij ./kT
where gij = e = radial distribution function.
W = potential of mean force.
ui. = pair potential.
B.. = sum of all bridge diagrams also known
as elementary clusters.
Although equation (31) is an exact expression for
the direct correlation function, it is of little practical
value because the bridge diagrams cannot be summed analyti
cally. This series is
Sn n
B.. f.f f .f.f. dr drd d
Bl 2'! Okp fikfk fZjfifkj dkd wkd
k=l =1
+ (32)
for a system consisting of n species.
u. ij./kT
where f.. = e 1 = Mayer bond function.
To obtain the hypernetted chain (HNC) approximation
(Rowlinson, 1965) all of the bridge diagrams are neglected
(B.. = 0). This introduces an error which is second order
1j
in density and ignores some four body contributions. It
is, therefore, exact up to the order of a third virial
coefficient. Thus, the HNC direct correlation function
is
cHNC
ij
= gi
ij
 1 ng. 
1ij kT
From the definition of the radial distribution
function,
W..
Z.n g..= 11
1] kT
(33)
(34)
W.. W .
giJ 1 1 kT ) 2
13kT 72T kT
3
1 W.
3! kT
which on insertion into equation (33) gives,
2 3
HNC u W.. W..
Cij T 1 3! 1 ) +
ij IT 2! kT 3! kT "
To apply equation (36) requires at least approximate
expressions for the potential of mean force in terms of
measurable variables. Such expressions, valid in the limit
(35)
(36)
of zero salt concentration and large separation between
the two interacting species, are available for ionion
interactions from the DebyeHuckel theory and for iondipole
and dipoledipole interactions from more recent work (H'ye
and Stell, 1978; Chan, Mitchell, and Ninham, 1979) which
yields results identical to those of Debye and Huckel for
ionic activities. Thus, the long range direct correlation
function is based on these potentials of mean force, WLR.
Then, our HNC approximation is
LIM
HNC LR
I*o c.. c.. (37)
r ij _0_00 3 13
12
LR 2 LR 3
LR 1 W1 W
cL. + 1 + (38)
13 kT 2! kT 3! kT
The potentials of mean force, however, are unphysical
inside the hard core of the molecules and must be set equal
to zero.
LR
W R = 0 r.. < a..
i] 1] ji
(39)
LR LR
W.. = W.. r.. > a..
ij ij] 3 13
1
where aij = (aii + a..) = distance of closest approach
of species i and j
At contact and inside the core of the molecules, the
direct correlation function is dominated by a very strong
repulsion which is modelled as a hard sphere interaction.
To obtain the appropriate expressions for the hard sphere
direct correlation functions, the PercusYevick theory
(1958) was used since it has been shown to give a compres
sibility equation of state which is in good agreement with
simulation results for hard spheres (Reed and Gubbins,
1973). The PercusYevick (PY) microscopic direct correlation
function for hard spheres is zero outside the core. Thus,
HS PYHS
c.. = c.. (310)
HS
uS/kT
PYHS HS (1 ijkT
cij = gij (1 e ) (311)
u. = 0 r.. > a..
13 13
where
HS
u.. = r.. < a..
1] 1] 13
Although the PY microscopic direct correlation function
is formally used in the development that follows, it was
not actually employed in obtaining the final expressions
for the DCFI's. Rather, the expression for the hard sphere
chemical potential as derived from PercusYevick theory
through the compressibility equation was used together
with equation (21) to obtain the desired relation (see
Appendix A). Although the more exact CarnahanStarling
(Carnahan and Starling, 1969; Mansoori, Carnahan, Starling,
and Leland, 1971) expression could have been used, it is
somewhat more complex and relatively little improvement
in accuracy would be expected.
At this point, we have established a viable, albeit
traditional, theory for the behavior of the direct correla
tion function as r.. m and at r.. < a... However, many
1] I] J1
interactions which are important in aqueous electrolyte
systems such as hydration of ions by water, hydrogen bonding
between water molecules, and ion pairing are strongest
at r.. just outside the core. Further, this is that kind
of interaction for which liquid state theory is not well
developed. Therefore, we attempt here to develop a method
for interpolation of the direct correlation function between
long and short range. Because generally available theory
offers little guidance, the method can at best be semiempiri
cal. For this purpose, the RusbrookeScoins expansion
of direct correlation function (Reichl, 1980; Croxton,
1975) for species i and j in a system of n kinds of species
is now introduced.
cij(T,p,r,rj ,wi'O.) = fij(T) +
n
+ Z pk ijk(T) + (312)
k=l
where (T) = J f. fi fj dr
ijk ij ik jk dk k
Since equation (312) represents the entire direct
LR HS
correlation function including cij and cij, these two must
be subtracted to obtain the interpolating function. There
fore, the complete model for the microscopic direct correla
tion function for species i and j in a system of n species
is
c.. = c.H + c. + Ac.. (313)
] i i] 1]
HS LR
where cij is defined by equation (311), c by equation
(38), and
HS LR
Ac.. = c c.. c. (314)
13 1] 13 3
which is approximated by the RushbrookeScoins expansion
as
HS LR n
Ac.. = (fi.. f f ) + I (p
] i] ij k= k ijk
HS o LR
P H P o L (315)
k ijk k ijk (315)
LIM
o LR LIM
where = r.. P
k ijk 13 k ijk
I o
The series in equation (312) is truncated at the
first order term in density to be consistent with the HNC
theory and because inclusion of the more complex higher
order terms was empirically unnecessary.
For the sake of simplicity in notation equation (315)
is expressed as
Ac = Af.. + n po LR
Skl k ijk ijk316)
where Af.. = f.. fHS fLR
13 i] ij ij
HS
A
No attempt was made in this work to analytically calcu
late the coefficients in equation (316); rather, their
spatial integrals were fitted to data. The importance
of equation (316), however, is in providing a theoretical
framework for describing the properties for a class of molecu
lar interactions which are not well understood. Thus,
the first term represents the contribution of pairing or
repulsion in the case of ion pairs, solvation in the case
of ionsolvent pairs, and hydrogen bonding in the case
of solvent pairs. The second term represents the effect
of a third body (k) on the direct correlation between species
i and j. If one or two of the three are solvent and the
rest ions, then this term is dominated by hydration. If
all three species are ions, then this term is dominated
by ion association or repulsion. The physical significance
of these terms will be discussed further below.
As pointed out in Chapter 2, solution properties are
related to spatial integrals of the direct correlation
function. In order to relate this model to thermodynamic
properties, equation (313) is integrated over angles first
and separated later. Thus
i c. d dW. (317)
C (T,p) = 47'p f r2. dr.. (318)
S o 0 ij 13
Cj(T,P) = CHS + CLR + C. (319)
ij 13 1j 13
HS
where C is obtained directly from the chemical potential
i3
as shown in Appendix A. Thus, CLR is defined by
ij
LR 47p LR 2
C1 = f r.. dr..
Ci kT o i] ) ij iJ
2 p LR 2 LR2 2dp LR 3 2
+2p> <(w ) 2> r2 dr.. 2 f <(w ) > r dr.
kT o j 13ij 3kT 0o ij J
+ (320)
Lastly, AC.j is defined by formally integrating equation
(316).
n LR
ACij = pAFij(T) + p (kAO(T) pk(T)) (321)
k=l ijk ijk
where AF(T) = 47 J r.2 dr.
ij o ] W 3 13
A(T) = 47 f r.2 dr..
ijk o ijk 13
LR 2
LR > 2
(T) = 47 r. dr.ij
ijk o
Equations (319), (320), (321), and the expression
for C. from Appendix A are the general forms of the model
for species direct correlation function integrals. To
obtain practical expressions one needs merely to introduce
the appropriate pair potential and potential of mean force
into equation (320) and perform the indicated integration
as illustrated in the sections that follow.
Since the coefficients in equation (321) are fitted
to data rather than evaluated analytically, it is of
importance to develop mixing rules to reduce the amount
of data necessary to model multicomponent systems. The
aim here is to predict all the coefficients from quantities
associated with no more than two different species so that
only binary or commonion solution data would be required.
For aqueous electrolytes, the situation can be improved
due to the relative simplicity of ionion interactions
which can be generally scaled with the ionic charge (Kusalik
and Patey, 1983). Thus, two and three ion coefficients
are expressed from quantities related to a single ion.
If i, j, and k are ions, then
AF (T) 1 (AFi + AF ) (322)
ij 2 ii ji
AQ(T) (A + Aj + AD ) (323)
3 iii jjj kkk
ijk
LR
L(T) (L + .. + LR ) (324)
ijk 3 11i 33 kkk
If one or two of the species i, j, and k are solvents while
the remainder are ions, then the mixing rule must be
expressed from quantities involving each of the species
and water. The reason for this is that ionsolvent inter
actions cannot possibly be predicted from solventsolvent
and ionion interactions separately. Therefore, if i is
an ion and j a solvent, then
AF..(T) = AF.. (325)
If i and j are ions while k is a solvent, then
AO(T) = (A ii + A k) (326)
ijk 2
LR
(T) ( + LR (327)
i 2 iik jjk
ijk
If i is an ion and j and k are solvents, then
AM(T) = Ajk (328)
ijk
LR LR
$(T) = D(T) (329)
ijk ijk
Lastly, if i, j, and k are all solvents, then
AF(T) = AF. (330)
ij 13
AI(T) = Ai. (331)
ijk ijk
LR
LR
((T) = ijk (332)
ijk
It should be noted that these additive mixing rules
are not the only possible ones. In fact, theory would
suggest that geometric mean type mixing rules might be
more appropriate. Geometric mean rules, however, only
work for positive quantities which turned out not to be
the case with our empirically fitted coefficients. This
situation is further discussed in Appendix C.
The last point that needs to be addressed here is
the extension of the model to multisolvent systems. First,
the extension of the expression for c. is well known.
LR
Second, the extension of equation (320) for cLR requires
potentials of mean force applicable to the system. Assuming
all solvents are dipolar requires only knowing the dipole
moment of each of the solvent molecules and the dielectric
constant of the solvent mixture. Neither of these are
expected to present a problem in general. Third, the exten
sion of equation (321) for Ac.. involves a few more coeffi
cients and slightly different mixing rules for some three
body terms. Thus, while equations (322) to (327) would
remain the same for all solvents, equations (328) and
(329) where i is an ion and j, k solvents would be altered
to
ijk
D(T) = (LR. + LR (334)
ijk 2 ii3 ikk
ijk
which reduce to the previous result only when j and k are
equal. Here, any nonadditive interaction between j and
k has been tacitly ignored because the difference in the
interactions between different solvents is likely to be
less important to direct correlation function integrals
than that from the much stronger ionsolvent interactions.
This assumption is based on previous investigation of
solventsolvent interactions which are dominated by angle
independent forces (Brelvi, 1973; Mathias, 1978; Telotte,
1985; Campanella, 1983; Gubbins and O'Connell, 1974; Brelvi
and O'Connell, 1975). Finally, equations (330), (331),
and (331) where i, j, and k are solvents would become
AF (T) (AF + AF ) (335)
ij 2 ii Fj
1
AM (T) = (At. + A + ) (336)
ijk 111 331 kkk
LR
LR 1 LR LR LR
f(T) = i ii + + k) (337)
ijk 3 111i 333 kkk
ijk
The above mixing rules for an aqueous system (single
solvent) have been tested against data for a number of
salts and may be regarded as established. The rules for
a multisolvent system, however, have not been tested.
They can only be seen as physically reasonable in the light
of previous experience but still tentative.
The next two sections deal with the application of
the theory developed here to specific interaction in order
to construct practical expressions.
Expression for SaltSalt DCFI
The saltsalt direct correlation function integral
(C a) can be expressed as a stoichiometric sum of ionion
DCFI's (c. ) given by equation (211).
n n v. v (1C..)
1 C = 1 la 18 (211)
i=l j=l ae
It is, thus, only necessary to develop general and practical
expressions for the ionion DCFI's and insert these into
equation (211) to obtain a general expression for the
saltsalt DCFI. The basic model for ionion DCFI's is
represented by equation (319). The expression for CHS
ij
has been developed in Appendix A and that for AC.. is given
by equation (321). This section is then chiefly concerned
with performing the integration in equation (320) to
LR
obtain an expression for CLR
The pair potential between two ions is given by
Z.Z .e2
LR z _
u. L3 (338)
13 r
Here the potential of mean force is approximated by a gener
alized form given by the DebyeHuckel theory.
Z.e K(a. .r. ij)
"LR = i e r > a.. (339a)
ij kTr.. D(1+Ka..) l]
LR
W. = 0 r < a.. (339b)
ij 13
where
K2 4ne2 n 2
K DkT zi Pi = DebyeHuckel
i=l
inverse length.
D = the dielectric constant of the solvent
or mixture of solvents.
Insertion of equations (338) and (339) into equation
(320) gives
2
LR 4TpZ Z e 2
C f r.i d r..
ij kT i
2 2 4
2rpZ Z .e 2Ka..
+ 1 1 e 13 J
(DkT) (1+Ka..) a
336 3Ka..
2JrpZ.Z .e 3Ka
i e ] f
3 3
3(DkT) (1+Ka..) a..
1J 1J
2Kr..
e 13 dr.. 
3Kr.
17
e i rj
e dr.. +
r.. 13
i]
The first term of equation (340) contains a divergent
integral. However, when it is introduced into equation
(211) which relates it to thermodynamic properties, electro
neutrality makes the coefficients of the integrals sums
to exactly zero.
(340)
2 n n
SkTi 1 Z vjZ f r.. dr.. = 0 (341)
v V kT i=la i j=1 1
n
where v. Z. = o
i=l ic 1
The second term of equation (340) is integrable and
contains the implications for DCFI's of the DebyeHuckel
limiting and extended laws (see equation 254). Then,
27rpZ2 2e 2Ka.. m 2Kr..
i e 13 e dr..
(DkT)2(1+Ka..)2 a. 1J
2 2 1/2
Z.Z.S pI
= 1 'Y (342)
1/2 2
(l+a..IB 1/2)
where 2e6 1/2
where S = ( 2 )
Y D3k3T3
K = B I1/2
Y
e2 1/2
B =K I /2 DkT
1 2
1 2 Zi Pi
i=l
The third term of equation (340) is also integrable
but more complex. The integral is the first order member
of a class of functions known as the exponential integrals.
These cannot be evaluated explicitly but a number of
asymptotic expansions and numerical approximations are
available (see Appendix D). It is convenient to express
the integral in dimensionless form.
Letting X = r/ai. then
3Kr.
e 13
a idr..
r.. 13
(3Kai)x
= e dx = E (3Ka..)
1 x 1 l]
(343)
where E1(3Ka ij) = the first exponential integral
The third term in equation (340) becomes
i ]
27rpZ.Z .e
3(DkT) (l+Ka..)
3Ka..
3Kr..
e 13
r.. dr..
a.. 13
a.3
3a..B II/2
3 3 2 3a. y12
Z.Z Spe Pe
S(l+a. .B E (3a..B 11/2)
3 1/2 13
(l+a..iB I
1] Y
(344)
which contains the implications for DCFI's to a higher
order limiting law for unsymmetric electrolytes (Friedman,
1962). Because of electroneutrality, this term, when
inserted into equation (211), is always very small for
symmetric electrolytes, and it approaches zero as the con
centration of salt decreases. For unsymmetric electrolytes,
however, the sum over the ionic charges is not small and
this term actually diverges logarithmically as the salt
concentration decreases. To further explore the relation
of (344) to Friedman's limiting law and to elucidate the
low salt concentration behavior, the exponential integral
(E1) can be expanded for low values of the ionic strength
(I 0).
E (3ai.B I1/2) = n(3aiB I1/2) a + O(I1/2) (345)
where a = 0.5772 = Euler's constant.
This expansion is valid only at extremely low ionic
strength. Equation (344) then becomes
ji S2p PE (3aB 11/2
3 Y olEl(3ai ) Y
z33
Z 1
3 ol 2 n + + n 3a
+ 0(11/2) (346)
where InI diverges as I 0 while a + In 3a..B are all
13 Y
constant.
The contribution of Friedman's limiting law to the
activity coefficient of a salt (a) is
1 n .3 2
V x ia i
FLL 1 _i=l 2
FLLn y1 i= Z S2 IknI (347)
a 3 n 2 Y
i=l La 1
and by taking the first derivative with respect to the
mole number of a salt 8 at I o,
FTL 2 P
Sn y Sypo n n 3
21L 01 3 ..zz
N 23v v X 3 3
oNo ITPN 3vaB i=l j=la Z
T,P,N
1 1
( An I + ) (348)
Z 2
Rearrangement of equation (224) gives
N any pV V0
N C oa oB (349)
v @N 0CaB$ v K TRT
S oBS a c aB
T,P,N 4
If equations (348) and (349) are compared, it is
clear that the contribution of Friedman's form of the limit
ing law to the saltsalt direct correlation function integral
is
2 P
cFLL S ol n n1
CF LLX z z ( Zn I + ) (350)
U 3vv ia j i j ]2 2
S 3vaeB i=1 j=1
Comparing equations (211) and (350) gives the ionion
DCFI.
3 3
CFLL S2 P 1 + 1) (351)
ij 3 y ol 2 2
Substitution of equation (346) in equation (340)
gives the expression for the limiting contribution of the
third term in equation (340) to the ionion DCFI.
3 3
SS2 P ( nI + + In 3a.. B) (352)
3 y o 2 13
Equations (351) and (352) have essentially the same
behavior as I 0 since they differ only by a small constant
which is negligible compared to ZnI as I 0. Therefore,
equation (344) contains the higher order limiting law.
LR
The general expression for C is
1/2 2 2
S pl12 n n n i. v Z.Z.
cLR 4v v 1/2 2
4t B i=l j=l (l+a..B I )
13 Y
1/2
S3a. .B I /
Sp n n i. Z3Ze E (3ai..B
3v V 1 1/2 3
3a i=l j=l (1 + a..B I2)
11 Y
(353)
The expression for CHS is
aB
HS 1 n n HS
CS 1 V V. V. C (354)
a ji=l j=l
Lastly, the expression for ACB is
n n
AcB = v I Via. AF. +
a B i=l j=l aB
n n n
+ v (p A po ) (355)
Va i=l j=l k=l a j k ijk kjk
Equations (353), (354), and (355) form the complete
model for the saltsalt DCFI.
S = CHS + CLR + AC (356)
CaB CaB aB
Since the limits of DCFI's as salt concentration
approaches zero are well defined, it is advantageous to
use equations (353) to (355) to model the deviations
from this limit. To this purpose, the infinite dilution
limit of the saltsalt DCFI is now explored. From equation
(242a)
N n p V
N1 N y = (1Co) s
S(1TC ) ola (242a)
S N a v vT,P, K RT
S oS T~~Noy a6
it is seen that the constant temperature and pressure limit
has divergent terms associated with the activity coefficient,
a first constant related to the partial molar volume, and
a second constant associated with the activity coefficient
and which is not so well defined. This second constant
is loosely related to a term linear in salt density which
often appears in empirical expressions for the salt activity
coefficient (Guggenheim and Turgen, 1955; Guggenheim and
Stokes, 1969). In the present model the divergent terms
are contained in equation (353). The first constant can
be calculated directly from infinite dilution partial molar
volumes and solvent quantities. The second constant must
be fitted to data using terms from equation (355) which
have only ionion and long range ionwater correlations.
This reflects the fact that triple ion direct correlations
are zero at infinite dilution and any contributing short
range ionsolvent correlations would generally be contained
in the first constant. Thus,
0 LIM LR TB
C XL1 (C aB A ) (357)
a X +91 aB ca aB
where TB n n n
where VAC = v. v jB ,ijk
SB i=l j=l k=l
p o_ o
p V V n n
1c oa oa B p (F
(1C) KRT v v v (AF
Sa i=l j=1
P LR
+ P 1l ) (358)
01 ijl
Finally, the general expression for the saltsalt
DCFI model including the infinite dilution limit is
m LR HS HSo'
1C = (C ) (C HS C
aB cB aaB ca6 )
TB TBm
(A AC ) (359)
where cHS = LIM HS
where C C
aB Xol l aB
TBC LIM TB
AC AC
aB X oll aB
P
Pol n n
= 01 y y v PA
V v I I itX jp o1 ijl
Va i=l j=l
Although equation (356) can be used in place of
equation (359), it was felt that the latter was more
appropriate for calculations at constant temperature and
pressure. Therefore, equation (359) was used in the com
parisons and correlations in this work. In calculations
where pressure varies, equation (356) would be more
convenient since it would eliminate the need to obtain
partial molar volumes as a function of pressure.
For illustrative purposes, equation (359) will now
be written for a binary system consisting of a solvent
(1) and a salt (2) which dissociates to formvy cations
and v anions.
n = 1 + v +
V = V +
v2 = + 
(1c22)LR HS
22 = 22 22 (C22
SHSc
22
TB TB
 (ACTB ACT )
22 22
(360)
where
TB TB C P
(AC22 AC ) 2
2
+ 2v+v _(plAl+
2
[v (p AD + p AD + p AD ) +
+ 1 1++ + +++ ++
+ p+A++
+ pA(_+ ) +
2
+ V (PlAl_ + p+AI+__ + pA___ )] +
P 2
(P p ) 2 9
01 [v 2A + 2v v A+ + v2A1 ]
2 + ++ + 1+ 1
V2
(361)
R S 1/2 2 4
cLR= ++ +
22 2 1/2 2
4v (l+a BI )
2 ++ y
22 24
2v _Z2Z2 v2 Z
+ ( BI1/22 + (1/2)2
(l+a+ BI ) (l+a BI )
+  y
S3a++B I1/2
2 6 + y
+ +
(1+a++B I/)
++ y
2v 3a B 1/2
3 3 3a+ B
2+ Z_ +
E (3a+ B 1/2)
1 /7
E (3a B 1 /2)
1 
2 6 3a B II/
2V  y
v_Z_e
(l+a B 11/2)3
2
P 2
SPol(Vo2) p 2
1 C22 [V (aF
SC22 =2 2 + ++
v2 1RT 2
+ 2v+v_ (AF_
(CHS HSm) 1
22 22 2
2
+ 2V v_ (CHS
+ +
P LR
 P l++) +
o1 1++
P LR 2 P LR
 Po (l ) + v (AF Po l )
S1+  01 1
[v2 (CHS ) +
+ ++ ++
CHS) + v2 (HS HSm)
+ + __
2
S3
v
3v2
(362)
(363)
(364)
(1 B1/2 3
(l+a BI )
+ Y
E (3a__ By1/2 )
Expression for SaltSolvent DCFI
The development in this section parallels that of
the previous one. Thus, a general expression for the
solvention DCFI is derived and then inserted in equation
(211) to yield the saltsolvent DCFI relation.
Although any type of interaction can, in principle,
be included, it was assumed here that ionsolvent interac
tions are dominated by dipolecharge forces at large separa
tion, and no other interactions were included. The pair
potential for an ion (i) and a dipolar solvent (1) is
LR l e
u = cos 6 (365)
ril
where pl = the dipole moment of solvent 1
in Debyes.
S= the Eulerian angle between dipole
and charge.
The potential of mean force is approximated by a func
tional form inspired by some recent applications of the
mean spherical approximation (Chan, Mitchell, and Ninham,
1979) and of perturbation theory (Hoye and Stell, 1978)
to nonprimitive electrolyte models.
wL lK(ailril
LR i 1 e il
il 2 (cos 0) e r > ail (366a)
kTri
LR
Wi = 0 r ail (366b)
iiil 1
where a is a universal constant that we have set equal
to 4.4 empirically.
Since equations (365) and (366) are functions of
orientation, it is necessary to first perform the integration
over angles as indicated in equation (320).
= u LR dw dwl (367)
il 2 i1 i 1
where dwo = sin e.di.d4.
1 1 1 1
S= f dw. = f sin 6.de. f2 7di = 47
1 0 1 1 0 i
When the integral in equation (367) is evaluated, it is
found that
LR
= 0 (368)
il W
The second term in equation (320) has
LR 2 1 I (W)LR2
<(W ) > (W) dw. dw (369)
il Q 2 il 1
After the integral in equation (369) is evaluated, it
gives
L2 2 2K(ailril)
2 Z ile )e
<(W i) > = T 4 (370)
ilDkT 4
r.l
The third term in equation (320) contains
LR 3 1 )3
<(W ) > ( dw d
ii 2 il 1
which also equals zero.
LR 3
<(W ) > = 0
Therefore, for iondipole pairs there is only one term
in equation (320).
i 2 Tr c 2 2Ka 2Kril
LR 2I 1I ) 2 il e e
il 3 P DkT ) e f 
aij il
1] 1
dril
(371)
(372)
(373)
The integral in equation (373) is also an exponential
integral (E2) which is expressed in dimensionless form
as before.
C 2Kr il (2Ka )X
S1 il E (2Kail )
2 e dr = 1 f e dX = ail
2 i1 a.i 1 a
ail ril
Equation (373) then becomes
2Ka
2 i1
iR 2T lZ el e E2(2Kail
11 3 DkT ail
(374)
(375)
Then, the general expression for the salt (a) and
solvent (1) DCFI is
S2a iB I1/2
2 11
LR 27p ea 2 2 n E. 1/2
Cl = ( ) E (2a il )
al 3v DkT i=l ail2 il y
a 1=1 11
(376)
The expression for CHS is given by
n
HS 1 HS
Cl v Via Cil
a i=l
and the relation for ACal is
n n n
AC v= P v. AF. + p vi
al a il a il V a il k la
a l=c a i=l k=l
(Pk Ailk
(377)
(378)
o LR
 p ik)
k ilk
Again, equations (376), (377), (378), and (379)
form the complete general model for the saltsolvent DCFI.
SCHS LR
C = CHS + C + AC
al al al al
(379)
As previously discussed, it is convenient, particularly
for isobaric calculations, to use the model only for
67
deviations from infinite dilution. (For nonisobaric calcu
lations, equation (379) would be more appropriate.) The
infinite dilution limit of Cal is given by equation (240)
LR
and that of CLR is (see Appendix D)
acl
P 2
P2eep 2 n y Z2
LRO LIM LR 2ol ( ea 2n i (380)
1C C = ) P 1 a1 (380)
al X ol+1 l 3v DkT 1 ail
ol a i=l a
HS
while the infinite dilution limit of C H is formally
HSm LIM HS (381)
C1 =X1 Cl (381)
Lastly, the infinite dilution limit of AC is
P
A LIM ol n P LR
al X o1 al i (iFil Pol ill
ol0 i=a
P
Pol n
+ 1 o' pnolA l (382)
iV Vi P1 ill
a i=l
The complete general relation for the saltsolvent
DCFI including the infinite dilution limit is
00
V
oc HS HS o LR LRm
1 IT RT Cl Cal) (C Cal
(ACal ACal) (383)
Finally, equation (383) will now be written for a
binary system consisting of solvent (1) and salt (2) with
v+ cations and v_ anions.
V o 02 (HS
21 v K2 RT 21
HS LR
 C2 ) (C2
21 21
(AC21 AC21)
where
HS _HS 1 HS
21 21 v2 +(C+1
"2
HS + (HS HSm)
+1 ) 1 1 )
2
CLR CLR 2 r ( ) e )
21 21 3v DkT
2
2
2 + +
1 [
a+1
2a +B I1/21/
(pe Y E2 (2a+1B
2 1y
\) z2
p +T
ol a_l
AC C = [v (AF+
1 C21 + +1
2
2a 1B BI/2
(pe E2(2a lBI2) p)]
(386)
P LR P LR
 i ) + _(AF Pol )]
01 +11 1 01 1
(p p ) +
+ ol + p+ p +
V2 + +1 1+1 + P+A++ + +
V2
SLR
21
(384)
(385)
+ v_(plA11_ + P+A+ + P_A __ )]
P 2
(Pol
[v+ &1+1 + v _l_ (387)
Expression for SolventSolvent DCFI
The solventsolvent direct correlation function integral
has the simplest relation since the solvent does not
dissociate so the species and component integrals are the
same.
As previously noted, any type of interaction can gen
erally be included in this theory, but it was assumed that
solventsolvent interactions at large separation are domi
nated only by dipoledipole forces. The solvent (1)solvent
(1) pair potential is
LR 1 1
u 3 1 (388)
rl
r11
where ( = 2 cos 611 cos e12
sin 011 sin 012 cos (11 012
611' 11 = Eulerian angles of solvent molecule
number 1.
612' 12 = Eulerian angles of solvent molecule
number 2.
The potential of mean force is approximated by a
function inspired by previously mentioned work (Chan,
Mitchell, and Ninham, 1979; Hzye and Stell, 1978).
K(a r)
LR 1 1 e 1
W = kr D
W11 3 D
kTr11
11
LR
W I 01
11
r > al
r < al
11
(389a)
(389b)
Again, equations (388) and (389) are inserted into
equation (320) and the required integration over angles
performed.
SLR = L l d11
11 Q 2 11 d11 12
dli= sin 1i deli d li
S= f d6l = f sin 6li dli
o
2Tr
f di = 47
0
The integration of equation (390) gives
LR
0
11 W
The second term in equation (320) has
LR 2 1 LR 2
<(W ) > f (W ) dw dw
<11 ( 2 (W11 d11 d12
which yields upon evaluation,
LR 2 1 i l11 2
<(W ) > (D )
11 w 3 DkT
2K(alrl)
2K(1111
e
6
11
(393)
where
(390)
(391)
(392)
The third term in equation (320) has
LR 3 1 LR 3
(W ) > (W ) d
<11 2 11 1 12
which becomes upon integration
LR
<(W )> = o
11
(394)
(395)
Thus, for dipolar solvents only one term of equation
(320) remains after the angle integration.
R = 2 e2Kal 2Krl
1 3 (DkT e r 4
all rll
dr11
(396)
The integral in equation (396) is also an exponential
integral (E4) which can be expressed in dimensionless form.
2Krll (2Kall)X
S11 (2Ka11 )X E (2Ka )
e 1 e 4 11
d 4 dr l =  4 dX = 3
al r11 all 1 X all
(397)
Equation (396) is then transformed
2alB Il/2
LR 4 ( 1 (1 1)1
C11 3 DkT 3
all
E (2a B 11/2)
4 11
(398)
which is the general expression for the solventsolvent
DCFI.
Since the solvent does not dissociate, there is no
HS
summation over species in C11. However, AC11 does have
a sum over third bodies.
n
CFo R ( 9LR
AC11 = pA1 + (Pkllk llk (399)
k=1
Equations (398), (399), and (3100) form the complete
general model for the solventsolvent DCFI.
C = CHS + + AC1 (3100)
11 11 11 11
Again, the infinite dilution limit of C11 is introduced
so that for isobaric calculations the model need only account
for deviations from the infinite dilution value. Also,
equation (3100) would be more practical for nonisobaric
cases. The infinite dilution limit of C11 is the bulk
modulus of the pure solvent given by equation (241).
The infinite dilution limit of CLR is given by
11
P
LRm LIM cLR 4pol r T D 2
CLR XLIM CII l ( (3101)
11 X 01 11 3 fDkT )0
11
and that for C S is formally
11
HS LIM HS
C =1 Xol1 CI (3102)
11 X 1 11 (3102)
ol
The infinite dilution limit for AC11 is given by
AC LIM ACP (AF P LR P 2 A
S11= Xol 11 01ol 11 ol 111 ll
(3103)
Finally, the complete general expression for the
solventsolvent DCFI including the infinite dilution limit
is
1 HS HS" LR LR"
1 C (Cl l ) (C C )
polKRT
11 11 11 11
(ACll ll) (3104)
Again, the application of equation (3104) to a binary
system consisting of solvent (1) and a salt (2) with v
+
cations and V_ anions is shown. However, for the solvent
solvent DCFI all of the terms except AC11 appear similar
to the general case since they have no summations over
species. Thus, only AC11 is illustrated below.
oo P P LR
AC ACI = (P p )(AF Po +
11 11 01 11 P ll
+ P(P1 111 + P+ A)11 + pA11) 
(p P)2 A (3105)
01o 111
Summary
A general statistical mechanical model of the direct
correlation function has been presented. In principle
it is applicable to any system, but it has been specialized
here to treat strong electrolyte solutions. The next chapter
shows the application of this model to six aqueous strong
electrolyte binary solutions. As a preview to the calcula
tions, the relative magnitude of the three contributions
HS LR
to the DCFI (C C R AC ) will now be discussed, the
model parameters will be listed, and the sensitivity of
solution properties to parameter value considered.
The saltsalt DCFI is dominated at very low salt con
LR
centration by C n which contains the long ranged electro
LR
static interactions. However, the magnitude of C 8 decreases
very fast as the salt concentration increases so that above
2M or so in salt density the dominant term becomes C HS
This reflects the increasing shielding of electrostatic
forces by more ions that more frequently repel each other.
AC makes a contribution that is generally not dominant
in either regime but is always numerically significant
above 0.5M.
HS
The saltsolvent DCFI is always dominated by C H with
LR
Cl making a small but not negligible contribution. Due
to the relative strength of the short ranged hydration
interactions, ACal makes the largest contribution after
CHS
Cl*
"al"
The solventsolvent DCFI is also dominated by CHS
over the entire range of salt concentration up to about
LR
6M. C11 makes a negligible contribution reflecting the
relative weakness of long range dipoledipole interactions.
HS
Again, the largest term after CI is AC which contains
the short ranged hydrogen bonding between solvent molecules
and the hydration related effect of an ion on two solvent
molecules at short range.
The parameters of the model are species specific and
universal. It is, therefore, necessary to build only a
relatively small set of parameter values to predict the
behavior of a large number of systems. Thus, a hard sphere
diameter (a. ) for each species is required for CHS and
11 a
LR
CaB (where a, B can be salts or solvents). To avoid con
fusion, the parameters for ACa will be those of a system
with one solvent (1), one salt (2), and many ions (i, j).
Then, AC involves AF PLR which is ion independent,
11 11 ol 111
A11 which is usually neglected, and Al for each ion.
111 ill
AC has AF POaLR i l and A. AC22 includes
1A2 i 01o lil 1il' 1i22
AF. p A and AoP.... This totals to two solvent
11 ol iin, ii, 111
specific parameters if AO11 is neglected and six parameters
for each ion (note that A lii = A.il and AD. = Al )
lil ill 111 ii
three of which involve solvention pairs.
Properties predicted with the model are most sensitive
HS
to the value of the hard sphere diameters because the C
is a very strong function of the diameters. But it is
76
not as sensitive as is the case with other models. This
is due to the fact that the two body coefficients AF..
13
are fitted to infinite dilution quantities that include
HSm
C so there is a degree of compensation for changes in
the diameters. The sensitivity of the results to the value
of the coefficients in ACa is generally small since they
make a small contribution to the DCFI's.
CHAPTER 4
APPLICATION OF THE MODEL TO AQUEOUS STRONG ELECTROLYTES
Introduction
In Chapter 2, the formal relations between DCFI's and
thermodynamic properties were introduced. In Chapter 3, a
model expressing the DCFI's in terms of measurable variables
was constructed. In the present chapter we illustrate the
use of the formal relations and the model in the calcula
tion of thermodynamic properties. We also explore the
scheme used to fit model parameters; further we compare
calculated values to experimental ones for the saltsalt,
saltsolvent, and solventsolvent DCFI's and for the bulk
modulus, partial molar volume, and salt activity coeffi
cient. Finally, a discussion of the above results and a few
conclusions are presented.
The use of Fluctuation Theory in general fluid phase
equilibria problems has been treated in detail by O'Connell
(1981). The specific case of liquids containing super
critical components has been addressed by Mathias and
O'Connell (1981) and Mathias (1978). The present treatment
generally follows these developments, but there are
important differences for the present case of electrolytes.
Calculation of Solution Properties
from the Model
The formal relations between solution properties and
DCFI's are given by equations (212), (234), (237), and
(238) for a system consisting of no components, salts (a,B)
and one solvent (1).
a2enyc
pv
ac ap~ ~
ooY
= V v (1C )
V n
V o
OT = Ea Vx (1Ca
ST =1C
n n
aP/RT o o
PT = E Z v VX X (1C
Tp a=l =l a oa oB aB
a=1 Bl
T,X
NVa aQtny
PK TRT a@i
oT TIPN0
n n
o o
v v V Y v vvX X 0
a y=1 6=1 6 oy 06
[(1C )(1C a) (1Cy )(1C )]
*Y
(212)
(234)
(237)
(238)
ae P/RT 1
where = 
ap I PK RT
ST,N T
In order to evaluate the change in solution density
with pressure while the composition and temperature are
constant, one needs to integrate equation (237) from a
known reference density (pR) at the reference pressure (pR)
at the temperature and composition (mole fraction) of the
F
system up to the desired density (p ) at the system pressure
(P).
n n F
R o o P (T,P,X)
v V X X (1C )dp (41)
RT E =i =l oa o R R T,N (
p R(T,R ,X)
Equation (41) represents an implicit equation for the
F
unknown density (p ) which can only be solved numerically
with realistic models.
It should be appreciated that equation (41) cannot be
applied to an isobaric change because that would imply that
pressure, as well as temperature and composition, were
F R
held constant. Then p would be the same as p so the state
of the system would not vary at all.
To evaluate the change in solution density isothermally
with varying composition, a different approach is required.
To develop the necessary relations we start by considering
that in Fluctuation Theory the pressure is treated as the
80
dependent variable, a function of temperature, density, and
mole fraction.
P = P (T,p,X) (42)
Taking the total differential of pressure gives
aP aP
dP dp+ dT
ap T,TN Tp,
T,N p,N
n
o
z aX
a=2 oa
dX
oa
T,pX
oyfa
If the change is isothermal and if we divide by RT,
1 dP
T
1 aP
1 3 dp
RT 3p T
T,N
1
+ 
RT
C=
o p
S ax
:2 oc
dX
TP oa
T,pX
oyfa
By making some identifications we obtain
n
o
B=I
DP
ax
oY
T,V,X /
oy#a
aN
ax a
Ny4 a
VKT
VK T
DN 0
o3
oa N
oy/8
N
v X
Inserting equations (45) and (46) into equation (44)
gives
dP 1 aP dp +
RTT RT ap T
T a TN
n
0
ct2
n 
SK RT
8=1 T
dX
6 vX
B oa
6eS Boa
(47)
(43)
(44)
(45)
(46)
We next insert equations (234) and (237) into
equation (47).
n n
1 dP = [= V V X (1C ) dp +
RT T c=l 8=1 a oaoB 0
n n n
+ 0 V X (1C pdXo (48)
a=2 8=1 y=l Y 6 vX
c G Buoc
Equation (48) permits us to evaluate the change in
solution density with both pressure and composition along an
isotherm. This equation is also applicable to an isobaric
and isothermal process where the solution density changes as
a function of composition only.
To obtain the density (p ) of a given solution at a
known temperature, pressure, and composition (X ), we
o
isothermally integrate equation (48) from the known
reference density (p ) at a system temperature and a conven
iently chosen reference pressure (P R) and composition (X )
o
F F
up to the desired density and composition (p and X ). It
o
is suggested that for aqueous electrolytes the reference
density be chosen to be that of pure saturated water at the
system temperature.
82
F F
SAT n n P (T,P,X )
PP o 0 o
R = I v B SAT X X (1Ca )) dp +
RT SAT oc oaS
Ra=l =1 Poi (T) T
F
n n n X
o o o oa pdX
+ I I oy(C) (49)
a=2 8=1 y=l Y oy a6 BXa T
In evaluating the integrals of equation (49) each
integral involves variables appearing in other integrals.
To explicitly find pF requires further manipulations
discussed below.
The activity coefficient on the mole fraction scale for
any component (a) can be obtained by integration of equation
(312) from the reference molar density (P R) to the molar
density of each component (PF ) at constant temperature.
F
Y n p 1C
n v R dP oS (410)
R R R p a T
7 8=1 8 Po T
a P05
where Po5 = Xo8P
Equation (410) is applicable to any isothermal change,
isobaric or not, and the reference state composition where
YaR = 1 need not be that chosen. However, for aqueous
electrolytes it is natural to choose pure saturated liquid
water at the system temperature as was done for equation
(49).
F F
P (T,P,X)I n P (T,P,X)
ZnYc = al dol + Z f 1Ca dp
pSATT T 8=2 o p OT
ol
(411)
Equation (411) can be used for either isobaric or
nonisobaric changes.
In equations (49) and (411) one can use the DCFI
model represented by equations (359), (383), and (3104)
for isobaric integration. But, for nonisobaric integra
tions with equations (41), (49), and (411) the DCFI model
of equations (356), (379), and (3100) will be more
applicable because the pressure behavior of the DCFI
infinite dilution limits, some of which involve salt partial
molar volumes, is not generally available.
The composition behavior of component activity coeffi
cients on the mole fraction scale at constant temperature
and pressure could also be obtained from equation (238)
with composition expressed as mole fractions. Thus, we
express the differential of the activity coefficient of a
component (a) as
no 8nya
dnya = z X dX (412)
ST,P B=2 a TPX
TIIoyX$
,nya n
eny o 8 ny aN
= 3o o N(413)
06 T,P, Xo l T,P,Noa 0 N
OY 8 oy/c OY7c
oo N
= N (414)
ax 6 v x
N OY4
oyf 
Inserting equations (414) and (238) into (413) and
then putting the resulting expression into equation (412)
gives
n n n n
o o PK RT o o
diny = I dX I Z v TV
T,P 6=2 =1 B O 0 Y=l 6=1
XoyXo (1C 6)(Ca) (Cy) (1C6)] (415)
To obtain the activity coefficient, equation (415) is
isothermally and isobarically integrated from the reference
to the desired state.
F
n n n n X
o o o o o pK RT
S= I R do 6 v IX iV 6
a=2 B=l y=l C=1 X a 6 oX oB Y
OB T,P
Xo Xo[(1C 6)(1C ) (1C )(1C6) ]
(416)
n n
e o o
where = v v X Xo (1C )
SRT = 8=1 B a o
6 = Kroniker Delta
Equation (416) can only be used for isothermal,
isobaric changes and thus either the DCFI model of equations
(359), (383), and (3104) or that of equations (356),
(379), and (3100) may be used.
Equations (41), (49), (411), and (416) express
integration of the DCFI model formally. However, these
cannot be explicitly evaluated because of the multiple
variables involved in the integrals. To actually evaluate
these integrals requires a change of variables as discussed
by Mathias (1979) and O'Connell (1981). Rather than give
their formal equations, we now give the above relations with
explicit expressions for the present DCFI model. Those
parts that are analytically integrable have been evaluated
while simplified integrals are given for the others. The
DCFI model used is that of equations (356), (379), and
(3100) which does not contain the DCFI infinite dilution
limits. This form of the model yields simpler expressions
which can be applied to both isobaric and nonisobaric
changes. We start by rewriting equation (41) as
_pR pPYHS(pX) F PYHSpRX)
RT RT RT
F R
 (p p )
n n n
n o 0 1 LR
S 0 0 v v X X X J C L (t)dt 
= oa 0 oy a
U=l =l y=1 o
F F RR n
p p p p
2 .
i=l
F F FR R R
p P p p p p
3
n
j=1
n
i=l
FF P,F RR P,R
P P P pp p
ol ol
n
where p(t) = I(pR +
y=l1
X.X. AF.. 
1 3 13
n n
SI X.X X Aj +
j=1 k=l 1 3 k
n n
S z x.x. ,LR
i=1 j=1 ijl
(417)
n
F R o
(P FP R)t)= I v p (t)
oy oy Y Y
Equation (49) can also be changed to
SAT PYHS F F PYHS R R
PP TT) P ( ,X ) P (p ,X)
1 o 0
RT RT RT
n n n LR
0 FF RR a C t)
v v (X p X p ) fp (t)p (t) p dt 
= 8 = oY oy o o o p(t)p(t)
a=l 8=1 y=1 o
n n n n 1
oo o o 1 1C (t)
2 ll yl =1 Y o P (t) (t) F (t) dt 
a=2 6=1 Y=I 6=1 0 oy poa(t) a
6 v p(
rB 8 p(t)
n
F FFF R RRR
S(X. X p p X X. pp ) AF. 
j=l 1 3 1 3
n n
F F FFFF R R RRRR
S(X. /X. X ppX. X. A4. +
j=1 k= i j P P Xi j X P p p ) A ijk
j=l k=l
n
n F F F F P,F R R R R P,R LR
(X. X. p p p X Xj p p ) p
j= p ol P ijl
j=1
n
1 n
2
2 i=w
n
1
3 i=
n
1
3.
1+
where
R P,R SAT
S=P = T)
R
ol ol
R
X = 1
X R =0 a 1
oU
n
o
X. = v. X
1i L )i Xot
a=l
RR FF R R
p (t) = X p + (X p X p )t
oa oa oa oa
x FF R R
X p X p
F (t) op(t)
a p(t)
n
0 FF R R
p o (t) (X p XR p )
oa p oB O
t= )
2
p(t)
(418)
88
The expression for PYHS is given by equation (Al) and
LR
that for C by equations (353), (376), and (398).
In a similar fashion we transform equation (411) to
1
nya =
a
n
i V.
i=l a
F
Pi
i
 n ) (p
R
p
i
n
o
6=2
F
n
Ij=
j=1
n
1 1
a 1=1
PYHS TpF PYHS pR
i T, ) pi TT, )
(1 
RT RT
LR
1 C LR(t)
F R) cl d
ol 1o p(t)
1 CLRt) F
 ) a dt + n 
Sp(t)
o p
SF
ia j
R
 pj)
AF.
n
j=l
n
j=l
n F F
I .. (PijP
k=l la 3 k
F P,F
S. (p .p
i cc ol0
n
o
Pi = p v iaX
a=1
R R
h j ijk
R P,R LR
j ol )ijl
n
o
= v. P
l ia oa
a=1
and the other quantities have been defined above.
n
a i=
1 n
2v
+ n
2v i
a i=1
where
(419)

Full Text 
Af .
1J
ID
HS LR
f^j f^j = differences of microscopic
twobody coefficient.
pair distribution function.
1 n 2
f p i = ionic strength
i=l
K
8iTe'
DkT
I = DebyeHuckel inverse length.
k
N
N.
i
N
o
N
oa
n
n
o
P
I NT
r,
r .
i
r .
ID
S
Y
T
u .
i
V
V .
Boltzmann 1s constant.
total number of moles of all species.
total number of moles of species i.
total number of moles of all components.
total number of moles of component a.
number of different species, integer greater
than one.
number of different components,
pressure.
internal partition function,
separation between species i and j.
position vector of i.
fi 1/2
2iTe&
(oo) = DebyeHuckel limiting law
D k TJ
o
efficient.
temperature.
pair potential.
total system volume.
partial molar volume of species i.
Vll
28
Insertion of equation (253) into equation (251a)
gives
vV1/2
Y ol
4v?
n
I
n
l v v Z2z2
j=l 12 ]2 i ]
+
ol
_ CO 2
(Vo2}
v2
= (lc
22
(254)
which approaches negative infinity as the salt concentration
approaches zero.
In order to construct a model capable of correlating
and predicting the solution properties of electrolytes,
it is helpful to calculate the experimental behavior of
the DCFI's from solution properties. To that purpose,
equations (243), (244), and (245) have been inverted
so that the three DCFI's can be calculated from
1 c = 5^ [1 X V p]2 +
11 xo2pktrt 02 02
X
ol
9 lny.
V
V 2 2 9N o
ol T,P,N
ol
(255)
1 C12 v2X < RT [1 Xo2Vo2P^
4 ol T
Xo2 31ny2
N
X 9N _
ol o2
T, P,N
ol
(256)
113
contributions into a simple additive form as above. How
ever, at very low salt mole fraction the contributions from
the waterwater DCFI tend to be selfcancelling, and those
of the saltwater DCFI remain finite. But the contributions
from the saltsalt DCFI are not bounded and thus become
dominant. At high salt mole fraction, the contributions
from each of the DCFI types are approximately equal and not
separable.
Conclusions
Although the usual solution properties such as density
or salt activity coefficient have not been calculated, it
was felt that testing the model with derivative properties
such as the bulk modulus (p< RT), the salt partial molar
volume group (V^/k^RT) and the salt activity coefficient
derivative (N9Â£,ny9/3N _I ) provide an even more
^ OZ 1 f xr f JN _
Ol
sensitive test of model adequacy.
Because the parameter values used in the calculations
are not optimized, the model does not fit within experi
mental error all of the above properties for all the salt
solutions over the entire range of salt composition.
Lastly, it must be mentioned that all the model
parameters are ionically additive and valid for all solu
tions. It should, therefore, be possible to predict the
behavior of complex multicomponent systems using parameters
obtained from solutions of single salt in water.
20
n
l N. ^
. i 3N .
i=l o
= V
T, V, N
3P
3N .
D
T'v'Vj
and by insertion of the equation (21),
(226 )
n
RT l (6.. X.C..)
i=l ID i ID
= V
3P
9N .
D
T, V, N
(227)
1 3P
RT 3p .
D
n
= 1 l X. C..
T,p
i=l
i ID
(228 )
Equation (228) is the general multicomponent compres
sibility equation expressed in terms of species quantities,
This relation is now transformed to one in terms of com
ponents by performing a summation over species j and use
of equations (29) and (229).
n
o
x, = I V X_R
i ip op
(229 )
1 n
3P
RT .L. jot 3p .
D=1 D
T, p
n n
= l~l l X.v C. .
i=l j=l 1 =>a ^
(230)
1 3P
RT 3p
oa
T rP,
Y^B
n
o n
n
1 
l x a l v. 0 l v C. .
61 oSi=l lBjil ]a 13
(231)
62
pLR
2 2
S pi
Y
1/2
4v'
2 4
V + Z +
U+a++ByI1/2)2
+
+
2v^z+Z
(l+a+_Bfll/2)2
24
v Z
d+a__B^ll/2)2
] 
2 26 3a++BYI
S p v Z e Y
V [
3v t
1/2
Ei(3a++B,i1/2
(1+a B I
++ y
1/2,3
,a R Â¡1/2
3 3 Byl 1/2
2v v zfz Y E,(3a B I )
+ + 1 I y
(1+a B I1/2)3
+ y
+
 e 3a B I1/2 i ,,
v2Z^e Y E1(3a__B^I1/2)
(1+a B I
Y
1/2,3
(362)
1 
22
P OO
pol(Vo2)
v2KlRT
[v+ (AF++
P, LR,
pol*1++1 +
+ 2v+v_ (AF+_
 pol4/> + V(iF
 V/l11
(363)
,HS HS
1 r .2 /r.HS HSC
(cr c) = ~ [v, (C1 c
'2 2 2 2
+ ++ ++
+
+ 2V+V_ (Cf cf") + v2
(364)
BIOGRAPHICAL SKETCH
Heriberto Cabezas, Jr., was born Heriberto Cabezas y
Fernandez on December 8, 1952, in Esperanza, Cuba. After
serving four years in the U.S. Navy (19711975), he entered
the New Jersey Institute of Technology from which he
received a degree of Bachelor of Science in chemical
engineering in May of 1980. He started graduate studies in
the Chemical Engineering Department of the University of
Florida in September of 1980 with the intent of studying
thermodynamics. He received a degree of Master of Science
from the same university in August of 1981.
193
76
not as sensitive as is the case with other models. This
is due to the fact that the two body coefficients Af. .
ID
are fitted to infinite dilution quantities that include
Hs
Cag so there is a degree of compensation for changes in
the diameters. The sensitivity of the results to the value
of the coefficients in AC is generally small since they
make a small contribution to the DCFI's.
14
Ui(T,P) = UV(T) + RT ln XiYi(T,P_)
23
where
P = the reference chemical potential.
1 Ni
N = mole fraction of species i.
Y^ = the activity coefficient of species i,
P_ = the vector of species mole densities.
By differentiating equation (23) with respect to
the number of moles of species j, we obtain
i_
RT 3N .
3
T, V, N
SlnY^
3N
ii 1
j T, V, N
N.
l
N
k^j
which upon insertion in equation (21) gives
31ny .
i
3N .
3
1 C. .
iJ.
N
T, V, N
k^j
(24)
(25)
and when multiplied by the system volume on both sides
of the equation,
31nyi
1 C. .
u
(26)
T,p
k^j
N
Pj = y= molar density of species i.
where
96
oo p p T.D
AC = p f(vjAF^Tp rVrT) +
al Hol tet +1 Kol +11
P T R
+ v_a(AF_iPol*ir)) +
P ^
(Pnl )
+ ^ (v + v Ai> )
2 +a +11 a 11
v
a
Equation (429) is now changed to
v+a(AF+l pol$+ll) + va(AFl pol$ll}
1
P
P
ol
V
oa
RT
a 1
1 + C
HS
al
+ C
LR.
al
+
v
a
(v. + n
+a +11
+ v A$ )
a 11
(430)
Equation (430) was used to obtain values of the sum of
the parameters on the lefthand side. This was repeated for
each salt (a). To calculate values of parameters involving
one ion only, a scale was built based on lithium. Again, a
finite value was chosen to allow for geometric mean mixing
rules in later analyses.
LR
+
Lill
600 ML MOL
(431)
80
dependent variable, a function of temperature, density, and
mole fraction.
P = P (T, p X)
(42)
Taking the total differential of pressure gives
n
3P
dP = ^
3p
, 3P
dp+
T,N 3T
dT + l
3P
T o 3X
p,N a=2 oa
dX
oa
(43)
T, pX
oy^a
If the change is isothermal and if we divide by RT,
n
RT
dP
1 3P
RT 3p
dp + ^ I
T,N
3P
n 3X
a=2 oa
T, p X
dXoa (44)
oy^a
By making some identifications we obtain
3P
3X
oa
n
o V
Vk
T, V, X
oy^a
3N
I
8=1 W^T
oB
oa
N
oy^B
(45)
3N
oB
3X
oa
N
N
oy^B
6 a~voX
aB B oa
(46)
Inserting equations (45) and (46) into equation (44)
gives
dP
RT
_1_ 3P
T RT 3p
no no pV dX
dp + l l 21 2L
T,N
a=2 6=1
(47)
41
At contact and inside the core of the molecules, the
direct correlation function is dominated by a very strong
repulsion which is modelled as a hard sphere interaction.
To obtain the appropriate expressions for the hard sphere
direct correlation functions, the PercusYevick theory
(1958) was used since it has been shown to give a compres
sibility equation of state which is in good agreement with
simulation results for hard spheres (Reed and Gubbins,
1973). The PercusYevick (PY) microscopic direct correlation
function for hard spheres is zero outside the core. Thus,
HS
c .
13
PYHS
c .
13
PYHS
c .
13
HS ,, m
u.,/kT
e 11 )
HS n
Uij =
r . > a .
13 13
where
HS
u. =
13
r. < a. .
13 ~ 13
(310)
(311)
Although the PY microscopic direct correlation function
is formally used in the development that follows, it was
not actually employed in obtaining the final expressions
for the DCFI's. Rather, the expression for the hard sphere
chemical potential as derived from PercusYevick theory
through the compressibility equation was used together
with equation (21) to obtain the desired relation (see
84
3 Â£ny
a
3X
06
T, P, X
no 3Â£ny
= y L
r1 3N r
?=1 oÂ£
3N
OC
3X
oy^B
T, P, N
oy^C
oB
(413)
N
oy^?
3N
oC
3X
oB
N
N
oy^C
hrVo,
(414 )
Inserting equations (414) and (238) into (413) and
then putting the resulting expression into equation (412)
gives
dÂ£ny
no no PK RT
= I dX J
T P
B=2 wÂ£=l 6CB Vo?XoB y=l 6=1
n n
o o
I I V
L L y 6
XOYXoi! UCa^) UCoy)(1C4?)] (415)
To obtain the activity coefficient, equation (415) is
isothermally and isobarically integrated from the reference
to the desired state.
n n n n X a
o o o o OP
Â£nYa = l l l I / R dXoB
a a=2 6=1 y=l ?=1 X
oB
p < RT
T
T, P
?B Vo?XoB
VrV
? y o
x x r [ (ic
oy o o y<5
(1Cay)(1'
(416)
150
The direct correlation function integral for hard
sphere species i and j can be obtained from equations (A 3)
and (A 4) by means of equation (21).
_1_
RT
T, P, N.
k^i
C. .
N
(21)
Thus, insertion of the PercusYevick compressibility
expression for the chemical potential into equation (21)
gives
CHSPY
ij
(a.+a )
i 1_
1,
3^ (a.a .)
i J
(a.+a .)
i 1
(l53)
+
3a1a.C2[(ai+a.)2+aia.l + (a^.)3^
(1SJ2
9(a.a e )3
J Â£
(ls3 )4
(A5)
Finally, the corresponding expression from the Carnahan
Starling equation is
CHSCS
i i
CHSPY
^2(aiaj}
(l53)3
6 + (953 15)
(a+a.) C2 (6+C3 [12C315])
118
X 02
Figure 10. Water (1) Activity Coefficient Derivative
in Aqueous NaCL (2) at 25, 1 ATM. For
data sources see Table 425.
58
H S
The expression for C is
a B
,HS
 i i cHS
'aB Viiiw i6
(354
Lastly, the expression for AC is
aB
AC
P
n n
l v. v AF. +
B Vs i1 jl 'ia jB "'ij
n n n
P., 1 Â£ Â£ v.v. (p,A4>.., p4>LR )
aVB i=l j=l kl lct ^ k ^k k ^k
(355)
Equations (353), (354), and (355) form the complete
model for the saltsalt DCFI.
C = CESr + CLR + AC
a B aB a 8 a 8
(356)
Since the limits of DCFI1s as salt concentration
approaches zero are well defined, it is advantageous to
use equations (353) to (355) to model the deviations
from this limit. To this purpose, the infinite dilution
limit of the saltsalt DCFI is now explored. From equation
(24 2a)
N 3 in y
ol 1
CO CO
OL
3N
oB
T,P,N
= (lc R)
a B
oy^B
p ?V V
ol oa oB
v v k RT
a B 1
CO
(242a)
70
W
LR
Vl a1"311'111
" kTrll D
ip r > a
11
(38 9a)
"Â£?
r X all
(38 9b'
Again, equations (388) and (389) are inserted into
equation (320) and the required integration over angles
performed.
. LR. L r JjK ,
= o I u, i da) do).
11 u p2 11 11 12
1 r LR
(390)
where
dw. = sin 0. d0. d4> .
li li lx li
2ir
Â¡2 = { d<5 = J sin 0. d0. J d(f>. = 4r
J _Ll J ll ll J ll
The integration of equation (390) gives
. LR. n
=0
11 0)
(391)
The second term in equation (320) has
y iT7LRi 2. 1 r,T7LR\2 .
<(W11) >0) = 72 / (wn } dwn dw
n
11' 11 12
(392)
which yields upon evaluation,
<(WLR)2> = 1
M ll' w 3
y1y1 2
,2K(airrn
DkT
11
(393)
27
activity coefficient of an electrolyte must approach the
DebyeHuckel Limiting Law at very low salt concentration.
Thus, the mean activity coefficient of a salt on the mole
fraction scale is given by this law as
lnY2
1 n
L I
'2 i=l
z2) i1/2
(252)
where
S =
Y
97r 6 1/2
2ie ^
3 3 3 J ~
D kJT
DebyeHuckel limiting law coefficient.
e = the electronic charge.
= pure solvent dielectric constant.
k = Boltzmann's constant.
T = temperature.
Z. = valence of ion i.
x
1 V 72 .
I =2 L Zipi = lonic strength.
i=l
and when the proper derivative is taken,
N 81ny?
? t,p,noi
ol
3N
o2
P 1/2
S p I
_ Y ol
4V?
n n
I I
i=l j=l
2 2
v._v Z.Z .
i2 j2 i j
(253)
92
PK ^RT
= 1 C
HS
11
 C
LR
11
 AC
11
(423)
where
,HSC
'11
LIM
Xol"1
PYHS , ,
C^ = PY hard sphere
ol x
DCFI given by equation (A5) at the
nure water limit.
4tt
9a
P1M1
DkT
2
)
AC
11
P.. P. LR.
"pol(AFll"p0l*lll)
Because we expect that the contribution of A4>^^^ will
be small and can be absorbed by the other parameters, we set
it equal to zero.
A4>
111
= 0
(424)
AF
11
PA LR
Pol$lll
ol
PKTRT
 1
t CHS t
t
plpl
(
DkT
2
) )
(425)
49
HS
the extension of the expressin for is well known.
Second, the extension of equation (320) for requires
potentials of mean force applicable to the system. Assuming
all solvents are dipolar requires only knowing the dipole
moment of each of the solvent molecules and the dielectric
constant of the solvent mixture. Neither of these are
expected to present a problem in general. Third, the exten
sion of equation (321) for involves a few more coeffi
cients and slightly different mixing rules for some three
body terms. Thus, while equations (322) to (327) would
remain the same for all solvents, equations (328) and
(329) where i is an ion and j, k solvents would be altered
to
A$(T)
ijk
^ ( A$. . +
2 133
AW
(333)
ijk
 ( <3?LR 4 <>LR )
2 ij j ikk'
(334)
which reduce to the previous result only when j and k are
equal. Here, any nonadditive interaction between j and
k has been tacitly ignored because the difference in the
interactions between different solvents is likely to be
less important to direct correlation function integrals
than that from the much stronger ionsolvent interactions.
This assumption is based on previous investigation of
9
(Molality) 2
Figure 1. Salt Activity Coefficient in Water at
25C, 1 ATM. Data of Hamer and Wu
(1972 ) .
Ill
the fit of the model to the salt activity coefficient
derivative was qualitatively good (1.614%) but well outside
experimental error (11.5%) in many cases.
To understand the above observations it is necessary to
consider the sensitivity of each type of DCFI to its
parameters and also the sensitivity of each group of prop
erties to each type of DCFI.
The saltsalt DCFI is extremely sensitive to the value
of the species hard sphere diameters at both low and high
salt mole fraction since the dominant contributions in both
cases are strong functions of species size.
The saltwater DCFI is most sensitive to the value of
the species hard sphere diameters at high salt mole frac
tion. At low salt mole fraction, the saltwater DCFI is
relatively insensitive to the species sizes since the
infinite dilution limit is dominant.
The waterwater DCFI is relatively insensitive to the
ionic hard sphere diameters but highly sensitive to the
water diameter at high salt mole fraction. Again, it is
insensitive to species size at low salt mole fraction
because its infinite dilution limit is well defined and
dominant.
The model fitted the experimental values of the water
water DCFI with the species diameters adopted from Marcus
(1983). However, the model fitted the saltwater DCFI less
well and the saltsalt DCFI somewhat worse. This may
5
Boston, and Evans, 1979) in nonprimitive models of electro
lyte solutions. Cruz and Renon separate the Gibbs energy
into three additive terms: an elecrostatic term from the
DebyeHuckel theory, a DebyeMcAulay contribution to correct
for the change in solvent dielectric constant due to the
ions, and an NRTL term for all the short range intermolecular
forces. Chen et al. adopted a DebyeHuckel contribution
and an NRTL term for the Gibbs energy but no DebyeMcAulay
term. More recently, the UNIQUAC model for nonelectrolytes
has been modified for short range intermolecular forces
in electrolyte solutions (Sander, Fredenslund, and Rasmussen,
1984). The resulting UNIQUAC expression has been added
to an empirically modified PitzerDebyeHuckel type electro
static term to form the complete Gibbs energy model.
Although the two NRTL and the UNIQUAC models correlate
activity coefficient data reasonably well even in multi
solvent systems, they have to be regarded as mainly
empirical. First, their resolution of the Gibbs energy
into additive contributions from each different kind of
interaction is not rigorous. Second, the problems associated
with the mixture dielectric constant are resolved in an
empirical and somewhat arbitrary fashion. As a result,
such models add little to our understanding of these systems
and may not be reliable for extension and extrapolation.
Of the various empirical methods developed, two have
been chosen to be discussed here because they represent
64
where a is a universal constant that we have set equal
to 4.4 empirically.
Since equations (365) and (366) are functions of
orientation, it is necessary to first perform the integration
over angles as indicated in equation (320).
(367)
where dw. = sin 0.d0.dd>.
i ill
When the integral in equation (367) is evaluated, it is
found that
(368)
The second term in equation (320) has
(369)
After the integral in equation (369) is evaluated, it
gives
(
DkT
(370)
TABLE 422
SALT
SALT DCFI
FOR KBr (2) IN
WATER (1) AT 25C, 1 ATM
X
o2
^1C22 *
rLR
22
(CHScHS ) 
v 22 22 1
TB TB
(ac22ac22)
(ic22)CALC*
EXP.
^22'
11
0
1
M
NJ
18.4325
2.1907x10
6 0.0000
0.0000
2.1907xl06
2.1907X106
1.5247x10
3
18.4333 
32.4553
0.2504
0.0013
13.768
12.805
3.0705x10
3
18.4337 
18.8995
0.4989
0.0011
0.0342
0.2969
9.4793x10
3
18.4350
7.0852
1.5106
0.0075
12.852
12.436
0.019825
18.4344
3.4346
3.1037
0.0413
18.062
17.655
0.035183
18.4300
1.8700
5.4463
0.1197
21.886
21.663
0.048075
18.4230
1.3214
7.4168
0.2042
24.314
24.123
0.070088
18.4086
0.8565
10.7711
0.3850
27.938
27.712
0.083974
18.3970
0.6919
12.9163
0.5141
30.107
30.107
(IC22 from equation (362).
EXP
(1C~) from equation (257) using the same sources of experimental data as
for Figure 3.
140
93
Equation (428) was used to calculate the solvent
P LR
parameters (AF.^ p ^$^^). We have left the parameter
LR
4> together with AF^ because we expect its contribution
to be sufficiently small.
The three body parameters involving two waters and one
ion are calculated from equations (3104) at finite salt
P T D
concentration once AF.. p ., is known.
11 ol 111
1 C
11
olKlRT
,rHS_rHS>
^11 C11 1
 (AC AC )
' 11 11;
(C
LR_rLR
11 U11
(3104)
where
HS PYHS
C, = Cnn = PY hard sphere DCFI given by
'11
11
equation (A5).
1/2
2ailB 1
LR = 4Â£Â£ Ulyl 2 6 Y
11 3 ^ DkT 3
E4(2auV
1/2
11
Acn= p(AFn p0I*m) +
+ pp (v, A , + v A $ )
HKo +a +11 a 11
Equation (3104) can be rearranged to give
43
Since equation (312) represents the entire direct
LR HS
correlation function including c. and c.these two must
13 13
be subtracted to obtain the interpolating function. There
fore, the complete model for the microscopic direct correla
tion function for species i and j in a system of n species
is
HS LR L A
c. = c. + c. + Ac. .
13 1: id
13
(313)
HS LR
where c^j is defined by equation (311), c^ by equation
( 38 ) and
A HS LR
Ac. = c. c. c. .
13 13 13 13
(314)
which is approximated by the RushbrookeScoins expansion
as
Â£HS rLR. r ,
13 id ID 13 k yi3k
k=l
aHS
pk ^ijk
o LR .
pk *ijk>
(315
where
o LR
Pk ^ijk
LIM
IT >00
i j
I o
pk ^ijk
The series in equation (312) is truncated at the
first order term in density to be consistent with the HNC
Two Body Parameters
(1 WATER)
186
(ML/MOL)
1
1
75.0237
Li +
1
600.0000
Na +
1
731.1595
K+
1
629.8387
CL_
1
422.1072
br"
1
287.6151
Li +
Li +
300.0000
Na+
Na +
149.1514
K+
K+
46.1979
CL
cl"
64.4814
BR~
br"
112.7558
Three Body
Parameters
(1 
WATER)
i
j
k
A<5 .
13k
(LML/MOL)
1
1
1
0
Li +
1
1
 4.0000
Na+
1
1
 4.9306
K+
1
1
 3.2584
CL~
1
1
 5.8246
br"
1
1
 3.6430
Li +
Li +
1
 3.0000
4
of the modern "theoretical" models is widely used in
engineering practice.
The most successful of the semiempirical models is
that due to Pitzer and coworkers (Pitzer, 1973; Pitzer
and Mayorga, 1973; Pitzer and Mayorga, 1974; Pitzer, 1974;
Pitzer and Silvester, 1976). Model parameters for activity
coefficients have been evaluated for a large number of
aqueous salt solutions, but volumetric properties and multi
solvent systems have not been treated. To construct the
model, Pitzer adopted the "Primitive Model" and inserted
the DebyeHuckel radial distribution function for ions
into the osmotic virial expansion from the McMillanMayer
formalism. This latter is analogous to using the "Pressure
Equation" of statistical mechanics (Pitzer, 1977). The
resulting expression contains the correct limiting law.
He then added empirical second and third virial coefficients
which are salt and solvent specific. Although Pitzer's
model correlates aqueous activity coefficients superbly,
it does not add to the fundamental understanding of these
solutions; further, its extension to multisolvent systems
would pose some serious problems associated with the mixture
dielectric constant as has been recently pointed out (Sander,
Fredenslund, and Rasumussen, 1984). Another semiempirical
approach uses the NRTL model for solutions of nonelectrolytes
(Renon and Prausnitz, 1968) adapted for short range ion
and solvent interactions (Cruz and Renon, 1978; Chen, Britt,
TABLE 423
SALT
WATER DCFI
FOR KBr
(2) IN WATER (1)
AT 25C, 1 ATM
!N
0
X
oo
uci2)
PLR
U12
. HS HS
'''12 12
^ (AC12AC12)
(ic12)CALC*
(ic )EXP*
C12;
1012
15.0354
0.0000
0.0000
0.0000
15.035
15.035
1.5247xl03
15.0354
0.4241
0.1578
0.0202
15.637
15.615
3.0705xl03
15.0354
0.4835
0.3143
0.0408
15.874
15.917
9.4 7 9 3xl0~3
15.0354
0.5768
0.9510
0.1274
17.529
16.899
0.019825
15.0354
0.6330
1.9524
0.2594
17.880
18.224
0.035183
15.0354
0.6727
3.4221
0.4829
19.613
19.922
0.048075
15.0354
0.6927
4.6562
0.6630
21.047
21.192
0.070088
15.0354
0.7151
6.7524
0.9712
23.474
23.266
0.083974
15.0354
0.7252
8.0904
1.1645
25.015
24.710
(1C^2)^ALC* from equation (384).
EXP
(12) from equation (256) using the same sources of experimental data as
for Figure 4.
141
8
functions. Chapter 3 has the full development of the new
model. Chapter 4 shows the application of the model to
solutions of aqueous strong electrolytes and the calculation
of solution properties. Chapter 5 has suggestions for further
work and conclusions.
151
a.a.f 2(6 + r [r (2614C )21] )
+ 1 3 2 3 2 ] +
6C2(aiai)2ta(lC3)(C3[ai+a.]C2+aia.22Â¡1
(A6 )
By comparing equations (A 5) and (A6) one can appreci
ate the fact that the expression or the hard sphere direct
correlation function integral from the CarnahanStarling
equtaion of state is significantly more complex than that
from PercusYevick compressibility equation of state.
APPENDIX C
RELATION OF DENSITY EXPANSION OF THE DIRECT CORRELATION
FUNCTION TO VIRIAL EQUATION OF STATE:
ALTERNATE MIXING RULES
The RushbrookeScoins density expansion of the direct
correlation function (Reichl, 1980; Croxton, 1975) for
species i and
j in a system of n species is given by equa
tion (312 ) .
cij(T,p)
n
= f (T) + I p,
ij k=l K
where f. (T) =
13
u. ./kT
in
 e J 1 = Mayer bond function
hjk'1
= S > fijfikfjkdÂ£kd"k
3
t
II
Integration of equation (312) over molecular orienta
tion gives
=
i j oo
J c. dw.dw. (Cl)
a2 13 1 J
=
1 j CO
n
+ l p <(p. > + ... (C2)
13 w k1 k 13k CO
172
179
were treated as parameters to be fitted to experimental
data. In fact, the virial series has been used here only in
a very formal fashion as in the osmotic virial expansion.
TABLE 47
SALT
WATER DCFI
FOR LiBr
(2) IN WATER (1)
AT 25C, 1 ATM
Xo2
(ici2)
pLR
U12
, HS HS
'^12 12
) 
Me \ CALC
ir '12
EXP.
U ^12
to"12
10.6224
0.0000
0.0000
0.0000
10.622
10.622
2.0866x10
3
10.6224
0.4893
0.1800
0.0955
11.196
11.174
4.1983x10
3
10.6224
0.5573
0.3611
0.1925
11.348
11.412
8.4962x10
3
10.6224
0.6250
0.7295
0.3482
11.628
11.790
0.017411
10.6224
0.6904
1.4986
0.8102
12.001
12.430
0.036619
10.6224
0.7520
3.2146
1.7381
12.850
13.556
0.057919
10.6224
0.7861
5.2701
2.8078
13.870
14.535
0.075482
10.6224
0.8042
7.1184
3.7221
14.822
15.105
0.10833
10.6224
0.8270
11.1497
5.5200
17.079
15.374
(IC12 ) from equation (384).
EXP
(1Cj2) from equation (256) using the same sources of experimental data as
for Figure 4.
24
Nv 91ny
a a
P
T, P, N
n n
o o
= V v l l V V X X .
8 6Â£l y 6 oy o5
Yt8
[(1CY6)(1Ca6) (1CaY)(1C66)]
(238)
1 
N N
l l
i=l j=l
v v (1C. )
ia 16 ii
v v
a 3
(211)
Useful bounds on the value of the direct correlation
function integrals as the system approaches infinite
dilution in all components except one (usually the solvent)
can be deduced from the preceding relations. Thus, by
taking the limit of pure solvent (component 1), one obtains
V
91 = (lc )
v kRT v al
a 1
(240)
polKlRT
UCll)
241
N 31ny
ol ^_a
*8 3NoB
= (1C
a8
T, P, N
PCO CO
polVo2VoB
VaVBKlRT
242a)
VB
N 31ny
ol a
VB 3NoB
T,P,N
= (1C .)
a 3
(1C ) (1Cfl1
al 31
CO
dCu
(24 2b)
include hydration and hydrogen bonding are modelled with a
density expansion of the direct correlation function. The
model requires six parameters for each ion and two for
water. The ionic parameters are valid for all solutions
and those for water are universal.
The model has been used to calculate derivative prop
erties for six 1:1 electrolytes in water at 25C, 1 ATM. The
calculated properties have been compared to experimentally
determined values in order to confirm the adequacy of the
model.
Xll
density of species i.
oa
$. .. =
xjk
A$ =
i]k
^ijk =
A*ijk =
Q
OJ ,
i
0)
Y
F
FLL
HNC
HS
KB
LR
MM
P
PY
N.
l
V
N
oa
V
= density of component a.
spatial integral of
spatial integral of
microscopic threebody coefficient.
HS
^ijk "" ^ijk = difference of microscopic three
body coefficients.
orientation dependence of dipoledipole
interaction.
/ d
angular orientation coordinates of i.
1 n
1 Z v. z2.
2 i=l ^ 1
Superscripts
Final.
Friedman's limiting law.
hypernetted chain,
hard sphere.
KirkwoodBuff.
long range or field type correlations or
interactions, LewisRandall.
McMillanMayer.
Pure component.
PercusYevick.
IX
56
The contribution of Friedman's limiting law to the
activity coefficient of a salt (a) is
FIjL
Â£n y =
a
l n
( I v.
1 va i=l ia
Zi)2
V 2
y v. z.
i=i ia 1
sy I&nl
(347)
and by taking the first derivative with respect to the
mole number of a salt 3 at I + o,
3 In y.
FLL
3N
o3
T,P,N a
OY / 3
_,2 P
S p n n o
X_2i Y Y v. v Z3z .
3Vb il jil ia 1 ]
(4 An I +
(348)
Rearrangement of equation (224) gives
N
3 Any
a
3N
o3
= 1C
PV v _
oa o3
a3 VQVp
T, P, N
oy 3
(349)
If equations (348) and (349) are compared, it is
clear that the contribution of Friedman's form of the limit
ing law to the saltsalt direct correlation function integral
is
154
Equation (B 4) relates the change in the chemical
potential of a component a in a KB system to the respective
change in an LR system when the mole number of a component 6
is changed. The quantities on the righthand side of equa
tion (B4) are all LR properties.
Next we consider a system consisting of one solvent and
several solutes. To this system we apply the McMillan
Mayer theory (MM) which is defined for a system whose
boundary is permeable to the solvent but not to any solute.
Further, the system temperature and the solvent chemical
potential are constant while the mole numbers of solute
components may vary. Conversions of thermodynamic quanti
ties from MM to LR have been considered in the literature
(Pailthorpe, Mitchell, and Ninham, 1984; Friedman, 1972).
We start our analysis by considering the total differ
ential of the LR chemical potential of the solvent (1).
d
LR
^1
3T
dT
P,N
o
9y
LR
+
3 P
dP
T,N
o
+
+
n
o
I
6=1
3y
LR
3N
oB
d N.
T,P,N
OYt^B
(B5 )
In accordance with the MM theory, we prescribe that y
LR
1
be constant at constant temperature.
19
Additionally, the partial molar volume of a component a
or 8 is expressed as a sum of the species partial molar
volumes.
 9V
v =
Vi 9N.
9V
T,P,N.
k^i
Vi 8
9N
08
T, P, N
Yt*3
(222
V
9V
08 9N
08
n
= l v. V.
18 i
T,P,N
i=l
Yt^B
Equation (220) is now transformed to
(223)
Nv 91ny
a 'a
pK RT 9N _
T 08
T, P, N
v v (1C .
a 8 a8
Vb
V
oa
V
08
p
k RT k RT
T T
(224
To make further progress, the relationship of the
7
oc
bulk modulus of the solution (pk RT) and the group,
to the direct correlation function integrals must be found.
First, the compressibility equation is derived from the
basic fluctuation theory result of equation (21) starting
with the GibbsDuhem equation for an isothermal but
nonisobaric process.
n
I
i=l
N.dU. = VdP
i l
(225)
Upon differentiation of equation (225) with respect
to the mole number of an arbitrary component j,
26
1 C
22
v+2 (^M1 +2v+2v2 < ,+v2 < !C
V.
(248)
and the respective infinite dilution limits are
oo
Vo2
v2
dc21r
1
p f k,RT
ol 1
oo
* (1cn
Kol 31nY2
CO
V2 3No2
T,p,Noi
(ic r pf
v2 Kirt
Nol 31nY2
v2 3No2
T,P,Noi
2
(1C
, 2
(1c22) 
12
(1C11)
(249)
(250)
(251a)
(251b)
The significance of equations (251) can be further
understood by realizing that any correct model for the
57
rFLL=
aB
P
ol
n n
I I
i=l j=l
V V
la
Z373(
j3 iZj(2
Â£n
1 +
(350)
Comparing equations (211) and (350) gives the ionion
DCFI.
3 3
rr Z Z
CFLL= x., 3
11 3
*nl + i)
(351)
Substitution of equation (346) in equation (340)
gives the expression for the limiting contribution of the
third term in equation (340) to the ionion DCFI.
Z3z3
i 1
Y
ol
(I
Â£nl +
a
+ In 3a
il
B )
Y
(352)
Equations (351) and (352) have essentially the same
behavior as I * 0 since they differ only by a small constant
which is negligible compared to &nl as I + 0. Therefore,
equation (344) contains the higher order limiting law.
The general expression for CFF is
,LR
'aB
S pi 1//2 n
Y
2 2
n v. v Z7Z .
Â£ Â£ la IB 1 1
4vaV B i = l j=l (1+a. .B l1/2)2
il Y
n
_ 3a. .B I
n v. v Z3Z3 e ^l Y
1/2
3vaVB i=l j=l
l l la IB i j
(1 + a..B I1/2)J
il Y
E1(3a. .B I1/2)
1 ,1J. Y
+
(353)
15
By performing a sum over all species i and j on equation
(26)
. n n 3lny.
1 I I V. v.
. , ia _
v v N S 'iot' j6 3p .
a B i=l j = l J
T,p
, n n 1C
i l l V. V .
VB i=l j1 101 33 p
(27 )
where
= number of species i in component a.
= total number of species in component a.
By noting the definition of the mean activity
coefficient of a component a,
1 n
lny = J v lny.
ot v ia i
a i = l
(28
and also assuming that species j is formed from an arbitrary
component 8 so that
pj = Vj 3 Po B
29 )
one then arrives at the first relation
, 3lny
1 a
vb 3poe
T, p,
y^3
i n n 1C. .
y v v. v. ii
P L1 , rajB v v
i=l j=l a B
(210)
82
PP
SAT
1
RT
n
o
I
a=l
n
o
I
3=1
a 8
PF(T,P,X^)
SAT XoaXoB(1_Ca8)
pol (T)
+
n n n X
o o o oa
+ I l I v v / X (1C
a=2 6=1 Yl 8 r o OY
pdX
oa
8y' 6 v X
a 8 8 oa
(49)
In evaluating the integrals of equation (49) each
integral involves variables appearing in other integrals.
F
To explicitly find p requires further manipulations
discussed below.
The activity coefficient on the mole fraction scale for
any component (a) can be obtained by integration of equation
(312) from the reference molar density (Pg ) to the molar
F
density of each component (Pg ) at constant temperature.
Jin
V V fP6 j
Y R 61 6 0 R P dP6
Ya B_1 po8
T
(410)
where P a = X flP
OP OP
Equation (410) is applicable to any isothermal change,
isobaric or not, and the reference state composition where
Ya =1 need not be that chosen. However, for aqueous
electrolytes it is natural to choose pure saturated liquid
water at the system temperature as was done for equation
(49).
100
where
(1C ) =
P oo oo
p :v v
ol oa oa P
aa
v k RT
a 1
v
2 p T P
v +v v ) ( af p : $ f) +
+ a + a a ++ Mol ++1'
P TP
+ (v + v V ) (AF p J ,))
a +a a ol 1
LR
Caa = long range DCFI given by
HS HSC
aa Laa
equation (3 63b)
HS
difference of C given by
aa 3 2
equation (364)
actb actbC
aa aa
P P
pp p ,p 0
ol ol ol / / 2 ...
9 V +
2 +a +a a ++1
v
a
(v +v v ) A4> ) + pp (v +
a +a a 1 oa +a +++
+ v A4> )
a
The above expression has been simplified by use of the
mixing rules given by equations (322), (323), (324), and
(326) in order to eliminate any parameters involving more
than one kind of ion.
We next rearrange equation (362) to give
Species Density (Mpl/ml)
Figure 2. Species Density in Aqueous Electrolytes
at 25C, 1 ATM. For data sources see
text.
TABLE 424
WATERWATER DCFI FOR KBr (2) IN WATER (1) AT 25C, 1 ATM
Xo2
oo
(1_cll)
PLR
C11
(cHScHS )
leu )
(AC11AC11)
(1_C11)CALC
EXP.
U ^11;
10"12
16.1084
0.0000
0.0000
0.0000
16.108
16.108
1.5247x10
3
16.1084
0.0005
0.0992
0.0316
16.239
16.253
3.0705x10
3
16.1084
0.0006
0.1975
0.0638
16.370
16.397
9.4793x10
3
16.1084
0.0009
0.5972
0.1983
16.904
16.973
0.019825
16.1084
0.0011
1.2250
0.4177
17.752
17.858
0.035183
16.1084
0.0013
2.1448
0.7453
18.999
19.094
0.048075
16.1084
0.0014
2.9159
1.0205
20.046
20.081
0.07008
16.1084
0.0015
4.2229
1.4882
21.821
21.683
0.083974
16.1084
0.0015
5.0555
1.7802
22.945
22.596
(1C^)CALC* from equation (3104).
EXP
(lC^) from equation (255) using the same sources of experimental data
as for Figure 5.
142
67
deviations from infinite dilution. (For nonisobaric calcu
lations, equation (379) would be more appropriate.) The
infinite dilution limit of is given by equation (240)
LR
and that of Ca^ is (see Appendix D)
LR LIM LR 2^Pol ea 2 2 ? ViaZi
al X ,+l al 3v DkT Ml.. a..
ol a i = l ll
(380)
while the infinite dilution limit of C is formally
al 1
HS00 LIM HS
Lai X .>1 Sal
ol
(381)
Lastly, the infinite dilution limit of AC is
1 al
a LIM _
AC =v AC .
al X .>1 al
ol
> n
Ol r
) .
a i=l
v a a (AF,
il p ol
P LR ,
ni
+
P
p n D
ol r P.
1. via polA<5ill
v ^, la
a i=l
(382)
The complete general relation for the saltsolvent
DCFI including the infinite dilution limit is
V
oa
1 v k,RT
a 1
(cHS cHS~)
' 'a l 'a l '
(cLR cLR)
lLal ua 1 '
(AC AC .
a 1 a 1
(383)
85
n n
1 o o
where = = Y Y v v x X (1C )
pKTRT aÂ£1 a goa oB ag;
6 = Kroniker Delta
Equation (416) can only be used for isothermal,
isobaric changes and thus either the DCFI model of equations
(359), (383), and (3104) or that of equations (356),
(379), and (3100) may be used.
Equations (41), (49), (411), and (416) express
integrations of the DCFI model formally. However, these
cannot be explicitly evaluated because of the multiple
variables involved in the integrals. To actually evaluate
these integrals requires a change of variables as discussed
by Mathias (1979) and O'Connell (1981). Rather than give
their formal equations, we now give the above relations with
explicit expressions for the present DCFI model. Those
parts that are analytically integrable have been evaluated
while simplified integrals are given for the others. The
DCFI model used is that of equations (356), (379), and
(3100) which does not contain the DCFI infinite dilution
limits. This form of the model yields simpler expressions
which can be applied to both isobaric and nonisobaric
changes. We start by rewriting equation (41) as
36
In summary, there are three distinct classes of inter
action: ionion, ionsolvent, solventsolvent. Each class
has unique contributions from longrange, fieldtype forces,
shortrange, repulsive forces, and intermediate range forces.
Traditionally, models have been written for the excess
Gibbs or Helmholtz energy of the system by adding contribu
tions from some of the above forces in an ad hoc and, gener
ally, nonrigorous fashion. The fact that free energy
contributions do not naturally separate into the types
of forces and that experimental values for each cannot
be separately determined has caused many of these models
to be complex and/or inconsistent. Further, they do not
yield volumetric properties along with the activities.
Within the framework of Fluctuation Solution Theory,
the contributions of the pair correlations to the thermo
dynamic properties can be rigorously added. Thus, there
are terms from the saltsalt, saltsolvent, and solvent
solvent DCFI's, as shown in Chapter 2. Further, the experi
mental behavior of each of the three DCFI types can be
separately calculated from solution data as seen in the
previous chapter. It is then possible to construct separate
and accurate models for each one of the DCFI's. These
models can later be manipulated to yield thermodynamic
properties.
As may be inferred from the above discussion, each
of the three types of DCFI's contains long range, short
TABLE 48
WATERWATER
DCFI FOR LiBr
(2) IN WATER
(1) AT 25 C, 1
ATM
Xo2
rLR
U11
(CHScHS )
lUll C11 1
(AC11AC11)
1 i p \ CALC.
U cll'
EXP.
U Cll'
io"12
16.1084
0.0000
0.0000
0.0000
16.108
16.108
2.0866x10
3
16.1084
0.0005
0.1445
0.0479
16.301
16.303
4.1983x10
3
16.1084
0.0007
0.2897
0.0967
16.495
16.498
8.4962x10
3
16.1084
0.0009
0.5852
0.1974
16.891
16.895
0.017411
16.1084
0.0011
1.2016
0.4105
17.721
17.724
0.036619
16.1084
0.0013
2.5749
0.8880
19.572
19.562
0.057919
16.1084
0.0014
4.2163
1.4435
21.769
21.704
0.075482
16.1084
0.0015
5.6893
1.9216
23.720
23.579
0.10833
16.1084
0.0015
8.8935
2.8657
27.869
27.509
(1C^ ) from equation (3104).
EXP
(1C^) from equation (255) using the same sources of experimental data
as for Figure 5.
126
TABLE 43
SALT
WATER DCFI
FOR LiCL
(2 ) IN WATER (1)
AT 25C, 1 ATM
Xo2
OC
(1_C12)
LR
12
, HS HS
1 12 12
> <4C12AC12>
(1_C12)CALC*
(1C )EXP'
lo"12
7.5556
0.0000
0.0000
0.0000
7.5556
7.5556
4.2558x10
3
7.5556
0.5636
0.3709
0.1621
8.328
8.336
8.5247x10
3
7.5556
0.6308
0.7413
0.3276
8.600
8.713
0.025729
7.5556
0.7298
2.2724
1.0204
9.537
10.021
0.043142
7.556
0.7706
3.9125
1.7628
10.4759
11.208
0.069664
7.556
0.8046
6.6600
2.9729
12.047
12.632
0.087620
7.556
0.8194
8.7446
3.8480
13.271
13.287
0.10580
7.556
0.8309
11.1003
4.7826
14.7042
13.786
(1C^2)CALC" from equation (384).
EXP
(l~Cj2) from equation (256) using the same sources of experimental data as
for Figure 4.
which is the general expression for the solventsolvent
DCFI.
72
Since the solvent does not dissociate, there is no
HS
summation over species in C.^. However, Ac.^ does have
a sum over third bodies.
ACn = pAFn + p l (pkA*llk P *k) (399
k=l
Equations (398), (399), and (3100) form the complete
general model for the solventsolvent DCFI.
C = CHS + c^R + /\c
11 11 '11 A 11
(3100)
Again, the infinite dilution limit of C^ is introduced
so that for isobaric calculations the model need only account
for deviations from the infinite dilution value. Also,
equation (3100) would be more practical for nonisobaric
cases. The infinite dilution limit of is the bulk
modulus of the pure solvent given by equation (241).
TR
The infinite dilution limit of C.^ is given by
LR LIM LR 471(3
11 X ,*1 11
ol 9 a
L t _L)
3 vDkT '
11
(3101)
HS
and that for C"^ is formally
,HS LIM HS
11 X +1 11
ol
(3102)
177
B. = (B. B . )1/'2 (C16)
11 ll 11
LR
B.. and B.. would be obtained from binary solution
1X11
data.
LR
Since and are three body quantities, the use
of mixing rules is important in all cases. Using equation
(C8) in equations (C10) and (Cll) gives
(T)
ijk
> (T)
ijk
3(_ Â£ \
J i jk ijk
 3CLRv
ilk
(C17)
(C18)
A rigorous expression for the hard sphere third virial
coefficient exists. But, a much simpler approximate
expression was considered adequate.
where
= (cHS.
in
rHS = 1
iii 8
cHS. c
111
HS
kkk
(
3
2ttn a .
a i
3
1/3
2
)
(C19)
The structure of equation (C8) suggests that and
LR
could be expressed empirically as a product of two body
quantities.
C. = C. C., C
13k 13 lk 3k
(C2 0)
161
Equations (B22) and (B23) contain no MM quantities
which indicates that the MM theory does not make any net
contributions to solventsolvent or to solventsolute direct
correlation function integrals (DCFI's). It would, there
fore, be theoretically incorrect to use MM quantities such
as chemical potentials from the DebyeHuckel theory in the
derivation of any expression for solventsolvent or solvent
solute DCFI's.
Equation (B24), however, does contain an MM quantity
MM
(y ) so that MM theory does make a contribution to solute
solute DCFI's. Further, as the concentration of all solutes
approaches zero (or Xq1 * 1), one would expect that
LIM
X + 1
ol
TT = 0
LIM P+7T 9VLR
a
Xol + 1 R
9N
o3
d P =0
T,P,N
oy^B
LIM P+^ 9VRR
X > 1 P
o
9N
oB
d P = 0
T, P, N
oy^B
(B25)
(B26)
(B27)
Also, the observed experimental behavior of the solvent
chemical potential as embodied in models for both
117
X 02
Figure 9. Contributions to the Water (l)Water (1)
DCFI in Aqueous NaCL at 25C, 1 ATM.
For data sources see Table 412.
167
1C = (1C ) cLR (cHS CHSo)
aB aB aB 3 aB
mn mpa)
 (Ac AcJ )
aB aB
(359)
As the zero ionic strength limit is approached, (1Cag)
LR
goes to a constant, C diverges, and the other terms
approach zero and we will, therefore, concentrate on the
first two terms.
The general expression for the saltsalt long range
DCFI is given by equation (353).
,LR
a B
V1
4Vb
1/2
2 2
n n v v Z. z .
V y J6. i 1
l L 1/2 2
i=l j=l (1+a. .B Ix/)
J U Y
S 2p
_Y
3v v
JVa B i=l j=l
. 3a. .B l1/2 ,
n n v. v. z.3z.3e 13 YE.(3a. ,B I1/2
l l 1.0t 3B 1 3 _A_ JJ Y
(1+a. B I
il Y
1/2
(353)
To obtain the limit as the ionic strength approaches
zero, we use equations (B38), (B39), and (B41) on equa
tion (353) and expand in ionic strength.
22
and when applied to equation (234)
3P/RT
9p
T,N
n n
&
(237)
which is the second necessary relation.
Substitution of equation (234) and equation (237)
into equation (224) gives
Nv 31ny
a a
PKTRT 3NoB
T,P,N
o
Yt^B
v v
a
B
n
o
l
Y=1
n
o
l
6=1
v v X
Y 6 oy o
6
[ (1C
Y
uc fl)
a B
(1C,
ay
)(1"C6B
(238)
) ]
which is further transformed by substituting for the bulk
modulus
Nv
a
31ny
a
3N
OB
T,P,NoY^6
v v
a
B
n
l
Y=1
n
o
I
6=1
V V X X
Y 6 Oy o
6
[ (1 Cy 6
1C Q)(lC
a B ay
d c6g)]
o
l
y=i
n
o
l
6=1
v v X
y o oy
(1C
Y 6
(239)
Equation (239) relates changes in the activity coefficient
of any component a with changes in the mole number of any
component B where the process occurs at constant pressure
2
there are models based on relatively rigorous statistical
mechanical results which can be called "theoretical."
Second, there are those composed of a mixture of rigorous
theory and empirical corrections which can be named "semi
empirical." Third, there are those models which directly
correlate experimental data and are thus termed "empirical."
Neither this classification nor the following list pretends
to be either unique or allinclusive.
Among the "theoretical" models, the earliest and still
the most widely accepted is the theory of Debye and Huckel
(1923) which gives the rigorous relation at very low salt
concentration (the limiting law) for salt activity coeffi
cients but fails at higher salt concentration. This theory
has been amply treated in the literature (Davidson, 1962;
Harned and Owen, 1958). The DebyeHuckel theory considers
an electrolyte solution as a collection of charged hard
spherical ions embedded in a dielectric solvent which is
continuous and devoid of structure. This is the physical
picture generally called the "Primitive Model." The correct
formalism for the application of modern statistical mechani
cal techniques to the "Primitive Model" is given by the
McMillanMayer theory (1945). A major method developed
for this formalism is a resummed hypernetted chain approxima
tion to the direct correlation function. This, together
with the OrnsteinZernike equation (1914), forms a solvable
integral equation for the primitive model ionion
52
where
K
4Tre2
DkT
n
I
i=l
DebyeHuckel
inverse length.
D = the dielectric constant of the solvent
or mixture of solvents.
Insertion of equations (338) and (339) into equation
(320) gives
CLR =
13
4ttp Z Z .Q
^ J r . d r .
kT * i] i]
o J J
+
2 2 4
2ttpZ Z e
J J
2Ka. 2Kr. .
! DkT)2 (1+Ka j)2
13
aij
e dr. 
13
3 3 6
2iTp z Z .e
1 J
3(DkT)3 (1+Ka. )3
13
3Ka. .
13
m 3Kr. .
 e ^
r. .
aij ^
dr. +
13
(340)
The first term of equation (340) contains a divergent
integral. However, when it is introduced into equation
(211) which relates it to thermodynamic properties, electro
neutrality makes the coefficients of the integrals sums
to exactly zero.
35
multipoles are not included, because their contribution
is expected to be numerically insignificant in an aqueous
system. The short range interactions are treated as hard
sphere repulsion. Intermediate range forces for the ion
solvent case are very important because they include solva
tion which makes a larger contribution than the long range
chargemultipole forces. Solvation of the ion by the solvent
is intimately related to the partial molar volume of the
salt and must be incorporated if there is to be any hope
of correlating and predicting the volumetric behavior of
the solution. As for ions, the long and short range intera
tions are treated theoretically while the intermediate
range forces are incorporated semiempirically.
The forces between solvent molecules at long range
can be considered to be those of dipoles in a dielectric
medium which has an ionic atmosphere. Higher order multi
poles may again be neglected because their contributions
are less important and can be covered in other ways. The
short range forces are again treated as hard sphere repul
sions. The intermediate range interactions for the solvent
solvent case are dominated by association type forces such
as hydrogen bonding which make a larger contribution than
the long range dipoledipole term. As above, the long
and short range interactions are treated theoretically
while the effects of the intermediate range forces are
included semiempirically.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS
KEY TO SYMBOLS vi
ABSTRACT xi
CHAPTERS
1 INTRODUCTION 1
2 FLUCTUATION THEORY FOR STRONG ELECTROLYTE
SOLUTIONS 11
Introduction 11
Thermodynamic Property Derivatives and
Direct Correlation Function Integrals... 12
Direct Correlation Function Integrals from
Solution Properties 23
Summary 2 9
3 A MODEL FOR DIRECT CORRELATION FUNCTION
INTEGRALS IN STRONG ELECTROLYTE SOLUTIONS.. 33
Introduction 33
Philosophy of the Model 33
Statistical Mechanical Basis 37
Expression for SaltSalt DCFI 51
Expression for SaltSolvent DCFI 63
Expression for SolventSolvent DCFI 69
Summary 7 4
4 APPLICATION OF THE MODEL TO AQUEOUS STRONG
ELECTROLYTES 77
Introduction 77
Calculation of Solution Properties from
the Model 78
Model Parameters from Experimental Data.... 90
Comparison of Calculated Properties with
Experimental Properties 104
IV
17
T, N
V.
l
(214)
3 P
3V
V .
3P
T,N 1
3N .
3N .
V K_
1
T'V'Vj 3V2
1
T'P'Nk^j
Then equations (21), (214), and (215) are inserted into
equation (213) to obtain
RT 3N .
1
where
T, P, N
k^j
6. C.
= _a:
N. N
l
V. V .
 1 3
Vk RT
(216)
V. =
i 3N.
= partial molar volume
T,P,Nk^i
volume of species i.
V = the system volume.
1 3V
KT V 3P
= isothermal compressibility.
T, N
To develop a relation in terms of activity coefficients,
the chemical potential is written as in equation (23)
and the proper constant pressure derivative is taken.
RT
T, P N
3lnYi
3Nj
+
T, P N
k^j
11 I
N N
(217)
From equations (216) and (217),
R
Reference.
SAT
Saturated.
TB
Three body.
o,00 =
infinite dilution in salt.
Subscripts
!/]/
= species.
/ B f
= components.
1 = solvent,
o = component.
Special Symbol
< > = integration over orientation,
w
x
TABLE 413
SOLUTION PROPERTIES FOR NaBr (2) IN WATER (1) AT 2 5C, 1 ATM
1
V o
o2
1/2 3inY2
v M N ^
PKTRT
Exp.
Xo2 2N3N _
Xo2
Calc.
Calc.
Exp.
o2
Calc.
T,P,Noi
Exp.
io12
16.108
16.108
20.951
20.951
8.763
8.763
9.9900x10
4
16.191
16.196
21.764
21.629
4.921
4.386
4.9751x10
3
16.523
16.541
22.842
22.807
1.953
1.764
8.9197x10
3
16.843
16.885
23.489
23.766
0.542
0.675
0.029126
18.555
18.652
27.470
28.204
2.886
2.247
0.052258
20.607
20.731
32.193
33.019
5.021
4.635
0.078001
23.081
23.175
37.960
38.334
6.680
6.688
0.10904
26.490
26.558
46.028
45.848
8.119
8.056
Calc, properties from equations (244), (245), (246), (362), (384), and
(3104).
Experimental data sources are those of Figures 3, 4, and 5.
89
PYHS
The expression for y^ is given in equation (A3).
LR
Note that the integrals of in equations (417),
(418), and (419) require that the pressure behavior of the
solvent dielectric constant be included for nonisobaric
integrations.
Explicit evaluation of these integrals does not in
general yield analytical forms. However, for isobaric
integrations it is possible to write simple analytical forms
for those cases where exponential integrals are not
involved. One example is the DebyeHuckel limiting law in
equation (419).
The last relation that must be reshaped to a more
tractable form is equation (416).
S,ny
a
n n n
o o o
t l l
6=2 Â£=2 y=l
n
I
6 = 1
1
/ F (t)dt
a
o
pKTRT(t)
PB(t>
SCB VC p(t)
pOY(t) po6(t)
VCVYV6 p(t) p(t)
(UcY6(t))(ica?(t)) 
(lCaY(t))(lC6c(t)))
(420)
p
n n P (t) P_(t)
" I l V8 a,t,2 PS
a=l 6=1 p(t)
where
63
Expression for SaltSolvent DCFI
The development in this section parallels that of
the previous one. Thus, a general expression for the
solvention DCFI is derived and then inserted in equation
(211) to yield the saltsolvent DCFI relation.
Although any type of interaction can, in principle,
be included, it was assumed here that ionsolvent interac
tions are dominated by dipolecharge forces at large separa
tion, and no other interactions were included. The pair
potential for an ion (i) and a dipolar solvent (1) is
LR
u.
i
z y. e
i 1
cos
0
(365)
where y^ = the dipole moment of solvent 1
in Debyes.
9 = the Eulerian angle between dipole
and charge.
The potential of mean force is approximated by a func
tional form inspired by some recent applications of the
mean spherical approximation (Chan, Mitchell, and Ninham,
1979) and of perturbation theory (H0ye and Stell, 1978)
to nonprimitive electrolyte models.
LR Z.y.ae
wu r (cos 01 5
kTr .
il
K(a..r )
il il
r > a^ (366a
WLR
wii =
r < a .
il
0
(36 6b)
164
and to remain close to the subject of this work which is
aqueous strong electrolytes. We start with equation (234)
VLR
a
v X .
a ol
(1C .)
al
n
+ Va 62 Xg(1
(234)
Our complete model for the saltsolvent DCFI is con
tained in equation (383) where the terms are defined by
equations (376) to (378) and (380) to (382).
1C
al
V'LR
a
V v k RT
a 1
al al al
 C
LRC
al
) 
 ( AC
al
OO
(383)
We now focus on the initial deviation from the infinite
dilution limit which is dominated by the long range field
term and ignore all the others.
cLR cLR<
2 7T ea
'al al 3v
a
DkT
) nr i
n v Z
a i
1 . a i
1 = 1 ll
2a B I1/2
(pe 11 Y E(2a..B I1^2
2 ll y

(B3 7
To explore the pure solvent limit, we expand equation
(B37) in ionic strength (I).
P
Pol + 0(I)
(B3 8)
156
MM
Pa1Jyi(T,P+Tr,X)
yRR(T,p,x) + J vRR
P
a
dP
(B9)
The next task is to relate the chemical potential in
the KB system to that in the MM system. To that purpose,
equation (B9) is rearranged and differentiated to obtain
LR
3y
n. MM
3y
P + TT
3VLR
a
a
 J
T'p'fWeP
a
3N
oB
3N
T, P, N B
oy^B
3KoS
dP 
T, P, N
oy^B
 v 31
a 3N
oB
T, P, N
P+tt 3V
oy^B
Equation (B7) is differentiated to obtain
LR
3N
P oB
n LR 3 7T
dp + V1 w
T, P N
oB
oy^B
T, P, N
n N a ~
o oB 3y.
3N
oB
{ I I
B=2 o
3N
oB
T, P, N
oy^B
d N 0 }
oB J
oy^B T,P,N
oy^B
(B10 )
(Bll
The righthand side of equation (Bll) is further
simplified by use of equations (B7) and (B9).
P+tt T T,
/ vf dP =
P
n N LR
o oB 3m,
y
3N o
B=2 o oB
d N
oB
T, P, N
oy^B
(B12 )
TABLE 420
WATERWATER DCFI FOR KCL (2) IN WATER (1) AT 25C, 1 ATM
X
0
K)
(1Cn>
LR
11
(CHSCHS )
lcll 11 1
(Ac Ac )
1 C11 ll'
M p CALC.
' '11 >
(1"C11)EXP
1012
16.1084
0.0000
0.0000
0.0000
16.108
16.108
9.9900x10
4
16.1084
0.0004
0.0753
0.02538
16.209
16.220
4.9751x10
3
16.1084
0.0007
0.3668
0.1297
16.605
16.651
8.8482x10
3
16.1084
0.0009
0.6457
0.2323
16.987
17.062
0.025637
16.1084
0.0012
1.8561
0.6870
18.652
18.775
0.041316
16.1084
0.0013
2.9843
1.1238
20.217
20.290
0.055992
16.1084
0.0014
4 0476
1.5405
21.697
21.674
0.069758
16.1084
0.0014
5.0432
1.9363
23.089
23.086
(1Cj^ )
FXP
(1c^) from equation (255) using the same sources of experimental data
as for Figure 5.
166
Equation (B42) expresses the composition behavior of
the saltsolvent DCFI at very low ionic strength or near the
pure solvent limit.
Now, we change equation (B42) to express the composi
tion in terms of mole fractions and then consider the salt
solvent DCFI contribution to equation (234).
I
1
2
Y z2p
i1 1 1
P
n
o
I
Y=1
(0
Y
where co
Y
v.
x Y
Z .
l
2
(B43)
v'LR
a
k^RT
4tt
DkT
2 p
U1 By Pol
3/2
( l co X )1/2 l v. Z.2(y +
, Y Y
Y=1
i=l
a i
+
Â£n(2a.,B
xl y
co X
Y Y
1/2
)
+ 0(X^ )
oy
(B44)
Our complete model for a general saltsalt DCFI is
expressed by equation (359) where the various contributions
are defined by equations (353) to (355) and (357),
(358) .
48
LR
$(T)
ijk
1 ( 4,LR + $LR
2 11k JDk'
If i is an ion and j and k are solvents, then
(327)
A$(T) = A$iik (328)
ijk J
LR LR
$(T) = $(T) (329)
ijk ijk
Lastly, if i, j, and k are all solvents, then
AF(T) = AF. (330)
ij 1D
A$(T) = A4>. .. (331)
ijk
LR LR
$(T) = <^k (332)
ijk J
It should be noted that these additive mixing rules
are not the only possible ones. In fact, theory would
suggest that geometric mean type mixing rules might be
more appropriate. Geometric mean rules, however, only
work for positive quantities which turned out not to be
the case with our empirically fitted coefficients. This
situation is further discussed in Appendix C.
The last point that needs to be addressed here is
the extension of the model to multisolvent systems. First,
106
transition in between. Further, at even higher salt mole
fractions (X^ > 0.12) a third regime of solution behavior
becomes apparent due to the scarcity of solvent (water).
The present model is not intended to be used for salt mole
fractions much above 0.10 or so. Thus, it will not yield
good results in the third regime so it will not be further
discussed here.
In order to discuss and compare solution behavior
calculated from the model and its various contributions to
experimental observations, NaCL (2) in water (1) is chosen
as prototype electrolyte solution whose properties are
illustrated in Figures 6 to 10 and Table 49. The major
differences between this 11 salt and others is that the
field terms will be more significant as with higher
charges. In addition, the interaction terms could also be
changed in importance.
First we look at the saltsalt DCFI in Figure 7. At
infinite dilution of salt, the saltsalt DCFI magnitude and
behavior are dominated by the long range ionion electro
LiR
static correlations in C22* But as the mole fraction of
CO
salt increases (0 < Xq2 < 0.02), (1C22) which includes
short range ionion and ionwater correlations significantly
contributes to the magnitude but not to the behavior while
the hard sphere (2 ~ (~22 ^ anc^ triple in ^^22 ~ ^<~22 ^
correlations are small and approximately cancel each other.
Once the mole fraction of salt is large (X 2
> 0.05),
98
+ ( (v + v ) AD +
v +a 2 a ++1
a
(v + v ) AD ) pp
a 2 a 1 oa
In the above expression, the mixing rule of equation
(326) to obtain AD__^ (i ^ j) from AD__^ and A$_._.^ has been
used to simplify equation (387).
Equation (384) is now changed to
PP
((v + v ) AD. + (v + i v ) AD )
v +a 2 a ++1 a 2 a 1
a
1C
 Va + / pHS pHS00,
al v < RT 1 al al '
a T
/ ,,LR _,LR > 1 P LR,
+ (C C ) + (v (AF p D ) +
a al v +a +1 ol +11
a
+ v_a ( 'P'Kol1 +
+ T (v+ai4+ll + vaMll) (ppol polpoI>
a
(432)
As previously, we label the righthand side of equation
(432) as Fla(p_).
PP
((v + 4 v )AD + (v + \ v ) AD ) = F, (p)
v +a 2 a ++1 a 2 a 1 la
a
(433)
68
Finally, equation (383) will now be written for a
binary system consisting of solvent (1) and salt (2) with
v+ cations and v_ anions.
V
1 C
o2
21
v2
(CHS cHS)
^21 ^21 1
, LR LRC
V 21 U21
~ (AC21 AC21}
(384)
where
HS HS<
21 C21
L [v (CHS CHS~) + V (CHS cHS)j
v2 L + ^+l *+l 1
(385)
,LR _LR 2tt
C C
21 21
/ e a 2 j.
3v 1 DkT yl L
v Z2 2a B ll/2
(Pe Y V2a+lV
1/2,
P VZ
Po!} + i"
(pe
2a .B I
1 y
1
1/2
E0 (2a .B I
2 1 y
1/2
" Pol} 1
(386)
A^i
 AC.
21
[v+(aF+i
 P
ol
$ LR)
+ 11'
+ V ( AF
1
 Po*?
(p
+
+ ^ [v+(plA$l+l +
p A4> +
K+ ++1
p_A$_+1) +
18
31nyi
3N .
3
T,P,N
1 C. V. V .
H i 1
N VKtRT
(218)
k^j
By multiplying equation (218) by the system volume
and rearranging, one finds
3 lny.
N x
pKTRT SNj
T, P, N
1 C.. V. V.
hi 1 1.
pKTRT
KmRT < RT
T T
(219)
k*j
upon which a double summation over species i and j is
performed to obtain
N
n n
31nyi
pKmRT v v L ,L 'ia'jB 3N.
T a 8 i=l j=l J i
l l v. v.
. ,Ln a i
T,P,Nk^j
n n
 y y v. v. (ic ) 
PKTRT Vb i=i j=i 101 ^ ci^
i i n n
^~2 I I ViaViB ViVi
(VT) vaVB 1=1 j = 1 B
(220
Equation (220) must be simplified so that all the properties
appear as component rather than species quantities. This
is done with the aid of equations (28), (211), and the
assumption that species j is formed from an arbitrary
component B so
N = v n o
] ]D OP
(221)
65
The third term in equation (320) contains
<(wii)3>, k I (wii>3 d"i di (371>
o 6
which also equals zero.
< (W
LR, 3
il1
>
U)
0
(372)
Therefore, for iondipole pairs there is only one term
in equation (320).
PLR
il
21 (
3 P (
Vi
DkT
a
2Ka
2Kr
il
e
~2~
i. r. .
il il
il
dr
il
(373)
The integral in equation (373) is also an exponential
integral (Â£Â¡3^ which is expressed in dimensionless form
as before.
f
J 2
a.,r,,
il xl
2Kr.,
il
dr.1 = J
11 aii 1 x2
(2Ka1)X
dX =
E2(2Kail)
ail
(374)
Equation (373) then becomes
2TTp
3
Z.y.ea
1 1
DkT
0 2Ka.,
e 11 E2(2Kail)
17
(375)
192
Reichl, L.E., A Modern Course in Statistical Physics,
University of Texas Press, Austin, TX (1980).
Renon, H., Foundations of ComputerAided Chemical Process
Design (Edited by R.S.H. Mah and W.D. Swider), II,
Amer. Inst. Chem. Eng., New York, NY (1981).
Renon, H. and J.M. Prausnitz, AIChEJ, 1_4, 135 ( 1968 ).
Robinson, R.A., and R.H. Stokes, Electrolyte Solutions, 2nd
ed., Butterworths, London (1970).
Romankiw, L.A. and I.M. Chou, J. Chem. Eng. Data, 2j3, 300
(1983 ) .
Rowlinson, J.S., Repts. Prog. Phys. 2Q_, 169 (1965 ).
Sander,B., A. Fredenslund, and P. Rasmussen, AIChE Annual
Meeting, San Francisco, CA (1984).
Telotte, J.C., Ph.D. Dissertation, University of Florida
(1985 ) .
Triolo, R., J.R. Grigera, and L. Blum, J. Phys. Chem. 80,
1858 (1976).
Uematsu, M. and E.U. Franck, J. Phys. Chem. Ref. Data 9_,
1291 (1980).
Vericat, F. and L. Blum, J. Stat. Phys. 2_2, 593 (1980 ).
Washburn, E.W. (Ed.), International Critical Tables of
Numerical Data, Physics, Chemistry, and Technology,
McGrawHill, New York, NY (1928).
Watanasiri, S., M.R. Brule, and L.L. Lee, J. Phys. Chem. 86,
292 (1982).
Watts, R.O., and McGee, Liquid State Chemical Physics, John
Wiley and Sons, New York, NY (1976).
Weast, R.C. (Ed.), Handbook of Chemistry and Physics, 57th
Edition, CRC Press, Cleveland, OH (1977).
47
due to the relative simplicity of ionion interactions
which can be generally scaled with the ionic charge (Kusalik
and Patey, 1983). Thus, two and three ion coefficients
are expressed from quantities related to a single ion.
If i, j, and k are ions, then
(322)
(323)
(324)
If one or two of the species i, j, and k are solvents while
the remainder are ions, then the mixing rule must be
expressed from quantities involving each of the species
and water. The reason for this is that ionsolvent inter
actions cannot possibly be predicted from solventsolvent
and ionion interactions separately. Therefore, if i is
an ion and j a solvent, then
AF. (T) = AF. .
i;j 13
(325)
If i and j are ions while k is a solvent, then
(326)
169
Lastly, we note that the ionic strength expansion of
the isothermal compressibility is of the form
< ip = < + 0(1)
(B47)
Finally, we insert equations (B44), (B46), and (B47)
into equation (234) to obtain an expansion for the salt
partial molar volume (V ) based on KB theory and our micro
scopic model for DCFI's. Note that the quantity k^RT is a
constant.
VLR LR
a V
k1RT < RT
4tt ea > 2 p .
T < DkT b BY(ol)
3/2
n
o 1/2 n
( l u X ) i v Z. (y
T Y oY a i
Y=1 1=1
n
1/2
+ Â£n (2a.,B (pP V w X ) )
il Y ol ^ y OY
c, p ,1/2 n n
Sv(p*\ ) o 1/2 o n n
7 ( l u X ) IXogl l v. v Z2Z2
4 Y=1 7 y 6=2 i=l j=l la 1 3
o 2 p n
S p^ o n n 0
3 l l l v. v ,Dz3z3
6=2 i=1 j=l la 1,6 1 J
n
Â£n (3a. .B (pP [ co X )1/2) + 0(X a:
il Y ol y OY o6
(B48
165
2a. B I1//2
e 11 Y = 1 + 2a. B I1/2 +0(1
ll Y
(B
2a. B I1/2
E (2ailByI1/2) = e 11 Y 2ailByI1/2E1(2a1By 1/2
(B
From equation (D8) we have the low ionic strength
expansion for E^.
El(2ailByll/2) = Y "^n(2ailByI1/2)
 2a.,B I1//2 + 0(1)
l y
(B
where y = 0.57721 = Euler's Constant.
Inserting equations (B38) to (B41) into equation
(B37) and collecting terms up to half power in ionic
strength we obtain
CLR CLR
cxl al
4tt ea 2 20 _l/2 p
3va DkT M1 ByJ p o1
n
y v. Z ,2(Y+Â£n(2a.,B I1/2)) + 0(1)
ia i ii y
39 )
40 )
41)
(B42)
60
(1C Q)
a 6
P CO CO
p ,V V .
ol oa og
v v k rt
a g 1
n
n
y y v. v (Af
Vg i=i j=i ia ^ ^
+
*LR
111
(358)
Finally, the general expression for the saltsalt
DCFI model including the infinite dilution limit is
1C
a g
= (1C )" cLR (CHS cHS)
1 ag; ag lCag Cag ;
 (Ac
TB
a g
AC
TB
6
where
CHS = LIM HS
ag X ,kL ag
ol
AC
TB
a g
LIM
X +1
ol
AC
TB
ag
(359)
Mol
V VD
a g
n
I
i=l
n
I
j=l
v. v p AO. ..
ia ]g ol ljl
Although equation (356) can be used in place of
equation (359), it was felt that the latter was more
appropriate for calculations at constant temperature and
pressure. Therefore, equation (359) was used in the com
parisons and correlations in this work. In calculations
TABLE 417
SOLUTION PROPERTIES FOR KCL (2) IN WATER (1) AT 25C, 1 ATM
1
pKTRT
Exp.
V o
o2
1/2 9Â£nY2
X v N
o2 2 3N _
o2
Calc.
Xo2
Calc.
Calc.
Exp.
T,P,Noi
Exp.
io12
16.108
16.108
23.937
23.937
8.763
8.763
9.9900x10
4
16.194
16.205
24.794
24.776
5.312
5.018
4.9751x10
3
16.535
16.583
25.967
26.296
2.940
2.787
8.8482x10
3
16.867
16.947
26.987
27.415
1.978
1.947
0.025637
18.358
18.483
31.281
31.555
0.158
0.293
0.041316
19.816
19.888
35.269
35.399
0.607
0.644
0.055992
21.239
21.215
39.041
38.990
1.001
1.384
0.069758
22.625
22.524
42.591
41.900
1.221
1.999
Calc, properties from equations (244), (245), (246), (362), (384), and
(3104).
Experimental data sources are those of Figures 3, 4, and 5.
95
bring all ion values to almost the same result rather than
having cations and anions possessing very different
parameter values.
A $ + = 4 ML M 1 (428)
Lili
Then the values for the ions N+, K+, CL Br were obtained
a
from C11 data for LiCL, LIBr, NaCL, NaBr, KCL, and KBr.
The twobody parameters involving one ion and a water
together with the long range threebody parameter involving
p
an ion and two waters are lumped as (AFiiP0i$iH) an<^
obtained from equations (240) and (379) at the pure
solvent or infinite dilution limit for each salt (a).
CO
a i rHS_ LR A
vaK1RT 1 cai ual Cal
(429 )
. HS00
where C =
al
LIM PYHS
Xoi! al
LIM
X +1
ol
V
+ CPyH? V CPrHS
+a +1 a 1
v
a
p yHS
C^ = PY hard sphere DCFI given by equation (A5)
LR LR
= infinite dilution given by equation
(380)
54
of a class of functions known as the exponential integrals.
These cannot be evaluated explicitly but a number of
asymptotic expansions and numerical approximations are
available (see Appendix D). It is convenient to express
the integral in dimensionless form.
Letting X = r/a^ then
e~3Krij (3Kaij)x
dr = J ^ dx = Ex(3Ka ..) (343)
aij ij J 1 J
where E.(3Ka. .) = the
1 13
The third term in equation
first exponential
(340) becomes
integral
2rrpZ3Z3e6
i 3
3 (DkT)3 (1+Ka^ )3
3Ka.
* 13 J
a .
13
3Kr. .
e xj
r .
13
dr. .
13
0 0 3a..B I
Z3z3 S2pe 13 Y
= i 3 _J
1/2
(1+a. .B I
13 Y
yz,T E. ( 3a .B I
1/2,4 i i] y
1/2
(344)
which contains the implications for DCFI's to a higher
order limiting law for unsymmetric electrolytes (Friedman,
1962). Because of electroneutrality, this term,
when
APPENDIX E
MODEL PARAMETERS
The application of any model requires numerical values
for certain input quantities loosely called parameters. In
the case of the present work, some of the input quantities
are actually properties whose value we have adopted from the
literature. But other quantities are true parameters
representing certain molecular correlations whose value we
have obtained from fitting the experimental values of the
DCFI's. What follows is a listing of the numerical values
for all the quantities used in the calculations presented in
Chapter IV.
Pure Water Properties
( 25 C, 1 ATM)
WATER DENSITY = 0.997048 GM/ML+
+
WATER COMPRESSIBILITY tc
45.248 x 10
6
BAR
1
1
WATER DIELECTRIC CONSTANT D = 78.4472
+ +
WATER DIPOLE MOMENT 0.
1.87 DEBYES
+++
1
+ FINE AND MILLERO, 1973.
++ UEMATSU AND FRANK, 1980
+++ WEAST, 1977.
184
16
which upon identification of
1 
n
n
l l v.
il jl 101
1C. .
v.0
30 vaV 3
assumes the simpler form
(211)
3lny
pv
a
a
3p
06
T / P o
= v v (1C )
a 6 ct 6
Yt^B
(212)
Equation (212) relates the DCFI to the derivative
of the activity coefficient of any component a (salt or
solvent) with respect to the molar density of any component
6 (salt or solvent) at constant volume, temperature, and
mole number of all components other than B.
Because most experiments and many practical calculations
are performed at constant pressure rather than constant
volume, it is of interest to derive a relation between
the activity coefficient derivatives at constant pressure
and direct correlation function integrals. First, a change
of variables is executed.
3p.
9N .
D
T, V, N
k^j
T,P,Nk?ij
+
3P
T, N
T, V, N
k^j
and the following identifications are made,
(213)
162
electrolyte and nonelectrolyte solutions always follows an
expression which near the pure solvent limit reduces to
y
LR
1
X
o
+ .
n > 1
(B28)
3y
LR
3N
oÂ£
LIM
a X
_nl
oB
T,P,N
3 y
0y/B
LR
X 1 v 1
ol
3N
oB
= 0
T, P, N
oy/S
n1 > 0
(B2 9)
(B3 0
But, for at least the case of a salt ( a ) in water (1) with
other salts (B) we have
o, X1/2 +
a oB
3y
MM
a
3N
oB
a x
1/2
oB
T, P, N
oy/B
LIM
3 y
MM
a
Xol ^ 1 3Ne T, P, N
Oy/B
(B31)
(B3 2)
(B33)
Lastly, LR and MM partial molar volumes are equivalent in
the pure solvent limit. To prove this, we differentiate
equation (B9) with respect to pressure and insert equation
(B2) in the resulting expression to get
yMM
a
VLR
a
P+TT
/
P
3VLR
a
3 P
d P
T, N
(B3 4)
6
distinct approaches. First, there is the method of Meissner
(1980) which is a correlation for the salt activity coeffi
cient in terms of a family of curves that are functions
of the ionic strength and a single parameter which can
be selected from a single data point. This method has
been extended to multicomponent electrolyte solutions and
is useful over a wide range of salt concentration (0.120
MOLAL), though it is not very accurate. Second, there
is the method of Hala (1969) which is more conventional
in that it consists of a purely empirical model for the
Gibbs energy of the solution. This method is an excellent
correlational tool, but it is not predictive. It has four
parameters per saltsolvent pair.
The existence of so many models to correlate and predict
the thermodynamic behavior of electrolyte solutions is
indicative of the complexity of these systems and, perhaps,
the relatively poor state of the art.
As examples of the physical complexity of electrolyte
solutions, the composition behavior of the salt activity
coefficient (Figure 1) and of the species (ions and solvents)
density (Figure 2) is presented. Figure 1 shows the large
deviation from ideal solution behavior (y= 1) even at
very low salt concentration for all salts. Second, it
indicates that salts of the same charge type show similar
behavior at low salt concentration but are widely different
at higher salt concentration. In Figure 2, the difference
ACKNOWLEDGMENTS
I would like to express my sincere gratitude to
Professor J.P. O'Connell, a man of wisdom and knowledge, for
his guidance and encouragement during the course of this
work.
I also wish to thank Drs. G.B. WestermannClark and
C.F. Hooper, Jr. for serving on the supervisory committee
and for making very pertinent suggestions regarding this
work.
It is a pleasure to thank Mrs. Smerage for her
excellent typing and patience and Mrs. Piercey for her help
with the figures.
Finally, I am grateful to the Chemical Engineering
Department of the University of Florida for financial sup
port and for providing the kind of intellectual environment
in which this work could take place. I am also grateful to
the National Science Foundation for providing the financial
support that made this work possible.
in
103
methods. First, the ionic parameters for a large number of
different salts could be fitted simultaneously in the hope
that overconstraining the problem would force the fitting
routine to choose ionic parameter values that would be valid
for many types of salts. Second, the fitting could be done
in a stepwise fashion. For example, the two and three body
parameters could be obtained assuming given values for the
hard sphere diameters for each salt separately. The species
diameters could then be optimized subject to the previously
calculated values for the two and three body coefficients
for all the salts simultaneously. The process could be
repeated until satisfactory results were obtained. The
second fitting scheme should be easier to implement since it
is mathematically simpler and since it is expected that few
iterations are required.
The present results given in Appendix E used the hard
sphere diameters of Marcus (1983 ) for all ions but Lx.
o
Marcus' value of aTff = 1.36 A could not yield accurate
LiLi J
results for LiCL and LiBr nor were the CL and Br values
+ +
obtained consistent with those from Na and K salts. The
present value of aLL = 145 A was obtained by a rough fit
+
of the Li salt data and then the values for the other
parameters for the other salts were found from the Na+ and K+
salt data. The difference between Marcus' and the present
O
hard sphere value lies within the variation (+ 0.12 A or 9%)
that he attributes to the different data for Li. It is
Author:
Cobcci$j tcrt bPC''Q
Application of fluctuation solution theory to
strong electrolyte solutions / (record number:
Title: 880471)
Publication 1985
Date
1, HgIZi&EiZTo ChGttAZ 3g as copyright holder for the
aforementioned dissertation, hereby grant specific and limited archive and distribution rights to
the Board of Trustees of the University of Florida and its agents. I authorize the University of
Florida to digitize and distribute the dissertation described above for nonprofit, educational
purposes via the Internet or successive technologies.
This is a nonexclusive grant of permissions for specific offline and online uses for an
indefinite term. Offline uses sha.!I be limited to those specifically allowed bv "Fair Use" as
prescribed by the terms of United States copyright legislation (cf, Title 17, U.S. Code) as well as
to the maintenance and preservation of a digital archive copy. Digitization allows the University
of Florida to generate image and textbased versions as appropriate and to provide and enhance
access using search software.
This grant of permissions prohibits use of the digitized versions for commercial use or profit.
Signature of Copyright Holder
HERi f? EF riT CA 6 c Z A $ F
Printed or Tvoed Name of Convrisht Holder/Licensee
Personal Information Blurred
Printed or Typed Phone Number and Email Address of Copyright Holder/Licensee
06 /lO/&og
Date of Signature
Please print, sign and return to:
Cathleen Martyniak
UF Dissertation Project
Preservation Department
University of Florida Libraries
P.O. Box 117007
Gainesville, FL 326117007
155
3u
LR
0 =
3P
dP
T,N
o
n LR
o 3y.
1 ^
d N
oB
(B6)
T, P, N
OYt8
Next, equation (B2) is used on (B6) and the resulting
expression integrated over a variation in the mole numbers
of each solute (8) from zero to NQg and a corresponding
change in system pressure from P to P+7T (note that since
P, = constant, dN = 0).
1 ol
P + TT
0 = J VRR dP +
P 1 B=2 o
n N 0 ^ LR
o oB 3 yT
Â£ J 3
oB
d N
oB
(B7)
T, P, N
oy^B
where tt = osmotic pressure.
Equation (B7) indicates that if the solvent chemical
potential is to be constant while solute is added, then the
system must be kept at total pressure equal to osmotic
pressure plus whatever pressure the pure solvent as under.
Since the pressure in an LR system remains constant at P as
solute is added, the only difference between a thermodynamic
property in an LR system and one in an MM system is the
system pressure if both are at the same temperature and
composition. This suggests that equation (B2) can be
integrated to relate the chemical potential of any component
(a) in the LR system to that in the MM system.
P + TT P + TT
J dn = / va dP
P U P
(B8 )
66
Then, the general expression for the salt (a) and
solvent (1) DCFI is
2rrp
3v
a
2a B I1//2
2 7 n v Z2e 1 Y /(J
<Â§T>  l 1 a, E2(2a,,B..I1/2
1 = 1 ll
xi r
(376)
HS .
The expression for C ^ is given by
n
HS 1 v HS
C i = / v. r. .
al v L, la nl
a 1=1
(377)
and the relation for AC is
al
n n
ac = y v. af., + y y v.
al v ia ll , L, la
a 1=1 a i=l k=l
( p, A$ p. $ ,. )
k ilk Kk ilk
(378)
Again, equations (376), (377), (378), and (379)
form the complete general model for the saltsolvent DCFI.
C = CH + CL* + AC 1
al al al al
(379)
As previously discussed, it is convenient, particularly
for isobaric calculations, to use the model only for
116
0.00 0.02 0.04 0.06 0.08 0.10
X 02
Figure 8.
Contributions to the Salt (2)Water (1)
DCFI in Aqueous NaCL at 25C, 1 ATM.
For data sources see Table 411.
39
in density and ignores some four body contributions. It
is, therefore, exact up to the order of a third virial
coefficient. Thus, the HNC direct correlation function
is
HNC no Ui
= g. 1 in g. 7
13 ^13 ^13 kT
(33)
From the definition of the radial distribution
function,
W. .
gij = is1
(34)
g. 1 =
yi3
2 3
1 W .
kT + 2T ^kT ) Urn ) +
3 ^ kT
+
(35)
which on insertion into equation (33) gives,
c
HNC
i j
(36)
To apply equation (36) requires at least approximate
expressions for the potential of mean force in terms of
measurable variables. Such expressions, valid in the limit
TABLE 416
WATERWATER DCFI FOR NaBr (2) IN WATER (1) AT 25C, 1 ATM
Xo2
oo
UCn)
PLR
U11
_(CHScHS )

(1C )CALC*
U Cll'
(lc )EXP*
U cll;
lo"12
16.1084
0.0000
0.0000
0.0000
16.108
16.108
9.9900x10
4
16.1084
0.0004
0.0806
0.0235
16.212
16.217
4.9751x10
3
16.1084
0.0007
0.3980
0.1180
16.625
16.644
8.9197x10
3
16.1084
0.0009
0.7053
0.2137
17.028
17.067
0.029126
16.1084
0.0012
2.3427
0.7206
19.172
19.232
0.052258
16.1084
0.0014
4.3052
1.3337
21.748
21.798
0.078001
16.1084
0.0015
6.6671
2.0495
24.826
24.885
0.10904
16.1084
0.0016
9.8782
2.9539
28.942
29.098
(1CQ )CALC' from equation (3104).
EXP
(1C^) from equation (255) using the same sources of experimental data
as for Figure 5.
TABLE 45
SOLUTION PROPERTIES FOR LiBr (2) IN WATER (1) AT 25C, 1 ATM
1
P K ipBT
ktrt
1/2 3Â£nY2
X v N 
o2 2 N _
o2
T,P,N
ol
X _
o2
Calc.
Exp.
Calc.
Exp.
Calc.
Exp.
12
10
16.108
16.108
21.244
21.244
8.763
8.763
2.0866xl0_3
16.258
16.260
22.222
22.206
3.056
2.445
4.1983.10"3
16.408
16.413
22.549
22.701
1.334
0.972
8.4 9 62x10 ~3
16.713
16.722
23.210
23.535
0.818
0.803
0.017411
17.334
17.365
24.245
25.049
3.700
3.230
0.036619
18.661
18.749
26.733
28.043
7.906
7.213
0.057919
20.169
20.259
29.890
31.140
11.624
11.072
0.075482
21.483
21.460
32.916
33.478
14.343
14.144
0.10833
24.343
23.508
40.092
37.091
18.781
20.250
Calc, properties from equations (244), (245), (246), (362), (384), and
(3104).
Experimental data sources are those of Figures 3, 4, and 5.
123
114
0.00 0.02 0.04 0.06 0.08 0.10
X 02
Figure 6. Salt (2) Activity Coefficient Derivative in
Aqueous Electrolyte Solutions at 25C, 1 ATM.
For data sources see Tables 45 and 49.
74
Summary
A general statistical mechanical model of the direct
correlation function has been presented. In principle
it is applicable to any system, but it has been specialized
here to treat strong electrolyte solutions. The next chapter
shows the application of this model to six aqueous strong
electrolyte binary solutions. As a preview to the calcula
tions, the relative magnitude of the three contributions
to the DCFI (CHq, CL^, Ac q) will now be discussed, the
a p a p a p
model parameters will be listed, and the sensitivity of
solution properties to parameter value considered.
The saltsalt DCFI is dominated at very low salt con
LR
centration by Cag which contains the long ranged electro
,LR
static interactions. However, the magnitude of Cag decreases
very fast as the salt concentration increases so that above
2M or so in salt density the dominant term becomes C^g.
This reflects the increasing shielding of electrostatic
forces by more ions that more frequently repel each other.
ACag makes a contribution that is generally not dominant
in either regime but is always numerically significant
above 0.5M.
HS
The saltsolvent DCFI is always dominated by Cag with
,LR
CQ^ making a small but not negligible contribution. Due
to the relative strength of the short ranged hydration
interactions, AC^ makes the largest contribution after
cHS
Lal
APPLICATION OF FLUCTUATION SOLUTION THEORY TO
STRONG ELECTROLYTE SOLUTIONS
By
HERIBERTO CABEZAS, JR.
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE
UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1985
To Flor Maria
ACKNOWLEDGMENTS
I would like to express my sincere gratitude to
Professor J.P. O'Connell, a man of wisdom and knowledge, for
his guidance and encouragement during the course of this
work.
I also wish to thank Drs. G.B. WestermannClark and
C.F. Hooper, Jr. for serving on the supervisory committee
and for making very pertinent suggestions regarding this
work.
It is a pleasure to thank Mrs. Smerage for her
excellent typing and patience and Mrs. Piercey for her help
with the figures.
Finally, I am grateful to the Chemical Engineering
Department of the University of Florida for financial sup
port and for providing the kind of intellectual environment
in which this work could take place. I am also grateful to
the National Science Foundation for providing the financial
support that made this work possible.
in
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS
KEY TO SYMBOLS vi
ABSTRACT xi
CHAPTERS
1 INTRODUCTION 1
2 FLUCTUATION THEORY FOR STRONG ELECTROLYTE
SOLUTIONS 11
Introduction 11
Thermodynamic Property Derivatives and
Direct Correlation Function Integrals... 12
Direct Correlation Function Integrals from
Solution Properties 23
Summary 2 9
3 A MODEL FOR DIRECT CORRELATION FUNCTION
INTEGRALS IN STRONG ELECTROLYTE SOLUTIONS.. 33
Introduction 33
Philosophy of the Model 33
Statistical Mechanical Basis 37
Expression for SaltSalt DCFI 51
Expression for SaltSolvent DCFI 63
Expression for SolventSolvent DCFI 69
Summary 7 4
4 APPLICATION OF THE MODEL TO AQUEOUS STRONG
ELECTROLYTES 77
Introduction 77
Calculation of Solution Properties from
the Model 78
Model Parameters from Experimental Data.... 90
Comparison of Calculated Properties with
Experimental Properties 104
IV
Discussion 105
Conclusions 113
5 CONCLUSIONS AND RECOMMENDATIONS 144
APPENDICES
A HARD SPHERE DIRECT CORRELATION FUNCTION
INTEGRAL FROM VARIOUS MODELS 148
B RELATION OF McMILLANMAYER THEORY TO
KIRKWOODBUFF THEORY 152
C RELATION OF DENSITY EXPANSION OF THE
DIRECT CORRELATION FUNCTION TO VIRIAL
EQUATION OF STATE: ALTERNATE MIXING
RULES 17 2
D EXPONENTIAL INTEGRALS 18 0
E MODEL PARAMETERS 18 4
REFERENCES 18 8
BIOGRAPHICAL SKETCH 193
V
KEY TO SYMBOLS
a .
l
'a 6
C. ..
ink
AC. .
13
D
D
E
1
n
e
f .
13
hard sphere diameter of species i.
distance of closest approach of species i and j.
k/I1/2
sum of all bridge diagrams, second virial
coefficient.
mixture third virial coefficient.
direct correlation function integral for species
i and j; twobody factor in third virial
coefficient.
direct correlation function integral for
components a and 8.
third virial coefficient for i, j, k.
short range direct correlation function
integral.
direct correlation function.
short range direct correlation function.
dielectric constant of solvent or solvent mixture.
pure solvent dielectric constant.
exponential integral or order n.
electronic charge.
spatial integral of Af ^ j
u. ./kT
1
e J 1 = Mayer bond functions.
vi
Af .
1J
ID
HS LR
f^j f^j = differences of microscopic
twobody coefficient.
pair distribution function.
1 n 2
f p i = ionic strength
i=l
K
8iTe'
DkT
I = DebyeHuckel inverse length.
k
N
N.
i
N
o
N
oa
n
n
o
P
I NT
r,
r .
i
r .
ID
S
Y
T
u .
i
V
V .
Boltzmann 1s constant.
total number of moles of all species.
total number of moles of species i.
total number of moles of all components.
total number of moles of component a.
number of different species, integer greater
than one.
number of different components,
pressure.
internal partition function,
separation between species i and j.
position vector of i.
fi 1/2
2iTe&
(oo) = DebyeHuckel limiting law
D k TJ
o
efficient.
temperature.
pair potential.
total system volume.
partial molar volume of species i.
Vll
V
oa
partial molar volume of component a.
W. .
13
potential of mean force.
X.
l
Kh/N = mole fraction of species i.
X
oa
N
= mole fraction of component a on a
species basis.
Z
dimensionless parameter in exponential integral.
Z .
i
valence of ion i.
a =
Euler's constant, empirical universal constant
for ionsolvent correlations.
Yi
activity coefficient of species i.
Ya
activity coefficient of component a.
6 .
13
Kroniker delta.
0
Eulerian angle between a charge and a dipole.
eii'*ii
= Eulerian angles of dipole of solvent molecule i
kt
isothermal compressibility.
K1
isothermal compressibility of pure solvent (1).
A .
i
ideal gas partition function.
M
l
chemical potential of species i.
II
i1
a
dipole moment of solvent.
v =
a
number of species i in component a.
v =
a
total number of species in component a.
Â£ p aK = reduced density.
6 i=i 1 1
TT =
P^, osmotic pressure.
P
N
= density of all species.
Â£
vector of species densities.
VI11
density of species i.
oa
$. .. =
xjk
A$ =
i]k
^ijk =
A*ijk =
Q
OJ ,
i
0)
Y
F
FLL
HNC
HS
KB
LR
MM
P
PY
N.
l
V
N
oa
V
= density of component a.
spatial integral of
spatial integral of
microscopic threebody coefficient.
HS
^ijk "" ^ijk = difference of microscopic three
body coefficients.
orientation dependence of dipoledipole
interaction.
/ d
angular orientation coordinates of i.
1 n
1 Z v. z2.
2 i=l ^ 1
Superscripts
Final.
Friedman's limiting law.
hypernetted chain,
hard sphere.
KirkwoodBuff.
long range or field type correlations or
interactions, LewisRandall.
McMillanMayer.
Pure component.
PercusYevick.
IX
R
Reference.
SAT
Saturated.
TB
Three body.
o,00 =
infinite dilution in salt.
Subscripts
!/]/
= species.
/ B f
= components.
1 = solvent,
o = component.
Special Symbol
< > = integration over orientation,
w
x
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of
Doctor of Philosophy
APPLICATION OF FLUCTUATION SOLUTION THEORY TO
STRONG ELECTROLYTE SOLUTIONS
By
Heriberto Cabezas, Jr.
August, 1985
Chairman: Dr. J.P. O'Connell
Major Department: Chemical Engineering
Fluctuation solution theory relates derivatives of the
thermodynamic properties to spatial integrals of the direct
correlation functions. This formalism has been used as the
basis for a model of aqueous strong electrolyte solutions
which gives both volumetric properties and activities.
The main thrust of the work has been the construction
of a microscopic model for the direct correlation func
tions. This model contains the correlations due to the hard
core repulsion, long range field interactions, and short
range forces. The hard core correlations are modelled with
a hard sphere expression derived from the PercusYevick
theory. The long range field correlations are accounted for
by using asymptotic potentials of mean force and the hyper
netted chain equation. The short range correlations which
xi
include hydration and hydrogen bonding are modelled with a
density expansion of the direct correlation function. The
model requires six parameters for each ion and two for
water. The ionic parameters are valid for all solutions
and those for water are universal.
The model has been used to calculate derivative prop
erties for six 1:1 electrolytes in water at 25C, 1 ATM. The
calculated properties have been compared to experimentally
determined values in order to confirm the adequacy of the
model.
Xll
CHAPTER 1
INTRODUCTION
Aqueous electrolytes are present in many natural and
artificial chemical systems. For example, the chemical
processes of life occur in an aqueous electrolyte medium.
All natural waters contain salts in concentrations ranging
from very low for fresh water to near saturation for geo
thermal brines. Industrially, electrolytes are used in
azeotropic distillation, electrical storage batteries and
fuel cells, liquidliquid separations, drilling muds, and
many other processes. Since a quantitative description
of the properties of these systems is required for under
standing, design, and simulation, the ability to predict
and correlate the solution properties of electrolytes is
both scientifically and technologically important.
In attempting to fill this need, many models of aqueous
salt solutions have been developed. Essentially all describe
only activities of the components but ignore the volumetric
properties. Several extensive reviews of electrolyte solu
tion models are available in the literature (Pytkowicz,
1979; Mauer, 1983; Renon, 1981). To be concise, the various
models have been classified here into three general cate
gories and a few examples of each briefly discussed. First,
1
2
there are models based on relatively rigorous statistical
mechanical results which can be called "theoretical."
Second, there are those composed of a mixture of rigorous
theory and empirical corrections which can be named "semi
empirical." Third, there are those models which directly
correlate experimental data and are thus termed "empirical."
Neither this classification nor the following list pretends
to be either unique or allinclusive.
Among the "theoretical" models, the earliest and still
the most widely accepted is the theory of Debye and Huckel
(1923) which gives the rigorous relation at very low salt
concentration (the limiting law) for salt activity coeffi
cients but fails at higher salt concentration. This theory
has been amply treated in the literature (Davidson, 1962;
Harned and Owen, 1958). The DebyeHuckel theory considers
an electrolyte solution as a collection of charged hard
spherical ions embedded in a dielectric solvent which is
continuous and devoid of structure. This is the physical
picture generally called the "Primitive Model." The correct
formalism for the application of modern statistical mechani
cal techniques to the "Primitive Model" is given by the
McMillanMayer theory (1945). A major method developed
for this formalism is a resummed hypernetted chain approxima
tion to the direct correlation function. This, together
with the OrnsteinZernike equation (1914), forms a solvable
integral equation for the primitive model ionion
3
distribution function which has been used to calculate
the properties of electrolytes up to 1 M salt concentration
(Rasaiah and Friedman, 1968; Friedman and Ramanathan, 1970;
Rasaiah, 1969). This method requires tedious numerical
calculations to obtain the properties. A simpler and more
generalizadle approach is the Mean Spherical Approximation
(MSA) which has been applied to both primitive (Blum, 1980;
Triolo, Grigera, and Blum, 1976; Watanasiri, Brule, and
Lee, 1982) and nonprimitive (Vericat and Blum, 1980;
PerezHernandez and Blum, 1981; Planche and Renon, 1981)
electrolyte models. The MSA method essentially consists
of solving the OrnsteinZernike (1914) equation for the
distribution functions subject to the boundary conditions
that the total correlation function is minus one inside
the hard core and that the direct correlation function
equals the pair potential outside the hard core. This
is equivalent to the PercusYevick method for rigid nonionic
systems (Lebowitz, 1964). The MSA generally gives good
thermodynamic properties if these are calculated from the
"Energy Equation" (Blum, 1980). It does not yield good
correlation functions and further suffers from the need
to numerically solve complex nonlinear relations for the
value of the shielding parameter at each set of conditions.
This last problem grows progressively worse as the sophisti
cation of the model increases. Due to their complexity none
4
of the modern "theoretical" models is widely used in
engineering practice.
The most successful of the semiempirical models is
that due to Pitzer and coworkers (Pitzer, 1973; Pitzer
and Mayorga, 1973; Pitzer and Mayorga, 1974; Pitzer, 1974;
Pitzer and Silvester, 1976). Model parameters for activity
coefficients have been evaluated for a large number of
aqueous salt solutions, but volumetric properties and multi
solvent systems have not been treated. To construct the
model, Pitzer adopted the "Primitive Model" and inserted
the DebyeHuckel radial distribution function for ions
into the osmotic virial expansion from the McMillanMayer
formalism. This latter is analogous to using the "Pressure
Equation" of statistical mechanics (Pitzer, 1977). The
resulting expression contains the correct limiting law.
He then added empirical second and third virial coefficients
which are salt and solvent specific. Although Pitzer's
model correlates aqueous activity coefficients superbly,
it does not add to the fundamental understanding of these
solutions; further, its extension to multisolvent systems
would pose some serious problems associated with the mixture
dielectric constant as has been recently pointed out (Sander,
Fredenslund, and Rasumussen, 1984). Another semiempirical
approach uses the NRTL model for solutions of nonelectrolytes
(Renon and Prausnitz, 1968) adapted for short range ion
and solvent interactions (Cruz and Renon, 1978; Chen, Britt,
5
Boston, and Evans, 1979) in nonprimitive models of electro
lyte solutions. Cruz and Renon separate the Gibbs energy
into three additive terms: an elecrostatic term from the
DebyeHuckel theory, a DebyeMcAulay contribution to correct
for the change in solvent dielectric constant due to the
ions, and an NRTL term for all the short range intermolecular
forces. Chen et al. adopted a DebyeHuckel contribution
and an NRTL term for the Gibbs energy but no DebyeMcAulay
term. More recently, the UNIQUAC model for nonelectrolytes
has been modified for short range intermolecular forces
in electrolyte solutions (Sander, Fredenslund, and Rasmussen,
1984). The resulting UNIQUAC expression has been added
to an empirically modified PitzerDebyeHuckel type electro
static term to form the complete Gibbs energy model.
Although the two NRTL and the UNIQUAC models correlate
activity coefficient data reasonably well even in multi
solvent systems, they have to be regarded as mainly
empirical. First, their resolution of the Gibbs energy
into additive contributions from each different kind of
interaction is not rigorous. Second, the problems associated
with the mixture dielectric constant are resolved in an
empirical and somewhat arbitrary fashion. As a result,
such models add little to our understanding of these systems
and may not be reliable for extension and extrapolation.
Of the various empirical methods developed, two have
been chosen to be discussed here because they represent
6
distinct approaches. First, there is the method of Meissner
(1980) which is a correlation for the salt activity coeffi
cient in terms of a family of curves that are functions
of the ionic strength and a single parameter which can
be selected from a single data point. This method has
been extended to multicomponent electrolyte solutions and
is useful over a wide range of salt concentration (0.120
MOLAL), though it is not very accurate. Second, there
is the method of Hala (1969) which is more conventional
in that it consists of a purely empirical model for the
Gibbs energy of the solution. This method is an excellent
correlational tool, but it is not predictive. It has four
parameters per saltsolvent pair.
The existence of so many models to correlate and predict
the thermodynamic behavior of electrolyte solutions is
indicative of the complexity of these systems and, perhaps,
the relatively poor state of the art.
As examples of the physical complexity of electrolyte
solutions, the composition behavior of the salt activity
coefficient (Figure 1) and of the species (ions and solvents)
density (Figure 2) is presented. Figure 1 shows the large
deviation from ideal solution behavior (y= 1) even at
very low salt concentration for all salts. Second, it
indicates that salts of the same charge type show similar
behavior at low salt concentration but are widely different
at higher salt concentration. In Figure 2, the difference
7
in the salt composition behavior of the species density
is obvious even for relatively similar salts, i.e., the
solution seems to expand for KBr while it seems to contract
for all other salts. The activity coefficient data were
taken from the compilation by Hamer and Wu (1972). For
NaCl and NaBr the density data of Gibson and Loeffler (1948)
were used. For LiCl, LiBr, and KBr the density data were
taken from the International Critical Tables. For KC1
the density data of Romankiw and Chou (1983) were used.
In the hope of improving the situation for obtaining
properties of solutions, a new model of strong aqueous
electrolyte solutions is presented here. This model has
been carefully constructed so that it overcomes a number
of the deficiencies of previous methods. For example,
this model is simple enough for economical engineering
calculations, yet sufficiently sophisticated to rigorously
include all the different interactions (ionion, ionsolvent,
solventsolvent) and the principal physical effects (electro
static, hard core repulsion, hydration, etc.) that contribute
to each interaction. The model is also extendable to multi
salt and multisolvent systems in a straightforward fashion.
Finally, it addresses both activity and volumeric
properties.
In the chapters that follow, a detailed development
of the new model is presented. Chapter 2 has the general
relations between solution properties and correlation
8
functions. Chapter 3 has the full development of the new
model. Chapter 4 shows the application of the model to
solutions of aqueous strong electrolytes and the calculation
of solution properties. Chapter 5 has suggestions for further
work and conclusions.
9
(Molality) 2
Figure 1. Salt Activity Coefficient in Water at
25C, 1 ATM. Data of Hamer and Wu
(1972 ) .
Species Density (Mpl/ml)
Figure 2. Species Density in Aqueous Electrolytes
at 25C, 1 ATM. For data sources see
text.
CHAPTER 2
FLUCTUATION THEORY FOR STRONG ELECTROLYTE SOLUTIONS
Introduction
There are three general relations among the thermodynamic
properties of a solution and statistical mechanical correlation
functions. The first two are the socalled "Energy Equation"
and "Pressure Equation" which are obtained from the canonical
ensemble with the assumption of pairwise additivity of inter
molecular forces. These equations relate the configurational
internal energy and the pressure respectively to spatial
integrals involving the intermolecular pair potential and
the radial distribution function (Reed and Gubbins, 1973;
McQuarrie, 1976). The third relation is the socalled "Com
pressibility Equation" which is derived in the grand canonical
ensemble without the need to assume pairwise additivity of
intermolecular forces. This equation relates concentration
derivatives of the chemical potential to spatial integrals
of the total correlation function (Kirkwood and Buff, 1951)
and to spatial integrals of the direct correlation function
(O'Connell, 1971; O'Connell, 1981). This last method is
generally known as Fluctuation Solution Theory.
Fluctuation solution theory has been applied to the
case of a general reacting system (Perry, 1980; Perry and
11
12
O'Connell, 1984), and the formalism has also been adapted
to treat strong electrolyte solutions which are considered
as systems where the reaction has gone to completion (Perry,
Cabezas, and O'Connell, 1985). The main body of this chapter
consists of a derivation of the general fluctuation solution
theory. Although the final results are identical to those
previously obtained by Perry (1980), the development is
more intuitive and mathematically simpler, though less
general. The remainder of the chapter illustrates the
calculation of direct correlation function integrals (DCFI)
from solution properties and sets theoretically rigorous
infinite dilution limits on the DCFI's.
Thermodynamic Property Derivatives and Direct
Correlation Function Integrals
A general multicomponent electrolyte solution, contain
ing n species (ions and solvents) formed from nQ components
(salts and solvents) by the dissociation of the salts into
ions, is not composed of truly independent species due
to the stoichiometric relations among ions originating
from the same salt. It is, therefore, not possible to
change the number of ions of one kind independently of
all the other ions. However, the independence of ions
has been assumed traditionally for theoretical derivations,
and it will lead us to the correct results by a relatively
simple mathematical route. Thus, with the assumption that
any two species i and j are independent of all other species,
13
Fluctuation Solution Theory gives the following well known
result (O'Connell, 1971; O'Connell, 1981):
RT 9N .
3
T, V, N
(21)
where
N.
i
N
P
the chemical potential per mole of
species i.
the number of moles of species i.
the total number of moles of all species.
the Kroniker delta.
CO
2
4up J r dr = spatial integral of
J 1 1 co
0 J
the direct correlation function.
^ = molecular density of all species.
The microscopic direct correlation function
c 1 j CO
is an angle averaged direct correlation function defined
by
=
^ co
'13
dco dco
where
= J dco. dco .
J i 1
(22)
In order to arrive at the first and simplest of the
desired relations, we define the activity coefficient for
species i on the mole fraction scale as
14
Ui(T,P) = UV(T) + RT ln XiYi(T,P_)
23
where
P = the reference chemical potential.
1 Ni
N = mole fraction of species i.
Y^ = the activity coefficient of species i,
P_ = the vector of species mole densities.
By differentiating equation (23) with respect to
the number of moles of species j, we obtain
i_
RT 3N .
3
T, V, N
SlnY^
3N
ii 1
j T, V, N
N.
l
N
k^j
which upon insertion in equation (21) gives
31ny .
i
3N .
3
1 C. .
iJ.
N
T, V, N
k^j
(24)
(25)
and when multiplied by the system volume on both sides
of the equation,
31nyi
1 C. .
u
(26)
T,p
k^j
N
Pj = y= molar density of species i.
where
15
By performing a sum over all species i and j on equation
(26)
. n n 3lny.
1 I I V. v.
. , ia _
v v N S 'iot' j6 3p .
a B i=l j = l J
T,p
, n n 1C
i l l V. V .
VB i=l j1 101 33 p
(27 )
where
= number of species i in component a.
= total number of species in component a.
By noting the definition of the mean activity
coefficient of a component a,
1 n
lny = J v lny.
ot v ia i
a i = l
(28
and also assuming that species j is formed from an arbitrary
component 8 so that
pj = Vj 3 Po B
29 )
one then arrives at the first relation
, 3lny
1 a
vb 3poe
T, p,
y^3
i n n 1C. .
y v v. v. ii
P L1 , rajB v v
i=l j=l a B
(210)
16
which upon identification of
1 
n
n
l l v.
il jl 101
1C. .
v.0
30 vaV 3
assumes the simpler form
(211)
3lny
pv
a
a
3p
06
T / P o
= v v (1C )
a 6 ct 6
Yt^B
(212)
Equation (212) relates the DCFI to the derivative
of the activity coefficient of any component a (salt or
solvent) with respect to the molar density of any component
6 (salt or solvent) at constant volume, temperature, and
mole number of all components other than B.
Because most experiments and many practical calculations
are performed at constant pressure rather than constant
volume, it is of interest to derive a relation between
the activity coefficient derivatives at constant pressure
and direct correlation function integrals. First, a change
of variables is executed.
3p.
9N .
D
T, V, N
k^j
T,P,Nk?ij
+
3P
T, N
T, V, N
k^j
and the following identifications are made,
(213)
17
T, N
V.
l
(214)
3 P
3V
V .
3P
T,N 1
3N .
3N .
V K_
1
T'V'Vj 3V2
1
T'P'Nk^j
Then equations (21), (214), and (215) are inserted into
equation (213) to obtain
RT 3N .
1
where
T, P, N
k^j
6. C.
= _a:
N. N
l
V. V .
 1 3
Vk RT
(216)
V. =
i 3N.
= partial molar volume
T,P,Nk^i
volume of species i.
V = the system volume.
1 3V
KT V 3P
= isothermal compressibility.
T, N
To develop a relation in terms of activity coefficients,
the chemical potential is written as in equation (23)
and the proper constant pressure derivative is taken.
RT
T, P N
3lnYi
3Nj
+
T, P N
k^j
11 I
N N
(217)
From equations (216) and (217),
18
31nyi
3N .
3
T,P,N
1 C. V. V .
H i 1
N VKtRT
(218)
k^j
By multiplying equation (218) by the system volume
and rearranging, one finds
3 lny.
N x
pKTRT SNj
T, P, N
1 C.. V. V.
hi 1 1.
pKTRT
KmRT < RT
T T
(219)
k*j
upon which a double summation over species i and j is
performed to obtain
N
n n
31nyi
pKmRT v v L ,L 'ia'jB 3N.
T a 8 i=l j=l J i
l l v. v.
. ,Ln a i
T,P,Nk^j
n n
 y y v. v. (ic ) 
PKTRT Vb i=i j=i 101 ^ ci^
i i n n
^~2 I I ViaViB ViVi
(VT) vaVB 1=1 j = 1 B
(220
Equation (220) must be simplified so that all the properties
appear as component rather than species quantities. This
is done with the aid of equations (28), (211), and the
assumption that species j is formed from an arbitrary
component B so
N = v n o
] ]D OP
(221)
19
Additionally, the partial molar volume of a component a
or 8 is expressed as a sum of the species partial molar
volumes.
 9V
v =
Vi 9N.
9V
T,P,N.
k^i
Vi 8
9N
08
T, P, N
Yt*3
(222
V
9V
08 9N
08
n
= l v. V.
18 i
T,P,N
i=l
Yt^B
Equation (220) is now transformed to
(223)
Nv 91ny
a 'a
pK RT 9N _
T 08
T, P, N
v v (1C .
a 8 a8
Vb
V
oa
V
08
p
k RT k RT
T T
(224
To make further progress, the relationship of the
7
oc
bulk modulus of the solution (pk RT) and the group,
to the direct correlation function integrals must be found.
First, the compressibility equation is derived from the
basic fluctuation theory result of equation (21) starting
with the GibbsDuhem equation for an isothermal but
nonisobaric process.
n
I
i=l
N.dU. = VdP
i l
(225)
Upon differentiation of equation (225) with respect
to the mole number of an arbitrary component j,
20
n
l N. ^
. i 3N .
i=l o
= V
T, V, N
3P
3N .
D
T'v'Vj
and by insertion of the equation (21),
(226 )
n
RT l (6.. X.C..)
i=l ID i ID
= V
3P
9N .
D
T, V, N
(227)
1 3P
RT 3p .
D
n
= 1 l X. C..
T,p
i=l
i ID
(228 )
Equation (228) is the general multicomponent compres
sibility equation expressed in terms of species quantities,
This relation is now transformed to one in terms of com
ponents by performing a summation over species j and use
of equations (29) and (229).
n
o
x, = I V X_R
i ip op
(229 )
1 n
3P
RT .L. jot 3p .
D=1 D
T, p
n n
= l~l l X.v C. .
i=l j=l 1 =>a ^
(230)
1 3P
RT 3p
oa
T rP,
Y^B
n
o n
n
1 
l x a l v. 0 l v C. .
61 oSi=l lBjil ]a 13
(231)
21
which by use of equation (211) becomes
n
_1_ _3P_
RT 3p
oa
T,P,
o
= i v y v n x D c Q
a o_i p 3 a8
p1
(232 )
Y^a
Equation (232) is the multicomponent compressibility
equation expressed in terms of components. The density
derivative of pressure is related to the partial molar
volume as
3P
3p
oa
= V
3P
3V
3V
T, p,
3N
T,N oa
V
oa
(233
Y^a
T, P, N
which when inserted in equation (232) gives one of the
desired relations.
V
oa
ktRT Va
n
o
I v
8=1
8 Xo6
ac fl)
a8
(234)
In order to relate the bulk modulus to direct correla
tion function integrals, the total volume is related to
the partial molar volumes.
V
0
l
a=l
3 V
3N
oa
N
oa
T,P,N
5Y^a
n
o
l
a=l
V N
oa oa
(235)
Dividing equation (235) by the mole number of species
(N) yields
1 = V
p N
n
l
a=l
V
oa
X
oa
(236)
22
and when applied to equation (234)
3P/RT
9p
T,N
n n
&
(237)
which is the second necessary relation.
Substitution of equation (234) and equation (237)
into equation (224) gives
Nv 31ny
a a
PKTRT 3NoB
T,P,N
o
Yt^B
v v
a
B
n
o
l
Y=1
n
o
l
6=1
v v X
Y 6 oy o
6
[ (1C
Y
uc fl)
a B
(1C,
ay
)(1"C6B
(238)
) ]
which is further transformed by substituting for the bulk
modulus
Nv
a
31ny
a
3N
OB
T,P,NoY^6
v v
a
B
n
l
Y=1
n
o
I
6=1
V V X X
Y 6 Oy o
6
[ (1 Cy 6
1C Q)(lC
a B ay
d c6g)]
o
l
y=i
n
o
l
6=1
v v X
y o oy
(1C
Y 6
(239)
Equation (239) relates changes in the activity coefficient
of any component a with changes in the mole number of any
component B where the process occurs at constant pressure
and temperature to sums of direct correlation function
integrals for components.
23
In summary, it should be noted that of the various
relations developed in this section, only a few are of prac
tical importance in relation to this work. These are listed
at the beginning of the next section.
Direct Correlation Function Integrals from
Solution Properties
The previous section consists of a relatively simple
but lengthy derivation of several basic relations between
solution properties and direct correlation function
integrals. The relations that are of most importance to
this work are listed below.
T,p
(212)
Y^6
V
oa
n
o
(234)
9P/RT
9p
(237)
24
Nv 91ny
a a
P
T, P, N
n n
o o
= V v l l V V X X .
8 6Â£l y 6 oy o5
Yt8
[(1CY6)(1Ca6) (1CaY)(1C66)]
(238)
1 
N N
l l
i=l j=l
v v (1C. )
ia 16 ii
v v
a 3
(211)
Useful bounds on the value of the direct correlation
function integrals as the system approaches infinite
dilution in all components except one (usually the solvent)
can be deduced from the preceding relations. Thus, by
taking the limit of pure solvent (component 1), one obtains
V
91 = (lc )
v kRT v al
a 1
(240)
polKlRT
UCll)
241
N 31ny
ol ^_a
*8 3NoB
= (1C
a8
T, P, N
PCO CO
polVo2VoB
VaVBKlRT
242a)
VB
N 31ny
ol a
VB 3NoB
T,P,N
= (1C .)
a 3
(1C ) (1Cfl1
al 31
CO
dCu
(24 2b)
25
where dC 6>" "xl( ^oB1
ol
and where equations (242) represent a constant pressure
limit on the DCFI. A corresponding constant volume limit
can be obtained from equation (212).
For a binary system consisting of one solvent (1)
and one salt (2), the fluctuation relations become
31ny
3p
o2
T,p
ol
(243)
V
o2
K RT
T
V2Xol(1 C12}
+ V^X (1C
2 o2 22
(244)
3P/RT
3p
1 2
= = X n (1Cn ) +
T,K PKTRT 01 11
2 2
vX X _(1C10) + v_X _(1C )
2 ol o2 12 2 o2 22
(245)
Nv ^ 31nY2
Hs^
T,P,N
ol
v^X2. [(1C..)(1C__) (lc,
2 ol 11 22 12
(246 )
V +2 ^ Cl+) + V2 (1 Cl'>
'12
1
(247)
26
1 C
22
v+2 (^M1 +2v+2v2 < ,+v2 < !C
V.
(248)
and the respective infinite dilution limits are
oo
Vo2
v2
dc21r
1
p f k,RT
ol 1
oo
* (1cn
Kol 31nY2
CO
V2 3No2
T,p,Noi
(ic r pf
v2 Kirt
Nol 31nY2
v2 3No2
T,P,Noi
2
(1C
, 2
(1c22) 
12
(1C11)
(249)
(250)
(251a)
(251b)
The significance of equations (251) can be further
understood by realizing that any correct model for the
27
activity coefficient of an electrolyte must approach the
DebyeHuckel Limiting Law at very low salt concentration.
Thus, the mean activity coefficient of a salt on the mole
fraction scale is given by this law as
lnY2
1 n
L I
'2 i=l
z2) i1/2
(252)
where
S =
Y
97r 6 1/2
2ie ^
3 3 3 J ~
D kJT
DebyeHuckel limiting law coefficient.
e = the electronic charge.
= pure solvent dielectric constant.
k = Boltzmann's constant.
T = temperature.
Z. = valence of ion i.
x
1 V 72 .
I =2 L Zipi = lonic strength.
i=l
and when the proper derivative is taken,
N 81ny?
? t,p,noi
ol
3N
o2
P 1/2
S p I
_ Y ol
4V?
n n
I I
i=l j=l
2 2
v._v Z.Z .
i2 j2 i j
(253)
28
Insertion of equation (253) into equation (251a)
gives
vV1/2
Y ol
4v?
n
I
n
l v v Z2z2
j=l 12 ]2 i ]
+
ol
_ CO 2
(Vo2}
v2
= (lc
22
(254)
which approaches negative infinity as the salt concentration
approaches zero.
In order to construct a model capable of correlating
and predicting the solution properties of electrolytes,
it is helpful to calculate the experimental behavior of
the DCFI's from solution properties. To that purpose,
equations (243), (244), and (245) have been inverted
so that the three DCFI's can be calculated from
1 c = 5^ [1 X V p]2 +
11 xo2pktrt 02 02
X
ol
9 lny.
V
V 2 2 9N o
ol T,P,N
ol
(255)
1 C12 v2X < RT [1 Xo2Vo2P^
4 ol T
Xo2 31ny2
N
X 9N _
ol o2
T, P,N
ol
(256)
29
1 C
22
(257)
Figures 35 show the results of equations (255) and
(257) for six different salts at 1 ATM and 25C. The
compressibility data used were those of Gibson and Loeffler
(1948) for NaCL and NaBR. For LiCL, LiBR, KCL and KBR
the compressibilities of Allam (1963) were used. The
activity coefficient data were taken from the compilation by
Hamer and Wu (1972). The density data of Gibson and Loeffler
(1948) were again used for NaCL and NaBR. For LiCL, LiBR,
and KBR the density data were taken from the International
Critical Tables. The newer density data of Romankiw and
Chou (1983) were used for KCL. The pure water data were
those of Fine and Millero (1973). The infinite dilution
partial molar volumes were also from Millero (1972).
Summary
The present chapter has introduced the basic relations
of interest, has shown how they have been used to calculate
the experimental behavior of the DCFI's, and has given
some bounds on the values of the DCFI's. The next chapter
introduces a model for correlating the observed experimental
behavior of the DCFI's.
X 02
Figure 3. Salt (2)Salt (2) DCFI in Aqueous
Electrolyte Solutions at 25C, 1 ATM.
For data sources see text.
31
X 05
Figure 4.
Salt (2)Water (1) DCFI in Aqueous
Electrolyte Solutions at 25C, 1 ATM.
For data sources see text.
Figure 5. Water (l)Water (1) DCFI in Aqueous
Electrolyte Solutions at 25C, 1 ATM.
For data sources see text.
CHAPTER 3
A MODEL FOR DIRECT CORRELATION FUNCTION INTEGRALS
IN STRONG ELECTROLYTE SOLUTIONS
Introduction
In order for the formalism introduced in the previous
chapter to be of practical value, a model to express direct
correlation function integrals in terms of measurable
quantities (p, T, x) must be constructed. The present
chapter describes such a model. First, a general physical
picture of electrolyte solutions and its relation to micro
scopic direct correlation functions is discussed. Second,
a rigorous statistical mechanical basis is laid for the
microscopic direct correlation functions and their spatial
integrals. Third, equations are given for each type of
pair correlations in the system (ionioin, ionsolvent,
solventsolvent). Lastly, a summary is presented of the
model parameters and estimated sensitivity of results to
their values.
Philosophy of the Model
The complex thermodynamic behavior of liquid electro
lytes is the observable result of the very complex interac
tions between the species in solution, i.e., the ions and
33
34
solvent molecules. In the absence of a complete understand
ing of all these forces, models use simpler or, at least
tractable, interactions which may have the essential charac
teristics of the real forces. In addition, some semiempiri
cal terms are used to account for those interactions that
cannot be simply approximated.
Thus the interactions between the ions at long distances
are modeled as those of charges in a dielectric medium
containing a diffuse atmosphere of charges. At very short
range, however, the dominant interaction becomes a hard
spherelike repulsion. There exist rigorous statistical
mechanical methods to treat these two types of interactions,
but these two are not adequate to correlate and predict
the solution behavior with sufficient accuracy. Interactions
that are important at intermediate ionion ranges must
be incorporated. Unfortunately, these intermediate range
forces cannot be simplistically approximated because they
involve strong manybody effects such as dielectric satura
tion, ionpairing, polarization, etc., which are not well
understood. In the present model the ionic and hard sphere
interactions are treated theoretically while the rest are
included in a semiempirical fashion.
The interactions between ions and solvent molecules
at large separation can be treated as those of charges and
multipoles in a dielectric medium containing an ionic
atmosphere. In general, quadrupoles and higher order
35
multipoles are not included, because their contribution
is expected to be numerically insignificant in an aqueous
system. The short range interactions are treated as hard
sphere repulsion. Intermediate range forces for the ion
solvent case are very important because they include solva
tion which makes a larger contribution than the long range
chargemultipole forces. Solvation of the ion by the solvent
is intimately related to the partial molar volume of the
salt and must be incorporated if there is to be any hope
of correlating and predicting the volumetric behavior of
the solution. As for ions, the long and short range intera
tions are treated theoretically while the intermediate
range forces are incorporated semiempirically.
The forces between solvent molecules at long range
can be considered to be those of dipoles in a dielectric
medium which has an ionic atmosphere. Higher order multi
poles may again be neglected because their contributions
are less important and can be covered in other ways. The
short range forces are again treated as hard sphere repul
sions. The intermediate range interactions for the solvent
solvent case are dominated by association type forces such
as hydrogen bonding which make a larger contribution than
the long range dipoledipole term. As above, the long
and short range interactions are treated theoretically
while the effects of the intermediate range forces are
included semiempirically.
36
In summary, there are three distinct classes of inter
action: ionion, ionsolvent, solventsolvent. Each class
has unique contributions from longrange, fieldtype forces,
shortrange, repulsive forces, and intermediate range forces.
Traditionally, models have been written for the excess
Gibbs or Helmholtz energy of the system by adding contribu
tions from some of the above forces in an ad hoc and, gener
ally, nonrigorous fashion. The fact that free energy
contributions do not naturally separate into the types
of forces and that experimental values for each cannot
be separately determined has caused many of these models
to be complex and/or inconsistent. Further, they do not
yield volumetric properties along with the activities.
Within the framework of Fluctuation Solution Theory,
the contributions of the pair correlations to the thermo
dynamic properties can be rigorously added. Thus, there
are terms from the saltsalt, saltsolvent, and solvent
solvent DCFI's, as shown in Chapter 2. Further, the experi
mental behavior of each of the three DCFI types can be
separately calculated from solution data as seen in the
previous chapter. It is then possible to construct separate
and accurate models for each one of the DCFI's. These
models can later be manipulated to yield thermodynamic
properties.
As may be inferred from the above discussion, each
of the three types of DCFI's contains long range, short
37
range, and intermediate range interactions. These can
be theoretically separated into a simple additive form
as will be shown in the next section of this chapter.
It is important to note that the separation is first
developed at the level of microscopic direct correlation
functions which are later integrated to obtain the DCFI's.
Although our particular additive separation of the micro
scopic direct correlation function is not fully rigorous,
we believe it is more reasonable than a similar resolution
of the radial distribution function into an additive form
(Planche and Renon, 1981). In fact, the radial distribution
function can naturally be resolved only into a multiplicative
rather than an additive form. The intermolecular potential
and, consequently, the potential of mean force can be
approximately decomposed into additive contributions from
interactions of different characteristic range, but this
potential appears in an exponential in the radial distribu
tion function. Thus, resolving the radial distribution
function into additive contributions is quite inappropriate.
Statistical Mechanical Basis
The above philosophy is a qualitatiave concept which
must be expressed in quantitative terms. To this end,
we now establish a rigorous statistical mechanical basis
for a model of microscopic direct correlation functions.
First, consider the diagrammatic expansion of the direct
38
correlation function (Reichl, 1980; Croxton, 1975) for
species i and j,
c. .
11
(T,P,ri,rj,
where
u. .
il
B. .
il
Â£n g
il
+ B. .
il
(31)
W. ./kT
i j
= e = radial distribution function.
= potential of mean force.
= pair potential.
= sum of all bridge diagrams also known
as elementary clusters.
Although equation (31) is an exact expression for
the direct correlation function, it is of little practical
value because the bridge diagrams cannot be summed analyti
cally. This series is
_1_
2 !
n
l
k=l
pkpÂ£ J fikfkÂ£ fÂ£jfiÂ£fkj dikd^dV^Â£
(32)
for a
where
system consisting of
f. = e
il
u. ./kT
il
n species.
1 = Mayer bond function.
To obtain the hypernetted chain (HNC) approximation
(Rowlinson, 1965) all of the bridge diagrams are neglected
(B^j = 0). This introduces an error which is second order
39
in density and ignores some four body contributions. It
is, therefore, exact up to the order of a third virial
coefficient. Thus, the HNC direct correlation function
is
HNC no Ui
= g. 1 in g. 7
13 ^13 ^13 kT
(33)
From the definition of the radial distribution
function,
W. .
gij = is1
(34)
g. 1 =
yi3
2 3
1 W .
kT + 2T ^kT ) Urn ) +
3 ^ kT
+
(35)
which on insertion into equation (33) gives,
c
HNC
i j
(36)
To apply equation (36) requires at least approximate
expressions for the potential of mean force in terms of
measurable variables. Such expressions, valid in the limit
40
of zero salt concentration and large separation between
the two interacting species, are available for ionion
interactions from the DebyeHuckel theory and for iondipole
and dipoledipole interactions from more recent work (H^ye
and Stell, 1978; Chan, Mitchell, and Ninham, 1979) which
yields results identical to those of Debye and Huckel for
ionic activities. Thus, the long range direct correlation
LR
function is based on these potentials of mean force, W^.
Then, our HNC approximation is
LIM
I*o
r. .
il
cHNC
il
c
LR
i j
(37)
(38)
The potentials of mean force, however, are unphysical
inside the hard core of the molecules and must be set equal
to zero.
T7lr
w. .
11
= 0 r. < a. .
il 13
(39)
T 7 LR
w. .
11
ttLR
= W . r . > a .
il il il
a.. (a . + a .)
il 11 11
= distance of closest
of species i and j
approach
where
41
At contact and inside the core of the molecules, the
direct correlation function is dominated by a very strong
repulsion which is modelled as a hard sphere interaction.
To obtain the appropriate expressions for the hard sphere
direct correlation functions, the PercusYevick theory
(1958) was used since it has been shown to give a compres
sibility equation of state which is in good agreement with
simulation results for hard spheres (Reed and Gubbins,
1973). The PercusYevick (PY) microscopic direct correlation
function for hard spheres is zero outside the core. Thus,
HS
c .
13
PYHS
c .
13
PYHS
c .
13
HS ,, m
u.,/kT
e 11 )
HS n
Uij =
r . > a .
13 13
where
HS
u. =
13
r. < a. .
13 ~ 13
(310)
(311)
Although the PY microscopic direct correlation function
is formally used in the development that follows, it was
not actually employed in obtaining the final expressions
for the DCFI's. Rather, the expression for the hard sphere
chemical potential as derived from PercusYevick theory
through the compressibility equation was used together
with equation (21) to obtain the desired relation (see
42
Appendix A). Although the more exact CarnahanStarling
(Carnahan and Starling, 1969; Mansoori, Carnahan, Starling,
and Leland, 1971) expression could have been used, it is
somewhat more complex and relatively little improvement
in accuracy would be expected.
At this point, we have established a viable, albeit
traditional, theory for the behavior of the direct correla
tion function as r. 00 and at r. < a. .. However, many
interactions which are important in aqueous electrolyte
systems such as hydration of ions by water, hydrogen bonding
between water molecules, and ion pairing are strongest
at r^j just outside the core. Further, this is that kind
of interaction for which liquid state theory is not well
developed. Therefore, we attempt here to develop a method
for interpolation of the direct correlation function between
long and short range. Because generally available theory
offers little guidance, the method can at best be semiempiri
cal. For this purpose, the RusbrookeScoins expansion
of direct correlation function (Reichl, 1980; Croxton,
1975) for species i and j in a system of n kinds of species
is now introduced.
c..(T,p,r.,r .,oj.,(*).) = f..(T) +
lj 1j 1 3 lj
n
+ Â£ p,(. (T) + ... (312)
k=l K 1:,K;
hjk(T) = I 1 fij fik Â£jk dikdk
where
43
Since equation (312) represents the entire direct
LR HS
correlation function including c. and c.these two must
13 13
be subtracted to obtain the interpolating function. There
fore, the complete model for the microscopic direct correla
tion function for species i and j in a system of n species
is
HS LR L A
c. = c. + c. + Ac. .
13 1: id
13
(313)
HS LR
where c^j is defined by equation (311), c^ by equation
( 38 ) and
A HS LR
Ac. = c. c. c. .
13 13 13 13
(314)
which is approximated by the RushbrookeScoins expansion
as
Â£HS rLR. r ,
13 id ID 13 k yi3k
k=l
aHS
pk ^ijk
o LR .
pk *ijk>
(315
where
o LR
Pk ^ijk
LIM
IT >00
i j
I o
pk ^ijk
The series in equation (312) is truncated at the
first order term in density to be consistent with the HNC
44
theory and because inclusion of the more complex higher
order terms was empirically unnecessary.
For the sake of simplicity in notation equation (315)
is expressed as
Ac_ = Afj + (Pk A*jk P +k)
13
where Af.. = f.. fHS fLR
11 il il il
(316)
HS
A(J)ijk fijk ijk
No attempt was made in this work to analytically calcu
late the coefficients in equation (316); rather, their
spatial integrals were fitted to data. The importance
of equation (316), however, is in providing a theoretical
framework for describing the properties for a class of molecu
lar interactions which are not well understood. Thus,
the first term represents the contribution of pairing or
repulsion in the case of ion pairs, solvation in the case
of ionsolvent pairs, and hydrogen bonding in the case
of solvent pairs. The second term represents the effect
of a third body (k) on the direct correlation between species
i and j. If one or two of the three are solvent and the
rest ions, then this term is dominated by hydration. If
all three species are ions, then this term is dominated
45
by ion association or repulsion. The physical significance
of these terms will be discussed further below.
As pointed out in Chapter 2, solution properties are
related to spatial integrals of the direct correlation
function. In order to relate this model to thermodynamic
properties, equation (313) is integrated over angles first
and separated later. Thus
(317)
CO
(318)
(319)
+
(320)
Lastly, A j is defined by formally integrating equation
(316 ) .
46
where
= p AF j (T)
AF ( T) = 4 tt
i j
n LR
+ p l (pvA$(T) p?$(T))
k=l ijk K ijk
< Af. .>
, i: <*>
(321)
A4> (T)
ijk
4TT
CO
J
< A .
ijk
>
00
LR
$ (T)
ijk
4tt J
,,LR n
<0 >
ilk oo
'13
Equations (319), (320), (321), and the expression
for from Appendix A are the general forms of the model
for species direct correlation function integrals. To
obtain practical expressions one needs merely to introduce
the appropriate pair potential and potential of mean force
into equation (320) and perform the indicated integrations
as illustrated in the sections that follow.
Since the coefficients in equation (321) are fitted
to data rather than evaluated analytically, it is of
importance to develop mixing rules to reduce the amount
of data necessary to model multicomponent systems. The
aim here is to predict all the coefficients from quantities
associated with no more than two different species so that
only binary or commonion solution data would be required.
For aqueous electrolytes, the situation can be improved
47
due to the relative simplicity of ionion interactions
which can be generally scaled with the ionic charge (Kusalik
and Patey, 1983). Thus, two and three ion coefficients
are expressed from quantities related to a single ion.
If i, j, and k are ions, then
(322)
(323)
(324)
If one or two of the species i, j, and k are solvents while
the remainder are ions, then the mixing rule must be
expressed from quantities involving each of the species
and water. The reason for this is that ionsolvent inter
actions cannot possibly be predicted from solventsolvent
and ionion interactions separately. Therefore, if i is
an ion and j a solvent, then
AF. (T) = AF. .
i;j 13
(325)
If i and j are ions while k is a solvent, then
(326)
48
LR
$(T)
ijk
1 ( 4,LR + $LR
2 11k JDk'
If i is an ion and j and k are solvents, then
(327)
A$(T) = A$iik (328)
ijk J
LR LR
$(T) = $(T) (329)
ijk ijk
Lastly, if i, j, and k are all solvents, then
AF(T) = AF. (330)
ij 1D
A$(T) = A4>. .. (331)
ijk
LR LR
$(T) = <^k (332)
ijk J
It should be noted that these additive mixing rules
are not the only possible ones. In fact, theory would
suggest that geometric mean type mixing rules might be
more appropriate. Geometric mean rules, however, only
work for positive quantities which turned out not to be
the case with our empirically fitted coefficients. This
situation is further discussed in Appendix C.
The last point that needs to be addressed here is
the extension of the model to multisolvent systems. First,
49
HS
the extension of the expressin for is well known.
Second, the extension of equation (320) for requires
potentials of mean force applicable to the system. Assuming
all solvents are dipolar requires only knowing the dipole
moment of each of the solvent molecules and the dielectric
constant of the solvent mixture. Neither of these are
expected to present a problem in general. Third, the exten
sion of equation (321) for involves a few more coeffi
cients and slightly different mixing rules for some three
body terms. Thus, while equations (322) to (327) would
remain the same for all solvents, equations (328) and
(329) where i is an ion and j, k solvents would be altered
to
A$(T)
ijk
^ ( A$. . +
2 133
AW
(333)
ijk
 ( <3?LR 4 <>LR )
2 ij j ikk'
(334)
which reduce to the previous result only when j and k are
equal. Here, any nonadditive interaction between j and
k has been tacitly ignored because the difference in the
interactions between different solvents is likely to be
less important to direct correlation function integrals
than that from the much stronger ionsolvent interactions.
This assumption is based on previous investigation of
50
solventsolvent interactions which are dominated by angle
independent forces (Brelvi, 1973; Mathias, 1978; Telotte,
1985; Campanella, 1983; Gubbins and O'Connell, 1974; Brelvi
and O'Connell, 1975). Finally, equations (330), (331),
and (331) where i, j, and k are solvents would become
(335)
(336)
(337)
The above mixing rules for an aqueous system (single
solvent) have been tested against data for a number of
salts and may be regarded as established. The rules for
a multisolvent system, however, have not been tested.
They can only be seen as physically reasonable in the light
of previous experience but still tentative.
The next two sections deal with the application of
the theory developed here to specific interaction in order
to construct practical expressions.
51
Expression for SaltSalt DCFI
The saltsalt direct correlation function integral
(C .) can be expressed as a stoichiometric sum of ionion
a 8
DCFI's (c^j) given by equation (211).
1
n n
I l
i=l j=l
V V
ia
IB
(1Cii)
v v
a
8
(211)
It is, thus, only necessary to develop general and practical
expressions for the ionion DCFI's and insert these into
equation (211) to obtain a general expression for the
saltsalt DCFI. The basic model for ionion DCFI's is
HS
represented by equation (319). The expression for C^j
has been developed in Appendix A and that for AC^j is given
by equation (321). This section is then chiefly concerned
with performing the integrations in equation (320) to
LR
obtain an expression for C^.
The pair potential between two ions is given by
u
LR
i j
Z Z .e2
1 3
r
(338 )
Here the potential of mean force is approximated by a gener
alized form given by the DebyeHuckel theory.
T7LR
w. .
13
Z Z .e
i 3
kTr. .
13
e
K ( a .
13
D(1+Ka. )
13
r>a^j (339a)
W
LR
13
0
r < a. .
 13
(339b)
52
where
K
4Tre2
DkT
n
I
i=l
DebyeHuckel
inverse length.
D = the dielectric constant of the solvent
or mixture of solvents.
Insertion of equations (338) and (339) into equation
(320) gives
CLR =
13
4ttp Z Z .Q
^ J r . d r .
kT * i] i]
o J J
+
2 2 4
2ttpZ Z e
J J
2Ka. 2Kr. .
! DkT)2 (1+Ka j)2
13
aij
e dr. 
13
3 3 6
2iTp z Z .e
1 J
3(DkT)3 (1+Ka. )3
13
3Ka. .
13
m 3Kr. .
 e ^
r. .
aij ^
dr. +
13
(340)
The first term of equation (340) contains a divergent
integral. However, when it is introduced into equation
(211) which relates it to thermodynamic properties, electro
neutrality makes the coefficients of the integrals sums
to exactly zero.
53
4 it o e
n
n
vavB
, m i v Z. y vZ. f r..dr..=0
i=l
j = l
(341)
n
where
l V. Z. =
i=l la 1
The second term of equation (340) is integrable and
contains the implications for DCFI's of the DebyeHuckel
limiting and extended laws (see equation 254). Then,
2 2 4
2iTpZ z e
2Ka. 2Kr. .
(DkT)2(l+KaiJ2
13
/
13
a .
13
dr. .
13
22 1/2
0 0 JL
1 1 Y
(l+a. .B I1/2)
13 y
(342)
where S = ( 2^e 3 )
Y D k T
K = B I
Y
1/2
B = K I
Y
1/2 ( 8r e
2 1/2
DkT
1 ^ 2
1 = 2 zi
1=1
The third term of equation (340) is also integrable
but more complex. The integral is the first order member
54
of a class of functions known as the exponential integrals.
These cannot be evaluated explicitly but a number of
asymptotic expansions and numerical approximations are
available (see Appendix D). It is convenient to express
the integral in dimensionless form.
Letting X = r/a^ then
e~3Krij (3Kaij)x
dr = J ^ dx = Ex(3Ka ..) (343)
aij ij J 1 J
where E.(3Ka. .) = the
1 13
The third term in equation
first exponential
(340) becomes
integral
2rrpZ3Z3e6
i 3
3 (DkT)3 (1+Ka^ )3
3Ka.
* 13 J
a .
13
3Kr. .
e xj
r .
13
dr. .
13
0 0 3a..B I
Z3z3 S2pe 13 Y
= i 3 _J
1/2
(1+a. .B I
13 Y
yz,T E. ( 3a .B I
1/2,4 i i] y
1/2
(344)
which contains the implications for DCFI's to a higher
order limiting law for unsymmetric electrolytes (Friedman,
1962). Because of electroneutrality, this term,
when
55
inserted into equation (211), is always very small for
symmetric electrolytes, and it approaches zero as the con
centration of salt decreases. For unsymmetric electrolytes,
however, the sum over the ionic charges is not small and
this term actually diverges logarithmically as the salt
concentration decreases. To further explore the relation
of (344) to Friedman's limiting law and to elucidate the
low salt concentration behavior, the exponential integral
(E^) can be expanded for low values of the ionic strength
(I + 0).
E1(3aijB I17"2) = in(3aijB I1^2 ) a + 0(I1//2) (345)
where a = 0.5772 = Euler's constant.
This expansion is valid only at extremely low ionic
strength. Equation (344) then becomes
z3z3
3 1
S2P ^E, (3a. .B I1//2) =
Y ol 1 13 Y
z3z3
31 S2Pq^ (2 ini + a + In 33.^^
+ 0(I1/2)
(346)
where lnl diverges as I * 0 while a + In 3a. .B are all
y 13 Y
constant.
56
The contribution of Friedman's limiting law to the
activity coefficient of a salt (a) is
FIjL
Â£n y =
a
l n
( I v.
1 va i=l ia
Zi)2
V 2
y v. z.
i=i ia 1
sy I&nl
(347)
and by taking the first derivative with respect to the
mole number of a salt 3 at I + o,
3 In y.
FLL
3N
o3
T,P,N a
OY / 3
_,2 P
S p n n o
X_2i Y Y v. v Z3z .
3Vb il jil ia 1 ]
(4 An I +
(348)
Rearrangement of equation (224) gives
N
3 Any
a
3N
o3
= 1C
PV v _
oa o3
a3 VQVp
T, P, N
oy 3
(349)
If equations (348) and (349) are compared, it is
clear that the contribution of Friedman's form of the limit
ing law to the saltsalt direct correlation function integral
is
57
rFLL=
aB
P
ol
n n
I I
i=l j=l
V V
la
Z373(
j3 iZj(2
Â£n
1 +
(350)
Comparing equations (211) and (350) gives the ionion
DCFI.
3 3
rr Z Z
CFLL= x., 3
11 3
*nl + i)
(351)
Substitution of equation (346) in equation (340)
gives the expression for the limiting contribution of the
third term in equation (340) to the ionion DCFI.
Z3z3
i 1
Y
ol
(I
Â£nl +
a
+ In 3a
il
B )
Y
(352)
Equations (351) and (352) have essentially the same
behavior as I * 0 since they differ only by a small constant
which is negligible compared to &nl as I + 0. Therefore,
equation (344) contains the higher order limiting law.
The general expression for CFF is
,LR
'aB
S pi 1//2 n
Y
2 2
n v. v Z7Z .
Â£ Â£ la IB 1 1
4vaV B i = l j=l (1+a. .B l1/2)2
il Y
n
_ 3a. .B I
n v. v Z3Z3 e ^l Y
1/2
3vaVB i=l j=l
l l la IB i j
(1 + a..B I1/2)J
il Y
E1(3a. .B I1/2)
1 ,1J. Y
+
(353)
58
H S
The expression for C is
a B
,HS
 i i cHS
'aB Viiiw i6
(354
Lastly, the expression for AC is
aB
AC
P
n n
l v. v AF. +
B Vs i1 jl 'ia jB "'ij
n n n
P., 1 Â£ Â£ v.v. (p,A4>.., p4>LR )
aVB i=l j=l kl lct ^ k ^k k ^k
(355)
Equations (353), (354), and (355) form the complete
model for the saltsalt DCFI.
C = CESr + CLR + AC
a B aB a 8 a 8
(356)
Since the limits of DCFI1s as salt concentration
approaches zero are well defined, it is advantageous to
use equations (353) to (355) to model the deviations
from this limit. To this purpose, the infinite dilution
limit of the saltsalt DCFI is now explored. From equation
(24 2a)
N 3 in y
ol 1
CO CO
OL
3N
oB
T,P,N
= (lc R)
a B
oy^B
p ?V V
ol oa oB
v v k RT
a B 1
CO
(242a)
59
it is seen that the constant temperature and pressure limit
has divergent terms associated with the activity coefficient,
a first constant related to the partial molar volume, and
a second constant associated with the activity coefficient
and which is not so well defined. This second constant
is loosely related to a term linear in salt density which
often appears in empirical expressions for the salt activity
coefficient (Guggenheim and Turgen, 1955; Guggenheim and
Stokes, 1969). In the present model the divergent terms
are contained in equation (353). The first constant can
be calculated directly from infinite dilution partial molar
volumes and solvent quantities. The second constant must
be fitted to data using terms from equation (355) which
have only ionion and long range ionwater correlations.
This reflects the fact that triple ion direct correlations
are zero at infinite dilution and any contributing short
range ionsolvent correlations would generally be contained
in the first constant. Thus,
c" = LIM (C cLR actb)
cct3 X .,>1 iLctB aB ca6;
ol
(357)
actb = p y y y v. v.^a*..,
aB v v L .L L lot ]BMk ljk
a B i=l j=l k=l J J
n n n
where
60
(1C Q)
a 6
P CO CO
p ,V V .
ol oa og
v v k rt
a g 1
n
n
y y v. v (Af
Vg i=i j=i ia ^ ^
+
*LR
111
(358)
Finally, the general expression for the saltsalt
DCFI model including the infinite dilution limit is
1C
a g
= (1C )" cLR (CHS cHS)
1 ag; ag lCag Cag ;
 (Ac
TB
a g
AC
TB
6
where
CHS = LIM HS
ag X ,kL ag
ol
AC
TB
a g
LIM
X +1
ol
AC
TB
ag
(359)
Mol
V VD
a g
n
I
i=l
n
I
j=l
v. v p AO. ..
ia ]g ol ljl
Although equation (356) can be used in place of
equation (359), it was felt that the latter was more
appropriate for calculations at constant temperature and
pressure. Therefore, equation (359) was used in the com
parisons and correlations in this work. In calculations
61
where pressure varies, equation (356) would be more
convenient since it would eliminate the need to obtain
partial molar volumes as a function of pressure.
For illustrative purposes, equation (359) will now
be written for a binary system consisting of a solvent
(1) and a salt (2) which dissociates to formv+ cations
and v_ anions.
n = 1 + v+ + v_
v
2
1 C22 (1 C22^
CO
LR / pHS HS
^22 ^22 C22
(360)
where
P
2
v
+ 2v+v_(A$^H + p + A$+H + p_A$_+_) +
1
(361)
62
pLR
2 2
S pi
Y
1/2
4v'
2 4
V + Z +
U+a++ByI1/2)2
+
+
2v^z+Z
(l+a+_Bfll/2)2
24
v Z
d+a__B^ll/2)2
] 
2 26 3a++BYI
S p v Z e Y
V [
3v t
1/2
Ei(3a++B,i1/2
(1+a B I
++ y
1/2,3
,a R Â¡1/2
3 3 Byl 1/2
2v v zfz Y E,(3a B I )
+ + 1 I y
(1+a B I1/2)3
+ y
+
 e 3a B I1/2 i ,,
v2Z^e Y E1(3a__B^I1/2)
(1+a B I
Y
1/2,3
(362)
1 
22
P OO
pol(Vo2)
v2KlRT
[v+ (AF++
P, LR,
pol*1++1 +
+ 2v+v_ (AF+_
 pol4/> + V(iF
 V/l11
(363)
,HS HS
1 r .2 /r.HS HSC
(cr c) = ~ [v, (C1 c
'2 2 2 2
+ ++ ++
+
+ 2V+V_ (Cf cf") + v2
(364)
63
Expression for SaltSolvent DCFI
The development in this section parallels that of
the previous one. Thus, a general expression for the
solvention DCFI is derived and then inserted in equation
(211) to yield the saltsolvent DCFI relation.
Although any type of interaction can, in principle,
be included, it was assumed here that ionsolvent interac
tions are dominated by dipolecharge forces at large separa
tion, and no other interactions were included. The pair
potential for an ion (i) and a dipolar solvent (1) is
LR
u.
i
z y. e
i 1
cos
0
(365)
where y^ = the dipole moment of solvent 1
in Debyes.
9 = the Eulerian angle between dipole
and charge.
The potential of mean force is approximated by a func
tional form inspired by some recent applications of the
mean spherical approximation (Chan, Mitchell, and Ninham,
1979) and of perturbation theory (H0ye and Stell, 1978)
to nonprimitive electrolyte models.
LR Z.y.ae
wu r (cos 01 5
kTr .
il
K(a..r )
il il
r > a^ (366a
WLR
wii =
r < a .
il
0
(36 6b)
64
where a is a universal constant that we have set equal
to 4.4 empirically.
Since equations (365) and (366) are functions of
orientation, it is necessary to first perform the integration
over angles as indicated in equation (320).
(367)
where dw. = sin 0.d0.dd>.
i ill
When the integral in equation (367) is evaluated, it is
found that
(368)
The second term in equation (320) has
(369)
After the integral in equation (369) is evaluated, it
gives
(
DkT
(370)
65
The third term in equation (320) contains
<(wii)3>, k I (wii>3 d"i di (371>
o 6
which also equals zero.
< (W
LR, 3
il1
>
U)
0
(372)
Therefore, for iondipole pairs there is only one term
in equation (320).
PLR
il
21 (
3 P (
Vi
DkT
a
2Ka
2Kr
il
e
~2~
i. r. .
il il
il
dr
il
(373)
The integral in equation (373) is also an exponential
integral (Â£Â¡3^ which is expressed in dimensionless form
as before.
f
J 2
a.,r,,
il xl
2Kr.,
il
dr.1 = J
11 aii 1 x2
(2Ka1)X
dX =
E2(2Kail)
ail
(374)
Equation (373) then becomes
2TTp
3
Z.y.ea
1 1
DkT
0 2Ka.,
e 11 E2(2Kail)
17
(375)
66
Then, the general expression for the salt (a) and
solvent (1) DCFI is
2rrp
3v
a
2a B I1//2
2 7 n v Z2e 1 Y /(J
<Â§T>  l 1 a, E2(2a,,B..I1/2
1 = 1 ll
xi r
(376)
HS .
The expression for C ^ is given by
n
HS 1 v HS
C i = / v. r. .
al v L, la nl
a 1=1
(377)
and the relation for AC is
al
n n
ac = y v. af., + y y v.
al v ia ll , L, la
a 1=1 a i=l k=l
( p, A$ p. $ ,. )
k ilk Kk ilk
(378)
Again, equations (376), (377), (378), and (379)
form the complete general model for the saltsolvent DCFI.
C = CH + CL* + AC 1
al al al al
(379)
As previously discussed, it is convenient, particularly
for isobaric calculations, to use the model only for
67
deviations from infinite dilution. (For nonisobaric calcu
lations, equation (379) would be more appropriate.) The
infinite dilution limit of is given by equation (240)
LR
and that of Ca^ is (see Appendix D)
LR LIM LR 2^Pol ea 2 2 ? ViaZi
al X ,+l al 3v DkT Ml.. a..
ol a i = l ll
(380)
while the infinite dilution limit of C is formally
al 1
HS00 LIM HS
Lai X .>1 Sal
ol
(381)
Lastly, the infinite dilution limit of AC is
1 al
a LIM _
AC =v AC .
al X .>1 al
ol
> n
Ol r
) .
a i=l
v a a (AF,
il p ol
P LR ,
ni
+
P
p n D
ol r P.
1. via polA<5ill
v ^, la
a i=l
(382)
The complete general relation for the saltsolvent
DCFI including the infinite dilution limit is
V
oa
1 v k,RT
a 1
(cHS cHS~)
' 'a l 'a l '
(cLR cLR)
lLal ua 1 '
(AC AC .
a 1 a 1
(383)
68
Finally, equation (383) will now be written for a
binary system consisting of solvent (1) and salt (2) with
v+ cations and v_ anions.
V
1 C
o2
21
v2
(CHS cHS)
^21 ^21 1
, LR LRC
V 21 U21
~ (AC21 AC21}
(384)
where
HS HS<
21 C21
L [v (CHS CHS~) + V (CHS cHS)j
v2 L + ^+l *+l 1
(385)
,LR _LR 2tt
C C
21 21
/ e a 2 j.
3v 1 DkT yl L
v Z2 2a B ll/2
(Pe Y V2a+lV
1/2,
P VZ
Po!} + i"
(pe
2a .B I
1 y
1
1/2
E0 (2a .B I
2 1 y
1/2
" Pol} 1
(386)
A^i
 AC.
21
[v+(aF+i
 P
ol
$ LR)
+ 11'
+ V ( AF
1
 Po*?
(p
+
+ ^ [v+(plA$l+l +
p A4> +
K+ ++1
p_A$_+1) +
69
+ v_(p1Acf>1_i + P+A$+_1 + P_A>__1)]
(387)
Expression for SolventSolvent DCFI
The solventsolvent direct correlation function integral
has the simplest relation since the solvent does not
dissociate so the species and component integrals are the
same.
As previously noted, any type of interaction can gen
erally be included in this theory, but it was assumed that
solventsolvent interactions at large separation are domi
nated only by dipoledipole forces. The solvent (l)solvent
(1) pair potential is
(388)
where
sin 0j^ sin 0^2 cos $12
0 cj> = Eulerian angles of solvent molecule
number 1.
0, p.. = Eulerian angles of solvent molecule
i 3 i
number 2.
The potential of mean force is approximated by a
function inspired by previously mentioned work (Chan,
Mitchell, and Ninham, 1979 ; Hs^ye and Stell, 1978).
70
W
LR
Vl a1"311'111
" kTrll D
ip r > a
11
(38 9a)
"Â£?
r X all
(38 9b'
Again, equations (388) and (389) are inserted into
equation (320) and the required integration over angles
performed.
. LR. L r JjK ,
= o I u, i da) do).
11 u p2 11 11 12
1 r LR
(390)
where
dw. = sin 0. d0. d4> .
li li lx li
2ir
Â¡2 = { d<5 = J sin 0. d0. J d(f>. = 4r
J _Ll J ll ll J ll
The integration of equation (390) gives
. LR. n
=0
11 0)
(391)
The second term in equation (320) has
y iT7LRi 2. 1 r,T7LR\2 .
<(W11) >0) = 72 / (wn } dwn dw
n
11' 11 12
(392)
which yields upon evaluation,
<(WLR)2> = 1
M ll' w 3
y1y1 2
,2K(airrn
DkT
11
(393)
71
The third term in equation (320) has
<(Wl)3>03 ~2 I
394)
which becomes upon integration
= o
(395)
Thus, for dipolar solvents only one term of equation
(320) remains after the angle integrations.
iÂ£Â£ (ifl,2 e2Kan f
3 DkT 1
2Kr
11
dr
all rll
11
(396)
The integral in equation (396) is also an exponential
integral (E^) which can be expressed in dimensionless form.
2Kr
/
11
dr
^lrll
11
1
T~
lll
(2Ka..)X
e 1X
T
x
dX =
E (2Kan.)
4 11
11
397)
Equation (396) is then transformed
,LR 4tt0 (ylyl}
'11 3 DkT
2 e
2ailB 1
1/2
E4(2anB I
1/2
L11
(398)
which is the general expression for the solventsolvent
DCFI.
72
Since the solvent does not dissociate, there is no
HS
summation over species in C.^. However, Ac.^ does have
a sum over third bodies.
ACn = pAFn + p l (pkA*llk P *k) (399
k=l
Equations (398), (399), and (3100) form the complete
general model for the solventsolvent DCFI.
C = CHS + c^R + /\c
11 11 '11 A 11
(3100)
Again, the infinite dilution limit of C^ is introduced
so that for isobaric calculations the model need only account
for deviations from the infinite dilution value. Also,
equation (3100) would be more practical for nonisobaric
cases. The infinite dilution limit of is the bulk
modulus of the pure solvent given by equation (241).
TR
The infinite dilution limit of C.^ is given by
LR LIM LR 471(3
11 X ,*1 11
ol 9 a
L t _L)
3 vDkT '
11
(3101)
HS
and that for C"^ is formally
,HS LIM HS
11 X +1 11
ol
(3102)
73
The infinite dilution limit for Ac^ is given by
Ac = lim Â£
U11 xq1+1 11
p R ( Af p R>RR ) + (p P)2 AO
oll *11 Pol 111' 1 ol' 111
(3103)
Finally, the complete general expression for the
solventsolvent DCFI including the infinite dilution limit
is
1 C
11
P ,
ol 1
 (C
HS
11
 C
HSC
11
) (C
LR
11
cLR)
U11
 ( Ac Ac )
1 ii ir
(3104)
Again, the application of equation (3104) to a binary
system consisting of solvent (1) and a salt (2) with
cations and v_ anions is shown. However, for the solvent
solvent DCFI all of the terms except AC.^ appear similar
to the general case since they have no summations over
species. Thus, only Ac^ is illustrated below.
ACn ACIi <0 PoI)(AFn PoKn1 +
+ + P+A$ + n + P_A
 (AoI)2a*111
(3105)
74
Summary
A general statistical mechanical model of the direct
correlation function has been presented. In principle
it is applicable to any system, but it has been specialized
here to treat strong electrolyte solutions. The next chapter
shows the application of this model to six aqueous strong
electrolyte binary solutions. As a preview to the calcula
tions, the relative magnitude of the three contributions
to the DCFI (CHq, CL^, Ac q) will now be discussed, the
a p a p a p
model parameters will be listed, and the sensitivity of
solution properties to parameter value considered.
The saltsalt DCFI is dominated at very low salt con
LR
centration by Cag which contains the long ranged electro
,LR
static interactions. However, the magnitude of Cag decreases
very fast as the salt concentration increases so that above
2M or so in salt density the dominant term becomes C^g.
This reflects the increasing shielding of electrostatic
forces by more ions that more frequently repel each other.
ACag makes a contribution that is generally not dominant
in either regime but is always numerically significant
above 0.5M.
HS
The saltsolvent DCFI is always dominated by Cag with
,LR
CQ^ making a small but not negligible contribution. Due
to the relative strength of the short ranged hydration
interactions, AC^ makes the largest contribution after
cHS
Lal
75
The solventsolvent DCFI is also dominated by C
HS
11
over the entire range of salt concentration up to about
LR
6M. C^ makes a negligible contribution reflecting the
relative weakness of long range dipoledipole interactions.
HS
Again, the largest term after C^ is AC^ which contains
the short ranged hydrogen bonding between solvent molecules
and the hydration related effect of an ion on two solvent
molecules at short range.
The parameters of the model are species specific and
universal. It is, therefore, necessary to build only a
relatively small set of parameter values to predict the
behavior of a large number of systems. Thus, a hard sphere
HS
diameter (a.^) for each species is required for CQg and
C"a8 (where a, 8 can be salts or solvents). To avoid con
fusion, the parameters for ACag will be those of a system
with one solvent (1), one salt (2), and many ions (i, j).
Then, Ac^ involves Af^ which is ion independent,
Aim which is usually neglected, and AO.,., for each ion.
ill
P T R
AC. 0 has Af. p Or1. AO and AO..... AC0 includes
12 li ol 111 111 ill 22
P
AF.. p AO..., and AO ... This totals to two solvent
11 ol 111 111 111
specific parameters if AO^ is neglected and six parameters
for each ion (note that AO... = AO... and AO... = AO...)
Ill ill ill In
three of which involve solvention pairs.
Properties predicted with the model are most sensitive
.HS
to the value of the hard sphere diameters because the C
is a very strong function of the diameters. But it is
a 8
76
not as sensitive as is the case with other models. This
is due to the fact that the two body coefficients Af. .
ID
are fitted to infinite dilution quantities that include
Hs
Cag so there is a degree of compensation for changes in
the diameters. The sensitivity of the results to the value
of the coefficients in AC is generally small since they
make a small contribution to the DCFI's.
CHAPTER 4
APPLICATION OF THE MODEL TO AQUEOUS STRONG ELECTROLYTES
Introduction
In Chapter 2, the formal relations between DCFI's and
thermodynamic properties were introduced. In Chapter 3, a
model expressing the DCFI's in terms of measurable variables
was constructed. In the present chapter we illustrate the
use of the formal relations and the model in the calcula
tion of thermodynamic properties. We also explore the
scheme used to fit model parameters; further we compare
calculated values to experimental ones for the saltsalt,
saltsolvent, and solventsolvent DCFI1s and for the bulk
modulus, partial molar volume, and salt activity coeffi
cient. Finally, a discussion of the above results and a few
conclusions are presented.
The use of Fluctuation Theory in general fluid phase
equilibria problems has been treated in detail by O'Connell
(1981). The specific case of liquids containing super
critical components has been addressed by Mathias and
O'Connell (1981) and Mathias (1978). The present treatment
generally follows these developments, but there are
important differences for the present case of electrolytes.
77
Calculation of Solution Properties
from the Model
78
The formal relations between solution properties and
DCFI's are given by equations (212), (234), (237), and
(238) for a system consisting of nQ components, salts (cuB)
and one solvent (1).
9 Â£ny
pv
a
a 9p
oB
= v v (1C _)
a B aB
T, p
oy^B
(212)
V
oa
n
Va VBXoB(1 CaB
(234)
9P/RT
9p
T, X
n n
o o
Z Z
a=l B=1
v v x
a B oa
(1C.a)
at
(237)
NV
a
9Â£ny
n
n
a
PKTRT 9Nq3
T, P, N
= v v Z z
a B y=l 6=1
v X
y 0 Oy oo
oy^B
 (1C ) (1C ) ]
ay yB
[(1C .)(1C Q)
y o aB
(238)
79
where
3P/RT
9p
T, N
In order to evaluate the change in solution density
with pressure while the composition and temperature are
constant, one needs to integrate equation (237) from a
R R
known reference density (p ) at the reference pressure (P )
at the temperature and composition (mole fraction) of the
F
system up to the desired density (p ) at the system pressure
(P) .
PP
RT
R
n n
o o
I Z v v y X
a=l B=1 a 6 oa oB
PF(T,P,X)
J (1C )dp
R R P
pK(T,P ,X)
T, N
(41)
Equation (41) represents an implicit equation for the
F
unknown density (p ) which can only be solved numerically
with realistic models.
It should be appreciated that equation (41) cannot be
applied to an isobaric change because that would imply that
pressure, as well as temperature and composition, were
F R
held constant. Then p would be the same as p so the state
of the system would not vary at all.
To evaluate the change in solution density isothermally
with varying composition, a different approach is required.
To develop the necessary relations we start by considering
that in Fluctuation Theory the pressure is treated as the
80
dependent variable, a function of temperature, density, and
mole fraction.
P = P (T, p X)
(42)
Taking the total differential of pressure gives
n
3P
dP = ^
3p
, 3P
dp+
T,N 3T
dT + l
3P
T o 3X
p,N a=2 oa
dX
oa
(43)
T, pX
oy^a
If the change is isothermal and if we divide by RT,
n
RT
dP
1 3P
RT 3p
dp + ^ I
T,N
3P
n 3X
a=2 oa
T, p X
dXoa (44)
oy^a
By making some identifications we obtain
3P
3X
oa
n
o V
Vk
T, V, X
oy^a
3N
I
8=1 W^T
oB
oa
N
oy^B
(45)
3N
oB
3X
oa
N
N
oy^B
6 a~voX
aB B oa
(46)
Inserting equations (45) and (46) into equation (44)
gives
dP
RT
_1_ 3P
T RT 3p
no no pV dX
dp + l l 21 2L
T,N
a=2 6=1
(47)
81
We next insert equations (234) and (237) into
equation (47 ) .
JL_
RT
n n
o o
[ y v V X X (1C
a^2_ a 6 oa oB a8
dp +
n n n
o o o
+ l l l VRVvX0
a=2 6=1 Y=1 6 Y Y
pdX
oa
6 o~v oX
a B B oa
( 48
Equation (48) permits us to evaluate the change in
solution density with both pressure and composition along an
isotherm. This equation is also applicable to an isobaric
and isothermal process where the solution density changes as
a function of composition only.
F
To obtain the density (p ) of a given solution at a
F
known temperature, pressure, and composition (X ), we
isothermally integrate equation (48) from the known
reference density (p ) at a system temperature and a conven
R R
iently chosen reference pressure (P ) and composition (Xq )
F F
up to the desired density and composition (p and XQ ) It
is suggested that for aqueous electrolytes the reference
density be chosen to be that of pure saturated water at the
system temperature.
82
PP
SAT
1
RT
n
o
I
a=l
n
o
I
3=1
a 8
PF(T,P,X^)
SAT XoaXoB(1_Ca8)
pol (T)
+
n n n X
o o o oa
+ I l I v v / X (1C
a=2 6=1 Yl 8 r o OY
pdX
oa
8y' 6 v X
a 8 8 oa
(49)
In evaluating the integrals of equation (49) each
integral involves variables appearing in other integrals.
F
To explicitly find p requires further manipulations
discussed below.
The activity coefficient on the mole fraction scale for
any component (a) can be obtained by integration of equation
(312) from the reference molar density (Pg ) to the molar
F
density of each component (Pg ) at constant temperature.
Jin
V V fP6 j
Y R 61 6 0 R P dP6
Ya B_1 po8
T
(410)
where P a = X flP
OP OP
Equation (410) is applicable to any isothermal change,
isobaric or not, and the reference state composition where
Ya =1 need not be that chosen. However, for aqueous
electrolytes it is natural to choose pure saturated liquid
water at the system temperature as was done for equation
(49).
83
iHYc =
Poi(T,p,X)i_c
al
dp
ol
n P (T,P,X)
+ I VR J CaB
T 6=2
dp
oB
T
(411)
Equation (411) can be used for either isobaric or
nonisobaric changes.
In equations (49) and (411) one can use the DCFI
model represented by equations (359), (383), and (3104)
for isobaric integrations. But, for nonisobaric integra
tions with equations (41), (49), and (411) the DCFI model
of equations (356), (379), and (3100) will be more
applicable because the pressure behavior of the DCFI
infinite dilution limits, some of which involve salt partial
molar volumes, is not generally available.
The composition behavior of component activity coeffi
cients on the mole fraction scale at constant temperature
and pressure could also be obtained from equation (238)
with composition expressed as mole fractions. Thus, we
express the differential of the activity coefficient of a
component ( a) as
n
dÂ£ny
a
o 9 Â£ny
= y Ji
1 a Y
T,P 6=2 OB
dX
oB
T, P, X
oy^B
(412)
84
3 Â£ny
a
3X
06
T, P, X
no 3Â£ny
= y L
r1 3N r
?=1 oÂ£
3N
OC
3X
oy^B
T, P, N
oy^C
oB
(413)
N
oy^?
3N
oC
3X
oB
N
N
oy^C
hrVo,
(414 )
Inserting equations (414) and (238) into (413) and
then putting the resulting expression into equation (412)
gives
dÂ£ny
no no PK RT
= I dX J
T P
B=2 wÂ£=l 6CB Vo?XoB y=l 6=1
n n
o o
I I V
L L y 6
XOYXoi! UCa^) UCoy)(1C4?)] (415)
To obtain the activity coefficient, equation (415) is
isothermally and isobarically integrated from the reference
to the desired state.
n n n n X a
o o o o OP
Â£nYa = l l l I / R dXoB
a a=2 6=1 y=l ?=1 X
oB
p < RT
T
T, P
?B Vo?XoB
VrV
? y o
x x r [ (ic
oy o o y<5
(1Cay)(1'
(416)
85
n n
1 o o
where = = Y Y v v x X (1C )
pKTRT aÂ£1 a goa oB ag;
6 = Kroniker Delta
Equation (416) can only be used for isothermal,
isobaric changes and thus either the DCFI model of equations
(359), (383), and (3104) or that of equations (356),
(379), and (3100) may be used.
Equations (41), (49), (411), and (416) express
integrations of the DCFI model formally. However, these
cannot be explicitly evaluated because of the multiple
variables involved in the integrals. To actually evaluate
these integrals requires a change of variables as discussed
by Mathias (1979) and O'Connell (1981). Rather than give
their formal equations, we now give the above relations with
explicit expressions for the present DCFI model. Those
parts that are analytically integrable have been evaluated
while simplified integrals are given for the others. The
DCFI model used is that of equations (356), (379), and
(3100) which does not contain the DCFI infinite dilution
limits. This form of the model yields simpler expressions
which can be applied to both isobaric and nonisobaric
changes. We start by rewriting equation (41) as
86
PpR PPYHS(pF,X)
RT
RT
dPYHS. R
P (p ,X)
RT
n n n
pi "P O O O Lyp
(p p ) l l I V V X XX J C o (t)dt
a=l 6=1 y=l a 6 oa 03 OY o aB
F F R R n n
p p PP y y x.x. af. .
2 ii j=i 1 3
FFF RRR n n n
p p p ? p p I l l XXX A*
3 i=l j=l k=l 1 3 k 13lc
+
F F P, F R R P, R
p p Pol p p PQl
n n
, LR
l l X.x.
nil jil 1 3 131
(417)
n n
where p(t) ^vy = vypoy(t)
Equation (49) can also be changed to
PP.
SA
T
T)
RT
ppyhs(pF,xF)
RT
ppYHS R R)
o
RT
n
o
 I
a=l
n n
o o
r r F F R Rv
y y v v (x px p
yÂ£x a 6 OY oyM
1 c (t)
tm,t",o6lt) pTtTpTtT
O
dt
87
n n n n
o o o o
l l l l vv__ / n 1"CBY(t)
a=2 8=1 Y
L XL 8yJ p (t)
=1 6=1 M o oy
~v
(t) a
F (t) dt 
oa
a8 8 P(t)
1 v p , F F F F R R R R, A
J l I (X X p p X X. p p ) AF 
i=l j=l 1 1 1
1? ? ?,VFFFFFF R R R R R R
T l l I (X X. X p p p X. X. X p p p ) +
J i=l j=lbl 1 ] k 1 J k
1 V *v /v F F F P'F v Rv R R R RrRN ,1SLR
+ ) > (X. X. p p p X. X. p p p )..,
3 .f, x j ol i j ol ljl
i=l j=l
(418)
where
PR =P P,R =P SAT(T)
ol ol
X R = 1
ol
X R = 0
oa
a Â¥ 1
n
o
x. = y v. x
i ^ ia
a=l
oa
. . ,rRR, /VFF v R R > ,
p (t) = X p + (X p X D )t
oa oa oa oa
F(t)
v F F R R
X p x p
oa oa
n
O p p R R
p (t) l (X Rp X Rp )
oa 8=1 06
p (t)
p (t)
88
PYHS
The expression for P is given by equation (Al) and
T R
that for by equations (353), (376), and (398).
In a similar fashion we transform equation (411) to
n
Â£ny =  y v (
ra v .ia RT
a i=l
^YH?T,pF) yfH?T, R
P )
RT
P
,n_ )
Pi
i LR \
, F R, f1 cal(t
Pol pol) s0 JTtT dt
n
6=2 (S
R 1 CaR(t) nF
po6> !o Hm dt + *
1 V r F R
L l v (p.p.)AF..
v L1 ,L1 ia ] j 13
a 1=1 j=l J J J
1 r Â£ y F F R R
2v Via pjpk phpj ijk
a i=l ]=1 k=l J J J
n n
1 r r F P,F R P,Rn LR
L L v P P i p p ) 4 .,
L L ] ol ijl
2v L, .L, ia j ol
a i=l j = l J
(419)
n
n
where p. = P l v X = l v. p
1 ia oo la <
a=l a=l
and the other quantities have been defined above.
89
PYHS
The expression for y^ is given in equation (A3).
LR
Note that the integrals of in equations (417),
(418), and (419) require that the pressure behavior of the
solvent dielectric constant be included for nonisobaric
integrations.
Explicit evaluation of these integrals does not in
general yield analytical forms. However, for isobaric
integrations it is possible to write simple analytical forms
for those cases where exponential integrals are not
involved. One example is the DebyeHuckel limiting law in
equation (419).
The last relation that must be reshaped to a more
tractable form is equation (416).
S,ny
a
n n n
o o o
t l l
6=2 Â£=2 y=l
n
I
6 = 1
1
/ F (t)dt
a
o
pKTRT(t)
PB(t>
SCB VC p(t)
pOY(t) po6(t)
VCVYV6 p(t) p(t)
(UcY6(t))(ica?(t)) 
(lCaY(t))(lC6c(t)))
(420)
p
n n P (t) P_(t)
" I l V8 a,t,2 PS
a=l 6=1 p(t)
where
90
F
Finally, once p has been obtained from either equation
(417) or (418), the compressibility of the solution and
the partial molar volume of any component can be calculated
from
n n
o o
< = 1/p RT l l v v XX (1C .)
T a=l 8=1 a B oa op a8
(421)
V
oa
n
(422)
The above equations give the method to be used in the
calculation of solution properties using Fluctuation Theory
and our model. However, work remains to be done on the
practical application of these expressions.
Model Parameters from Experimental Data
The chief aim of this chapter is to demonstrate the
ability of the model to correlate solution properties for
different salts in water. This requires obtaining model
parameters for different systems. As an expedient, those
parameters which could be independently obtained from
literature sources were adopted rather than calculated from
solution data. We hoped then to avoid obscuring any model
inadequancies by parameter fitting. Thus, we used litera
ture values for the pure water density, compressibility,
dielectric constant, and dipole moment and for the ionic
91
partial molar volumes at infinite dilution. In addition,
the hard sphere diameters for all the species were taken
from Marcus (1983) who used neutron and Xray scattering
among other sources. Marcus' diameters (Appendix E) are
close to the Pauling crystal radii, and we feel that they
represent physically meaningful bare ion diameters. This
last point is truly important because hard sphere calcula
tions are very sensitive to the species diameters.
The ability of the model to fit experimental solution
data with physically meaningful bare ion diameters consti
tutes a very sensitive test of the capability of the model
to represent intermolecular correlations present in the
system. However, because they have not been treated any
where, the two and three body parameters listed in Appendix
E were fitted to solution data in the rough fashion
discussed below.
This model is designed such that all of the parameters
are ionically additive, and we hope to obtain the empirical
parameters solely from single salt data. We will, there
fore, limit the discussion to systems containing a salt ( )
in water (1).
The twobody parameters are where i and j include
the ions and water and the threebody parameters are A
and $....
ill
Those parameters involving water only are obtained from
equations (241) and (3100) at the pure solvent limit.
92
PK ^RT
= 1 C
HS
11
 C
LR
11
 AC
11
(423)
where
,HSC
'11
LIM
Xol"1
PYHS , ,
C^ = PY hard sphere
ol x
DCFI given by equation (A5) at the
nure water limit.
4tt
9a
P1M1
DkT
2
)
AC
11
P.. P. LR.
"pol(AFll"p0l*lll)
Because we expect that the contribution of A4>^^^ will
be small and can be absorbed by the other parameters, we set
it equal to zero.
A4>
111
= 0
(424)
AF
11
PA LR
Pol$lll
ol
PKTRT
 1
t CHS t
t
plpl
(
DkT
2
) )
(425)
93
Equation (428) was used to calculate the solvent
P LR
parameters (AF.^ p ^$^^). We have left the parameter
LR
4> together with AF^ because we expect its contribution
to be sufficiently small.
The three body parameters involving two waters and one
ion are calculated from equations (3104) at finite salt
P T D
concentration once AF.. p ., is known.
11 ol 111
1 C
11
olKlRT
,rHS_rHS>
^11 C11 1
 (AC AC )
' 11 11;
(C
LR_rLR
11 U11
(3104)
where
HS PYHS
C, = Cnn = PY hard sphere DCFI given by
'11
11
equation (A5).
1/2
2ailB 1
LR = 4Â£Â£ Ulyl 2 6 Y
11 3 ^ DkT 3
E4(2auV
1/2
11
Acn= p(AFn p0I*m) +
+ pp (v, A , + v A $ )
HKo +a +11 a 11
Equation (3104) can be rearranged to give
94
pp (v, Ai> + v AO .. ) = 1 C. .
oa +a +11 a 11 11
PolKlRT
r,HS Pw P .LR \
(C11 C11 } (p~pol)(AFll Pollll)
4tt ,
+ V (
Vi 2
1/2
9a
11
2ailBYr 1/2 P
DkT > (Pe R4 2ailBYI > Pol <426)
F11(P)
Where we label the righthand side of equation (429) as
*!!<Â£)
To obtain values for the three body parameters on the
lefthand side of equation (426), F, (p_) and pp were
calculated from known parameters and experimental values for
the component densities. Then, the slope was obtained by a
linear least squares routine over all the data points avail
able. The process was repeated for each salt of interest.
Thus, for each salt this slope (SFF) is
S
11
a
v AE> , +
+a +11
v .AO ..
a 11
(427)
The parameters A<^_ and AO ^ are properties of the
ions and water independent of the salt (a). To obtain
individual values for each AO. a scale was constructed
ill
starting with lithium. A finite value was chosen to allow
for geometric mean mixing rules in later analyses and to
95
bring all ion values to almost the same result rather than
having cations and anions possessing very different
parameter values.
A $ + = 4 ML M 1 (428)
Lili
Then the values for the ions N+, K+, CL Br were obtained
a
from C11 data for LiCL, LIBr, NaCL, NaBr, KCL, and KBr.
The twobody parameters involving one ion and a water
together with the long range threebody parameter involving
p
an ion and two waters are lumped as (AFiiP0i$iH) an<^
obtained from equations (240) and (379) at the pure
solvent or infinite dilution limit for each salt (a).
CO
a i rHS_ LR A
vaK1RT 1 cai ual Cal
(429 )
. HS00
where C =
al
LIM PYHS
Xoi! al
LIM
X +1
ol
V
+ CPyH? V CPrHS
+a +1 a 1
v
a
p yHS
C^ = PY hard sphere DCFI given by equation (A5)
LR LR
= infinite dilution given by equation
(380)
96
oo p p T.D
AC = p f(vjAF^Tp rVrT) +
al Hol tet +1 Kol +11
P T R
+ v_a(AF_iPol*ir)) +
P ^
(Pnl )
+ ^ (v + v Ai> )
2 +a +11 a 11
v
a
Equation (429) is now changed to
v+a(AF+l pol$+ll) + va(AFl pol$ll}
1
P
P
ol
V
oa
RT
a 1
1 + C
HS
al
+ C
LR.
al
+
v
a
(v. + n
+a +11
+ v A$ )
a 11
(430)
Equation (430) was used to obtain values of the sum of
the parameters on the lefthand side. This was repeated for
each salt (a). To calculate values of parameters involving
one ion only, a scale was built based on lithium. Again, a
finite value was chosen to allow for geometric mean mixing
rules in later analyses.
LR
+
Lill
600 ML MOL
(431)
97
P LR
The values for AF.^ were obtained for the
DO
other ions (i) from C data for the above salts.
al
Once values for the above parameters have been
obtained, it is then possible to calculate the values for
the three body parameters involving two ions and one water,
A4>
These are determined from equation (384) for C^
at finite salt concentration for each salt (a).
co
V
1C = (CHSCHS") 
al v k RT vual cal 1
a 1
LR_ LR A _A
lCal Cal 1 al al
(384)
where
rHS _HSC
t'al cal
r,LR_r,LRoo
1 Sxl
HS
difference of Ca^ given by
equation (385)
LR
difference of Ca^ given by
equation (386)
*C1 ACJ r <',+a(4F+rpoIt> +
a
P T R P
+ ',a(iFrpoltl1))(ppol) +
+ V A4 + v A4> ) +
v. +a +11 a 11
a
+ (ppol polpol)+
98
+ ( (v + v ) AD +
v +a 2 a ++1
a
(v + v ) AD ) pp
a 2 a 1 oa
In the above expression, the mixing rule of equation
(326) to obtain AD__^ (i ^ j) from AD__^ and A$_._.^ has been
used to simplify equation (387).
Equation (384) is now changed to
PP
((v + v ) AD. + (v + i v ) AD )
v +a 2 a ++1 a 2 a 1
a
1C
 Va + / pHS pHS00,
al v < RT 1 al al '
a T
/ ,,LR _,LR > 1 P LR,
+ (C C ) + (v (AF p D ) +
a al v +a +1 ol +11
a
+ v_a ( 'P'Kol1 +
+ T (v+ai4+ll + vaMll) (ppol polpoI>
a
(432)
As previously, we label the righthand side of equation
(432) as Fla(p_).
PP
((v + 4 v )AD + (v + \ v ) AD ) = F, (p)
v +a 2 a ++1 a 2 a 1 la
a
(433)
99
In order to get values for the three body parameters on
the lefthand side of equation (433), F^ (p_) and PPQa were
calculated from previously obtained parameters and experi
mental values for the component densities. The slope
,1a
= ( (v
v +a
a
+ I
2
a
A4>
++1
+ (v
 a
a
1)
(434)
was calculated by a linear least squares routine over all
the data ponts. The same procedure was repeated for each
salt.
Individual values for the three body parameters in
Ia
equation (439) were extractd from the slope Sa by con
structing a scale based on lithium
A + + = 3 ML M
LiLil
(435)
Again the value was chosen for the reasons stated
above. Values for A$. were obtained from data on several
ill
salts.
The last set of parameters left consists of
and A$ ^ where i, j, k^lori, j, k are ions,
obtained from equation (362) for each salt (a).
a *LR
Af. $ .... ,
13 13I
These are
1c = (lc )" cLR (CHSCHS~)
aa v acr aa aa aa '
 (4CTBiCTB)
aa aa
(362)
100
where
(1C ) =
P oo oo
p :v v
ol oa oa P
aa
v k RT
a 1
v
2 p T P
v +v v ) ( af p : $ f) +
+ a + a a ++ Mol ++1'
P TP
+ (v + v V ) (AF p J ,))
a +a a ol 1
LR
Caa = long range DCFI given by
HS HSC
aa Laa
equation (3 63b)
HS
difference of C given by
aa 3 2
equation (364)
actb actbC
aa aa
P P
pp p ,p 0
ol ol ol / / 2 ...
9 V +
2 +a +a a ++1
v
a
(v +v v ) A4> ) + pp (v +
a +a a 1 oa +a +++
+ v A4> )
a
The above expression has been simplified by use of the
mixing rules given by equations (322), (323), (324), and
(326) in order to eliminate any parameters involving more
than one kind of ion.
We next rearrange equation (362) to give
101
_ i 2 p T p 9
~ ((v + v V )(AF p ,) + (v + v v )
2 +a +a a ++ ol ++1' a +a a
P T R
 (AF p n4> ,)) p (V A$ ^ + v A$ ) =
ol 1 Koa +a +++ a
1 p 00 oo
P p V V T U
1,, \ ol oa oa ~1 LR
p r~ + p Caa +
v k n RT
a 1
P P
+ p"1 (CHS CHS) PP.A.: ((V 2 + V v )
aa aa 2 +a +a a
va p
A<Â¡> n + (v + v v )A$ ) = I +
++1 a +a a 1 a
+ Saa p = F (p)
a oa aa
(436)
As previously done for other parameters, Faa(_P) was
calculated from known parameters and experimental values of
aa
component densities. The slope (S ) and the intercept (Ia)
of equation (442) were obtained by a linear least squares
routine over all the data points for each salt (a). Values
a
for the ionic parameters in Saa and Ia were extracted by
means of a scale based on lithium.
P LR
AF+ + p > + + = 30 0 ML/MOL (43 7)
LiLi LiLil
A$ + + + = 16 ML M1 (438)
Li Li Li
102
The values were chosen as before and the quantities for
other ions are given in Appendix E. The procedure for
obtaining model parameters described in this section is
strictly drawn from the theory and yields a unique set of
values for the parameters if independently determined values
for the species hard sphere diameters are available. The
minimal data required includes pure water density, pure
water isothermal compressibility, and aqueous salt partial
molar volume at infinite dilution. In addition, values for
the solution density, salt partial molar volume, compres
sibilty, and either the salt activity or the osmotic
coefficient derivative with respect to salt concentration
must all be available at two different solution
compositions.
As shown in the next section, the model fits solution
data adequately but not within experimental error with
parameters that have been evaluated as outlined above. It
is believed that this could be greatly improved by fitting
the hard sphere diameters along with the other parameters to
solution data. This scheme could cause the uniqueness of
the parameters to be lost. The lack of uniqueness may cause
problems for this model where the parameters are ionic
quantities that must be internally consistent. There is no
procedure capable of eliminating the lack of uniqueness
problem in a complex model such as the present one. How
ever, the difficulty may be ameliorated by either of two
103
methods. First, the ionic parameters for a large number of
different salts could be fitted simultaneously in the hope
that overconstraining the problem would force the fitting
routine to choose ionic parameter values that would be valid
for many types of salts. Second, the fitting could be done
in a stepwise fashion. For example, the two and three body
parameters could be obtained assuming given values for the
hard sphere diameters for each salt separately. The species
diameters could then be optimized subject to the previously
calculated values for the two and three body coefficients
for all the salts simultaneously. The process could be
repeated until satisfactory results were obtained. The
second fitting scheme should be easier to implement since it
is mathematically simpler and since it is expected that few
iterations are required.
The present results given in Appendix E used the hard
sphere diameters of Marcus (1983 ) for all ions but Lx.
o
Marcus' value of aTff = 1.36 A could not yield accurate
LiLi J
results for LiCL and LiBr nor were the CL and Br values
+ +
obtained consistent with those from Na and K salts. The
present value of aLL = 145 A was obtained by a rough fit
+
of the Li salt data and then the values for the other
parameters for the other salts were found from the Na+ and K+
salt data. The difference between Marcus' and the present
O
hard sphere value lies within the variation (+ 0.12 A or 9%)
that he attributes to the different data for Li. It is
104
possible to justify modifying his value for only Lx because
for the other ions, his variations were much smaller
O
(+ 0.06 A or 3%).
Comparison of Calculated Properties
to Experimental Properties
Extensive calculations of solution properties is beyond
the scope of this work. Thus, the calculations presented in
this section aim to prove the suitability of the model.
This consists of comparing values of the DCFI's, the solu
tion bulk modulus, the salt partial molar volume group, and
the salt activity coefficient derivative obtained from the
model to values for the same quantities calculated from
experimental solution properties. These are shown in Tables
41 to 424. The calculations from the model were performed
using the ionically additive parameters listed in Appendix
E. The six salts included in the calculation are LiCL,
LIBr, NaCL, NaBr, KCL, KBr. Graphical comparison of the
salt activity coefficient derivative as calculated from the
model and from experimental data are presented in Figure 6
for NaCL and LiBr. Also, the NaCL saltsalt, saltwater,
and waterwater DCFI's are presented in Figures 7, 8, and 9,
respectively, including the various contributions from the
model and the experimental values.
Additionally, the derivative of the water activity
coefficient was calculated from both the model by the use of
the relation
105
3Â£ny.
N
3N
o2
t,p,n
=v2xo1Xo2 (PKTRT) 1(1C12>2
ol
 (1C1X)(1C22,)
where
n n
o o
PKRT =1/1 l V V X X (1C )
c=l 8=1 e oo 08
'a 8
(439)
and from experimental data by use of the GibbsDuhem
relation ignoring enthalpy and volume changes.
3 Â£ny.
N
3N
o2
T,P, N
X 3Â£ny
= v N
2 X 3N _
ol o2
(440)
T, P, N
Ol ' Ol
Thus, the water activity coefficient derivative was
obtained from the analogous salt quantity. These results
are presented in Table 426 and Figure 10.
Discussion
The properties of electrolyte solutions at concentra
tions up to the solubility limits of salts generally show
two distinct regimes of behavior at low (Xq2 < 0.02) and
high (0.05
behavior of the solution at intermediate salt mole fractions
contains elements of both regimes. This can be clearly seen
in Figures 1 to 10. Thus, a solution model which claims to
include the physics of the system must smoothly follow
measured behavior over both regimes as well as the
106
transition in between. Further, at even higher salt mole
fractions (X^ > 0.12) a third regime of solution behavior
becomes apparent due to the scarcity of solvent (water).
The present model is not intended to be used for salt mole
fractions much above 0.10 or so. Thus, it will not yield
good results in the third regime so it will not be further
discussed here.
In order to discuss and compare solution behavior
calculated from the model and its various contributions to
experimental observations, NaCL (2) in water (1) is chosen
as prototype electrolyte solution whose properties are
illustrated in Figures 6 to 10 and Table 49. The major
differences between this 11 salt and others is that the
field terms will be more significant as with higher
charges. In addition, the interaction terms could also be
changed in importance.
First we look at the saltsalt DCFI in Figure 7. At
infinite dilution of salt, the saltsalt DCFI magnitude and
behavior are dominated by the long range ionion electro
LiR
static correlations in C22* But as the mole fraction of
CO
salt increases (0 < Xq2 < 0.02), (1C22) which includes
short range ionion and ionwater correlations significantly
contributes to the magnitude but not to the behavior while
the hard sphere (2 ~ (~22 ^ anc^ triple in ^^22 ~ ^<~22 ^
correlations are small and approximately cancel each other.
Once the mole fraction of salt is large (X 2
> 0.05),
107
LR
however, C22 does not make a large contribution to either
the magnitude or the behavior of the saltsalt DCFI. Here
00
the magnitude is dominated by (IC22) r the hard sphere
correlations, and the triple ion correlations while the
variations are dominated mostly by C22 ~ C^ with
TB TB
AC22 ~ ^*99 making a smaller contribution. At the highest
'22
HS
salt mole fraction (X 2 = 0.084953), C22 accounts for 77% of
the magnitude of the saltsalt DCFI.
The deviation of the calculated saltsalt DCFI from the
experimental values over the entire composition range is
mainly due to the choice of the ionic hard diameters
employed, since the diameters appear in the DebyeHuckel
type denominator in C^ and, naturally, in C^ C22
are the leading contributions at low and high salt mole
fraction, respectively.
The salt (2)water (1) DCFI1s behavior is illustrated
in Figure 8. At low salt mole fraction and down to infinite
dilution (0 < Xq2 _< 0.02), the magnitude is dominated by
CO
(lC^) which is a constant while the behavior is controlled
HS HS00
by the hard sphere correlations (C^ ^ anc* *on
LR
range field correlations (C^) As the salt mole fraction
LR
increses (0.02 < Xq2 < 0.05), the contribution of C^2
remains small, becomes nearly constant, and is largely
cancelled by AC^ ~ AC.2) Thus, the magnitude is dominated
00 HS
by (lC^) anc^ ^12 behavior is controlled by
HS
C12 At large salt mole fraction (X^ > 0.05), the
108
co pjCJ HSco
magnitude is again dominated by (lC^) and C^ ~ ^^.2
while the behavior is controlled by the hard sphere crrela
LR 00
tions since C^ and AC^ ~ ^*12 lar
For example, at the highest salt mole fraction (X 2 =
T p 00
0.084953 ) the sum of and Aci2 AC12 :''s _8*58
smaller values elsewhere.
The deviation of the calculated saltwater DCFI from
experimental values at low salt mole fraction is probably
due to the inadequacy of the functional form used to
represent the ionwater correlations. At high salt mole
fraction, however, the deviations are attributable to the
values used for the hard sphere diameters.
The behavior of the water (l)water (1) DCFI is illus
trated in Figure 9. For this case, the contribution of long
LR
range field correlations in C^ is negligible over the
entire composition range. Therefore, the magnitude of the
OO
waterwater DCFI is dominated by (1C^) the hard sphere
HS HS 00
correlations (C^ C^ ), and AC^ AC^ in order of
HS HS
importance. The behavior is controlled by and
OO
AC^ AC^ again in order of importance. For example, at
the highest salt mole fraction (X 2 =0.084953) the relative
00 hs HS00 00
contributions of (lC^) > C^i / and AC^ AC^ are
58%, 31%, and 11%, respectively, while the relative con
tributions to the deviation from the infinite dilution value
fl C U Cqo 00
are 75% for c" c and 25% for AC^ AC^.
109
The agreement of calculated values of the waterwater
DCFI with experimental ones is good for all salts. This
leads us to believe that the parameter values used, includ
ing the water hard sphere diameters, are appropriate. The
HS
hard sphere sizes for the ions play a secondary role in C^.
The solution bulk modulus (p
molar volume group (Vq2/ktrt) are generally well behaved
with salt mole fraction. The salt and the water activity
coefficient derivatives, illustrated in Figures 6 and 10,
undergo extremely rapid changes with salt mole fraction near
infinite dilution in salt. This is particularly evident for
the water activity coefficient. This behavior was observed
by Pitzer (1977) for water activity coefficients in
electrolytes.
Solution properties such as density and activity coef
ficients are, as discussed earlier, obtained from this model
by the integration of the model expressions. Because model
deviations from experimental values generally fluctuate from
positive to negative, the errors tend to be selfcancelling
rather than cumulative during the integrations. Therefore,
based on a propagation of errors analysis, it is expected
that the deviation of calcualted densities and activity
coefficients from experimental values will be at least a
factor of five less than that of the DCFI's at the same salt
mole fraction.
110
The accurate estimation of the uncertainties present in
experimentally determined values of the bulk modulus, the
salt partial molar volume group, the salt activity coeffi
cient derivative, and the saltsalt, saltwater, and water
water DCFI's was not possible because little information is
available regarding the errors present in the original
experiments. Nevertheless, based on a general knowledge of
the quality of the data obtained from the various experi
ments it was assumed that the uncertainty in the solution
compressibility (
(p) 0.02%, and that in the salt partial molar volume (V^)
0.20%. The uncertainty in the derivative of the salt
activity coefficient was assumed to be approximately ten
times that in the activity coefficient itself which would
place it at 1% or less.
By a propagation of errors analysis using the above
estimates, it was determined that the uncertainty in the
three DCFI's was approximately 2% and that in the three
solution property groups 11.5%.
The model fitted the solution bulk modulus within
experimental uncertainty for LiCL, LiBr, NaCL, NaBr, KCL,
and KBr at all compositions except for LICL and LIBr at salt
mole fractions above 0.1. The model fitted the salt partial
molar volume group adequately (23%) in most cases but
slightly outside of experimental error (11.5%). The fit
for the lithium salts was worse at a few points. Lastly,
Ill
the fit of the model to the salt activity coefficient
derivative was qualitatively good (1.614%) but well outside
experimental error (11.5%) in many cases.
To understand the above observations it is necessary to
consider the sensitivity of each type of DCFI to its
parameters and also the sensitivity of each group of prop
erties to each type of DCFI.
The saltsalt DCFI is extremely sensitive to the value
of the species hard sphere diameters at both low and high
salt mole fraction since the dominant contributions in both
cases are strong functions of species size.
The saltwater DCFI is most sensitive to the value of
the species hard sphere diameters at high salt mole frac
tion. At low salt mole fraction, the saltwater DCFI is
relatively insensitive to the species sizes since the
infinite dilution limit is dominant.
The waterwater DCFI is relatively insensitive to the
ionic hard sphere diameters but highly sensitive to the
water diameter at high salt mole fraction. Again, it is
insensitive to species size at low salt mole fraction
because its infinite dilution limit is well defined and
dominant.
The model fitted the experimental values of the water
water DCFI with the species diameters adopted from Marcus
(1983). However, the model fitted the saltwater DCFI less
well and the saltsalt DCFI somewhat worse. This may
112
indicate that the water diameter of Marcus (1983) is
adequate for this model but his ionic diameters are less
appropriate.
The solution bulk modulus is dominated by the water
water DCFI at low salt mole fraction. As the mole fraction
of salt increases, the other DCFI's make significant but
small contributions to the bulk modulus. For example, for
NaCL (2) in water (1) at the highest salt mole fraction
(X = 0.084953) the contribution of the waterwater DCFI is
o2
78%, that of the saltwater DCFI is 20%, and that of the
saltsalt DCFI 2%. Thus, the good fit of the bulk modulus
can be understood by the fact that its chief contribution is
due to the waterwater DCFI which the model fits best of all
the DCFI's.
The salt partial molar volume group is dominated by the
saltwater DCFI at low salt mole fraction. The saltsalt
DCFI makes a small but significant contribution at high salt
mole fraction. For example, for NaCL (2) in water (1) at
the highest salt mole fraction (X^ = 0.084953) the con
tribution of the saltwaer DCFI is 84% and that of the salt
salt DCFI is 16%. Thus, the accuracy of the fit for this
property is determined by the ability of the model to fit
the saltwater DCFI which the model could not fit as well.
The salt activity coefficient derivative includes
contributions from all three types of DCFI's in a functional
form which does not permit separation of the respective
113
contributions into a simple additive form as above. How
ever, at very low salt mole fraction the contributions from
the waterwater DCFI tend to be selfcancelling, and those
of the saltwater DCFI remain finite. But the contributions
from the saltsalt DCFI are not bounded and thus become
dominant. At high salt mole fraction, the contributions
from each of the DCFI types are approximately equal and not
separable.
Conclusions
Although the usual solution properties such as density
or salt activity coefficient have not been calculated, it
was felt that testing the model with derivative properties
such as the bulk modulus (p< RT), the salt partial molar
volume group (V^/k^RT) and the salt activity coefficient
derivative (N9Â£,ny9/3N _I ) provide an even more
^ OZ 1 f xr f JN _
Ol
sensitive test of model adequacy.
Because the parameter values used in the calculations
are not optimized, the model does not fit within experi
mental error all of the above properties for all the salt
solutions over the entire range of salt composition.
Lastly, it must be mentioned that all the model
parameters are ionically additive and valid for all solu
tions. It should, therefore, be possible to predict the
behavior of complex multicomponent systems using parameters
obtained from solutions of single salt in water.
114
0.00 0.02 0.04 0.06 0.08 0.10
X 02
Figure 6. Salt (2) Activity Coefficient Derivative in
Aqueous Electrolyte Solutions at 25C, 1 ATM.
For data sources see Tables 45 and 49.
115
X 02
Figure 7. Contributions to the Salt(2)Salt (2)
DCFI in Aqueous NaCL at 25C, 1 ATM.
For data sources see Table 410.
116
0.00 0.02 0.04 0.06 0.08 0.10
X 02
Figure 8.
Contributions to the Salt (2)Water (1)
DCFI in Aqueous NaCL at 25C, 1 ATM.
For data sources see Table 411.
117
X 02
Figure 9. Contributions to the Water (l)Water (1)
DCFI in Aqueous NaCL at 25C, 1 ATM.
For data sources see Table 412.
118
X 02
Figure 10. Water (1) Activity Coefficient Derivative
in Aqueous NaCL (2) at 25, 1 ATM. For
data sources see Table 425.
TABLE 41
SOLUTION PROPERTIES FOR LiCL (2) IN WATER (1) AT 25C, 1 ATM
1
pktRT
V
o2
ktRT
j/2 3)inL
Xo2 V2N3N
o2
T,P,Noi
Xo2
Calc.
Exp.
Calc.
Exp.
Calc.
Exp.
io"12
16.108
16.108
15.Ill
15.111
8.763
8.763
4.2558xl03
16.430
16.448
16.471
16.527
1.739
1.107
8.5247xl03
16.749
16.787
17.079
17.337
0.254
0.587
0.025729
18.028
18.151
19.366
20.233
4.601
4.003
0.043142
19.333
19.518
21.769
23.028
7.546
6.612
0.069664
21.398
21.514
25.889
26.970
11.253
10.876
0.087620
22.844
22.768
29.131
29.292
13.464
13.762
0.10580
24.581
23.946
32.924
31.310
15.596
16.113
Calc, properties from equations (244), (245), (246), (362), (384), and
(3104).
Experimental data sources are those of Figures 3, 4, and 5.
TABLE 42
SALTSALT
DCFI FOR
LiCL (2) IN
WATER (1) AT 25
C, 1 ATM
Xo2
^1_C22 ^
00 rLR
22
HS _HS
K 22 22
) ( Ac Ac TBT
' 22 22
(1"C22)CMjC*
uc22,exp
IQ"12
13.6300
2.1907x10
6 0.0000
0.0000
2.1907x10 6
2.1907x10
4.2558xl0~2
13.6741
16.5065
0.4047
01099
2.537
0.1625
8.5247xl02
13.7173
9.2628
0.8089
0.2206
5.042
6.065
0.025729
13.8855
3.3199
2.4814
0.6758
12.371
11.877
0.043142
14.0510
1.9617
4.2756
1.1542
15.210
14.750
0.069664
14.2996
1.1735
7.2865
1.9232
18.489
18.754
0.08762
14.4691
0.9091
9.5751
2.4754
20.659
21.044
0.1058
14.6442
0.7318
12.1653
3.0650
23.012
22.619
(1~C22 ^CAL< from equation (362).
EXP
(IC22) from equation (257) using the same sources of experimental data as
for Figure 3.
TABLE 43
SALT
WATER DCFI
FOR LiCL
(2 ) IN WATER (1)
AT 25C, 1 ATM
Xo2
OC
(1_C12)
LR
12
, HS HS
1 12 12
> <4C12AC12>
(1_C12)CALC*
(1C )EXP'
lo"12
7.5556
0.0000
0.0000
0.0000
7.5556
7.5556
4.2558x10
3
7.5556
0.5636
0.3709
0.1621
8.328
8.336
8.5247x10
3
7.5556
0.6308
0.7413
0.3276
8.600
8.713
0.025729
7.5556
0.7298
2.2724
1.0204
9.537
10.021
0.043142
7.556
0.7706
3.9125
1.7628
10.4759
11.208
0.069664
7.556
0.8046
6.6600
2.9729
12.047
12.632
0.087620
7.556
0.8194
8.7446
3.8480
13.271
13.287
0.10580
7.556
0.8309
11.1003
4.7826
14.7042
13.786
(1C^2)CALC" from equation (384).
EXP
(l~Cj2) from equation (256) using the same sources of experimental data as
for Figure 4.
TABLE 44
WATERWATER
DCFI FOR LiCL
(2) IN WATER
(1) AT 25 C, 1
ATM
Xo2
oo
(1CiP
rLR
U11
(CHScHS )
lcll 11
(AC11AC11)
(1~C11)CALC*
(ic11)EXP
IQ12
16.1084
0.0000
0.0000
0.0000
16.108
16.108
4.2 5 58x10 ~3
16.1084
0.0007
0.3398
0.1219
16.570
16.588
8.5247xl03
16.1084
0.0009
0.6789
0.2471
17.035
17.070
0.025729
16.1084
0.0011
2.0796
0.7774
18.966
19.052
0.043142
16.1084
0.0013
3.5780
1.3543
21.042
21.130
0.069664
16.1084
0.0014
6.0835
2.3092
24.502
24.462
0.087620
16.1084
0.0015
7.9811
3.0089
27.010
26.875
0.10580
16.1084
0.0015
10.1222
3.7637
29.995
29.496
(lC^. )CAL<" from equation (3104).
EXP
(lC^^) from equation (255) using the same sources of experimental data
as for Figure 5.
TABLE 45
SOLUTION PROPERTIES FOR LiBr (2) IN WATER (1) AT 25C, 1 ATM
1
P K ipBT
ktrt
1/2 3Â£nY2
X v N 
o2 2 N _
o2
T,P,N
ol
X _
o2
Calc.
Exp.
Calc.
Exp.
Calc.
Exp.
12
10
16.108
16.108
21.244
21.244
8.763
8.763
2.0866xl0_3
16.258
16.260
22.222
22.206
3.056
2.445
4.1983.10"3
16.408
16.413
22.549
22.701
1.334
0.972
8.4 9 62x10 ~3
16.713
16.722
23.210
23.535
0.818
0.803
0.017411
17.334
17.365
24.245
25.049
3.700
3.230
0.036619
18.661
18.749
26.733
28.043
7.906
7.213
0.057919
20.169
20.259
29.890
31.140
11.624
11.072
0.075482
21.483
21.460
32.916
33.478
14.343
14.144
0.10833
24.343
23.508
40.092
37.091
18.781
20.250
Calc, properties from equations (244), (245), (246), (362), (384), and
(3104).
Experimental data sources are those of Figures 3, 4, and 5.
123
TABLE 46
SALT
SALT DCFI FOR
LiBr (2)
IN WATER (1)
AT 25 C, 1
ATM
CM
O
X
(l<22 ^
cLR (C
22 1
,HS HS
22 22 '
TB TB
 ( Ac Ac )
' 22 22'
ac22)CALC
. EXP.
v 22
1012
18.4268
2.1907xl06
0.0000
0.0000
2.1907xl06
2.1907x10 6
2.08 66xl0~3
18.4422
27.7666
0.2239
0.0336
9.134
5.801
4.1983xl03
18.4575
16.2388
0.4492
0.0670
2.600
4.098
8.4962xl0~3
18.4880
8.9838
0.9077
0.1337
10.278
10.459
0.017411
18.5485
4.6576
1.8657
0.2686
15.488
15.153
0.036619
18.6723
2.2250
4.0065
0.5506
19.903
19.910
0.057919
18.8050
1.3714
6.5763
0.8608
23.149
23.467
0.075482
18.9130
1.0267
8.8919
1.1183
25.660
25.927
0.10833
19.1239
0.6793
13.9563
1.6276
30.773
30.011
PAT.P
(IC22) from equation (362).
EXP
(IC22) from equation (257) using the same sources of experimental data as
for Figure 3.
TABLE 47
SALT
WATER DCFI
FOR LiBr
(2) IN WATER (1)
AT 25C, 1 ATM
Xo2
(ici2)
pLR
U12
, HS HS
'^12 12
) 
Me \ CALC
ir '12
EXP.
U ^12
to"12
10.6224
0.0000
0.0000
0.0000
10.622
10.622
2.0866x10
3
10.6224
0.4893
0.1800
0.0955
11.196
11.174
4.1983x10
3
10.6224
0.5573
0.3611
0.1925
11.348
11.412
8.4962x10
3
10.6224
0.6250
0.7295
0.3482
11.628
11.790
0.017411
10.6224
0.6904
1.4986
0.8102
12.001
12.430
0.036619
10.6224
0.7520
3.2146
1.7381
12.850
13.556
0.057919
10.6224
0.7861
5.2701
2.8078
13.870
14.535
0.075482
10.6224
0.8042
7.1184
3.7221
14.822
15.105
0.10833
10.6224
0.8270
11.1497
5.5200
17.079
15.374
(IC12 ) from equation (384).
EXP
(1Cj2) from equation (256) using the same sources of experimental data as
for Figure 4.
TABLE 48
WATERWATER
DCFI FOR LiBr
(2) IN WATER
(1) AT 25 C, 1
ATM
Xo2
rLR
U11
(CHScHS )
lUll C11 1
(AC11AC11)
1 i p \ CALC.
U cll'
EXP.
U Cll'
io"12
16.1084
0.0000
0.0000
0.0000
16.108
16.108
2.0866x10
3
16.1084
0.0005
0.1445
0.0479
16.301
16.303
4.1983x10
3
16.1084
0.0007
0.2897
0.0967
16.495
16.498
8.4962x10
3
16.1084
0.0009
0.5852
0.1974
16.891
16.895
0.017411
16.1084
0.0011
1.2016
0.4105
17.721
17.724
0.036619
16.1084
0.0013
2.5749
0.8880
19.572
19.562
0.057919
16.1084
0.0014
4.2163
1.4435
21.769
21.704
0.075482
16.1084
0.0015
5.6893
1.9216
23.720
23.579
0.10833
16.1084
0.0015
8.8935
2.8657
27.869
27.509
(1C^ ) from equation (3104).
EXP
(1C^) from equation (255) using the same sources of experimental data
as for Figure 5.
126
TABLE 49
SOLUTION PROPERTIES FOR NaCL (2) IN WATER (1) AT 25C, 1 ATM
1
V o
o2
X1{2V N
o2 2
8 2
PktRT
Exp.
8N
o2
Xo2
Calc.
Calc.
ktrt
Exp.
Calc.
T,P,Noi
Exp.
io~12
16.108
16.108
14.817
14.817
8.763
8.763
9.9900x10
4
16.195
16.203
15.642
15.507
5.163
4
.576
4.9751x10
3
16.538
16.574
16.608
16.676
2.365
Â¡.113
9.9010x10
3
16.966
17.027
17.649
17.816
0.845
0.945
0.015554
17.478
17.539
18.844
19.024
0.270
0.099
0.032695
19.021
19.078
22.314
22.546
2.283
1.691
0.048670
20.527
20.509
25.635
25.821
3.448
2
.092
0.066607
22.364
22.183
29.660
29.529
4.386
4
. 560
0.084953
24.329
24.168
33.946
33.339
5.050
C
. 915
Calc, properties from equations (244), (245), (246), (362), (384), and
(3104).
Experimental data sources are those of Figures 3, 4, and 5.
127
TABLE 410
SALT
SALT DCFI
FOR NaCL (2)
IN WATER (1)
AT 25 C, 1
ATM
Xo2
*1C22 ^
00 pLR
22
(cHScHS )
1 22 22 '
(actbactbT
v 22 22;
(1_C22)CALC
lO"12
9.3190
2.1907x10
6 0.0000
0.0000
2.1907xl06
2.1907xl06
9.9900x10
4
9.3252
46.4674
0.1304
0.0479 
37.059
32.482
4.9751x10
3
9.3494
13.9618
0.6446
0.2432
4.211
3.295
9.9010x10
3
9.3783
7.7026
1.2843
0.4930
2.467
2.285
0.015554
9.4120
5.0615
2.0471
0.7864
5.621
4.961
0.032695
9.5048
2.4063
4.3497
1.7464
9.701
8.999
0.048670
9.5867
1.5721
6.6103
2.7133
11.911
11.630
0.066607
9.6779
1.1084
9.3846
3.8708
14.083
14.244
0.084953
9.7653
0.8380
12.3857
5.1393
16.173
16.571
(IC2 2 ) from equation (362).
EXP
(1C) from equation (257) using the same sources of experimental data as
for Figure 3.
TABLE 411
SALT
WATER DCFI
FOR NaCL (2)
IN WATER (1)
AT 25C, 1 ATM
X
0
NJ
(
UCl2)
30 PLR
i2
(CHScHS )
^12 C12 1
 (AC12AC12)
(1C )CALC*
' '"'12'
(1C )EXP
u u12'
1012
7.4085
0.0000
0.0000
0.0000
7.408
7.408
9. 9900xl0~4
7.4085
0.4086
0.1091
0.0145
7.911
4.834
4.9751xl03
7.4085
0.5558
0.5389
0.0727
8.430
8.455
9.9010xl0~3
7.4085
0.6173
1.0734
0.1455
8.953
9.042
0.015554
7.4085
0.6555
1.7104
0.2300
9.544
9.658
0.032695
7.4085
0.7134
3.6313
0.4938
11.259
11.432
0.048670
7.4085
0.7412
5.5143
0.7490
12.915
13.048
0.066607
7.4085
0.7613
7.8217
1.0456
14.954
14.845
0.084953
7.4085
0.7760
10.3137
1.3609
17.137
16.690
(lC^2 )* from equation (384).
EXP
(1C,) from equation (256) using the same sources of experimental data as
for Figure 4.
129
TABLE 412
WATERWATER
DCFI FOR NaCL
( 2 ) IN WATER
(1) AT 2 5 C, 1
ATM
X
0
K3
OO
dCll)
PLR
dl
(cHScHS )
ldl dl 1
(AcnAcn>
(1 j CALC.
U dd
(lc )EXP
1 Id
io12
16.1084
0.0000
0.0000
0.0000
16.108
16.108
9.9900x10
4
16.1084
0.0004
0.0912
0.0286
16.228
16.236
4.9751x10
3
16.1084
0.0007
0.4501
0.1443
16.703
16.739
9.9010x10
3
16.1084
0.0009
0.8963
0.2911
17.296
17.356
0.015554
16.1084
0.0010
1.4279
0.4634
18.000
18.058
0.032695
16.1084
0.0012
3.0290
1.0154
20.153
20.198
0.048670
16.1084
0.0013
4.5962
1.5639
22.269
22.221
0.066607
16.1084
0.0014
6.5138
2.2167
24.840
24.627
0.084953
16.1084
0.0015
8.5816
2.9237
27.615
27.548
(1C^^)CALC from equation (3104).
EXP
(lC^) from equation (255) using the same sources of experimental data
as for Figure 5.
130
TABLE 413
SOLUTION PROPERTIES FOR NaBr (2) IN WATER (1) AT 2 5C, 1 ATM
1
V o
o2
1/2 3inY2
v M N ^
PKTRT
Exp.
Xo2 2N3N _
Xo2
Calc.
Calc.
Exp.
o2
Calc.
T,P,Noi
Exp.
io12
16.108
16.108
20.951
20.951
8.763
8.763
9.9900x10
4
16.191
16.196
21.764
21.629
4.921
4.386
4.9751x10
3
16.523
16.541
22.842
22.807
1.953
1.764
8.9197x10
3
16.843
16.885
23.489
23.766
0.542
0.675
0.029126
18.555
18.652
27.470
28.204
2.886
2.247
0.052258
20.607
20.731
32.193
33.019
5.021
4.635
0.078001
23.081
23.175
37.960
38.334
6.680
6.688
0.10904
26.490
26.558
46.028
45.848
8.119
8.056
Calc, properties from equations (244), (245), (246), (362), (384), and
(3104).
Experimental data sources are those of Figures 3, 4, and 5.
TABLE 414
SALT
SALT DCFI FOR NaBr (2) IN WATER (1)
AT 2 5 C, 1
ATM
(N
0
X
^ ^~C22 ^
00 rLR
22
(CHScHS ) 
' '22 22 1
TB TB
(AC2T2AC22)
Uc22,
io~12
14.0597
2.1907x10
6 0.0000
0.0000
2.1907x10 6
2.1907xl06
9.9900x10
4
14.0646
45.7709
0.1472
0.0483
31.607
27.472
4.9751x10
3
14.0832
13.5940
0.7268
0.2424
0.973
1.610
8.9197x10
3
14.1002
8.1970
1.2888
0.4381
6.751
6.577
0.029126
14.1827
2.6084
4.2918
1.4718
14.394
13.953
0.052258
14.2649
1.3952
7.9103
2.7148
18.065
18.216
0.078001
14.3469
0.8881
12.2895
4.1607
21.587
21.838
0.10904
14.4395
0.5959
18.2805
5.9828
26.141
25.885
^('22^CALC* ^rom equation (362).
EXP
(1C9) from equation (257) using the same sources of experimental data as
for Figure 3.
TABLE 415
SALT
WATER DCFI
FOR NaBr
(2) IN WATER (1)
AT 25C, 1 ATM
Xo2
UCi2>"
rLR
12
HS HS
1 12 12
> (4C124C12)
(1~C12)CALC
EXP.
u 12'
io~12
10.4753
0.0000
0.0000
0.0000
10.475
10.475
9.9900xl0~4
10.4753
0.4051
0.1091
0.0220
10.967
10.891
4.9751xl0~3
10.4753
0.5506
0.5384
0.0375
11.526
11.502
8.9197xl0~3
10.4753
0.6025
0.9543
0.1971
11.835
11.979
0.029126
10.4753
0.6985
3.1737
0.6527
13.694
14.111
0.052258
10.4753
0.7395
5.8408
1.1882
15.867
16.310
0.078001
10.4753
0.7648
9.0595
1.8014
18.498
18.673
0.10904
10.4753
0.7841
13.4489
2.5654
22.142
22.098
(IC12' from equation (384).
EXP
(lCj^) from equation (256) using the same sources of experimental data as
for Figure 4.
133
TABLE 416
WATERWATER DCFI FOR NaBr (2) IN WATER (1) AT 25C, 1 ATM
Xo2
oo
UCn)
PLR
U11
_(CHScHS )

(1C )CALC*
U Cll'
(lc )EXP*
U cll;
lo"12
16.1084
0.0000
0.0000
0.0000
16.108
16.108
9.9900x10
4
16.1084
0.0004
0.0806
0.0235
16.212
16.217
4.9751x10
3
16.1084
0.0007
0.3980
0.1180
16.625
16.644
8.9197x10
3
16.1084
0.0009
0.7053
0.2137
17.028
17.067
0.029126
16.1084
0.0012
2.3427
0.7206
19.172
19.232
0.052258
16.1084
0.0014
4.3052
1.3337
21.748
21.798
0.078001
16.1084
0.0015
6.6671
2.0495
24.826
24.885
0.10904
16.1084
0.0016
9.8782
2.9539
28.942
29.098
(1CQ )CALC' from equation (3104).
EXP
(1C^) from equation (255) using the same sources of experimental data
as for Figure 5.
TABLE 417
SOLUTION PROPERTIES FOR KCL (2) IN WATER (1) AT 25C, 1 ATM
1
pKTRT
Exp.
V o
o2
1/2 9Â£nY2
X v N
o2 2 3N _
o2
Calc.
Xo2
Calc.
Calc.
Exp.
T,P,Noi
Exp.
io12
16.108
16.108
23.937
23.937
8.763
8.763
9.9900x10
4
16.194
16.205
24.794
24.776
5.312
5.018
4.9751x10
3
16.535
16.583
25.967
26.296
2.940
2.787
8.8482x10
3
16.867
16.947
26.987
27.415
1.978
1.947
0.025637
18.358
18.483
31.281
31.555
0.158
0.293
0.041316
19.816
19.888
35.269
35.399
0.607
0.644
0.055992
21.239
21.215
39.041
38.990
1.001
1.384
0.069758
22.625
22.524
42.591
41.900
1.221
1.999
Calc, properties from equations (244), (245), (246), (362), (384), and
(3104).
Experimental data sources are those of Figures 3, 4, and 5.
TABLE 418
SALT
SALT DCFI
FOR KCL (2)
IN WATER (1)
AT 25 C,
1 ATM
Xo2
( *'~<222 ^
rLR
22
, HS HS
V 22 *22
1 22 22J
1 ^1_C2 2 ^
CALC.
dc22)EXP
IQ"12
11.9555
2.1907x10
6 0.0000
0.0000
2.1907x10 6
2.1907x10 6
9.9900xl0~4
11.9570 
44.6388
0.1541
0.0055
32.522
30.223
4.9751xl0~3
11.9624 
12.9584
0.7520
0.0186
0.225
0.547
8.84 82x10 ~3
11.9671
7.7786
1.3244
0.0242
5.537
5.913
0.025637
11.9851
2.7250
3.8145
0.0041
13.078
13.010
0.041316
11.9983
1.6267
6.1437
0.0747
16.440
16.543
0.055992
12.0081
1.1545
8.3458
0.1893
19.010
19.377
0.069758
12.0151
0.8950
10.4132
0.3330
21.200
21.378
(IC22 )CAL<^* from equation (362).
EXP
(IC22) from equation (257) using the same sources of experimental data as
for Figure 3.
TABLE 419
SALT
WATER DCFI
FOR KCL
(2) IN WATER (1)
AT 2 5 C, 1 ATM
Xo2
(1C12>"
rLR
12
. HS HS
'12 12
) 
dc12)CALC*
EXP.
U C12;
io"12
11.9686
0.0000
0.0000
0.0000
11.969
11.969
9.9900xl0~4
11.9686
0.3922
0.1078
0.0185
12.487
12.473
4.5751xl0"3
11.9686
0.5294
0.5256
0.0929
13.116
13.274
8.84 82xl0~3
11.9686
0.5772
0.9255
0.1661
13.637
13.848
0.025637
11.9686
0.6580
2.6629
0.4895
15.779
15.927
0.041316
11.9686
0.6900
4.2851
0.7985
17.742
17.803
0.055992
11.9686
0.7089
5.8164
1.0198
19.585
19.510
0.069758
11.9686
0.7218
7.2521
1.3691
21.311
20.880
(1C^2)CALC* from equation (384).
EXP
(1C, 2) from equation (256) using the same sources of experimental data as
for Figure 4.
TABLE 420
WATERWATER DCFI FOR KCL (2) IN WATER (1) AT 25C, 1 ATM
X
0
K)
(1Cn>
LR
11
(CHSCHS )
lcll 11 1
(Ac Ac )
1 C11 ll'
M p CALC.
' '11 >
(1"C11)EXP
1012
16.1084
0.0000
0.0000
0.0000
16.108
16.108
9.9900x10
4
16.1084
0.0004
0.0753
0.02538
16.209
16.220
4.9751x10
3
16.1084
0.0007
0.3668
0.1297
16.605
16.651
8.8482x10
3
16.1084
0.0009
0.6457
0.2323
16.987
17.062
0.025637
16.1084
0.0012
1.8561
0.6870
18.652
18.775
0.041316
16.1084
0.0013
2.9843
1.1238
20.217
20.290
0.055992
16.1084
0.0014
4 0476
1.5405
21.697
21.674
0.069758
16.1084
0.0014
5.0432
1.9363
23.089
23.086
(1Cj^ )
FXP
(1c^) from equation (255) using the same sources of experimental data
as for Figure 5.
TABLE 421
SOLUTION PROPERTIES FOR KBr (2) IN WATER (1) AT 25C, 1 ATM
1
V o
o2
ktRT
Calc.
1/2 3Â£nY2
CM
0
X
Calc.
Pktrt
Exp.
Exp.
Xo2 V2 3N _
o2
Calc.
T,p,Noi
Exp.
1012
16.1084
16.1084
30.071
30.071
8.763
8.763
1.5247x10
3
16.235
16.249
31.095
31.057
4.476
4.318
3.0705x10
3
16.363
16.390
31.553
31.642
3.364
3.319
9.4793x10
3
16.926
16.969
34.881
33.630
1.993
1.646
0.019825
17.762
17.885
35.774
36.404
0.028
0.489
0.035183
19.0914
19.216
39.546
40.089
1.058
0.566
0.048075
20.259
20.312
42.722
42.948
1.571
1.246
0.070088
22.340
22.183
48.199
47.778
2.054
2.103
0.083974
23.726
23.399
51.740
51.232
2.200
2.392
Calc, properties from equations (244), (245), (246), (362), (384), and
(3104).
Experimental data sources are those of Figures 3, 4, and 5.
139
TABLE 422
SALT
SALT DCFI
FOR KBr (2) IN
WATER (1) AT 25C, 1 ATM
X
o2
^1C22 *
rLR
22
(CHScHS ) 
v 22 22 1
TB TB
(ac22ac22)
(ic22)CALC*
EXP.
^22'
11
0
1
M
NJ
18.4325
2.1907x10
6 0.0000
0.0000
2.1907xl06
2.1907X106
1.5247x10
3
18.4333 
32.4553
0.2504
0.0013
13.768
12.805
3.0705x10
3
18.4337 
18.8995
0.4989
0.0011
0.0342
0.2969
9.4793x10
3
18.4350
7.0852
1.5106
0.0075
12.852
12.436
0.019825
18.4344
3.4346
3.1037
0.0413
18.062
17.655
0.035183
18.4300
1.8700
5.4463
0.1197
21.886
21.663
0.048075
18.4230
1.3214
7.4168
0.2042
24.314
24.123
0.070088
18.4086
0.8565
10.7711
0.3850
27.938
27.712
0.083974
18.3970
0.6919
12.9163
0.5141
30.107
30.107
(IC22 from equation (362).
EXP
(1C~) from equation (257) using the same sources of experimental data as
for Figure 3.
140
TABLE 423
SALT
WATER DCFI
FOR KBr
(2) IN WATER (1)
AT 25C, 1 ATM
!N
0
X
oo
uci2)
PLR
U12
. HS HS
'''12 12
^ (AC12AC12)
(ic12)CALC*
(ic )EXP*
C12;
1012
15.0354
0.0000
0.0000
0.0000
15.035
15.035
1.5247xl03
15.0354
0.4241
0.1578
0.0202
15.637
15.615
3.0705xl03
15.0354
0.4835
0.3143
0.0408
15.874
15.917
9.4 7 9 3xl0~3
15.0354
0.5768
0.9510
0.1274
17.529
16.899
0.019825
15.0354
0.6330
1.9524
0.2594
17.880
18.224
0.035183
15.0354
0.6727
3.4221
0.4829
19.613
19.922
0.048075
15.0354
0.6927
4.6562
0.6630
21.047
21.192
0.070088
15.0354
0.7151
6.7524
0.9712
23.474
23.266
0.083974
15.0354
0.7252
8.0904
1.1645
25.015
24.710
(1C^2)^ALC* from equation (384).
EXP
(12) from equation (256) using the same sources of experimental data as
for Figure 4.
141
TABLE 424
WATERWATER DCFI FOR KBr (2) IN WATER (1) AT 25C, 1 ATM
Xo2
oo
(1_cll)
PLR
C11
(cHScHS )
leu )
(AC11AC11)
(1_C11)CALC
EXP.
U ^11;
10"12
16.1084
0.0000
0.0000
0.0000
16.108
16.108
1.5247x10
3
16.1084
0.0005
0.0992
0.0316
16.239
16.253
3.0705x10
3
16.1084
0.0006
0.1975
0.0638
16.370
16.397
9.4793x10
3
16.1084
0.0009
0.5972
0.1983
16.904
16.973
0.019825
16.1084
0.0011
1.2250
0.4177
17.752
17.858
0.035183
16.1084
0.0013
2.1448
0.7453
18.999
19.094
0.048075
16.1084
0.0014
2.9159
1.0205
20.046
20.081
0.07008
16.1084
0.0015
4.2229
1.4882
21.821
21.683
0.083974
16.1084
0.0015
5.0555
1.7802
22.945
22.596
(1C^)CALC* from equation (3104).
EXP
(lC^) from equation (255) using the same sources of experimental data
as for Figure 5.
142
TABLE 425
WATER ACTIVITY COEFFICIENT IN AQUEOUS NaCL AT 25C, 1 ATM
3 Â£ny.
N
X
3N
o2
o2
CALC.
3 &ny.
N
3N
T, P, N
o2
EXP.
ol
T,P,Noi
1012
0.0000
0.0000
9.9900xl0~4
4.837
4.567
4.9751xl0~3
0.168
0.151
9.9010x10_3
0.086
0.096
0.015554
0.035
0.013
0.032695
0.442
0.327
0.066607
1.306
1.358
0.084953
1.773
2.077
CALC.
3 Â£n y.
N
3N
o2
from equation (439)
T,P,Noi
3 dny.
N
3N
o2
T, P, N
ol
from equation (440)
using the data of
Hamer and Wu (1972)
143
CHAPTER 5
CONCLUSIONS AND RECOMMENDATIONS
The chief accomplishment and significance of the
present work is the construction of a microscopic statis
tical mechanical model of liquid electrolyte solutions which
has been shown capable of representing both volumetric and
activity behavior. All of the model parameters are obtain
able from one salt and one solvent data. These parameters
are ionically additive and applicable to all solutions.
Thus, the properties of a large number of systems can be
predicted using information obtained from a very limited
number of experimental measurements.
The model includes the effect of the various molecular
interactions such as hard core repulsion, long range field,
and short range forces in a theoretically sound fashion
which is simple enough for economical engineering calcula
tions. The inclusion of all the above was necessary in
order to properly represent all the solution properties
which are sensitive to different types of interaction. For
example, the density is mostly determined by the hard core
repulsive forces while the activity is dominated by long
range field interactions at low salt concentration.
144
145
An important theoretical byproduct of the present work
has been to show that expressions for the solvent chemical
potential derived from a McMillanMayer formalism such as
the DebyeHuckel theory are not appropriate for fluctuation
properties such as waterwater and watersalt DCFI's. In
this work, we have presented an alternate model for the
above DCFI's which leads to a limiting slope for the salt
partial molar volume which is different than that of the
traditional DebyeHuckel theory. The functional form of
this model includes the rigorous DebyeHuckel limiting law
for activity coefficients and a general form which contains
the higher order limiting law for the activity coefficient
of unsymmetric electrolytes first discovered by Friedman
(1962 ).
The calculation of solution properties using the model
has been extensively discussed in Chapter 4. Here we simply
restate that to obtain properties the model is integrated
over concentrations from the pure water limit. The result
ing integrals are not always simple or analytic but are
tractable using numerical methods and modern computers.
A benefit of the above is that errors due to model approxi
mations tend to be smoothed and largely selfcancelling
rather than cumulative.
Much work remains to be done in testing, developing,
and applying this model. What follows contains some sugges
tions to guide future work.
146
The model parameters, especially the species hard
sphere diameters, have not been properly optimized to fit
solution property data. Thus, all of the model parameters
should be fitted globally for the six 1:1 electrolytes
included in the present work and for a number of other
salts, especially higher charge types and unsymmetric
electrolytes.
Once parameter values have been obtained for several
different salts, the model should be tested against solution
properties such as density and activity coefficient rather
than derivative properties as done here. The reason is that
solution properties are of most interest in practice. It
would be very important to test the predictive ability of
the model with a few salts that were not included in the
fitting of the parameters.
A more severe test would be to compare solution prop
erties calculated from the model to experimental values in
aqueous solutions containing at least two salts. A more
sensitive test than density or activity would be provided by
the prediction of Harned coefficients.
Besides exploring the range of practical applicability,
it is also of importance to test the theoretical basis with
a view to simplification and improvement. First, the alter
nate mixing rules introduced in Appendix C should be tested
with better fitted parameter values. These alternate mixing
rules have a sounder theoretical basis than those of Chapter
147
3. Second, the possibility of using a density expansion of
the hard sphere DCFI up to second order in place of the
full expression should be explored. The resulting form
would be simpler and consistent with the Hypernetted Chain
approximation used for the long range field correlations.
Lastly, the multisolvent electrolyte problem should be
addressed. The model may avoid the difficulties encountered
by excess Gibbs energy models.
APPENDIX A
HARD SPHERE DIRECT CORRELATION FUNCTION INTEGRAL
FROM VARIOUS MODELS
PercusYevick theory (1958) yields through the "Com
pressibility Equation" an equation of state for a mixture of
n hard shere species which has the following form (Reed and
Gubbins, 1973 ) :
pPYHS
RT
6
IT
+
3 ^ 1 ^2
(1C3)2
+
3?.
de3)
where ^
n
r r
6 ^ i piai
i=l
k = 0,1,2,3
(A1)
The more exact mixture form of the CarnahanStarling
equation of state for hard spheres (Mansoori, Carnahan,
Starling, and Leland, 1971) gives a very similar expression
which compares slightly better with molecular dynamics
results for hard spheres (Reed and Gubbins, 1973).
pHSCS pHSPY
RT RT
(lÂ£3)3
(A2 )
The chemical potential for a species i in solution can
be obtained from equations (Al) and (A2) by standard
thermodynamic manipulations. Thus, the hard sphere chemical
148
149
potential from the PercusYevick compressibility equation of
state is
HSPY :
y p A .
i i i
RT n int
qi
 Â£n (1C ) +
ira
i PHSPY+ 3(ai*2 + ai *1}
6 RT 15,
+
9{22 ai2
2(l{3)2
(A3
h2 I/2 ...
where A = ( ) = ideal gas partition function
i 2tt m^kT
h = Planck's constant
= mass of species i
k = Boltzmann's constant
INT
q_^ = internal partition function.
The corresponding hard sphere chemical potential from the
CarnahanStarling equation of state is
HSCS HSPY 2 
b d + m (ie )(i2a> (32 [f] )
RT RT ^3 ?3 53
(^2ai)3 (2~^3) 53 2 + 3_
15
^2ai
15,
3^3
ai g2 g3
(l53)3
(A4 )
150
The direct correlation function integral for hard
sphere species i and j can be obtained from equations (A 3)
and (A 4) by means of equation (21).
_1_
RT
T, P, N.
k^i
C. .
N
(21)
Thus, insertion of the PercusYevick compressibility
expression for the chemical potential into equation (21)
gives
CHSPY
ij
(a.+a )
i 1_
1,
3^ (a.a .)
i J
(a.+a .)
i 1
(l53)
+
3a1a.C2[(ai+a.)2+aia.l + (a^.)3^
(1SJ2
9(a.a e )3
J Â£
(ls3 )4
(A5)
Finally, the corresponding expression from the Carnahan
Starling equation is
CHSCS
i i
CHSPY
^2(aiaj}
(l53)3
6 + (953 15)
(a+a.) C2 (6+C3 [12C315])
151
a.a.f 2(6 + r [r (2614C )21] )
+ 1 3 2 3 2 ] +
6C2(aiai)2ta(lC3)(C3[ai+a.]C2+aia.22Â¡1
(A6 )
By comparing equations (A 5) and (A6) one can appreci
ate the fact that the expression or the hard sphere direct
correlation function integral from the CarnahanStarling
equtaion of state is significantly more complex than that
from PercusYevick compressibility equation of state.
APPENDIX B
RELATION OF McMILLANMAYER THEORY TO
KIRKWOODBUFF THEORY
Fluctuation Solution Theory, also known as Kirkwood
Buff Theory (Kirkwood and Buff, 1951) is a statistical
mechanical formalism relating concentration derivatives of
the chemical potential to spatial integrals of correlation
functions in the grand canonical ensemble where the natural
independent variable are system temperature, volume, and
composition. However, practical calculations are normally
done in the LewisRandall system for which system tempera
ture, pressure, and composition are the independent vari
ables. A further complication is introduced by the fact
that some important theoretical results such as the Debye
Huckel (1923) limiting law for salt activity coefficients
can only be rigorously obtained from the McMillanMayer
theory (Friedman, 1962) for which the natural independent
variables are solvent chemical potential, system tempera
ture, and composition. Presently, the conversions of the
chemical potential of components a and 6 in system of nQ
components will be explored. We start by relating the
KirkwoodBuff (KB) chemical potential to that in the Lewis
Randall (LR) system.
152
153
3mKB(T,V,X)
a
9N
06
3 yBR(T,P,X:
a
3N
T, V, N
oy^B
oy
+
T,P,N
oy^B
+
3ULR(T,P,X)
a
3P
3p
3N
T, N
T, V, N
o ' oy^B
By making the following identifications we obtain
(Bl)
9y
LR
a
3P T,N
o
= VRR (T,P,X)
(B2 )
3P
3N
oB
T, V, N
VBR (T,P,X)
V
oy^B
(B3
where Va = partial molar volume of component a
V = system volume
kt = isothermal compressibility
and by inserting equations (B2) and (B3) into (Bl) we get
3 y KB ( t V, X)
a
3N
o3
3 yLR(T,P,X;
a
9N
T, V, N
oy^B
oB
T, P, N
oy^B
vLRvLR
_a B_
V
(B4 )
154
Equation (B 4) relates the change in the chemical
potential of a component a in a KB system to the respective
change in an LR system when the mole number of a component 6
is changed. The quantities on the righthand side of equa
tion (B4) are all LR properties.
Next we consider a system consisting of one solvent and
several solutes. To this system we apply the McMillan
Mayer theory (MM) which is defined for a system whose
boundary is permeable to the solvent but not to any solute.
Further, the system temperature and the solvent chemical
potential are constant while the mole numbers of solute
components may vary. Conversions of thermodynamic quanti
ties from MM to LR have been considered in the literature
(Pailthorpe, Mitchell, and Ninham, 1984; Friedman, 1972).
We start our analysis by considering the total differ
ential of the LR chemical potential of the solvent (1).
d
LR
^1
3T
dT
P,N
o
9y
LR
+
3 P
dP
T,N
o
+
+
n
o
I
6=1
3y
LR
3N
oB
d N.
T,P,N
OYt^B
(B5 )
In accordance with the MM theory, we prescribe that y
LR
1
be constant at constant temperature.
155
3u
LR
0 =
3P
dP
T,N
o
n LR
o 3y.
1 ^
d N
oB
(B6)
T, P, N
OYt8
Next, equation (B2) is used on (B6) and the resulting
expression integrated over a variation in the mole numbers
of each solute (8) from zero to NQg and a corresponding
change in system pressure from P to P+7T (note that since
P, = constant, dN = 0).
1 ol
P + TT
0 = J VRR dP +
P 1 B=2 o
n N 0 ^ LR
o oB 3 yT
Â£ J 3
oB
d N
oB
(B7)
T, P, N
oy^B
where tt = osmotic pressure.
Equation (B7) indicates that if the solvent chemical
potential is to be constant while solute is added, then the
system must be kept at total pressure equal to osmotic
pressure plus whatever pressure the pure solvent as under.
Since the pressure in an LR system remains constant at P as
solute is added, the only difference between a thermodynamic
property in an LR system and one in an MM system is the
system pressure if both are at the same temperature and
composition. This suggests that equation (B2) can be
integrated to relate the chemical potential of any component
(a) in the LR system to that in the MM system.
P + TT P + TT
J dn = / va dP
P U P
(B8 )
156
MM
Pa1Jyi(T,P+Tr,X)
yRR(T,p,x) + J vRR
P
a
dP
(B9)
The next task is to relate the chemical potential in
the KB system to that in the MM system. To that purpose,
equation (B9) is rearranged and differentiated to obtain
LR
3y
n. MM
3y
P + TT
3VLR
a
a
 J
T'p'fWeP
a
3N
oB
3N
T, P, N B
oy^B
3KoS
dP 
T, P, N
oy^B
 v 31
a 3N
oB
T, P, N
P+tt 3V
oy^B
Equation (B7) is differentiated to obtain
LR
3N
P oB
n LR 3 7T
dp + V1 w
T, P N
oB
oy^B
T, P, N
n N a ~
o oB 3y.
3N
oB
{ I I
B=2 o
3N
oB
T, P, N
oy^B
d N 0 }
oB J
oy^B T,P,N
oy^B
(B10 )
(Bll
The righthand side of equation (Bll) is further
simplified by use of equations (B7) and (B9).
P+tt T T,
/ vf dP =
P
n N LR
o oB 3m,
y
3N o
B=2 o oB
d N
oB
T, P, N
oy^B
(B12 )
157
.MM UK y i
1 ~ L i
1 1 6=2 O
. LR
no NoB 9y
LR
3N
06
d N
06
(B13)
T, P, N
oy^B
By inserting equation (B13) into equation (Bll) and
rearranging we obtain
v:
3 7T
MM
9mi
3N Q
oB
3N .
T P N
' oy^B
T,P,Noy^B
3NoB
T, P, N
OYt^B
P+tt 3VRR
! 1
3N
o6
d P
T, P,N
OY^B
(B14
Equation (B14) is used to substitute for the osmotic
pressure derivative in equation (B10) and the resulting
expression inserted into equation (B4) to give
~
3 U
~ MM
3u
P + TT
3VLR
a
a
 J
T,P,N P
a
3N
oB
3N .
T, V, N
3N
oB
oyt^B
oy^B
d P 
T, P, N
OYt^B
VLR
3y
MM
VRR 3NoB
+
Tp'tWe 1
3y
LR
vLR
a
VPR 3NoS
T, P, N
oy^B
VLR
a
LR
P + TT 3 V
J
p
LR
3N
oB
d P +
VLRVLR
a B
VKm
T,P,N
OYt^B
(B15)
158
Equation (B15) relates a change in the KB chemical
potential of any component (a) to the corresponding change
in the MM system when the mole number of any component (8)
is varied in a mixture containing one solvent (1) and (n 1)
solutes.
We now explore some specific cases. First, if both ot
and 8 are the solvent (a = 8 = 1), then equation (B15)
reduces to
3y
KB
3N
ol
3y
LR
3N
T, V, N
ol
^LRLR
V K
(B16)
T,P,N
oy^l ' oy^l
Second, if a is the solvent (a = 1) and 8 is a solute, then
3 M
kB
3N
3y
LR
3N
T, V, N
o8
vLRvLR
1 8
V k_
(B17 )
T,P,N
oy^8 ' oy^8
Third, if both a and 8 are solutes, then equation (B15)
gives
kB
3y
3N
o6
,, MM
3 y
P+TT
3vlr
a
r
a
3N
J
3N n
T V N
1,V,Noy^6
T'p'We
p
08
d P +
T, P, N
oy^8
LR LR
Va 8yi
vf 9No8
VÂ£R P + TT 3VRR
+ _SL_ / 1
vRR P
T'P'Noy^8 1
3N
08
d P +
T, P, N
oy^8
LRLR
a 8
V K
(B18 )
159
where
3N
oS
T,P,N
oy=3
0
since is a constant with respect to changes in the mole
number of solutes.
In order to relate the above results to direct correla
tion function integrals we use equation (212).
8 Â£ny
KB
pv
a 8p
oB
= vavB (1CaB}
(212)
T, p
oy^B
Now the chemical potential of any component (a) can be
written as
.KB
(T,V,X) = M,KB(T) + v RT *n y^B(T,V,X) (B19)
Ut LX O LX CX
Differentiation of equation (B19) gives
8y
KB
a
8N
oB
, 0 8 Â£n y
= V RT + v RT 5 
a N a 8N
T, V, N
a
oB
oy^B
T, V, N
oy^B
 v RT f
a N
(B20
where 6 = Kroniker delta
a B
v = no. species formed by the dissociation of
a
a which equals one if a does not dissociate.
160
By inserting equation (B20) into equation (212) we
obtain
~ KB
JL l
RT 3N
OB
T, V, N
= v V v C
a xoa a B aB
(B21)
oy^B
We next apply equation (B21) to each of the three
cases represented by equations (B16), (B17), and (B18).
First, if and 6 are solvents (a = B = 1), then
a LR
= _N_ h
'n Xol RT 3Nol
pvrrvrr
T, P, N
< RT
T
(B22
oy^l
Second, if a is the solvent (a = 1) and 3 is a solute, we
obtain
N
3y
LR
'IB
VT 3Noe
T, P, N
vBkTRT
oy^B
Third, if both a and 3 are solutes, then
(B23)
a B
N
3 Vi
MM
a
'aB
vcX
B oa
v v RT 3N n
a B oB
T, P, N
oy^B
N
v v RT
a B
P+tt 3VLR
I
P
3N
o B
d P 
T, P,N
oy^B
NVLR
a
3 vi
LR
v v RTVRR 9NoB
a B 1
T,P,NQy^B
NVLR
a
v v.RTV^ p
a B 1
P+tt 3vLR
I 1
3N
oB
d P 
T, P,N
oy^B
pvLRvf
a B
vaV BkTRT
(B24)
161
Equations (B22) and (B23) contain no MM quantities
which indicates that the MM theory does not make any net
contributions to solventsolvent or to solventsolute direct
correlation function integrals (DCFI's). It would, there
fore, be theoretically incorrect to use MM quantities such
as chemical potentials from the DebyeHuckel theory in the
derivation of any expression for solventsolvent or solvent
solute DCFI's.
Equation (B24), however, does contain an MM quantity
MM
(y ) so that MM theory does make a contribution to solute
solute DCFI's. Further, as the concentration of all solutes
approaches zero (or Xq1 * 1), one would expect that
LIM
X + 1
ol
TT = 0
LIM P+7T 9VLR
a
Xol + 1 R
9N
o3
d P =0
T,P,N
oy^B
LIM P+^ 9VRR
X > 1 P
o
9N
oB
d P = 0
T, P, N
oy^B
(B25)
(B26)
(B27)
Also, the observed experimental behavior of the solvent
chemical potential as embodied in models for both
162
electrolyte and nonelectrolyte solutions always follows an
expression which near the pure solvent limit reduces to
y
LR
1
X
o
+ .
n > 1
(B28)
3y
LR
3N
oÂ£
LIM
a X
_nl
oB
T,P,N
3 y
0y/B
LR
X 1 v 1
ol
3N
oB
= 0
T, P, N
oy/S
n1 > 0
(B2 9)
(B3 0
But, for at least the case of a salt ( a ) in water (1) with
other salts (B) we have
o, X1/2 +
a oB
3y
MM
a
3N
oB
a x
1/2
oB
T, P, N
oy/B
LIM
3 y
MM
a
Xol ^ 1 3Ne T, P, N
Oy/B
(B31)
(B3 2)
(B33)
Lastly, LR and MM partial molar volumes are equivalent in
the pure solvent limit. To prove this, we differentiate
equation (B9) with respect to pressure and insert equation
(B2) in the resulting expression to get
yMM
a
VLR
a
P+TT
/
P
3VLR
a
3 P
d P
T, N
(B3 4)
163
LIM
X 1
ol
V
:MM
a
LIM
X 1
ol
VLR + 0
a
(B3 5a)
_oo,MM ^,LR
(B3 5 b)
Taking the pure solvent limit of equation (B24) and
using the results of equations (B25) to (B35) gives
6a 6
N
3u
MM
a
a 3
v0X v v rt 9N q
3 oa a 3 oB
T, P, N
oy^B
p MM ~, MM
P i v V '
ol a 3
v v K RT
a 3 1
(B36
where C
a3
LIM
X ^ 1
ol
'a3
Therefore, the pure solvent or infinite dilution limit
of the solutesolute DCFI1s can very generally and rigor
ously be obtained from MM theory.
A rather interesting consequence of the above arguments
arises in relation to the composition behavior of the solute
partial molar volume near the pure solvent limit. Although
the arguments that follow are valid for any general system,
the discussion will be developed for a system consisting of
several salts (a,3) and one solvent (1) only. The intent is
to avoid any complexities that may obscure the central topic
164
and to remain close to the subject of this work which is
aqueous strong electrolytes. We start with equation (234)
VLR
a
v X .
a ol
(1C .)
al
n
+ Va 62 Xg(1
(234)
Our complete model for the saltsolvent DCFI is con
tained in equation (383) where the terms are defined by
equations (376) to (378) and (380) to (382).
1C
al
V'LR
a
V v k RT
a 1
al al al
 C
LRC
al
) 
 ( AC
al
OO
(383)
We now focus on the initial deviation from the infinite
dilution limit which is dominated by the long range field
term and ignore all the others.
cLR cLR<
2 7T ea
'al al 3v
a
DkT
) nr i
n v Z
a i
1 . a i
1 = 1 ll
2a B I1/2
(pe 11 Y E(2a..B I1^2
2 ll y

(B3 7
To explore the pure solvent limit, we expand equation
(B37) in ionic strength (I).
P
Pol + 0(I)
(B3 8)
165
2a. B I1//2
e 11 Y = 1 + 2a. B I1/2 +0(1
ll Y
(B
2a. B I1/2
E (2ailByI1/2) = e 11 Y 2ailByI1/2E1(2a1By 1/2
(B
From equation (D8) we have the low ionic strength
expansion for E^.
El(2ailByll/2) = Y "^n(2ailByI1/2)
 2a.,B I1//2 + 0(1)
l y
(B
where y = 0.57721 = Euler's Constant.
Inserting equations (B38) to (B41) into equation
(B37) and collecting terms up to half power in ionic
strength we obtain
CLR CLR
cxl al
4tt ea 2 20 _l/2 p
3va DkT M1 ByJ p o1
n
y v. Z ,2(Y+Â£n(2a.,B I1/2)) + 0(1)
ia i ii y
39 )
40 )
41)
(B42)
166
Equation (B42) expresses the composition behavior of
the saltsolvent DCFI at very low ionic strength or near the
pure solvent limit.
Now, we change equation (B42) to express the composi
tion in terms of mole fractions and then consider the salt
solvent DCFI contribution to equation (234).
I
1
2
Y z2p
i1 1 1
P
n
o
I
Y=1
(0
Y
where co
Y
v.
x Y
Z .
l
2
(B43)
v'LR
a
k^RT
4tt
DkT
2 p
U1 By Pol
3/2
( l co X )1/2 l v. Z.2(y +
, Y Y
Y=1
i=l
a i
+
Â£n(2a.,B
xl y
co X
Y Y
1/2
)
+ 0(X^ )
oy
(B44)
Our complete model for a general saltsalt DCFI is
expressed by equation (359) where the various contributions
are defined by equations (353) to (355) and (357),
(358) .
167
1C = (1C ) cLR (cHS CHSo)
aB aB aB 3 aB
mn mpa)
 (Ac AcJ )
aB aB
(359)
As the zero ionic strength limit is approached, (1Cag)
LR
goes to a constant, C diverges, and the other terms
approach zero and we will, therefore, concentrate on the
first two terms.
The general expression for the saltsalt long range
DCFI is given by equation (353).
,LR
a B
V1
4Vb
1/2
2 2
n n v v Z. z .
V y J6. i 1
l L 1/2 2
i=l j=l (1+a. .B Ix/)
J U Y
S 2p
_Y
3v v
JVa B i=l j=l
. 3a. .B l1/2 ,
n n v. v. z.3z.3e 13 YE.(3a. ,B I1/2
l l 1.0t 3B 1 3 _A_ JJ Y
(1+a. B I
il Y
1/2
(353)
To obtain the limit as the ionic strength approaches
zero, we use equations (B38), (B39), and (B41) on equa
tion (353) and expand in ionic strength.
168
o nP T1/2
IR Voj1
aB 4v
a 6
n n
I I v v +
i=l jl la
S 2pP n n
+ I I v. v. Z.3Z.3(y +
3Vs =i j=i ia]M 3
+ Un ( 3a .B I1//2 )
13 Y
+ 3a B I1//2 )
l Y
+ 0(1)
(B45:
Equation (B54) expresses the composition behavior of
saltsalt DCFI's near zero ionic strength or pure solvent.
We now change concentration scales from ionic strength
to mole fraction and then consider the contribution of the
saltsalt DCFI's to equation (234) noting that (1C^)
approaches a constant.
v v y Q(lC n )
a 6 oB aB
1/2
(
n
o
I
Y=1
co X
Y OY
1/2
X
oB
n n
I I
i=l j=l
V V
ia
P
ol
n
n
n
l I v. v Z.Jz.JÂ£n(3a. .B (pp. Â£ co X )1/2) +
> ia IB i 3 i] y ol L. y oY
i=l j=l
36
3 Y
Yl Y Y
+ 0(X
(B46)
169
Lastly, we note that the ionic strength expansion of
the isothermal compressibility is of the form
< ip = < + 0(1)
(B47)
Finally, we insert equations (B44), (B46), and (B47)
into equation (234) to obtain an expansion for the salt
partial molar volume (V ) based on KB theory and our micro
scopic model for DCFI's. Note that the quantity k^RT is a
constant.
VLR LR
a V
k1RT < RT
4tt ea > 2 p .
T < DkT b BY(ol)
3/2
n
o 1/2 n
( l u X ) i v Z. (y
T Y oY a i
Y=1 1=1
n
1/2
+ Â£n (2a.,B (pP V w X ) )
il Y ol ^ y OY
c, p ,1/2 n n
Sv(p*\ ) o 1/2 o n n
7 ( l u X ) IXogl l v. v Z2Z2
4 Y=1 7 y 6=2 i=l j=l la 1 3
o 2 p n
S p^ o n n 0
3 l l l v. v ,Dz3z3
6=2 i=1 j=l la 1,6 1 J
n
Â£n (3a. .B (pP [ co X )1/2) + 0(X a:
il Y ol y OY o6
(B48
170
Equation (B48) includes the contributions of the
DebyeHuckel limiting law and the higher order limiting law
for unsymmetric electrolytes first reported by Friedman.
The development of equation (B48) is rigorous in a statis
tical mechanical and a thermodynamic sense. However, it
appears to accurately fit experimental partial molar volume
data only at very low ionic strength (0.1 M) where uncer
tainty in the data is high.
If one assumes that salt activity coefficient is given
by the DebyeHuckel limiting law,
n
Â£n y =
a
2 N a/2
S ( l v Z ) I
y v ia i
(252)
a i=l
and then inserts this expression into the thermodynamic
relation,
9 Â£n y
_c
9P
T, N
VLR V'LR
a a
v RT
a
(B49)
then one obtains the RedlichMeyer equation (1964).
VÂ£R CLR +  sv < + 0II)
(B50)
yLR ,LR
a a
k^RT c^RT 2 ^RT
3 V Pl)1/2 n 2
+ 3 V _ol ( Â£ v. Z.2)( l w x
. ia i L
i=l
1/2
L. Y oY'
Y=1
+
+ 0(X
(B51)
171
<
1
where S.
3
T,N
Equation (B51) generally fits the partial molar volume
data for aqueous electrolytes at 25C (Millero, 1970) and
ionic strengths below 0.3 M. But, it has not been tested
for nonaqueous or for multisolvent systems. Further, equa
tion (B51) is thermodynamically but not statistical
mechanically rigorous because it is based on the Debye
Huckel theory which being a MM theory is not an adequate
vehicle to represent solvention interactions. Also, it
does include any contribution from Friedman's limiting law
which would be a significant omission for the case of
unsymmetric electrolytes.
The somewhat poorer agreement with experimental data of
equation (B48) in relation to equation (B51) most likely
indicates either inadequate fitting of the model parameters
which has not been carefully done in this work or at worst
incompleteness in the DCFI model. Although, equation (B48)
certainly represents a more complete approach.
APPENDIX C
RELATION OF DENSITY EXPANSION OF THE DIRECT CORRELATION
FUNCTION TO VIRIAL EQUATION OF STATE:
ALTERNATE MIXING RULES
The RushbrookeScoins density expansion of the direct
correlation function (Reichl, 1980; Croxton, 1975) for
species i and
j in a system of n species is given by equa
tion (312 ) .
cij(T,p)
n
= f (T) + I p,
ij k=l K
where f. (T) =
13
u. ./kT
in
 e J 1 = Mayer bond function
hjk'1
= S > fijfikfjkdÂ£kd"k
3
t
II
Integration of equation (312) over molecular orienta
tion gives
=
i j oo
J c. dw.dw. (Cl)
a2 13 1 J
=
1 j CO
n
+ l p <(p. > + ... (C2)
13 w k1 k 13k CO
172
173
When equation (C2) is inserted into the compressibil
ity equation (228) which is
J_ 3P_
RT 3p .
3
n
I
i=l
X.
i
c. .
13
where C. .
il
P
1 j to
dr.dr .
i j
(228 )
we obtain after integration
_1_ 9P
RT 3p .
3
= 1
T,p
n
I
i=l
pi
n
I P
i=l
I P
k=l
dr.dr .
ij (D l j
/ <$
ilk
dr.dr . .
to i j
(C3 )
To establish the desired relation we begin with the
general virial equation of state for a system of n species
(Reed and Gubbins, 1973).
P
RTp
1 + PB(T)
+
p2c(t)
+
(C4 )
n
where P = l p.
i = l X
n n
= I X. I X.
i=l Xjl 3
Bi.(T) =
BIT)
Second
Virial
Coefficient
174
n n n
c ( t ) = y x. y x. y x. c.., ( t ) =
. L1 i 3, k ilk
i=l 3=1 k=l
Third
Virial
Coefficient
Equation (C4) can be rearranged to
P
RT
n n n
[p. + I p I p B. (T) +
i=l 1 i=l 3 = 1 3 13
n n n
+ I Pi l Pi I Pk C. .,(T) +
i=l j=l 3k=l 3
(C5 )
By taking the first derivative of equation (C5) with
respect to the density of an arbitrary species j we obtain
1 3P
RT 3p
n
= 1 + 2 y p B. +
u i ii
T,p
i=l
1 11
+ 3 iLa/k +
(C6 )
Comparison of equations (C3) and (C6) and substitu
tion of
2B. .
1
.(T) = f dr. dr
1 J l] co 1
(C7 )
3C. ., (T) = f dr. dr .dr,
13k J 13 lk 3k a) 1 3 k
(C8 )
175
Equations (C7) and (C8) will now be used in the develop
ment of alternate mixing rules for the quantities A F^ .,
LR
A$ijk' an<3 ^ijk These are defined in equations (312),
(315), (316), and (321). We now restate these defini
tions in a different form.
Af. (T) = /
13 ico 13 w i 3
 / M dr.dr .
13 w i 3
 / M dr.dr .
ioo x3 w i 3
(C9 )
where = ^ J f .d co. d
^ n2 1
CO.
u. ./kT
i 1
f^j = e 1 = Mayer bond function
= J dco .
A$ (T) = f dr. dr .dr,
13k J i] k ]k di i j n
f ..HS.HS.HS. , ,
 j dr dr .dr,
J lj lk 3 k co i j k
:cio)
where = x f f . f ., f ., dco. dco .dco,
13 lk 3k co fi2 J i] ik ]k 13k
LR 00
$ (T) = f dr.dr .dr,
ijk o 13 lk 3k w _1 ~J ~k
(Cll)
176
From a practical point of view, the mixing rules
represent an effort to obtain the quantities AF^j, A^^,
LR
and using pure component and binary solution data. For
1
the case of aqueous strong electrolytes, the mixing rules of
AF^j are only important when both i and j are ions. This is
due to the fact that any ionsolvent AF^j can in principle
be obtained from one solventone salt systems but some
ionion AF^j (say for two anions or two cations) could only
be obtained from ternary solution data.
Therefore, when i is an ion and j is a solvent, the
quantity AF^j(T) is a parameter.
When i and j are ions, we use equation (C7) to obtain
AF. .(T) = 2 (BHS + BLR B. .)
il il il il
(C12)
The hard sphere second virial coefficient is rigorously
given by
2ttn a. .
bHS = *,iJ
i j 3
(C13)
where a.. = (a. + a.)
il 2 i l
= Avogadro1s number
(C14
LR
B^j and B^j are approximated for ions i and j as
= (blr blr)1/2
ii 11
blr
11
(C15)
177
B. = (B. B . )1/'2 (C16)
11 ll 11
LR
B.. and B.. would be obtained from binary solution
1X11
data.
LR
Since and are three body quantities, the use
of mixing rules is important in all cases. Using equation
(C8) in equations (C10) and (Cll) gives
(T)
ijk
> (T)
ijk
3(_ Â£ \
J i jk ijk
 3CLRv
ilk
(C17)
(C18)
A rigorous expression for the hard sphere third virial
coefficient exists. But, a much simpler approximate
expression was considered adequate.
where
= (cHS.
in
rHS = 1
iii 8
cHS. c
111
HS
kkk
(
3
2ttn a .
a i
3
1/3
2
)
(C19)
The structure of equation (C8) suggests that and
LR
could be expressed empirically as a product of two body
quantities.
C. = C. C., C
13k 13 lk 3k
(C2 0)
178
cLR = cLR cLR cLR
ilk 13 ik 3k
(C21)
All of the two body quantities (C^ ) in equations
(C20) and (C21) can in principle be obtained from one salt
and one solvent solution data except for the case of any two
k,j that are both cations or anions of different kinds.
Then, to avoid using ternary solution data, the following
mixing rule should be used.
c. =
(C. .
c..)1/2
11
11
11
cLR =
(CLR
CLR)i/2
11
11
j j
(C22)
(C23)
C.. and C.. can be obtained from binary solution data.
The above mixing rules were not used for calculations
in this work. However, they are theoretically sounder than
the mixing rules introduced in Chapter III which were found
adequate for the systems treated in this work. It is pos
sible that as more complex systems are treated with this
theory, the alternate mixing rules will be found to be more
appropriate.
Lastly, it should be understood that the validity of
the application of the virial expansion to a system at
liquid densities has been assumed and not proven. Although,
this should not be a serious problem since the virial
coefficients for all of the interactions except hard spheres
179
were treated as parameters to be fitted to experimental
data. In fact, the virial series has been used here only in
a very formal fashion as in the osmotic virial expansion.
APPENDIX D
EXPONENTIAL INTEGRALS
The exponential integrals form a class of mathematical
functions whose properties have been extensively studied
(Abramowitz and Stegun, 1973). Here we present a few of the
properties which are relevant to this work.
A general exponential integral, E (Z), of order n is
defined by
Zt
E (Z) = J dt (Dl)
n It
where n = 0, 1, 2, 3, . .
R(Z) > 0 (Real part of Z)
The integral of order zero (n = 0) is trivial. For
those of higher order there exists a recurrence relation
among integrals of adjacent order.
E(Z) = (e~Z ZE (Z)) (D2)
, ] n n
n+1
where n = 1, 2, 3, . .
180
181
In practice, equation (D2) allows one to express an
exponential integral of any order in terms of the
exponential integral of order one (n = 1). Thus
00 Zt
E, (Z) = / I dt
1 t
E2(Z) = eZ ZEX(Z)
E3(Z) =  e_Z Ze"Z + ~ Z2E1(Z)
E4(Z) = f e'Z f Ze~Z +
+ 6 z2e_Z 6 z3 E1(Z)
(D3 )
(D4 )
(D5)
(D6 )
The zeroth and first order
the parameters, Z, approaches
integrals are divergent as
zero, as can be seen from
Eo(z)
(D7)
Ex ( Z )
Y 2nZ
l
K = 1
(~l)KzK
K K !
(D8 )
where Y = 0.57721 = Euler's constant
ARG Z < ff
Integrals of second and higher order, however, approach
definite values as the parameter, Z, goes to zero. These
limits can be generally expressed as
182
1
n = 2, 3, 4
(D9 )
n1
Equations (D2) through (D6) indicate that in order to
obtain numerical values for an exponenteial integral of any
order, one only needs values of E^(Z ) To this purpose,
approximations to E^(Z) were taken from Abramowitz and
Stegun (1973). Unfortunately, no single approximation valid
over the full range of values for (o < Z < 00) was found.
Therefore, two different approximations were spliced
together. These are for 0 Â£ Z _< 1
(1)
E, (Z) = n Z + a + a. Z + a Z +
1 o 1 2
+ a3 Z3 + a4 Z4 + a5 Z5 + e(Z)
(D10)
where
Â£(Z)  < 2 x 10
and for 1 < Z <
(2)
Ex (Z)
+ Â£ ( Z )
(Dll)
Z 5 4 3 2
e (Z +b.Z^+bcZJ+b^Zz+bQZ)
D 0 / o
Z
I Â£ ( Z )  < 2 x 10 8 
where
183
E^1^ and e^ ^ were spliced together to calculate E^ as
E1(Z) = 2 E11^Z) (1 f tan"1 n(Zl)) +
+ j e{2!z) (1+  tan"1 n(Zl)) (D12)
where n = 10b(^
The numerical values of the parameters in equations
(D10) and (Dll) are
a =
o
0.57721566
bl
=
8.573328
al =
0.999999193
b2

18.059016
a2
0.24991055
b3
=
8.634760
II
m
rtf
0.05519968
b4
=
0.267773
a4 =
0.00976004
b5
=
9.573322
II
in
0
0.00107857
b6
=
25.63295
b7
=
21.09965
00
&
=
3.958496
APPENDIX E
MODEL PARAMETERS
The application of any model requires numerical values
for certain input quantities loosely called parameters. In
the case of the present work, some of the input quantities
are actually properties whose value we have adopted from the
literature. But other quantities are true parameters
representing certain molecular correlations whose value we
have obtained from fitting the experimental values of the
DCFI's. What follows is a listing of the numerical values
for all the quantities used in the calculations presented in
Chapter IV.
Pure Water Properties
( 25 C, 1 ATM)
WATER DENSITY = 0.997048 GM/ML+
+
WATER COMPRESSIBILITY tc
45.248 x 10
6
BAR
1
1
WATER DIELECTRIC CONSTANT D = 78.4472
+ +
WATER DIPOLE MOMENT 0.
1.87 DEBYES
+++
1
+ FINE AND MILLERO, 1973.
++ UEMATSU AND FRANK, 1980
+++ WEAST, 1977.
184
185
CO
Infinite Dilution Ion Partial Molar Volumes (V )
(25]C, 1 ATM)
ION
oo
V
(ML/MOL)
Li +
Na
.+
+
K
cl"
BR~
+ MILLERO, 1972,
0.88
1.21
9.02
17.83
24.71
Hard Sphere Diameters
WATER MOLECULE
all =
LITHIUM ION
aLiLi
SODIUM ION
aNaNa
POTASSIUM ION
a + +
K+K
CHLORINE ION
aCL CL
BROMINE ION
aBR BR
+ MARCUS, 1983
* FITTED TO OUR DATA. THE
IS 1.36 + 0.12 A.
2.78 6A+
*
= 1.45 A
+
= 1.96 A
+
= 2.68 A
+
= 3.66 A
+
= 3.88 A
VALUE GIVEN BY MARCUS, 1983,
Two Body Parameters
(1 WATER)
186
(ML/MOL)
1
1
75.0237
Li +
1
600.0000
Na +
1
731.1595
K+
1
629.8387
CL_
1
422.1072
br"
1
287.6151
Li +
Li +
300.0000
Na+
Na +
149.1514
K+
K+
46.1979
CL
cl"
64.4814
BR~
br"
112.7558
Three Body
Parameters
(1 
WATER)
i
j
k
A<5 .
13k
(LML/MOL)
1
1
1
0
Li +
1
1
 4.0000
Na+
1
1
 4.9306
K+
1
1
 3.2584
CL~
1
1
 5.8246
br"
1
1
 3.6430
Li +
Li +
1
 3.0000
187
Na+
Na+
1
11.4220
K+
K+
1
21.0341
CL~
cl
1
5.7911
BR~
br
1
9.5753
. +
Li
. +
Li
. +
Li
16.0000
Na +
Na +
Na +
23.6270
K+
K+
K+
1.9808
cl
cl
cl
7.7004
BR
br
br
8.4594
REFERENCES
Abramowitz, M. and I.A. Stegun, Handbook of Mathematical
Functions, Dover, New York, NY (1973).
Allam, D.S., Ph.D. Thesis, London (1963).
Ananthaswamym, J., and G. Atkinson, J. Soln. Chem. Id., 509
(1982 ) .
Ball, F.X., W. Furst, and H. Renon, AIChEJ, 31, 393 (1985).
Berne, B.J. (Ed.), Statistical Mechanics Part A;
Equilibrium Techniques, Plenum Press, New York, NY
(1977).
Blum, L., Theoretical Chemistry Advances and Perspectives.
(Edited by H. Eyring and D. Henderson) Academic Press,
New York, NY (1980), p. 1.
Brevli, S.W., Ph.D. Dissertation, University of Florida
(1973).
Brelvi, S.W., and J.P. O'Connell, AIChEJ, 21, 171 (1975).
Campanella, E.A., Ph.D. Dissertation, University of Florida
(1983).
Carnahan, N.F., and K.E. Starling, J. Chem. Phys., 5JL, 635
(1969).
Chan, D.Y.C., D.J. Mitchell, and B.W. Ninham, J. Chem.
Phys. 1Q_, 2946 ( 1979 ) .
Chen, C.C., H.I. Britt, J.F. Boston, and L.A. Evans, ACS
Symposium, Washington (1979).
Croxton, C., Introduction to Liquid State Physics, John
Wiley and Sons, New York, NY (1975).
Cruz, J.L., and H. Renon, AIChEJ, E4, 135 (1978).
Davidson, N., Statistical Mechanics, McGrawHill, New York,
NY (1962).
188
189
Debye, P. and E. Huckel, Phys. Zeitschrift, 2_4, 185206
(1923) .
Downes, C.J., J. Soln. Chem. _4 191 (1975 ).
Enderby, J.E., Ann. Rev. Phys. Chem., 3_4, 155 ( 1983 ).
Fine, R.A., and F.J. Millero, J. Chem. Phys., 5_9, 5529
(1973 ) .
Friedman, H.L., Ionic Solution Theory, Interscience, New
York, NY (1962).
Friedman, H.L., J. Soln. Chem., 5, 387 (1972).
Friedman, H.L., and P.S. Ramanathan, J. Phys. Chem., 74,
3756 (1970).
Gautschi, W., and W.F. Cahill, Handbook of Mathematical
Functions. (Edited by M. Abramowitz and I.A. Stegun)
Dover, New York, NY (1972).
Gibson, R.E., and O.H. Loeffler, Ann. N.Y. Acad. Sci., 51,
727 (1948).
Golberg, R.N., and R.L. Nuttall, J. Phys. Chem. Ref. Data,
1, 263 (1978).
Gubbins, K.E., and J.P. O'Connell, J. Chem. Phys., 6_0 3449
(1974 ) .
Guggenheim, E.A., and R.H. Stokes, Equilibrium of Single
Strong ElectrolytesInternational Encyclopedia of
Physical Chemistry and Chemical Physics, vol. 15:1,
Pergamon, Elmsford, NY (1969).
Guggenheim, E.A., and J.C. Turgeon, Trans. Faraday Soc., 51,
747 (1955).
Hala, J.H., Proc. Intern. Symp. Distillation 1969, Inst, of
Chem. Engrs. Symp. Series 32 (1969), 513.
Hamer, W.J., and Y.C. Wu, J. Phys. Chem. Ref. Data, JL, 1047
(1972) .
Harned, H.S., and B.B. Owen, The Physical Chemistry of
Electrolyte Solutions, 3rd Ed., Reinhold, New York,
NY (1958).
Hill, T.L., Statistical Mechanics: Principles and Selected
Applications, McGrawHill, New York, NY (1956).
190
H0ye, J.S., and G. Stell, J. Chem. Phys., 68, 4145 (1978).
Kirkwood, J.G., and F.B. Buff, J. Chem. Phys., 19_, 774
(1951).
Kusalik, G. and G.N. Patey, J. Chem. Phys., 7_9, 4468
(1983 ) .
Lebowitz, J.L., Phys. Rev., 133, A895 (1964).
Lobo, V.M.M., and J.L. Quaresma, Electrolyte Solutions:
Literature Data on Thermodynamic and Transport
Properties, vol. II, Coimbra Univ. Press, Coimbra,
Portugal (1981).
Mansoori, G.A., N.F. Carnahan, K.E. Starling, and
T.W. Leland, J. Chem. Phys., 5_4, 1523 (1971).
Marcus, Y., J. Soln. Chem., 12, 271 (1983).
Mathias, M., Ph.D. Dissertation, University of Florida
(1978 ) .
Mathias, P.M., and J.P. OConnell, Chem. Eng. Sci. 3_6' 1123
(1971) .
Mauer, G., Fluid Phase Equilibria 13, 269 (1983).
McMillan, W.G., and J.E. Meyer, J. Chem. Phys., 13, 276
(1945 ) .
McQuarrie, D.A., Statistical Mechanics, Harper and Row, New
York, NY (1976).
Meissner, H.P., Thermodynamics of Aqueous Systems with
Industrial Applications. (Edited by S.A. Newman) ACS
Symp. Ser. No. 133, Washington, DC (1980),
pp. 495511.
Millero, F.J., J. Phys. Chem. 74, 356 (1970).
MiHero, F.J., Water and Aqueous Solutions: Structure,
Thermodynamics, and Transport Processes. (Edited by
R.A. Horne) WileyInterscience, New York, NY (1972),
pp. 519565.
Newman, S.A. (Ed.), Thermodynamics of Aqueous Systems with
Industrial Applications, ACS Symposium Series No. 133,
Washington, DC (1980).
O Connell, J.P.,
Mol. Phys. 20, 27 (1971).
191
O'Connell, J.P., Fluid Phase Equil. _6, 21 (1981).
Ornstein, L.S., and F. Zernike, Proc. Acad. Sci. Amst. 17,
793 (1914).
Pailthorpe, B.A., D.J. Mitchell, and B.W. Ninhan, J. Chem.
Soc. Faraday Trans. jH), 115 (1984 ).
Percus, J.K., and G.J. Yevick, Phys. Rev. 110, 1 (1958).
PerezHernandez, W. and L. Blum, J. Stat. Phys. _24, 451
(1981).
Perry, R.L., Ph.D. Dissertation, University of Florida
(1980 ) .
Perry, R.L., H. Cabezas, and J.P. O'Connell, submitted to
J. Soln. Chem. (1985).
Perry, R.L., and J.P. O'Connell, Mol. Phys. 5_2, 137 (1984 ).
Pitzer,
K. S. ,
J.
Phys. Chem.
77,
268 (1973).
Pitzer,
K. S. ,
Acc
. Chem. Res.
l
i11
371 (1977)
Pitzer, K.S.,
(1973).
and
G. Mayorga,
J.
Phys. Chem.
77, 2300
Pitzer,
K. S. ,
and
G. Mayorga,
J.
Soln. Chem.
3, 539
(1974 ) .
Pitzer,
K. S. ,
and
L.F. Silvester,
J. Soln. Chem. 5,
269
(1976 ) .
Planche, H., and H. Renon, J. Phys. Chem. _8_5, 3924 (1981).
Pytcowizc, R.M. (Ed.), Activity Coefficients in Electrolyte
Solutions I, II. CRC Press, W. Palm Beach, FL (1979).
Rard, J.A., and D.G. Miller, J. Chem.
Eng.
Data
27, 342
(1982).
Rasaiah, J.C., and H.L.
Friedman, J.
Chem.
Phys.
, 48, 2742
(1966).
Rasaiah, J.C., and H.L.
Friedman, J.
Chem.
Phys.
, 50, 3965
(1969 ) .
Redlich, 0., and D.M. Meyer, Chem. Rev. 64_, 221 (1964).
Reed, T.M., and K.E. Gubbins, Applied Statistical Mechanics,
McGrawHill, New York, NY (1973).
192
Reichl, L.E., A Modern Course in Statistical Physics,
University of Texas Press, Austin, TX (1980).
Renon, H., Foundations of ComputerAided Chemical Process
Design (Edited by R.S.H. Mah and W.D. Swider), II,
Amer. Inst. Chem. Eng., New York, NY (1981).
Renon, H. and J.M. Prausnitz, AIChEJ, 1_4, 135 ( 1968 ).
Robinson, R.A., and R.H. Stokes, Electrolyte Solutions, 2nd
ed., Butterworths, London (1970).
Romankiw, L.A. and I.M. Chou, J. Chem. Eng. Data, 2j3, 300
(1983 ) .
Rowlinson, J.S., Repts. Prog. Phys. 2Q_, 169 (1965 ).
Sander,B., A. Fredenslund, and P. Rasmussen, AIChE Annual
Meeting, San Francisco, CA (1984).
Telotte, J.C., Ph.D. Dissertation, University of Florida
(1985 ) .
Triolo, R., J.R. Grigera, and L. Blum, J. Phys. Chem. 80,
1858 (1976).
Uematsu, M. and E.U. Franck, J. Phys. Chem. Ref. Data 9_,
1291 (1980).
Vericat, F. and L. Blum, J. Stat. Phys. 2_2, 593 (1980 ).
Washburn, E.W. (Ed.), International Critical Tables of
Numerical Data, Physics, Chemistry, and Technology,
McGrawHill, New York, NY (1928).
Watanasiri, S., M.R. Brule, and L.L. Lee, J. Phys. Chem. 86,
292 (1982).
Watts, R.O., and McGee, Liquid State Chemical Physics, John
Wiley and Sons, New York, NY (1976).
Weast, R.C. (Ed.), Handbook of Chemistry and Physics, 57th
Edition, CRC Press, Cleveland, OH (1977).
BIOGRAPHICAL SKETCH
Heriberto Cabezas, Jr., was born Heriberto Cabezas y
Fernandez on December 8, 1952, in Esperanza, Cuba. After
serving four years in the U.S. Navy (19711975), he entered
the New Jersey Institute of Technology from which he
received a degree of Bachelor of Science in chemical
engineering in May of 1980. He started graduate studies in
the Chemical Engineering Department of the University of
Florida in September of 1980 with the intent of studying
thermodynamics. He received a degree of Master of Science
from the same university in August of 1981.
193
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
7 ^ / yO y
f
John P. O'Connell, Chairman
Professor of Chemical
Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
rr. /
Charles F: Hooper, Jev
Professor of Physics
/
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Gerald B. WestermannClark
Assistant Professor of
Chemical Engineering
This dissertation was submitted to the Graduate Faculty of
the College of Engineering and to the Graduate School and
was accepted as partial fulfillment of the requirements for
the degree of Doctor of Philosophy.
August, 1985
Dean, College of Engineering
Dean, Graduate School
Author:
Cobcci$j tcrt bPC''Q
Application of fluctuation solution theory to
strong electrolyte solutions / (record number:
Title: 880471)
Publication 1985
Date
1, HgIZi&EiZTo ChGttAZ 3g as copyright holder for the
aforementioned dissertation, hereby grant specific and limited archive and distribution rights to
the Board of Trustees of the University of Florida and its agents. I authorize the University of
Florida to digitize and distribute the dissertation described above for nonprofit, educational
purposes via the Internet or successive technologies.
This is a nonexclusive grant of permissions for specific offline and online uses for an
indefinite term. Offline uses sha.!I be limited to those specifically allowed bv "Fair Use" as
prescribed by the terms of United States copyright legislation (cf, Title 17, U.S. Code) as well as
to the maintenance and preservation of a digital archive copy. Digitization allows the University
of Florida to generate image and textbased versions as appropriate and to provide and enhance
access using search software.
This grant of permissions prohibits use of the digitized versions for commercial use or profit.
Signature of Copyright Holder
HERi f? EF riT CA 6 c Z A $ F
Printed or Tvoed Name of Convrisht Holder/Licensee
Personal Information Blurred
Printed or Typed Phone Number and Email Address of Copyright Holder/Licensee
06 /lO/&og
Date of Signature
Please print, sign and return to:
Cathleen Martyniak
UF Dissertation Project
Preservation Department
University of Florida Libraries
P.O. Box 117007
Gainesville, FL 326117007
APPENDIX D
EXPONENTIAL INTEGRALS
The exponential integrals form a class of mathematical
functions whose properties have been extensively studied
(Abramowitz and Stegun, 1973). Here we present a few of the
properties which are relevant to this work.
A general exponential integral, E (Z), of order n is
defined by
Zt
E (Z) = J dt (Dl)
n It
where n = 0, 1, 2, 3, . .
R(Z) > 0 (Real part of Z)
The integral of order zero (n = 0) is trivial. For
those of higher order there exists a recurrence relation
among integrals of adjacent order.
E(Z) = (e~Z ZE (Z)) (D2)
, ] n n
n+1
where n = 1, 2, 3, . .
180
TABLE 421
SOLUTION PROPERTIES FOR KBr (2) IN WATER (1) AT 25C, 1 ATM
1
V o
o2
ktRT
Calc.
1/2 3Â£nY2
CM
0
X
Calc.
Pktrt
Exp.
Exp.
Xo2 V2 3N _
o2
Calc.
T,p,Noi
Exp.
1012
16.1084
16.1084
30.071
30.071
8.763
8.763
1.5247x10
3
16.235
16.249
31.095
31.057
4.476
4.318
3.0705x10
3
16.363
16.390
31.553
31.642
3.364
3.319
9.4793x10
3
16.926
16.969
34.881
33.630
1.993
1.646
0.019825
17.762
17.885
35.774
36.404
0.028
0.489
0.035183
19.0914
19.216
39.546
40.089
1.058
0.566
0.048075
20.259
20.312
42.722
42.948
1.571
1.246
0.070088
22.340
22.183
48.199
47.778
2.054
2.103
0.083974
23.726
23.399
51.740
51.232
2.200
2.392
Calc, properties from equations (244), (245), (246), (362), (384), and
(3104).
Experimental data sources are those of Figures 3, 4, and 5.
139
CHAPTER 2
FLUCTUATION THEORY FOR STRONG ELECTROLYTE SOLUTIONS
Introduction
There are three general relations among the thermodynamic
properties of a solution and statistical mechanical correlation
functions. The first two are the socalled "Energy Equation"
and "Pressure Equation" which are obtained from the canonical
ensemble with the assumption of pairwise additivity of inter
molecular forces. These equations relate the configurational
internal energy and the pressure respectively to spatial
integrals involving the intermolecular pair potential and
the radial distribution function (Reed and Gubbins, 1973;
McQuarrie, 1976). The third relation is the socalled "Com
pressibility Equation" which is derived in the grand canonical
ensemble without the need to assume pairwise additivity of
intermolecular forces. This equation relates concentration
derivatives of the chemical potential to spatial integrals
of the total correlation function (Kirkwood and Buff, 1951)
and to spatial integrals of the direct correlation function
(O'Connell, 1971; O'Connell, 1981). This last method is
generally known as Fluctuation Solution Theory.
Fluctuation solution theory has been applied to the
case of a general reacting system (Perry, 1980; Perry and
11
44
theory and because inclusion of the more complex higher
order terms was empirically unnecessary.
For the sake of simplicity in notation equation (315)
is expressed as
Ac_ = Afj + (Pk A*jk P +k)
13
where Af.. = f.. fHS fLR
11 il il il
(316)
HS
A(J)ijk fijk ijk
No attempt was made in this work to analytically calcu
late the coefficients in equation (316); rather, their
spatial integrals were fitted to data. The importance
of equation (316), however, is in providing a theoretical
framework for describing the properties for a class of molecu
lar interactions which are not well understood. Thus,
the first term represents the contribution of pairing or
repulsion in the case of ion pairs, solvation in the case
of ionsolvent pairs, and hydrogen bonding in the case
of solvent pairs. The second term represents the effect
of a third body (k) on the direct correlation between species
i and j. If one or two of the three are solvent and the
rest ions, then this term is dominated by hydration. If
all three species are ions, then this term is dominated
147
3. Second, the possibility of using a density expansion of
the hard sphere DCFI up to second order in place of the
full expression should be explored. The resulting form
would be simpler and consistent with the Hypernetted Chain
approximation used for the long range field correlations.
Lastly, the multisolvent electrolyte problem should be
addressed. The model may avoid the difficulties encountered
by excess Gibbs energy models.
112
indicate that the water diameter of Marcus (1983) is
adequate for this model but his ionic diameters are less
appropriate.
The solution bulk modulus is dominated by the water
water DCFI at low salt mole fraction. As the mole fraction
of salt increases, the other DCFI's make significant but
small contributions to the bulk modulus. For example, for
NaCL (2) in water (1) at the highest salt mole fraction
(X = 0.084953) the contribution of the waterwater DCFI is
o2
78%, that of the saltwater DCFI is 20%, and that of the
saltsalt DCFI 2%. Thus, the good fit of the bulk modulus
can be understood by the fact that its chief contribution is
due to the waterwater DCFI which the model fits best of all
the DCFI's.
The salt partial molar volume group is dominated by the
saltwater DCFI at low salt mole fraction. The saltsalt
DCFI makes a small but significant contribution at high salt
mole fraction. For example, for NaCL (2) in water (1) at
the highest salt mole fraction (X^ = 0.084953) the con
tribution of the saltwaer DCFI is 84% and that of the salt
salt DCFI is 16%. Thus, the accuracy of the fit for this
property is determined by the ability of the model to fit
the saltwater DCFI which the model could not fit as well.
The salt activity coefficient derivative includes
contributions from all three types of DCFI's in a functional
form which does not permit separation of the respective
CHAPTER 3
A MODEL FOR DIRECT CORRELATION FUNCTION INTEGRALS
IN STRONG ELECTROLYTE SOLUTIONS
Introduction
In order for the formalism introduced in the previous
chapter to be of practical value, a model to express direct
correlation function integrals in terms of measurable
quantities (p, T, x) must be constructed. The present
chapter describes such a model. First, a general physical
picture of electrolyte solutions and its relation to micro
scopic direct correlation functions is discussed. Second,
a rigorous statistical mechanical basis is laid for the
microscopic direct correlation functions and their spatial
integrals. Third, equations are given for each type of
pair correlations in the system (ionioin, ionsolvent,
solventsolvent). Lastly, a summary is presented of the
model parameters and estimated sensitivity of results to
their values.
Philosophy of the Model
The complex thermodynamic behavior of liquid electro
lytes is the observable result of the very complex interac
tions between the species in solution, i.e., the ions and
33
178
cLR = cLR cLR cLR
ilk 13 ik 3k
(C21)
All of the two body quantities (C^ ) in equations
(C20) and (C21) can in principle be obtained from one salt
and one solvent solution data except for the case of any two
k,j that are both cations or anions of different kinds.
Then, to avoid using ternary solution data, the following
mixing rule should be used.
c. =
(C. .
c..)1/2
11
11
11
cLR =
(CLR
CLR)i/2
11
11
j j
(C22)
(C23)
C.. and C.. can be obtained from binary solution data.
The above mixing rules were not used for calculations
in this work. However, they are theoretically sounder than
the mixing rules introduced in Chapter III which were found
adequate for the systems treated in this work. It is pos
sible that as more complex systems are treated with this
theory, the alternate mixing rules will be found to be more
appropriate.
Lastly, it should be understood that the validity of
the application of the virial expansion to a system at
liquid densities has been assumed and not proven. Although,
this should not be a serious problem since the virial
coefficients for all of the interactions except hard spheres
TABLE 41
SOLUTION PROPERTIES FOR LiCL (2) IN WATER (1) AT 25C, 1 ATM
1
pktRT
V
o2
ktRT
j/2 3)inL
Xo2 V2N3N
o2
T,P,Noi
Xo2
Calc.
Exp.
Calc.
Exp.
Calc.
Exp.
io"12
16.108
16.108
15.Ill
15.111
8.763
8.763
4.2558xl03
16.430
16.448
16.471
16.527
1.739
1.107
8.5247xl03
16.749
16.787
17.079
17.337
0.254
0.587
0.025729
18.028
18.151
19.366
20.233
4.601
4.003
0.043142
19.333
19.518
21.769
23.028
7.546
6.612
0.069664
21.398
21.514
25.889
26.970
11.253
10.876
0.087620
22.844
22.768
29.131
29.292
13.464
13.762
0.10580
24.581
23.946
32.924
31.310
15.596
16.113
Calc, properties from equations (244), (245), (246), (362), (384), and
(3104).
Experimental data sources are those of Figures 3, 4, and 5.
104
possible to justify modifying his value for only Lx because
for the other ions, his variations were much smaller
O
(+ 0.06 A or 3%).
Comparison of Calculated Properties
to Experimental Properties
Extensive calculations of solution properties is beyond
the scope of this work. Thus, the calculations presented in
this section aim to prove the suitability of the model.
This consists of comparing values of the DCFI's, the solu
tion bulk modulus, the salt partial molar volume group, and
the salt activity coefficient derivative obtained from the
model to values for the same quantities calculated from
experimental solution properties. These are shown in Tables
41 to 424. The calculations from the model were performed
using the ionically additive parameters listed in Appendix
E. The six salts included in the calculation are LiCL,
LIBr, NaCL, NaBr, KCL, KBr. Graphical comparison of the
salt activity coefficient derivative as calculated from the
model and from experimental data are presented in Figure 6
for NaCL and LiBr. Also, the NaCL saltsalt, saltwater,
and waterwater DCFI's are presented in Figures 7, 8, and 9,
respectively, including the various contributions from the
model and the experimental values.
Additionally, the derivative of the water activity
coefficient was calculated from both the model by the use of
the relation
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of
Doctor of Philosophy
APPLICATION OF FLUCTUATION SOLUTION THEORY TO
STRONG ELECTROLYTE SOLUTIONS
By
Heriberto Cabezas, Jr.
August, 1985
Chairman: Dr. J.P. O'Connell
Major Department: Chemical Engineering
Fluctuation solution theory relates derivatives of the
thermodynamic properties to spatial integrals of the direct
correlation functions. This formalism has been used as the
basis for a model of aqueous strong electrolyte solutions
which gives both volumetric properties and activities.
The main thrust of the work has been the construction
of a microscopic model for the direct correlation func
tions. This model contains the correlations due to the hard
core repulsion, long range field interactions, and short
range forces. The hard core correlations are modelled with
a hard sphere expression derived from the PercusYevick
theory. The long range field correlations are accounted for
by using asymptotic potentials of mean force and the hyper
netted chain equation. The short range correlations which
xi
29
1 C
22
(257)
Figures 35 show the results of equations (255) and
(257) for six different salts at 1 ATM and 25C. The
compressibility data used were those of Gibson and Loeffler
(1948) for NaCL and NaBR. For LiCL, LiBR, KCL and KBR
the compressibilities of Allam (1963) were used. The
activity coefficient data were taken from the compilation by
Hamer and Wu (1972). The density data of Gibson and Loeffler
(1948) were again used for NaCL and NaBR. For LiCL, LiBR,
and KBR the density data were taken from the International
Critical Tables. The newer density data of Romankiw and
Chou (1983) were used for KCL. The pure water data were
those of Fine and Millero (1973). The infinite dilution
partial molar volumes were also from Millero (1972).
Summary
The present chapter has introduced the basic relations
of interest, has shown how they have been used to calculate
the experimental behavior of the DCFI's, and has given
some bounds on the values of the DCFI's. The next chapter
introduces a model for correlating the observed experimental
behavior of the DCFI's.
171
<
1
where S.
3
T,N
Equation (B51) generally fits the partial molar volume
data for aqueous electrolytes at 25C (Millero, 1970) and
ionic strengths below 0.3 M. But, it has not been tested
for nonaqueous or for multisolvent systems. Further, equa
tion (B51) is thermodynamically but not statistical
mechanically rigorous because it is based on the Debye
Huckel theory which being a MM theory is not an adequate
vehicle to represent solvention interactions. Also, it
does include any contribution from Friedman's limiting law
which would be a significant omission for the case of
unsymmetric electrolytes.
The somewhat poorer agreement with experimental data of
equation (B48) in relation to equation (B51) most likely
indicates either inadequate fitting of the model parameters
which has not been carefully done in this work or at worst
incompleteness in the DCFI model. Although, equation (B48)
certainly represents a more complete approach.
79
where
3P/RT
9p
T, N
In order to evaluate the change in solution density
with pressure while the composition and temperature are
constant, one needs to integrate equation (237) from a
R R
known reference density (p ) at the reference pressure (P )
at the temperature and composition (mole fraction) of the
F
system up to the desired density (p ) at the system pressure
(P) .
PP
RT
R
n n
o o
I Z v v y X
a=l B=1 a 6 oa oB
PF(T,P,X)
J (1C )dp
R R P
pK(T,P ,X)
T, N
(41)
Equation (41) represents an implicit equation for the
F
unknown density (p ) which can only be solved numerically
with realistic models.
It should be appreciated that equation (41) cannot be
applied to an isobaric change because that would imply that
pressure, as well as temperature and composition, were
F R
held constant. Then p would be the same as p so the state
of the system would not vary at all.
To evaluate the change in solution density isothermally
with varying composition, a different approach is required.
To develop the necessary relations we start by considering
that in Fluctuation Theory the pressure is treated as the
86
PpR PPYHS(pF,X)
RT
RT
dPYHS. R
P (p ,X)
RT
n n n
pi "P O O O Lyp
(p p ) l l I V V X XX J C o (t)dt
a=l 6=1 y=l a 6 oa 03 OY o aB
F F R R n n
p p PP y y x.x. af. .
2 ii j=i 1 3
FFF RRR n n n
p p p ? p p I l l XXX A*
3 i=l j=l k=l 1 3 k 13lc
+
F F P, F R R P, R
p p Pol p p PQl
n n
, LR
l l X.x.
nil jil 1 3 131
(417)
n n
where p(t) ^vy = vypoy(t)
Equation (49) can also be changed to
PP.
SA
T
T)
RT
ppyhs(pF,xF)
RT
ppYHS R R)
o
RT
n
o
 I
a=l
n n
o o
r r F F R Rv
y y v v (x px p
yÂ£x a 6 OY oyM
1 c (t)
tm,t",o6lt) pTtTpTtT
O
dt
Discussion 105
Conclusions 113
5 CONCLUSIONS AND RECOMMENDATIONS 144
APPENDICES
A HARD SPHERE DIRECT CORRELATION FUNCTION
INTEGRAL FROM VARIOUS MODELS 148
B RELATION OF McMILLANMAYER THEORY TO
KIRKWOODBUFF THEORY 152
C RELATION OF DENSITY EXPANSION OF THE
DIRECT CORRELATION FUNCTION TO VIRIAL
EQUATION OF STATE: ALTERNATE MIXING
RULES 17 2
D EXPONENTIAL INTEGRALS 18 0
E MODEL PARAMETERS 18 4
REFERENCES 18 8
BIOGRAPHICAL SKETCH 193
V
TABLE 410
SALT
SALT DCFI
FOR NaCL (2)
IN WATER (1)
AT 25 C, 1
ATM
Xo2
*1C22 ^
00 pLR
22
(cHScHS )
1 22 22 '
(actbactbT
v 22 22;
(1_C22)CALC
lO"12
9.3190
2.1907x10
6 0.0000
0.0000
2.1907xl06
2.1907xl06
9.9900x10
4
9.3252
46.4674
0.1304
0.0479 
37.059
32.482
4.9751x10
3
9.3494
13.9618
0.6446
0.2432
4.211
3.295
9.9010x10
3
9.3783
7.7026
1.2843
0.4930
2.467
2.285
0.015554
9.4120
5.0615
2.0471
0.7864
5.621
4.961
0.032695
9.5048
2.4063
4.3497
1.7464
9.701
8.999
0.048670
9.5867
1.5721
6.6103
2.7133
11.911
11.630
0.066607
9.6779
1.1084
9.3846
3.8708
14.083
14.244
0.084953
9.7653
0.8380
12.3857
5.1393
16.173
16.571
(IC2 2 ) from equation (362).
EXP
(1C) from equation (257) using the same sources of experimental data as
for Figure 3.
191
O'Connell, J.P., Fluid Phase Equil. _6, 21 (1981).
Ornstein, L.S., and F. Zernike, Proc. Acad. Sci. Amst. 17,
793 (1914).
Pailthorpe, B.A., D.J. Mitchell, and B.W. Ninhan, J. Chem.
Soc. Faraday Trans. jH), 115 (1984 ).
Percus, J.K., and G.J. Yevick, Phys. Rev. 110, 1 (1958).
PerezHernandez, W. and L. Blum, J. Stat. Phys. _24, 451
(1981).
Perry, R.L., Ph.D. Dissertation, University of Florida
(1980 ) .
Perry, R.L., H. Cabezas, and J.P. O'Connell, submitted to
J. Soln. Chem. (1985).
Perry, R.L., and J.P. O'Connell, Mol. Phys. 5_2, 137 (1984 ).
Pitzer,
K. S. ,
J.
Phys. Chem.
77,
268 (1973).
Pitzer,
K. S. ,
Acc
. Chem. Res.
l
i11
371 (1977)
Pitzer, K.S.,
(1973).
and
G. Mayorga,
J.
Phys. Chem.
77, 2300
Pitzer,
K. S. ,
and
G. Mayorga,
J.
Soln. Chem.
3, 539
(1974 ) .
Pitzer,
K. S. ,
and
L.F. Silvester,
J. Soln. Chem. 5,
269
(1976 ) .
Planche, H., and H. Renon, J. Phys. Chem. _8_5, 3924 (1981).
Pytcowizc, R.M. (Ed.), Activity Coefficients in Electrolyte
Solutions I, II. CRC Press, W. Palm Beach, FL (1979).
Rard, J.A., and D.G. Miller, J. Chem.
Eng.
Data
27, 342
(1982).
Rasaiah, J.C., and H.L.
Friedman, J.
Chem.
Phys.
, 48, 2742
(1966).
Rasaiah, J.C., and H.L.
Friedman, J.
Chem.
Phys.
, 50, 3965
(1969 ) .
Redlich, 0., and D.M. Meyer, Chem. Rev. 64_, 221 (1964).
Reed, T.M., and K.E. Gubbins, Applied Statistical Mechanics,
McGrawHill, New York, NY (1973).
83
iHYc =
Poi(T,p,X)i_c
al
dp
ol
n P (T,P,X)
+ I VR J CaB
T 6=2
dp
oB
T
(411)
Equation (411) can be used for either isobaric or
nonisobaric changes.
In equations (49) and (411) one can use the DCFI
model represented by equations (359), (383), and (3104)
for isobaric integrations. But, for nonisobaric integra
tions with equations (41), (49), and (411) the DCFI model
of equations (356), (379), and (3100) will be more
applicable because the pressure behavior of the DCFI
infinite dilution limits, some of which involve salt partial
molar volumes, is not generally available.
The composition behavior of component activity coeffi
cients on the mole fraction scale at constant temperature
and pressure could also be obtained from equation (238)
with composition expressed as mole fractions. Thus, we
express the differential of the activity coefficient of a
component ( a) as
n
dÂ£ny
a
o 9 Â£ny
= y Ji
1 a Y
T,P 6=2 OB
dX
oB
T, P, X
oy^B
(412)
55
inserted into equation (211), is always very small for
symmetric electrolytes, and it approaches zero as the con
centration of salt decreases. For unsymmetric electrolytes,
however, the sum over the ionic charges is not small and
this term actually diverges logarithmically as the salt
concentration decreases. To further explore the relation
of (344) to Friedman's limiting law and to elucidate the
low salt concentration behavior, the exponential integral
(E^) can be expanded for low values of the ionic strength
(I + 0).
E1(3aijB I17"2) = in(3aijB I1^2 ) a + 0(I1//2) (345)
where a = 0.5772 = Euler's constant.
This expansion is valid only at extremely low ionic
strength. Equation (344) then becomes
z3z3
3 1
S2P ^E, (3a. .B I1//2) =
Y ol 1 13 Y
z3z3
31 S2Pq^ (2 ini + a + In 33.^^
+ 0(I1/2)
(346)
where lnl diverges as I * 0 while a + In 3a. .B are all
y 13 Y
constant.
53
4 it o e
n
n
vavB
, m i v Z. y vZ. f r..dr..=0
i=l
j = l
(341)
n
where
l V. Z. =
i=l la 1
The second term of equation (340) is integrable and
contains the implications for DCFI's of the DebyeHuckel
limiting and extended laws (see equation 254). Then,
2 2 4
2iTpZ z e
2Ka. 2Kr. .
(DkT)2(l+KaiJ2
13
/
13
a .
13
dr. .
13
22 1/2
0 0 JL
1 1 Y
(l+a. .B I1/2)
13 y
(342)
where S = ( 2^e 3 )
Y D k T
K = B I
Y
1/2
B = K I
Y
1/2 ( 8r e
2 1/2
DkT
1 ^ 2
1 = 2 zi
1=1
The third term of equation (340) is also integrable
but more complex. The integral is the first order member
189
Debye, P. and E. Huckel, Phys. Zeitschrift, 2_4, 185206
(1923) .
Downes, C.J., J. Soln. Chem. _4 191 (1975 ).
Enderby, J.E., Ann. Rev. Phys. Chem., 3_4, 155 ( 1983 ).
Fine, R.A., and F.J. Millero, J. Chem. Phys., 5_9, 5529
(1973 ) .
Friedman, H.L., Ionic Solution Theory, Interscience, New
York, NY (1962).
Friedman, H.L., J. Soln. Chem., 5, 387 (1972).
Friedman, H.L., and P.S. Ramanathan, J. Phys. Chem., 74,
3756 (1970).
Gautschi, W., and W.F. Cahill, Handbook of Mathematical
Functions. (Edited by M. Abramowitz and I.A. Stegun)
Dover, New York, NY (1972).
Gibson, R.E., and O.H. Loeffler, Ann. N.Y. Acad. Sci., 51,
727 (1948).
Golberg, R.N., and R.L. Nuttall, J. Phys. Chem. Ref. Data,
1, 263 (1978).
Gubbins, K.E., and J.P. O'Connell, J. Chem. Phys., 6_0 3449
(1974 ) .
Guggenheim, E.A., and R.H. Stokes, Equilibrium of Single
Strong ElectrolytesInternational Encyclopedia of
Physical Chemistry and Chemical Physics, vol. 15:1,
Pergamon, Elmsford, NY (1969).
Guggenheim, E.A., and J.C. Turgeon, Trans. Faraday Soc., 51,
747 (1955).
Hala, J.H., Proc. Intern. Symp. Distillation 1969, Inst, of
Chem. Engrs. Symp. Series 32 (1969), 513.
Hamer, W.J., and Y.C. Wu, J. Phys. Chem. Ref. Data, JL, 1047
(1972) .
Harned, H.S., and B.B. Owen, The Physical Chemistry of
Electrolyte Solutions, 3rd Ed., Reinhold, New York,
NY (1958).
Hill, T.L., Statistical Mechanics: Principles and Selected
Applications, McGrawHill, New York, NY (1956).
Figure 5. Water (l)Water (1) DCFI in Aqueous
Electrolyte Solutions at 25C, 1 ATM.
For data sources see text.
81
We next insert equations (234) and (237) into
equation (47 ) .
JL_
RT
n n
o o
[ y v V X X (1C
a^2_ a 6 oa oB a8
dp +
n n n
o o o
+ l l l VRVvX0
a=2 6=1 Y=1 6 Y Y
pdX
oa
6 o~v oX
a B B oa
( 48
Equation (48) permits us to evaluate the change in
solution density with both pressure and composition along an
isotherm. This equation is also applicable to an isobaric
and isothermal process where the solution density changes as
a function of composition only.
F
To obtain the density (p ) of a given solution at a
F
known temperature, pressure, and composition (X ), we
isothermally integrate equation (48) from the known
reference density (p ) at a system temperature and a conven
R R
iently chosen reference pressure (P ) and composition (Xq )
F F
up to the desired density and composition (p and XQ ) It
is suggested that for aqueous electrolytes the reference
density be chosen to be that of pure saturated water at the
system temperature.
CHAPTER 4
APPLICATION OF THE MODEL TO AQUEOUS STRONG ELECTROLYTES
Introduction
In Chapter 2, the formal relations between DCFI's and
thermodynamic properties were introduced. In Chapter 3, a
model expressing the DCFI's in terms of measurable variables
was constructed. In the present chapter we illustrate the
use of the formal relations and the model in the calcula
tion of thermodynamic properties. We also explore the
scheme used to fit model parameters; further we compare
calculated values to experimental ones for the saltsalt,
saltsolvent, and solventsolvent DCFI1s and for the bulk
modulus, partial molar volume, and salt activity coeffi
cient. Finally, a discussion of the above results and a few
conclusions are presented.
The use of Fluctuation Theory in general fluid phase
equilibria problems has been treated in detail by O'Connell
(1981). The specific case of liquids containing super
critical components has been addressed by Mathias and
O'Connell (1981) and Mathias (1978). The present treatment
generally follows these developments, but there are
important differences for the present case of electrolytes.
77
187
Na+
Na+
1
11.4220
K+
K+
1
21.0341
CL~
cl
1
5.7911
BR~
br
1
9.5753
. +
Li
. +
Li
. +
Li
16.0000
Na +
Na +
Na +
23.6270
K+
K+
K+
1.9808
cl
cl
cl
7.7004
BR
br
br
8.4594
61
where pressure varies, equation (356) would be more
convenient since it would eliminate the need to obtain
partial molar volumes as a function of pressure.
For illustrative purposes, equation (359) will now
be written for a binary system consisting of a solvent
(1) and a salt (2) which dissociates to formv+ cations
and v_ anions.
n = 1 + v+ + v_
v
2
1 C22 (1 C22^
CO
LR / pHS HS
^22 ^22 C22
(360)
where
P
2
v
+ 2v+v_(A$^H + p + A$+H + p_A$_+_) +
1
(361)
159
where
3N
oS
T,P,N
oy=3
0
since is a constant with respect to changes in the mole
number of solutes.
In order to relate the above results to direct correla
tion function integrals we use equation (212).
8 Â£ny
KB
pv
a 8p
oB
= vavB (1CaB}
(212)
T, p
oy^B
Now the chemical potential of any component (a) can be
written as
.KB
(T,V,X) = M,KB(T) + v RT *n y^B(T,V,X) (B19)
Ut LX O LX CX
Differentiation of equation (B19) gives
8y
KB
a
8N
oB
, 0 8 Â£n y
= V RT + v RT 5 
a N a 8N
T, V, N
a
oB
oy^B
T, V, N
oy^B
 v RT f
a N
(B20
where 6 = Kroniker delta
a B
v = no. species formed by the dissociation of
a
a which equals one if a does not dissociate.
KEY TO SYMBOLS
a .
l
'a 6
C. ..
ink
AC. .
13
D
D
E
1
n
e
f .
13
hard sphere diameter of species i.
distance of closest approach of species i and j.
k/I1/2
sum of all bridge diagrams, second virial
coefficient.
mixture third virial coefficient.
direct correlation function integral for species
i and j; twobody factor in third virial
coefficient.
direct correlation function integral for
components a and 8.
third virial coefficient for i, j, k.
short range direct correlation function
integral.
direct correlation function.
short range direct correlation function.
dielectric constant of solvent or solvent mixture.
pure solvent dielectric constant.
exponential integral or order n.
electronic charge.
spatial integral of Af ^ j
u. ./kT
1
e J 1 = Mayer bond functions.
vi
145
An important theoretical byproduct of the present work
has been to show that expressions for the solvent chemical
potential derived from a McMillanMayer formalism such as
the DebyeHuckel theory are not appropriate for fluctuation
properties such as waterwater and watersalt DCFI's. In
this work, we have presented an alternate model for the
above DCFI's which leads to a limiting slope for the salt
partial molar volume which is different than that of the
traditional DebyeHuckel theory. The functional form of
this model includes the rigorous DebyeHuckel limiting law
for activity coefficients and a general form which contains
the higher order limiting law for the activity coefficient
of unsymmetric electrolytes first discovered by Friedman
(1962 ).
The calculation of solution properties using the model
has been extensively discussed in Chapter 4. Here we simply
restate that to obtain properties the model is integrated
over concentrations from the pure water limit. The result
ing integrals are not always simple or analytic but are
tractable using numerical methods and modern computers.
A benefit of the above is that errors due to model approxi
mations tend to be smoothed and largely selfcancelling
rather than cumulative.
Much work remains to be done in testing, developing,
and applying this model. What follows contains some sugges
tions to guide future work.
40
of zero salt concentration and large separation between
the two interacting species, are available for ionion
interactions from the DebyeHuckel theory and for iondipole
and dipoledipole interactions from more recent work (H^ye
and Stell, 1978; Chan, Mitchell, and Ninham, 1979) which
yields results identical to those of Debye and Huckel for
ionic activities. Thus, the long range direct correlation
LR
function is based on these potentials of mean force, W^.
Then, our HNC approximation is
LIM
I*o
r. .
il
cHNC
il
c
LR
i j
(37)
(38)
The potentials of mean force, however, are unphysical
inside the hard core of the molecules and must be set equal
to zero.
T7lr
w. .
11
= 0 r. < a. .
il 13
(39)
T 7 LR
w. .
11
ttLR
= W . r . > a .
il il il
a.. (a . + a .)
il 11 11
= distance of closest
of species i and j
approach
where
102
The values were chosen as before and the quantities for
other ions are given in Appendix E. The procedure for
obtaining model parameters described in this section is
strictly drawn from the theory and yields a unique set of
values for the parameters if independently determined values
for the species hard sphere diameters are available. The
minimal data required includes pure water density, pure
water isothermal compressibility, and aqueous salt partial
molar volume at infinite dilution. In addition, values for
the solution density, salt partial molar volume, compres
sibilty, and either the salt activity or the osmotic
coefficient derivative with respect to salt concentration
must all be available at two different solution
compositions.
As shown in the next section, the model fits solution
data adequately but not within experimental error with
parameters that have been evaluated as outlined above. It
is believed that this could be greatly improved by fitting
the hard sphere diameters along with the other parameters to
solution data. This scheme could cause the uniqueness of
the parameters to be lost. The lack of uniqueness may cause
problems for this model where the parameters are ionic
quantities that must be internally consistent. There is no
procedure capable of eliminating the lack of uniqueness
problem in a complex model such as the present one. How
ever, the difficulty may be ameliorated by either of two
46
where
= p AF j (T)
AF ( T) = 4 tt
i j
n LR
+ p l (pvA$(T) p?$(T))
k=l ijk K ijk
< Af. .>
, i: <*>
(321)
A4> (T)
ijk
4TT
CO
J
< A .
ijk
>
00
LR
$ (T)
ijk
4tt J
,,LR n
<0 >
ilk oo
'13
Equations (319), (320), (321), and the expression
for from Appendix A are the general forms of the model
for species direct correlation function integrals. To
obtain practical expressions one needs merely to introduce
the appropriate pair potential and potential of mean force
into equation (320) and perform the indicated integrations
as illustrated in the sections that follow.
Since the coefficients in equation (321) are fitted
to data rather than evaluated analytically, it is of
importance to develop mixing rules to reduce the amount
of data necessary to model multicomponent systems. The
aim here is to predict all the coefficients from quantities
associated with no more than two different species so that
only binary or commonion solution data would be required.
For aqueous electrolytes, the situation can be improved
146
The model parameters, especially the species hard
sphere diameters, have not been properly optimized to fit
solution property data. Thus, all of the model parameters
should be fitted globally for the six 1:1 electrolytes
included in the present work and for a number of other
salts, especially higher charge types and unsymmetric
electrolytes.
Once parameter values have been obtained for several
different salts, the model should be tested against solution
properties such as density and activity coefficient rather
than derivative properties as done here. The reason is that
solution properties are of most interest in practice. It
would be very important to test the predictive ability of
the model with a few salts that were not included in the
fitting of the parameters.
A more severe test would be to compare solution prop
erties calculated from the model to experimental values in
aqueous solutions containing at least two salts. A more
sensitive test than density or activity would be provided by
the prediction of Harned coefficients.
Besides exploring the range of practical applicability,
it is also of importance to test the theoretical basis with
a view to simplification and improvement. First, the alter
nate mixing rules introduced in Appendix C should be tested
with better fitted parameter values. These alternate mixing
rules have a sounder theoretical basis than those of Chapter
90
F
Finally, once p has been obtained from either equation
(417) or (418), the compressibility of the solution and
the partial molar volume of any component can be calculated
from
n n
o o
< = 1/p RT l l v v XX (1C .)
T a=l 8=1 a B oa op a8
(421)
V
oa
n
(422)
The above equations give the method to be used in the
calculation of solution properties using Fluctuation Theory
and our model. However, work remains to be done on the
practical application of these expressions.
Model Parameters from Experimental Data
The chief aim of this chapter is to demonstrate the
ability of the model to correlate solution properties for
different salts in water. This requires obtaining model
parameters for different systems. As an expedient, those
parameters which could be independently obtained from
literature sources were adopted rather than calculated from
solution data. We hoped then to avoid obscuring any model
inadequancies by parameter fitting. Thus, we used litera
ture values for the pure water density, compressibility,
dielectric constant, and dipole moment and for the ionic
88
PYHS
The expression for P is given by equation (Al) and
T R
that for by equations (353), (376), and (398).
In a similar fashion we transform equation (411) to
n
Â£ny =  y v (
ra v .ia RT
a i=l
^YH?T,pF) yfH?T, R
P )
RT
P
,n_ )
Pi
i LR \
, F R, f1 cal(t
Pol pol) s0 JTtT dt
n
6=2 (S
R 1 CaR(t) nF
po6> !o Hm dt + *
1 V r F R
L l v (p.p.)AF..
v L1 ,L1 ia ] j 13
a 1=1 j=l J J J
1 r Â£ y F F R R
2v Via pjpk phpj ijk
a i=l ]=1 k=l J J J
n n
1 r r F P,F R P,Rn LR
L L v P P i p p ) 4 .,
L L ] ol ijl
2v L, .L, ia j ol
a i=l j = l J
(419)
n
n
where p. = P l v X = l v. p
1 ia oo la <
a=l a=l
and the other quantities have been defined above.
Calculation of Solution Properties
from the Model
78
The formal relations between solution properties and
DCFI's are given by equations (212), (234), (237), and
(238) for a system consisting of nQ components, salts (cuB)
and one solvent (1).
9 Â£ny
pv
a
a 9p
oB
= v v (1C _)
a B aB
T, p
oy^B
(212)
V
oa
n
Va VBXoB(1 CaB
(234)
9P/RT
9p
T, X
n n
o o
Z Z
a=l B=1
v v x
a B oa
(1C.a)
at
(237)
NV
a
9Â£ny
n
n
a
PKTRT 9Nq3
T, P, N
= v v Z z
a B y=l 6=1
v X
y 0 Oy oo
oy^B
 (1C ) (1C ) ]
ay yB
[(1C .)(1C Q)
y o aB
(238)
51
Expression for SaltSalt DCFI
The saltsalt direct correlation function integral
(C .) can be expressed as a stoichiometric sum of ionion
a 8
DCFI's (c^j) given by equation (211).
1
n n
I l
i=l j=l
V V
ia
IB
(1Cii)
v v
a
8
(211)
It is, thus, only necessary to develop general and practical
expressions for the ionion DCFI's and insert these into
equation (211) to obtain a general expression for the
saltsalt DCFI. The basic model for ionion DCFI's is
HS
represented by equation (319). The expression for C^j
has been developed in Appendix A and that for AC^j is given
by equation (321). This section is then chiefly concerned
with performing the integrations in equation (320) to
LR
obtain an expression for C^.
The pair potential between two ions is given by
u
LR
i j
Z Z .e2
1 3
r
(338 )
Here the potential of mean force is approximated by a gener
alized form given by the DebyeHuckel theory.
T7LR
w. .
13
Z Z .e
i 3
kTr. .
13
e
K ( a .
13
D(1+Ka. )
13
r>a^j (339a)
W
LR
13
0
r < a. .
 13
(339b)
185
CO
Infinite Dilution Ion Partial Molar Volumes (V )
(25]C, 1 ATM)
ION
oo
V
(ML/MOL)
Li +
Na
.+
+
K
cl"
BR~
+ MILLERO, 1972,
0.88
1.21
9.02
17.83
24.71
Hard Sphere Diameters
WATER MOLECULE
all =
LITHIUM ION
aLiLi
SODIUM ION
aNaNa
POTASSIUM ION
a + +
K+K
CHLORINE ION
aCL CL
BROMINE ION
aBR BR
+ MARCUS, 1983
* FITTED TO OUR DATA. THE
IS 1.36 + 0.12 A.
2.78 6A+
*
= 1.45 A
+
= 1.96 A
+
= 2.68 A
+
= 3.66 A
+
= 3.88 A
VALUE GIVEN BY MARCUS, 1983,
TABLE 418
SALT
SALT DCFI
FOR KCL (2)
IN WATER (1)
AT 25 C,
1 ATM
Xo2
( *'~<222 ^
rLR
22
, HS HS
V 22 *22
1 22 22J
1 ^1_C2 2 ^
CALC.
dc22)EXP
IQ"12
11.9555
2.1907x10
6 0.0000
0.0000
2.1907x10 6
2.1907x10 6
9.9900xl0~4
11.9570 
44.6388
0.1541
0.0055
32.522
30.223
4.9751xl0~3
11.9624 
12.9584
0.7520
0.0186
0.225
0.547
8.84 82x10 ~3
11.9671
7.7786
1.3244
0.0242
5.537
5.913
0.025637
11.9851
2.7250
3.8145
0.0041
13.078
13.010
0.041316
11.9983
1.6267
6.1437
0.0747
16.440
16.543
0.055992
12.0081
1.1545
8.3458
0.1893
19.010
19.377
0.069758
12.0151
0.8950
10.4132
0.3330
21.200
21.378
(IC22 )CAL<^* from equation (362).
EXP
(IC22) from equation (257) using the same sources of experimental data as
for Figure 3.
34
solvent molecules. In the absence of a complete understand
ing of all these forces, models use simpler or, at least
tractable, interactions which may have the essential charac
teristics of the real forces. In addition, some semiempiri
cal terms are used to account for those interactions that
cannot be simply approximated.
Thus the interactions between the ions at long distances
are modeled as those of charges in a dielectric medium
containing a diffuse atmosphere of charges. At very short
range, however, the dominant interaction becomes a hard
spherelike repulsion. There exist rigorous statistical
mechanical methods to treat these two types of interactions,
but these two are not adequate to correlate and predict
the solution behavior with sufficient accuracy. Interactions
that are important at intermediate ionion ranges must
be incorporated. Unfortunately, these intermediate range
forces cannot be simplistically approximated because they
involve strong manybody effects such as dielectric satura
tion, ionpairing, polarization, etc., which are not well
understood. In the present model the ionic and hard sphere
interactions are treated theoretically while the rest are
included in a semiempirical fashion.
The interactions between ions and solvent molecules
at large separation can be treated as those of charges and
multipoles in a dielectric medium containing an ionic
atmosphere. In general, quadrupoles and higher order
and temperature to sums of direct correlation function
integrals for components.
23
In summary, it should be noted that of the various
relations developed in this section, only a few are of prac
tical importance in relation to this work. These are listed
at the beginning of the next section.
Direct Correlation Function Integrals from
Solution Properties
The previous section consists of a relatively simple
but lengthy derivation of several basic relations between
solution properties and direct correlation function
integrals. The relations that are of most importance to
this work are listed below.
T,p
(212)
Y^6
V
oa
n
o
(234)
9P/RT
9p
(237)
12
O'Connell, 1984), and the formalism has also been adapted
to treat strong electrolyte solutions which are considered
as systems where the reaction has gone to completion (Perry,
Cabezas, and O'Connell, 1985). The main body of this chapter
consists of a derivation of the general fluctuation solution
theory. Although the final results are identical to those
previously obtained by Perry (1980), the development is
more intuitive and mathematically simpler, though less
general. The remainder of the chapter illustrates the
calculation of direct correlation function integrals (DCFI)
from solution properties and sets theoretically rigorous
infinite dilution limits on the DCFI's.
Thermodynamic Property Derivatives and Direct
Correlation Function Integrals
A general multicomponent electrolyte solution, contain
ing n species (ions and solvents) formed from nQ components
(salts and solvents) by the dissociation of the salts into
ions, is not composed of truly independent species due
to the stoichiometric relations among ions originating
from the same salt. It is, therefore, not possible to
change the number of ions of one kind independently of
all the other ions. However, the independence of ions
has been assumed traditionally for theoretical derivations,
and it will lead us to the correct results by a relatively
simple mathematical route. Thus, with the assumption that
any two species i and j are independent of all other species,
149
potential from the PercusYevick compressibility equation of
state is
HSPY :
y p A .
i i i
RT n int
qi
 Â£n (1C ) +
ira
i PHSPY+ 3(ai*2 + ai *1}
6 RT 15,
+
9{22 ai2
2(l{3)2
(A3
h2 I/2 ...
where A = ( ) = ideal gas partition function
i 2tt m^kT
h = Planck's constant
= mass of species i
k = Boltzmann's constant
INT
q_^ = internal partition function.
The corresponding hard sphere chemical potential from the
CarnahanStarling equation of state is
HSCS HSPY 2 
b d + m (ie )(i2a> (32 [f] )
RT RT ^3 ?3 53
(^2ai)3 (2~^3) 53 2 + 3_
15
^2ai
15,
3^3
ai g2 g3
(l53)3
(A4 )
190
H0ye, J.S., and G. Stell, J. Chem. Phys., 68, 4145 (1978).
Kirkwood, J.G., and F.B. Buff, J. Chem. Phys., 19_, 774
(1951).
Kusalik, G. and G.N. Patey, J. Chem. Phys., 7_9, 4468
(1983 ) .
Lebowitz, J.L., Phys. Rev., 133, A895 (1964).
Lobo, V.M.M., and J.L. Quaresma, Electrolyte Solutions:
Literature Data on Thermodynamic and Transport
Properties, vol. II, Coimbra Univ. Press, Coimbra,
Portugal (1981).
Mansoori, G.A., N.F. Carnahan, K.E. Starling, and
T.W. Leland, J. Chem. Phys., 5_4, 1523 (1971).
Marcus, Y., J. Soln. Chem., 12, 271 (1983).
Mathias, M., Ph.D. Dissertation, University of Florida
(1978 ) .
Mathias, P.M., and J.P. OConnell, Chem. Eng. Sci. 3_6' 1123
(1971) .
Mauer, G., Fluid Phase Equilibria 13, 269 (1983).
McMillan, W.G., and J.E. Meyer, J. Chem. Phys., 13, 276
(1945 ) .
McQuarrie, D.A., Statistical Mechanics, Harper and Row, New
York, NY (1976).
Meissner, H.P., Thermodynamics of Aqueous Systems with
Industrial Applications. (Edited by S.A. Newman) ACS
Symp. Ser. No. 133, Washington, DC (1980),
pp. 495511.
Millero, F.J., J. Phys. Chem. 74, 356 (1970).
MiHero, F.J., Water and Aqueous Solutions: Structure,
Thermodynamics, and Transport Processes. (Edited by
R.A. Horne) WileyInterscience, New York, NY (1972),
pp. 519565.
Newman, S.A. (Ed.), Thermodynamics of Aqueous Systems with
Industrial Applications, ACS Symposium Series No. 133,
Washington, DC (1980).
O Connell, J.P.,
Mol. Phys. 20, 27 (1971).
TABLE 425
WATER ACTIVITY COEFFICIENT IN AQUEOUS NaCL AT 25C, 1 ATM
3 Â£ny.
N
X
3N
o2
o2
CALC.
3 &ny.
N
3N
T, P, N
o2
EXP.
ol
T,P,Noi
1012
0.0000
0.0000
9.9900xl0~4
4.837
4.567
4.9751xl0~3
0.168
0.151
9.9010x10_3
0.086
0.096
0.015554
0.035
0.013
0.032695
0.442
0.327
0.066607
1.306
1.358
0.084953
1.773
2.077
CALC.
3 Â£n y.
N
3N
o2
from equation (439)
T,P,Noi
3 dny.
N
3N
o2
T, P, N
ol
from equation (440)
using the data of
Hamer and Wu (1972)
143
73
The infinite dilution limit for Ac^ is given by
Ac = lim Â£
U11 xq1+1 11
p R ( Af p R>RR ) + (p P)2 AO
oll *11 Pol 111' 1 ol' 111
(3103)
Finally, the complete general expression for the
solventsolvent DCFI including the infinite dilution limit
is
1 C
11
P ,
ol 1
 (C
HS
11
 C
HSC
11
) (C
LR
11
cLR)
U11
 ( Ac Ac )
1 ii ir
(3104)
Again, the application of equation (3104) to a binary
system consisting of solvent (1) and a salt (2) with
cations and v_ anions is shown. However, for the solvent
solvent DCFI all of the terms except AC.^ appear similar
to the general case since they have no summations over
species. Thus, only Ac^ is illustrated below.
ACn ACIi <0 PoI)(AFn PoKn1 +
+ + P+A$ + n + P_A
 (AoI)2a*111
(3105)
CHAPTER 5
CONCLUSIONS AND RECOMMENDATIONS
The chief accomplishment and significance of the
present work is the construction of a microscopic statis
tical mechanical model of liquid electrolyte solutions which
has been shown capable of representing both volumetric and
activity behavior. All of the model parameters are obtain
able from one salt and one solvent data. These parameters
are ionically additive and applicable to all solutions.
Thus, the properties of a large number of systems can be
predicted using information obtained from a very limited
number of experimental measurements.
The model includes the effect of the various molecular
interactions such as hard core repulsion, long range field,
and short range forces in a theoretically sound fashion
which is simple enough for economical engineering calcula
tions. The inclusion of all the above was necessary in
order to properly represent all the solution properties
which are sensitive to different types of interaction. For
example, the density is mostly determined by the hard core
repulsive forces while the activity is dominated by long
range field interactions at low salt concentration.
144
108
co pjCJ HSco
magnitude is again dominated by (lC^) and C^ ~ ^^.2
while the behavior is controlled by the hard sphere crrela
LR 00
tions since C^ and AC^ ~ ^*12 lar
For example, at the highest salt mole fraction (X 2 =
T p 00
0.084953 ) the sum of and Aci2 AC12 :''s _8*58
smaller values elsewhere.
The deviation of the calculated saltwater DCFI from
experimental values at low salt mole fraction is probably
due to the inadequacy of the functional form used to
represent the ionwater correlations. At high salt mole
fraction, however, the deviations are attributable to the
values used for the hard sphere diameters.
The behavior of the water (l)water (1) DCFI is illus
trated in Figure 9. For this case, the contribution of long
LR
range field correlations in C^ is negligible over the
entire composition range. Therefore, the magnitude of the
OO
waterwater DCFI is dominated by (1C^) the hard sphere
HS HS 00
correlations (C^ C^ ), and AC^ AC^ in order of
HS HS
importance. The behavior is controlled by and
OO
AC^ AC^ again in order of importance. For example, at
the highest salt mole fraction (X 2 =0.084953) the relative
00 hs HS00 00
contributions of (lC^) > C^i / and AC^ AC^ are
58%, 31%, and 11%, respectively, while the relative con
tributions to the deviation from the infinite dilution value
fl C U Cqo 00
are 75% for c" c and 25% for AC^ AC^.
TABLE 419
SALT
WATER DCFI
FOR KCL
(2) IN WATER (1)
AT 2 5 C, 1 ATM
Xo2
(1C12>"
rLR
12
. HS HS
'12 12
) 
dc12)CALC*
EXP.
U C12;
io"12
11.9686
0.0000
0.0000
0.0000
11.969
11.969
9.9900xl0~4
11.9686
0.3922
0.1078
0.0185
12.487
12.473
4.5751xl0"3
11.9686
0.5294
0.5256
0.0929
13.116
13.274
8.84 82xl0~3
11.9686
0.5772
0.9255
0.1661
13.637
13.848
0.025637
11.9686
0.6580
2.6629
0.4895
15.779
15.927
0.041316
11.9686
0.6900
4.2851
0.7985
17.742
17.803
0.055992
11.9686
0.7089
5.8164
1.0198
19.585
19.510
0.069758
11.9686
0.7218
7.2521
1.3691
21.311
20.880
(1C^2)CALC* from equation (384).
EXP
(1C, 2) from equation (256) using the same sources of experimental data as
for Figure 4.
TABLE 42
SALTSALT
DCFI FOR
LiCL (2) IN
WATER (1) AT 25
C, 1 ATM
Xo2
^1_C22 ^
00 rLR
22
HS _HS
K 22 22
) ( Ac Ac TBT
' 22 22
(1"C22)CMjC*
uc22,exp
IQ"12
13.6300
2.1907x10
6 0.0000
0.0000
2.1907x10 6
2.1907x10
4.2558xl0~2
13.6741
16.5065
0.4047
01099
2.537
0.1625
8.5247xl02
13.7173
9.2628
0.8089
0.2206
5.042
6.065
0.025729
13.8855
3.3199
2.4814
0.6758
12.371
11.877
0.043142
14.0510
1.9617
4.2756
1.1542
15.210
14.750
0.069664
14.2996
1.1735
7.2865
1.9232
18.489
18.754
0.08762
14.4691
0.9091
9.5751
2.4754
20.659
21.044
0.1058
14.6442
0.7318
12.1653
3.0650
23.012
22.619
(1~C22 ^CAL< from equation (362).
EXP
(IC22) from equation (257) using the same sources of experimental data as
for Figure 3.
7
in the salt composition behavior of the species density
is obvious even for relatively similar salts, i.e., the
solution seems to expand for KBr while it seems to contract
for all other salts. The activity coefficient data were
taken from the compilation by Hamer and Wu (1972). For
NaCl and NaBr the density data of Gibson and Loeffler (1948)
were used. For LiCl, LiBr, and KBr the density data were
taken from the International Critical Tables. For KC1
the density data of Romankiw and Chou (1983) were used.
In the hope of improving the situation for obtaining
properties of solutions, a new model of strong aqueous
electrolyte solutions is presented here. This model has
been carefully constructed so that it overcomes a number
of the deficiencies of previous methods. For example,
this model is simple enough for economical engineering
calculations, yet sufficiently sophisticated to rigorously
include all the different interactions (ionion, ionsolvent,
solventsolvent) and the principal physical effects (electro
static, hard core repulsion, hydration, etc.) that contribute
to each interaction. The model is also extendable to multi
salt and multisolvent systems in a straightforward fashion.
Finally, it addresses both activity and volumeric
properties.
In the chapters that follow, a detailed development
of the new model is presented. Chapter 2 has the general
relations between solution properties and correlation
175
Equations (C7) and (C8) will now be used in the develop
ment of alternate mixing rules for the quantities A F^ .,
LR
A$ijk' an<3 ^ijk These are defined in equations (312),
(315), (316), and (321). We now restate these defini
tions in a different form.
Af. (T) = /
13 ico 13 w i 3
 / M dr.dr .
13 w i 3
 / M dr.dr .
ioo x3 w i 3
(C9 )
where = ^ J f .d co. d
^ n2 1
CO.
u. ./kT
i 1
f^j = e 1 = Mayer bond function
= J dco .
A$ (T) = f dr. dr .dr,
13k J i] k ]k di i j n
f ..HS.HS.HS. , ,
 j dr dr .dr,
J lj lk 3 k co i j k
:cio)
where = x f f . f ., f ., dco. dco .dco,
13 lk 3k co fi2 J i] ik ]k 13k
LR 00
$ (T) = f dr.dr .dr,
ijk o 13 lk 3k w _1 ~J ~k
(Cll)
158
Equation (B15) relates a change in the KB chemical
potential of any component (a) to the corresponding change
in the MM system when the mole number of any component (8)
is varied in a mixture containing one solvent (1) and (n 1)
solutes.
We now explore some specific cases. First, if both ot
and 8 are the solvent (a = 8 = 1), then equation (B15)
reduces to
3y
KB
3N
ol
3y
LR
3N
T, V, N
ol
^LRLR
V K
(B16)
T,P,N
oy^l ' oy^l
Second, if a is the solvent (a = 1) and 8 is a solute, then
3 M
kB
3N
3y
LR
3N
T, V, N
o8
vLRvLR
1 8
V k_
(B17 )
T,P,N
oy^8 ' oy^8
Third, if both a and 8 are solutes, then equation (B15)
gives
kB
3y
3N
o6
,, MM
3 y
P+TT
3vlr
a
r
a
3N
J
3N n
T V N
1,V,Noy^6
T'p'We
p
08
d P +
T, P, N
oy^8
LR LR
Va 8yi
vf 9No8
VÂ£R P + TT 3VRR
+ _SL_ / 1
vRR P
T'P'Noy^8 1
3N
08
d P +
T, P, N
oy^8
LRLR
a 8
V K
(B18 )
170
Equation (B48) includes the contributions of the
DebyeHuckel limiting law and the higher order limiting law
for unsymmetric electrolytes first reported by Friedman.
The development of equation (B48) is rigorous in a statis
tical mechanical and a thermodynamic sense. However, it
appears to accurately fit experimental partial molar volume
data only at very low ionic strength (0.1 M) where uncer
tainty in the data is high.
If one assumes that salt activity coefficient is given
by the DebyeHuckel limiting law,
n
Â£n y =
a
2 N a/2
S ( l v Z ) I
y v ia i
(252)
a i=l
and then inserts this expression into the thermodynamic
relation,
9 Â£n y
_c
9P
T, N
VLR V'LR
a a
v RT
a
(B49)
then one obtains the RedlichMeyer equation (1964).
VÂ£R CLR +  sv < + 0II)
(B50)
yLR ,LR
a a
k^RT c^RT 2 ^RT
3 V Pl)1/2 n 2
+ 3 V _ol ( Â£ v. Z.2)( l w x
. ia i L
i=l
1/2
L. Y oY'
Y=1
+
+ 0(X
(B51)
This dissertation was submitted to the Graduate Faculty of
the College of Engineering and to the Graduate School and
was accepted as partial fulfillment of the requirements for
the degree of Doctor of Philosophy.
August, 1985
Dean, College of Engineering
Dean, Graduate School
APPENDIX B
RELATION OF McMILLANMAYER THEORY TO
KIRKWOODBUFF THEORY
Fluctuation Solution Theory, also known as Kirkwood
Buff Theory (Kirkwood and Buff, 1951) is a statistical
mechanical formalism relating concentration derivatives of
the chemical potential to spatial integrals of correlation
functions in the grand canonical ensemble where the natural
independent variable are system temperature, volume, and
composition. However, practical calculations are normally
done in the LewisRandall system for which system tempera
ture, pressure, and composition are the independent vari
ables. A further complication is introduced by the fact
that some important theoretical results such as the Debye
Huckel (1923) limiting law for salt activity coefficients
can only be rigorously obtained from the McMillanMayer
theory (Friedman, 1962) for which the natural independent
variables are solvent chemical potential, system tempera
ture, and composition. Presently, the conversions of the
chemical potential of components a and 6 in system of nQ
components will be explored. We start by relating the
KirkwoodBuff (KB) chemical potential to that in the Lewis
Randall (LR) system.
152
173
When equation (C2) is inserted into the compressibil
ity equation (228) which is
J_ 3P_
RT 3p .
3
n
I
i=l
X.
i
c. .
13
where C. .
il
P
1 j to
dr.dr .
i j
(228 )
we obtain after integration
_1_ 9P
RT 3p .
3
= 1
T,p
n
I
i=l
pi
n
I P
i=l
I P
k=l
dr.dr .
ij (D l j
/ <$
ilk
dr.dr . .
to i j
(C3 )
To establish the desired relation we begin with the
general virial equation of state for a system of n species
(Reed and Gubbins, 1973).
P
RTp
1 + PB(T)
+
p2c(t)
+
(C4 )
n
where P = l p.
i = l X
n n
= I X. I X.
i=l Xjl 3
Bi.(T) =
BIT)
Second
Virial
Coefficient
168
o nP T1/2
IR Voj1
aB 4v
a 6
n n
I I v v +
i=l jl la
S 2pP n n
+ I I v. v. Z.3Z.3(y +
3Vs =i j=i ia]M 3
+ Un ( 3a .B I1//2 )
13 Y
+ 3a B I1//2 )
l Y
+ 0(1)
(B45:
Equation (B54) expresses the composition behavior of
saltsalt DCFI's near zero ionic strength or pure solvent.
We now change concentration scales from ionic strength
to mole fraction and then consider the contribution of the
saltsalt DCFI's to equation (234) noting that (1C^)
approaches a constant.
v v y Q(lC n )
a 6 oB aB
1/2
(
n
o
I
Y=1
co X
Y OY
1/2
X
oB
n n
I I
i=l j=l
V V
ia
P
ol
n
n
n
l I v. v Z.Jz.JÂ£n(3a. .B (pp. Â£ co X )1/2) +
> ia IB i 3 i] y ol L. y oY
i=l j=l
36
3 Y
Yl Y Y
+ 0(X
(B46)
109
The agreement of calculated values of the waterwater
DCFI with experimental ones is good for all salts. This
leads us to believe that the parameter values used, includ
ing the water hard sphere diameters, are appropriate. The
HS
hard sphere sizes for the ions play a secondary role in C^.
The solution bulk modulus (p
molar volume group (Vq2/ktrt) are generally well behaved
with salt mole fraction. The salt and the water activity
coefficient derivatives, illustrated in Figures 6 and 10,
undergo extremely rapid changes with salt mole fraction near
infinite dilution in salt. This is particularly evident for
the water activity coefficient. This behavior was observed
by Pitzer (1977) for water activity coefficients in
electrolytes.
Solution properties such as density and activity coef
ficients are, as discussed earlier, obtained from this model
by the integration of the model expressions. Because model
deviations from experimental values generally fluctuate from
positive to negative, the errors tend to be selfcancelling
rather than cumulative during the integrations. Therefore,
based on a propagation of errors analysis, it is expected
that the deviation of calcualted densities and activity
coefficients from experimental values will be at least a
factor of five less than that of the DCFI's at the same salt
mole fraction.
181
In practice, equation (D2) allows one to express an
exponential integral of any order in terms of the
exponential integral of order one (n = 1). Thus
00 Zt
E, (Z) = / I dt
1 t
E2(Z) = eZ ZEX(Z)
E3(Z) =  e_Z Ze"Z + ~ Z2E1(Z)
E4(Z) = f e'Z f Ze~Z +
+ 6 z2e_Z 6 z3 E1(Z)
(D3 )
(D4 )
(D5)
(D6 )
The zeroth and first order
the parameters, Z, approaches
integrals are divergent as
zero, as can be seen from
Eo(z)
(D7)
Ex ( Z )
Y 2nZ
l
K = 1
(~l)KzK
K K !
(D8 )
where Y = 0.57721 = Euler's constant
ARG Z < ff
160
By inserting equation (B20) into equation (212) we
obtain
~ KB
JL l
RT 3N
OB
T, V, N
= v V v C
a xoa a B aB
(B21)
oy^B
We next apply equation (B21) to each of the three
cases represented by equations (B16), (B17), and (B18).
First, if and 6 are solvents (a = B = 1), then
a LR
= _N_ h
'n Xol RT 3Nol
pvrrvrr
T, P, N
< RT
T
(B22
oy^l
Second, if a is the solvent (a = 1) and 3 is a solute, we
obtain
N
3y
LR
'IB
VT 3Noe
T, P, N
vBkTRT
oy^B
Third, if both a and 3 are solutes, then
(B23)
a B
N
3 Vi
MM
a
'aB
vcX
B oa
v v RT 3N n
a B oB
T, P, N
oy^B
N
v v RT
a B
P+tt 3VLR
I
P
3N
o B
d P 
T, P,N
oy^B
NVLR
a
3 vi
LR
v v RTVRR 9NoB
a B 1
T,P,NQy^B
NVLR
a
v v.RTV^ p
a B 1
P+tt 3vLR
I 1
3N
oB
d P 
T, P,N
oy^B
pvLRvf
a B
vaV BkTRT
(B24)
174
n n n
c ( t ) = y x. y x. y x. c.., ( t ) =
. L1 i 3, k ilk
i=l 3=1 k=l
Third
Virial
Coefficient
Equation (C4) can be rearranged to
P
RT
n n n
[p. + I p I p B. (T) +
i=l 1 i=l 3 = 1 3 13
n n n
+ I Pi l Pi I Pk C. .,(T) +
i=l j=l 3k=l 3
(C5 )
By taking the first derivative of equation (C5) with
respect to the density of an arbitrary species j we obtain
1 3P
RT 3p
n
= 1 + 2 y p B. +
u i ii
T,p
i=l
1 11
+ 3 iLa/k +
(C6 )
Comparison of equations (C3) and (C6) and substitu
tion of
2B. .
1
.(T) = f dr. dr
1 J l] co 1
(C7 )
3C. ., (T) = f dr. dr .dr,
13k J 13 lk 3k a) 1 3 k
(C8 )
153
3mKB(T,V,X)
a
9N
06
3 yBR(T,P,X:
a
3N
T, V, N
oy^B
oy
+
T,P,N
oy^B
+
3ULR(T,P,X)
a
3P
3p
3N
T, N
T, V, N
o ' oy^B
By making the following identifications we obtain
(Bl)
9y
LR
a
3P T,N
o
= VRR (T,P,X)
(B2 )
3P
3N
oB
T, V, N
VBR (T,P,X)
V
oy^B
(B3
where Va = partial molar volume of component a
V = system volume
kt = isothermal compressibility
and by inserting equations (B2) and (B3) into (Bl) we get
3 y KB ( t V, X)
a
3N
o3
3 yLR(T,P,X;
a
9N
T, V, N
oy^B
oB
T, P, N
oy^B
vLRvLR
_a B_
V
(B4 )
To Flor Maria
163
LIM
X 1
ol
V
:MM
a
LIM
X 1
ol
VLR + 0
a
(B3 5a)
_oo,MM ^,LR
(B3 5 b)
Taking the pure solvent limit of equation (B24) and
using the results of equations (B25) to (B35) gives
6a 6
N
3u
MM
a
a 3
v0X v v rt 9N q
3 oa a 3 oB
T, P, N
oy^B
p MM ~, MM
P i v V '
ol a 3
v v K RT
a 3 1
(B36
where C
a3
LIM
X ^ 1
ol
'a3
Therefore, the pure solvent or infinite dilution limit
of the solutesolute DCFI1s can very generally and rigor
ously be obtained from MM theory.
A rather interesting consequence of the above arguments
arises in relation to the composition behavior of the solute
partial molar volume near the pure solvent limit. Although
the arguments that follow are valid for any general system,
the discussion will be developed for a system consisting of
several salts (a,3) and one solvent (1) only. The intent is
to avoid any complexities that may obscure the central topic
TABLE 415
SALT
WATER DCFI
FOR NaBr
(2) IN WATER (1)
AT 25C, 1 ATM
Xo2
UCi2>"
rLR
12
HS HS
1 12 12
> (4C124C12)
(1~C12)CALC
EXP.
u 12'
io~12
10.4753
0.0000
0.0000
0.0000
10.475
10.475
9.9900xl0~4
10.4753
0.4051
0.1091
0.0220
10.967
10.891
4.9751xl0~3
10.4753
0.5506
0.5384
0.0375
11.526
11.502
8.9197xl0~3
10.4753
0.6025
0.9543
0.1971
11.835
11.979
0.029126
10.4753
0.6985
3.1737
0.6527
13.694
14.111
0.052258
10.4753
0.7395
5.8408
1.1882
15.867
16.310
0.078001
10.4753
0.7648
9.0595
1.8014
18.498
18.673
0.10904
10.4753
0.7841
13.4489
2.5654
22.142
22.098
(IC12' from equation (384).
EXP
(lCj^) from equation (256) using the same sources of experimental data as
for Figure 4.
133
TABLE 411
SALT
WATER DCFI
FOR NaCL (2)
IN WATER (1)
AT 25C, 1 ATM
X
0
NJ
(
UCl2)
30 PLR
i2
(CHScHS )
^12 C12 1
 (AC12AC12)
(1C )CALC*
' '"'12'
(1C )EXP
u u12'
1012
7.4085
0.0000
0.0000
0.0000
7.408
7.408
9. 9900xl0~4
7.4085
0.4086
0.1091
0.0145
7.911
4.834
4.9751xl03
7.4085
0.5558
0.5389
0.0727
8.430
8.455
9.9010xl0~3
7.4085
0.6173
1.0734
0.1455
8.953
9.042
0.015554
7.4085
0.6555
1.7104
0.2300
9.544
9.658
0.032695
7.4085
0.7134
3.6313
0.4938
11.259
11.432
0.048670
7.4085
0.7412
5.5143
0.7490
12.915
13.048
0.066607
7.4085
0.7613
7.8217
1.0456
14.954
14.845
0.084953
7.4085
0.7760
10.3137
1.3609
17.137
16.690
(lC^2 )* from equation (384).
EXP
(1C,) from equation (256) using the same sources of experimental data as
for Figure 4.
129
21
which by use of equation (211) becomes
n
_1_ _3P_
RT 3p
oa
T,P,
o
= i v y v n x D c Q
a o_i p 3 a8
p1
(232 )
Y^a
Equation (232) is the multicomponent compressibility
equation expressed in terms of components. The density
derivative of pressure is related to the partial molar
volume as
3P
3p
oa
= V
3P
3V
3V
T, p,
3N
T,N oa
V
oa
(233
Y^a
T, P, N
which when inserted in equation (232) gives one of the
desired relations.
V
oa
ktRT Va
n
o
I v
8=1
8 Xo6
ac fl)
a8
(234)
In order to relate the bulk modulus to direct correla
tion function integrals, the total volume is related to
the partial molar volumes.
V
0
l
a=l
3 V
3N
oa
N
oa
T,P,N
5Y^a
n
o
l
a=l
V N
oa oa
(235)
Dividing equation (235) by the mole number of species
(N) yields
1 = V
p N
n
l
a=l
V
oa
X
oa
(236)
31
X 05
Figure 4.
Salt (2)Water (1) DCFI in Aqueous
Electrolyte Solutions at 25C, 1 ATM.
For data sources see text.
Integrals of second and higher order, however, approach
definite values as the parameter, Z, goes to zero. These
limits can be generally expressed as
182
1
n = 2, 3, 4
(D9 )
n1
Equations (D2) through (D6) indicate that in order to
obtain numerical values for an exponenteial integral of any
order, one only needs values of E^(Z ) To this purpose,
approximations to E^(Z) were taken from Abramowitz and
Stegun (1973). Unfortunately, no single approximation valid
over the full range of values for (o < Z < 00) was found.
Therefore, two different approximations were spliced
together. These are for 0 Â£ Z _< 1
(1)
E, (Z) = n Z + a + a. Z + a Z +
1 o 1 2
+ a3 Z3 + a4 Z4 + a5 Z5 + e(Z)
(D10)
where
Â£(Z)  < 2 x 10
and for 1 < Z <
(2)
Ex (Z)
+ Â£ ( Z )
(Dll)
Z 5 4 3 2
e (Z +b.Z^+bcZJ+b^Zz+bQZ)
D 0 / o
Z
I Â£ ( Z )  < 2 x 10 8 
where
97
P LR
The values for AF.^ were obtained for the
DO
other ions (i) from C data for the above salts.
al
Once values for the above parameters have been
obtained, it is then possible to calculate the values for
the three body parameters involving two ions and one water,
A4>
These are determined from equation (384) for C^
at finite salt concentration for each salt (a).
co
V
1C = (CHSCHS") 
al v k RT vual cal 1
a 1
LR_ LR A _A
lCal Cal 1 al al
(384)
where
rHS _HSC
t'al cal
r,LR_r,LRoo
1 Sxl
HS
difference of Ca^ given by
equation (385)
LR
difference of Ca^ given by
equation (386)
*C1 ACJ r <',+a(4F+rpoIt> +
a
P T R P
+ ',a(iFrpoltl1))(ppol) +
+ V A4 + v A4> ) +
v. +a +11 a 11
a
+ (ppol polpol)+
101
_ i 2 p T p 9
~ ((v + v V )(AF p ,) + (v + v v )
2 +a +a a ++ ol ++1' a +a a
P T R
 (AF p n4> ,)) p (V A$ ^ + v A$ ) =
ol 1 Koa +a +++ a
1 p 00 oo
P p V V T U
1,, \ ol oa oa ~1 LR
p r~ + p Caa +
v k n RT
a 1
P P
+ p"1 (CHS CHS) PP.A.: ((V 2 + V v )
aa aa 2 +a +a a
va p
A<Â¡> n + (v + v v )A$ ) = I +
++1 a +a a 1 a
+ Saa p = F (p)
a oa aa
(436)
As previously done for other parameters, Faa(_P) was
calculated from known parameters and experimental values of
aa
component densities. The slope (S ) and the intercept (Ia)
of equation (442) were obtained by a linear least squares
routine over all the data points for each salt (a). Values
a
for the ionic parameters in Saa and Ia were extracted by
means of a scale based on lithium.
P LR
AF+ + p > + + = 30 0 ML/MOL (43 7)
LiLi LiLil
A$ + + + = 16 ML M1 (438)
Li Li Li
59
it is seen that the constant temperature and pressure limit
has divergent terms associated with the activity coefficient,
a first constant related to the partial molar volume, and
a second constant associated with the activity coefficient
and which is not so well defined. This second constant
is loosely related to a term linear in salt density which
often appears in empirical expressions for the salt activity
coefficient (Guggenheim and Turgen, 1955; Guggenheim and
Stokes, 1969). In the present model the divergent terms
are contained in equation (353). The first constant can
be calculated directly from infinite dilution partial molar
volumes and solvent quantities. The second constant must
be fitted to data using terms from equation (355) which
have only ionion and long range ionwater correlations.
This reflects the fact that triple ion direct correlations
are zero at infinite dilution and any contributing short
range ionsolvent correlations would generally be contained
in the first constant. Thus,
c" = LIM (C cLR actb)
cct3 X .,>1 iLctB aB ca6;
ol
(357)
actb = p y y y v. v.^a*..,
aB v v L .L L lot ]BMk ljk
a B i=l j=l k=l J J
n n n
where
REFERENCES
Abramowitz, M. and I.A. Stegun, Handbook of Mathematical
Functions, Dover, New York, NY (1973).
Allam, D.S., Ph.D. Thesis, London (1963).
Ananthaswamym, J., and G. Atkinson, J. Soln. Chem. Id., 509
(1982 ) .
Ball, F.X., W. Furst, and H. Renon, AIChEJ, 31, 393 (1985).
Berne, B.J. (Ed.), Statistical Mechanics Part A;
Equilibrium Techniques, Plenum Press, New York, NY
(1977).
Blum, L., Theoretical Chemistry Advances and Perspectives.
(Edited by H. Eyring and D. Henderson) Academic Press,
New York, NY (1980), p. 1.
Brevli, S.W., Ph.D. Dissertation, University of Florida
(1973).
Brelvi, S.W., and J.P. O'Connell, AIChEJ, 21, 171 (1975).
Campanella, E.A., Ph.D. Dissertation, University of Florida
(1983).
Carnahan, N.F., and K.E. Starling, J. Chem. Phys., 5JL, 635
(1969).
Chan, D.Y.C., D.J. Mitchell, and B.W. Ninham, J. Chem.
Phys. 1Q_, 2946 ( 1979 ) .
Chen, C.C., H.I. Britt, J.F. Boston, and L.A. Evans, ACS
Symposium, Washington (1979).
Croxton, C., Introduction to Liquid State Physics, John
Wiley and Sons, New York, NY (1975).
Cruz, J.L., and H. Renon, AIChEJ, E4, 135 (1978).
Davidson, N., Statistical Mechanics, McGrawHill, New York,
NY (1962).
188
91
partial molar volumes at infinite dilution. In addition,
the hard sphere diameters for all the species were taken
from Marcus (1983) who used neutron and Xray scattering
among other sources. Marcus' diameters (Appendix E) are
close to the Pauling crystal radii, and we feel that they
represent physically meaningful bare ion diameters. This
last point is truly important because hard sphere calcula
tions are very sensitive to the species diameters.
The ability of the model to fit experimental solution
data with physically meaningful bare ion diameters consti
tutes a very sensitive test of the capability of the model
to represent intermolecular correlations present in the
system. However, because they have not been treated any
where, the two and three body parameters listed in Appendix
E were fitted to solution data in the rough fashion
discussed below.
This model is designed such that all of the parameters
are ionically additive, and we hope to obtain the empirical
parameters solely from single salt data. We will, there
fore, limit the discussion to systems containing a salt ( )
in water (1).
The twobody parameters are where i and j include
the ions and water and the threebody parameters are A
and $....
ill
Those parameters involving water only are obtained from
equations (241) and (3100) at the pure solvent limit.
157
.MM UK y i
1 ~ L i
1 1 6=2 O
. LR
no NoB 9y
LR
3N
06
d N
06
(B13)
T, P, N
oy^B
By inserting equation (B13) into equation (Bll) and
rearranging we obtain
v:
3 7T
MM
9mi
3N Q
oB
3N .
T P N
' oy^B
T,P,Noy^B
3NoB
T, P, N
OYt^B
P+tt 3VRR
! 1
3N
o6
d P
T, P,N
OY^B
(B14
Equation (B14) is used to substitute for the osmotic
pressure derivative in equation (B10) and the resulting
expression inserted into equation (B4) to give
~
3 U
~ MM
3u
P + TT
3VLR
a
a
 J
T,P,N P
a
3N
oB
3N .
T, V, N
3N
oB
oyt^B
oy^B
d P 
T, P, N
OYt^B
VLR
3y
MM
VRR 3NoB
+
Tp'tWe 1
3y
LR
vLR
a
VPR 3NoS
T, P, N
oy^B
VLR
a
LR
P + TT 3 V
J
p
LR
3N
oB
d P +
VLRVLR
a B
VKm
T,P,N
OYt^B
(B15)
APPENDIX A
HARD SPHERE DIRECT CORRELATION FUNCTION INTEGRAL
FROM VARIOUS MODELS
PercusYevick theory (1958) yields through the "Com
pressibility Equation" an equation of state for a mixture of
n hard shere species which has the following form (Reed and
Gubbins, 1973 ) :
pPYHS
RT
6
IT
+
3 ^ 1 ^2
(1C3)2
+
3?.
de3)
where ^
n
r r
6 ^ i piai
i=l
k = 0,1,2,3
(A1)
The more exact mixture form of the CarnahanStarling
equation of state for hard spheres (Mansoori, Carnahan,
Starling, and Leland, 1971) gives a very similar expression
which compares slightly better with molecular dynamics
results for hard spheres (Reed and Gubbins, 1973).
pHSCS pHSPY
RT RT
(lÂ£3)3
(A2 )
The chemical potential for a species i in solution can
be obtained from equations (Al) and (A2) by standard
thermodynamic manipulations. Thus, the hard sphere chemical
148
V
oa
partial molar volume of component a.
W. .
13
potential of mean force.
X.
l
Kh/N = mole fraction of species i.
X
oa
N
= mole fraction of component a on a
species basis.
Z
dimensionless parameter in exponential integral.
Z .
i
valence of ion i.
a =
Euler's constant, empirical universal constant
for ionsolvent correlations.
Yi
activity coefficient of species i.
Ya
activity coefficient of component a.
6 .
13
Kroniker delta.
0
Eulerian angle between a charge and a dipole.
eii'*ii
= Eulerian angles of dipole of solvent molecule i
kt
isothermal compressibility.
K1
isothermal compressibility of pure solvent (1).
A .
i
ideal gas partition function.
M
l
chemical potential of species i.
II
i1
a
dipole moment of solvent.
v =
a
number of species i in component a.
v =
a
total number of species in component a.
Â£ p aK = reduced density.
6 i=i 1 1
TT =
P^, osmotic pressure.
P
N
= density of all species.
Â£
vector of species densities.
VI11
25
where dC 6>" "xl( ^oB1
ol
and where equations (242) represent a constant pressure
limit on the DCFI. A corresponding constant volume limit
can be obtained from equation (212).
For a binary system consisting of one solvent (1)
and one salt (2), the fluctuation relations become
31ny
3p
o2
T,p
ol
(243)
V
o2
K RT
T
V2Xol(1 C12}
+ V^X (1C
2 o2 22
(244)
3P/RT
3p
1 2
= = X n (1Cn ) +
T,K PKTRT 01 11
2 2
vX X _(1C10) + v_X _(1C )
2 ol o2 12 2 o2 22
(245)
Nv ^ 31nY2
Hs^
T,P,N
ol
v^X2. [(1C..)(1C__) (lc,
2 ol 11 22 12
(246 )
V +2 ^ Cl+) + V2 (1 Cl'>
'12
1
(247)
TABLE 49
SOLUTION PROPERTIES FOR NaCL (2) IN WATER (1) AT 25C, 1 ATM
1
V o
o2
X1{2V N
o2 2
8 2
PktRT
Exp.
8N
o2
Xo2
Calc.
Calc.
ktrt
Exp.
Calc.
T,P,Noi
Exp.
io~12
16.108
16.108
14.817
14.817
8.763
8.763
9.9900x10
4
16.195
16.203
15.642
15.507
5.163
4
.576
4.9751x10
3
16.538
16.574
16.608
16.676
2.365
Â¡.113
9.9010x10
3
16.966
17.027
17.649
17.816
0.845
0.945
0.015554
17.478
17.539
18.844
19.024
0.270
0.099
0.032695
19.021
19.078
22.314
22.546
2.283
1.691
0.048670
20.527
20.509
25.635
25.821
3.448
2
.092
0.066607
22.364
22.183
29.660
29.529
4.386
4
. 560
0.084953
24.329
24.168
33.946
33.339
5.050
C
. 915
Calc, properties from equations (244), (245), (246), (362), (384), and
(3104).
Experimental data sources are those of Figures 3, 4, and 5.
127
TABLE 412
WATERWATER
DCFI FOR NaCL
( 2 ) IN WATER
(1) AT 2 5 C, 1
ATM
X
0
K3
OO
dCll)
PLR
dl
(cHScHS )
ldl dl 1
(AcnAcn>
(1 j CALC.
U dd
(lc )EXP
1 Id
io12
16.1084
0.0000
0.0000
0.0000
16.108
16.108
9.9900x10
4
16.1084
0.0004
0.0912
0.0286
16.228
16.236
4.9751x10
3
16.1084
0.0007
0.4501
0.1443
16.703
16.739
9.9010x10
3
16.1084
0.0009
0.8963
0.2911
17.296
17.356
0.015554
16.1084
0.0010
1.4279
0.4634
18.000
18.058
0.032695
16.1084
0.0012
3.0290
1.0154
20.153
20.198
0.048670
16.1084
0.0013
4.5962
1.5639
22.269
22.221
0.066607
16.1084
0.0014
6.5138
2.2167
24.840
24.627
0.084953
16.1084
0.0015
8.5816
2.9237
27.615
27.548
(1C^^)CALC from equation (3104).
EXP
(lC^) from equation (255) using the same sources of experimental data
as for Figure 5.
130
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
7 ^ / yO y
f
John P. O'Connell, Chairman
Professor of Chemical
Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
rr. /
Charles F: Hooper, Jev
Professor of Physics
/
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Gerald B. WestermannClark
Assistant Professor of
Chemical Engineering
45
by ion association or repulsion. The physical significance
of these terms will be discussed further below.
As pointed out in Chapter 2, solution properties are
related to spatial integrals of the direct correlation
function. In order to relate this model to thermodynamic
properties, equation (313) is integrated over angles first
and separated later. Thus
(317)
CO
(318)
(319)
+
(320)
Lastly, A j is defined by formally integrating equation
(316 ) .
107
LR
however, C22 does not make a large contribution to either
the magnitude or the behavior of the saltsalt DCFI. Here
00
the magnitude is dominated by (IC22) r the hard sphere
correlations, and the triple ion correlations while the
variations are dominated mostly by C22 ~ C^ with
TB TB
AC22 ~ ^*99 making a smaller contribution. At the highest
'22
HS
salt mole fraction (X 2 = 0.084953), C22 accounts for 77% of
the magnitude of the saltsalt DCFI.
The deviation of the calculated saltsalt DCFI from the
experimental values over the entire composition range is
mainly due to the choice of the ionic hard diameters
employed, since the diameters appear in the DebyeHuckel
type denominator in C^ and, naturally, in C^ C22
are the leading contributions at low and high salt mole
fraction, respectively.
The salt (2)water (1) DCFI1s behavior is illustrated
in Figure 8. At low salt mole fraction and down to infinite
dilution (0 < Xq2 _< 0.02), the magnitude is dominated by
CO
(lC^) which is a constant while the behavior is controlled
HS HS00
by the hard sphere correlations (C^ ^ anc* *on
LR
range field correlations (C^) As the salt mole fraction
LR
increses (0.02 < Xq2 < 0.05), the contribution of C^2
remains small, becomes nearly constant, and is largely
cancelled by AC^ ~ AC.2) Thus, the magnitude is dominated
00 HS
by (lC^) anc^ ^12 behavior is controlled by
HS
C12 At large salt mole fraction (X^ > 0.05), the
71
The third term in equation (320) has
<(Wl)3>03 ~2 I
394)
which becomes upon integration
= o
(395)
Thus, for dipolar solvents only one term of equation
(320) remains after the angle integrations.
iÂ£Â£ (ifl,2 e2Kan f
3 DkT 1
2Kr
11
dr
all rll
11
(396)
The integral in equation (396) is also an exponential
integral (E^) which can be expressed in dimensionless form.
2Kr
/
11
dr
^lrll
11
1
T~
lll
(2Ka..)X
e 1X
T
x
dX =
E (2Kan.)
4 11
11
397)
Equation (396) is then transformed
,LR 4tt0 (ylyl}
'11 3 DkT
2 e
2ailB 1
1/2
E4(2anB I
1/2
L11
(398)
37
range, and intermediate range interactions. These can
be theoretically separated into a simple additive form
as will be shown in the next section of this chapter.
It is important to note that the separation is first
developed at the level of microscopic direct correlation
functions which are later integrated to obtain the DCFI's.
Although our particular additive separation of the micro
scopic direct correlation function is not fully rigorous,
we believe it is more reasonable than a similar resolution
of the radial distribution function into an additive form
(Planche and Renon, 1981). In fact, the radial distribution
function can naturally be resolved only into a multiplicative
rather than an additive form. The intermolecular potential
and, consequently, the potential of mean force can be
approximately decomposed into additive contributions from
interactions of different characteristic range, but this
potential appears in an exponential in the radial distribu
tion function. Thus, resolving the radial distribution
function into additive contributions is quite inappropriate.
Statistical Mechanical Basis
The above philosophy is a qualitatiave concept which
must be expressed in quantitative terms. To this end,
we now establish a rigorous statistical mechanical basis
for a model of microscopic direct correlation functions.
First, consider the diagrammatic expansion of the direct
110
The accurate estimation of the uncertainties present in
experimentally determined values of the bulk modulus, the
salt partial molar volume group, the salt activity coeffi
cient derivative, and the saltsalt, saltwater, and water
water DCFI's was not possible because little information is
available regarding the errors present in the original
experiments. Nevertheless, based on a general knowledge of
the quality of the data obtained from the various experi
ments it was assumed that the uncertainty in the solution
compressibility (
(p) 0.02%, and that in the salt partial molar volume (V^)
0.20%. The uncertainty in the derivative of the salt
activity coefficient was assumed to be approximately ten
times that in the activity coefficient itself which would
place it at 1% or less.
By a propagation of errors analysis using the above
estimates, it was determined that the uncertainty in the
three DCFI's was approximately 2% and that in the three
solution property groups 11.5%.
The model fitted the solution bulk modulus within
experimental uncertainty for LiCL, LiBr, NaCL, NaBr, KCL,
and KBr at all compositions except for LICL and LIBr at salt
mole fractions above 0.1. The model fitted the salt partial
molar volume group adequately (23%) in most cases but
slightly outside of experimental error (11.5%). The fit
for the lithium salts was worse at a few points. Lastly,
13
Fluctuation Solution Theory gives the following well known
result (O'Connell, 1971; O'Connell, 1981):
RT 9N .
3
T, V, N
(21)
where
N.
i
N
P
the chemical potential per mole of
species i.
the number of moles of species i.
the total number of moles of all species.
the Kroniker delta.
CO
2
4up J r dr = spatial integral of
J 1 1 co
0 J
the direct correlation function.
^ = molecular density of all species.
The microscopic direct correlation function
c 1 j CO
is an angle averaged direct correlation function defined
by
=
^ co
'13
dco dco
where
= J dco. dco .
J i 1
(22)
In order to arrive at the first and simplest of the
desired relations, we define the activity coefficient for
species i on the mole fraction scale as
183
E^1^ and e^ ^ were spliced together to calculate E^ as
E1(Z) = 2 E11^Z) (1 f tan"1 n(Zl)) +
+ j e{2!z) (1+  tan"1 n(Zl)) (D12)
where n = 10b(^
The numerical values of the parameters in equations
(D10) and (Dll) are
a =
o
0.57721566
bl
=
8.573328
al =
0.999999193
b2

18.059016
a2
0.24991055
b3
=
8.634760
II
m
rtf
0.05519968
b4
=
0.267773
a4 =
0.00976004
b5
=
9.573322
II
in
0
0.00107857
b6
=
25.63295
b7
=
21.09965
00
&
=
3.958496
87
n n n n
o o o o
l l l l vv__ / n 1"CBY(t)
a=2 8=1 Y
L XL 8yJ p (t)
=1 6=1 M o oy
~v
(t) a
F (t) dt 
oa
a8 8 P(t)
1 v p , F F F F R R R R, A
J l I (X X p p X X. p p ) AF 
i=l j=l 1 1 1
1? ? ?,VFFFFFF R R R R R R
T l l I (X X. X p p p X. X. X p p p ) +
J i=l j=lbl 1 ] k 1 J k
1 V *v /v F F F P'F v Rv R R R RrRN ,1SLR
+ ) > (X. X. p p p X. X. p p p )..,
3 .f, x j ol i j ol ljl
i=l j=l
(418)
where
PR =P P,R =P SAT(T)
ol ol
X R = 1
ol
X R = 0
oa
a Â¥ 1
n
o
x. = y v. x
i ^ ia
a=l
oa
. . ,rRR, /VFF v R R > ,
p (t) = X p + (X p X D )t
oa oa oa oa
F(t)
v F F R R
X p x p
oa oa
n
O p p R R
p (t) l (X Rp X Rp )
oa 8=1 06
p (t)
p (t)
TABLE 414
SALT
SALT DCFI FOR NaBr (2) IN WATER (1)
AT 2 5 C, 1
ATM
(N
0
X
^ ^~C22 ^
00 rLR
22
(CHScHS ) 
' '22 22 1
TB TB
(AC2T2AC22)
Uc22,
io~12
14.0597
2.1907x10
6 0.0000
0.0000
2.1907x10 6
2.1907xl06
9.9900x10
4
14.0646
45.7709
0.1472
0.0483
31.607
27.472
4.9751x10
3
14.0832
13.5940
0.7268
0.2424
0.973
1.610
8.9197x10
3
14.1002
8.1970
1.2888
0.4381
6.751
6.577
0.029126
14.1827
2.6084
4.2918
1.4718
14.394
13.953
0.052258
14.2649
1.3952
7.9103
2.7148
18.065
18.216
0.078001
14.3469
0.8881
12.2895
4.1607
21.587
21.838
0.10904
14.4395
0.5959
18.2805
5.9828
26.141
25.885
^('22^CALC* ^rom equation (362).
EXP
(1C9) from equation (257) using the same sources of experimental data as
for Figure 3.
75
The solventsolvent DCFI is also dominated by C
HS
11
over the entire range of salt concentration up to about
LR
6M. C^ makes a negligible contribution reflecting the
relative weakness of long range dipoledipole interactions.
HS
Again, the largest term after C^ is AC^ which contains
the short ranged hydrogen bonding between solvent molecules
and the hydration related effect of an ion on two solvent
molecules at short range.
The parameters of the model are species specific and
universal. It is, therefore, necessary to build only a
relatively small set of parameter values to predict the
behavior of a large number of systems. Thus, a hard sphere
HS
diameter (a.^) for each species is required for CQg and
C"a8 (where a, 8 can be salts or solvents). To avoid con
fusion, the parameters for ACag will be those of a system
with one solvent (1), one salt (2), and many ions (i, j).
Then, Ac^ involves Af^ which is ion independent,
Aim which is usually neglected, and AO.,., for each ion.
ill
P T R
AC. 0 has Af. p Or1. AO and AO..... AC0 includes
12 li ol 111 111 ill 22
P
AF.. p AO..., and AO ... This totals to two solvent
11 ol 111 111 111
specific parameters if AO^ is neglected and six parameters
for each ion (note that AO... = AO... and AO... = AO...)
Ill ill ill In
three of which involve solvention pairs.
Properties predicted with the model are most sensitive
.HS
to the value of the hard sphere diameters because the C
is a very strong function of the diameters. But it is
a 8
99
In order to get values for the three body parameters on
the lefthand side of equation (433), F^ (p_) and PPQa were
calculated from previously obtained parameters and experi
mental values for the component densities. The slope
,1a
= ( (v
v +a
a
+ I
2
a
A4>
++1
+ (v
 a
a
1)
(434)
was calculated by a linear least squares routine over all
the data ponts. The same procedure was repeated for each
salt.
Individual values for the three body parameters in
Ia
equation (439) were extractd from the slope Sa by con
structing a scale based on lithium
A + + = 3 ML M
LiLil
(435)
Again the value was chosen for the reasons stated
above. Values for A$. were obtained from data on several
ill
salts.
The last set of parameters left consists of
and A$ ^ where i, j, k^lori, j, k are ions,
obtained from equation (362) for each salt (a).
a *LR
Af. $ .... ,
13 13I
These are
1c = (lc )" cLR (CHSCHS~)
aa v acr aa aa aa '
 (4CTBiCTB)
aa aa
(362)
105
3Â£ny.
N
3N
o2
t,p,n
=v2xo1Xo2 (PKTRT) 1(1C12>2
ol
 (1C1X)(1C22,)
where
n n
o o
PKRT =1/1 l V V X X (1C )
c=l 8=1 e oo 08
'a 8
(439)
and from experimental data by use of the GibbsDuhem
relation ignoring enthalpy and volume changes.
3 Â£ny.
N
3N
o2
T,P, N
X 3Â£ny
= v N
2 X 3N _
ol o2
(440)
T, P, N
Ol ' Ol
Thus, the water activity coefficient derivative was
obtained from the analogous salt quantity. These results
are presented in Table 426 and Figure 10.
Discussion
The properties of electrolyte solutions at concentra
tions up to the solubility limits of salts generally show
two distinct regimes of behavior at low (Xq2 < 0.02) and
high (0.05
behavior of the solution at intermediate salt mole fractions
contains elements of both regimes. This can be clearly seen
in Figures 1 to 10. Thus, a solution model which claims to
include the physics of the system must smoothly follow
measured behavior over both regimes as well as the
50
solventsolvent interactions which are dominated by angle
independent forces (Brelvi, 1973; Mathias, 1978; Telotte,
1985; Campanella, 1983; Gubbins and O'Connell, 1974; Brelvi
and O'Connell, 1975). Finally, equations (330), (331),
and (331) where i, j, and k are solvents would become
(335)
(336)
(337)
The above mixing rules for an aqueous system (single
solvent) have been tested against data for a number of
salts and may be regarded as established. The rules for
a multisolvent system, however, have not been tested.
They can only be seen as physically reasonable in the light
of previous experience but still tentative.
The next two sections deal with the application of
the theory developed here to specific interaction in order
to construct practical expressions.
42
Appendix A). Although the more exact CarnahanStarling
(Carnahan and Starling, 1969; Mansoori, Carnahan, Starling,
and Leland, 1971) expression could have been used, it is
somewhat more complex and relatively little improvement
in accuracy would be expected.
At this point, we have established a viable, albeit
traditional, theory for the behavior of the direct correla
tion function as r. 00 and at r. < a. .. However, many
interactions which are important in aqueous electrolyte
systems such as hydration of ions by water, hydrogen bonding
between water molecules, and ion pairing are strongest
at r^j just outside the core. Further, this is that kind
of interaction for which liquid state theory is not well
developed. Therefore, we attempt here to develop a method
for interpolation of the direct correlation function between
long and short range. Because generally available theory
offers little guidance, the method can at best be semiempiri
cal. For this purpose, the RusbrookeScoins expansion
of direct correlation function (Reichl, 1980; Croxton,
1975) for species i and j in a system of n kinds of species
is now introduced.
c..(T,p,r.,r .,oj.,(*).) = f..(T) +
lj 1j 1 3 lj
n
+ Â£ p,(. (T) + ... (312)
k=l K 1:,K;
hjk(T) = I 1 fij fik Â£jk dikdk
where
CHAPTER 1
INTRODUCTION
Aqueous electrolytes are present in many natural and
artificial chemical systems. For example, the chemical
processes of life occur in an aqueous electrolyte medium.
All natural waters contain salts in concentrations ranging
from very low for fresh water to near saturation for geo
thermal brines. Industrially, electrolytes are used in
azeotropic distillation, electrical storage batteries and
fuel cells, liquidliquid separations, drilling muds, and
many other processes. Since a quantitative description
of the properties of these systems is required for under
standing, design, and simulation, the ability to predict
and correlate the solution properties of electrolytes is
both scientifically and technologically important.
In attempting to fill this need, many models of aqueous
salt solutions have been developed. Essentially all describe
only activities of the components but ignore the volumetric
properties. Several extensive reviews of electrolyte solu
tion models are available in the literature (Pytkowicz,
1979; Mauer, 1983; Renon, 1981). To be concise, the various
models have been classified here into three general cate
gories and a few examples of each briefly discussed. First,
1
X 02
Figure 3. Salt (2)Salt (2) DCFI in Aqueous
Electrolyte Solutions at 25C, 1 ATM.
For data sources see text.
TABLE 44
WATERWATER
DCFI FOR LiCL
(2) IN WATER
(1) AT 25 C, 1
ATM
Xo2
oo
(1CiP
rLR
U11
(CHScHS )
lcll 11
(AC11AC11)
(1~C11)CALC*
(ic11)EXP
IQ12
16.1084
0.0000
0.0000
0.0000
16.108
16.108
4.2 5 58x10 ~3
16.1084
0.0007
0.3398
0.1219
16.570
16.588
8.5247xl03
16.1084
0.0009
0.6789
0.2471
17.035
17.070
0.025729
16.1084
0.0011
2.0796
0.7774
18.966
19.052
0.043142
16.1084
0.0013
3.5780
1.3543
21.042
21.130
0.069664
16.1084
0.0014
6.0835
2.3092
24.502
24.462
0.087620
16.1084
0.0015
7.9811
3.0089
27.010
26.875
0.10580
16.1084
0.0015
10.1222
3.7637
29.995
29.496
(lC^. )CAL<" from equation (3104).
EXP
(lC^^) from equation (255) using the same sources of experimental data
as for Figure 5.
69
+ v_(p1Acf>1_i + P+A$+_1 + P_A>__1)]
(387)
Expression for SolventSolvent DCFI
The solventsolvent direct correlation function integral
has the simplest relation since the solvent does not
dissociate so the species and component integrals are the
same.
As previously noted, any type of interaction can gen
erally be included in this theory, but it was assumed that
solventsolvent interactions at large separation are domi
nated only by dipoledipole forces. The solvent (l)solvent
(1) pair potential is
(388)
where
sin 0j^ sin 0^2 cos $12
0 cj> = Eulerian angles of solvent molecule
number 1.
0, p.. = Eulerian angles of solvent molecule
i 3 i
number 2.
The potential of mean force is approximated by a
function inspired by previously mentioned work (Chan,
Mitchell, and Ninham, 1979 ; Hs^ye and Stell, 1978).
TABLE 46
SALT
SALT DCFI FOR
LiBr (2)
IN WATER (1)
AT 25 C, 1
ATM
CM
O
X
(l<22 ^
cLR (C
22 1
,HS HS
22 22 '
TB TB
 ( Ac Ac )
' 22 22'
ac22)CALC
. EXP.
v 22
1012
18.4268
2.1907xl06
0.0000
0.0000
2.1907xl06
2.1907x10 6
2.08 66xl0~3
18.4422
27.7666
0.2239
0.0336
9.134
5.801
4.1983xl03
18.4575
16.2388
0.4492
0.0670
2.600
4.098
8.4962xl0~3
18.4880
8.9838
0.9077
0.1337
10.278
10.459
0.017411
18.5485
4.6576
1.8657
0.2686
15.488
15.153
0.036619
18.6723
2.2250
4.0065
0.5506
19.903
19.910
0.057919
18.8050
1.3714
6.5763
0.8608
23.149
23.467
0.075482
18.9130
1.0267
8.8919
1.1183
25.660
25.927
0.10833
19.1239
0.6793
13.9563
1.6276
30.773
30.011
PAT.P
(IC22) from equation (362).
EXP
(IC22) from equation (257) using the same sources of experimental data as
for Figure 3.
3
distribution function which has been used to calculate
the properties of electrolytes up to 1 M salt concentration
(Rasaiah and Friedman, 1968; Friedman and Ramanathan, 1970;
Rasaiah, 1969). This method requires tedious numerical
calculations to obtain the properties. A simpler and more
generalizadle approach is the Mean Spherical Approximation
(MSA) which has been applied to both primitive (Blum, 1980;
Triolo, Grigera, and Blum, 1976; Watanasiri, Brule, and
Lee, 1982) and nonprimitive (Vericat and Blum, 1980;
PerezHernandez and Blum, 1981; Planche and Renon, 1981)
electrolyte models. The MSA method essentially consists
of solving the OrnsteinZernike (1914) equation for the
distribution functions subject to the boundary conditions
that the total correlation function is minus one inside
the hard core and that the direct correlation function
equals the pair potential outside the hard core. This
is equivalent to the PercusYevick method for rigid nonionic
systems (Lebowitz, 1964). The MSA generally gives good
thermodynamic properties if these are calculated from the
"Energy Equation" (Blum, 1980). It does not yield good
correlation functions and further suffers from the need
to numerically solve complex nonlinear relations for the
value of the shielding parameter at each set of conditions.
This last problem grows progressively worse as the sophisti
cation of the model increases. Due to their complexity none
38
correlation function (Reichl, 1980; Croxton, 1975) for
species i and j,
c. .
11
(T,P,ri,rj,
where
u. .
il
B. .
il
Â£n g
il
+ B. .
il
(31)
W. ./kT
i j
= e = radial distribution function.
= potential of mean force.
= pair potential.
= sum of all bridge diagrams also known
as elementary clusters.
Although equation (31) is an exact expression for
the direct correlation function, it is of little practical
value because the bridge diagrams cannot be summed analyti
cally. This series is
_1_
2 !
n
l
k=l
pkpÂ£ J fikfkÂ£ fÂ£jfiÂ£fkj dikd^dV^Â£
(32)
for a
where
system consisting of
f. = e
il
u. ./kT
il
n species.
1 = Mayer bond function.
To obtain the hypernetted chain (HNC) approximation
(Rowlinson, 1965) all of the bridge diagrams are neglected
(B^j = 0). This introduces an error which is second order
115
X 02
Figure 7. Contributions to the Salt(2)Salt (2)
DCFI in Aqueous NaCL at 25C, 1 ATM.
For data sources see Table 410.
94
pp (v, Ai> + v AO .. ) = 1 C. .
oa +a +11 a 11 11
PolKlRT
r,HS Pw P .LR \
(C11 C11 } (p~pol)(AFll Pollll)
4tt ,
+ V (
Vi 2
1/2
9a
11
2ailBYr 1/2 P
DkT > (Pe R4 2ailBYI > Pol <426)
F11(P)
Where we label the righthand side of equation (429) as
*!!<Â£)
To obtain values for the three body parameters on the
lefthand side of equation (426), F, (p_) and pp were
calculated from known parameters and experimental values for
the component densities. Then, the slope was obtained by a
linear least squares routine over all the data points avail
able. The process was repeated for each salt of interest.
Thus, for each salt this slope (SFF) is
S
11
a
v AE> , +
+a +11
v .AO ..
a 11
(427)
The parameters A<^_ and AO ^ are properties of the
ions and water independent of the salt (a). To obtain
individual values for each AO. a scale was constructed
ill
starting with lithium. A finite value was chosen to allow
for geometric mean mixing rules in later analyses and to
APPLICATION OF FLUCTUATION SOLUTION THEORY TO
STRONG ELECTROLYTE SOLUTIONS
By
HERIBERTO CABEZAS, JR.
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE
UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1985
176
From a practical point of view, the mixing rules
represent an effort to obtain the quantities AF^j, A^^,
LR
and using pure component and binary solution data. For
1
the case of aqueous strong electrolytes, the mixing rules of
AF^j are only important when both i and j are ions. This is
due to the fact that any ionsolvent AF^j can in principle
be obtained from one solventone salt systems but some
ionion AF^j (say for two anions or two cations) could only
be obtained from ternary solution data.
Therefore, when i is an ion and j is a solvent, the
quantity AF^j(T) is a parameter.
When i and j are ions, we use equation (C7) to obtain
AF. .(T) = 2 (BHS + BLR B. .)
il il il il
(C12)
The hard sphere second virial coefficient is rigorously
given by
2ttn a. .
bHS = *,iJ
i j 3
(C13)
where a.. = (a. + a.)
il 2 i l
= Avogadro1s number
(C14
LR
B^j and B^j are approximated for ions i and j as
= (blr blr)1/2
ii 11
blr
11
(C15)

