APPLICATION OF FLUCTUATION SOLUTION THEORY TO
STRONG ELECTROLYTE SOLUTIONS
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
To Flor Maria
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
I also wish to thank Drs. G.B. Westermann-Clark and
C.F. Hooper, Jr. for serving on the supervisory committee
and for making very pertinent suggestions regarding this
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
KEY TO SYMBOLS...................................... vi
1 INTRODUCTION......... ...... .................. 1
2 FLUCTUATION THEORY FOR STRONG ELECTROLYTE
SOLUTIONS .................................. 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
Philosophy of the Model.................... 33
Statistical Mechanical Basis ............... 37
Expression for Salt-Salt DCFI.............. 51
Expression for Salt-Solvent DCFI............ 63
Expression for Solvent-Solvent DCFI........ 69
4 APPLICATION OF THE MODEL TO AQUEOUS STRONG
ELECTROLYTES ......... ............... ....... 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
A HARD SPHERE DIRECT CORRELATION FUNCTION
INTEGRAL FROM VARIOUS MODELS................ 148
B RELATION OF McMILLAN-MAYER THEORY TO
KIRKWOOD-BUFF THEORY...................... 152
C RELATION OF DENSITY EXPANSION OF THE
DIRECT CORRELATION FUNCTION TO VIRIAL
EQUATION OF STATE: ALTERNATE MIXING
RULES. ...................................... 172
D EXPONENTIAL INTEGRALS........................ 180
E MODEL PARAMETERS........................... 184
REFERENCES ........................................ 188
BIOGRAPHICAL SKETCH............................... 193
KEY TO SYMBOLS
ai = hard sphere diameter of species i.
a. = distance of closest approach of species i and j.
B = K/II/2
B. = sum of all bridge diagrams, second virial
C = mixture third virial coefficient.
C.. = direct correlation function integral for species
13 i and j; two-body factor in third virial
C =direct correlation function integral for
components a and 3.
C.i = third virial coefficient for i, j, k.
AC.. = short range direct correlation function
c.. = direct correlation function.
Ac.. = short range direct correlation function.
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...
f. = e -1 = Mayer bond functions.
Af. = f.. f.HS fLR = differences of microscopic
1] 13 jj ij
g = pair distribution function.
1 n 2
I Z. p = ionic strength
= -8T I = Debye-Huckel inverse length.
= 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
= number of different components.
= internal partition function.
= separation between species i and j.
= position vector of i.
= (2 3k3T3) = Debye-Huckel limiting law
D k T
= pair potential.
= total system volume.
= partial molar volume of species i.
Vo = partial molar volume of component a.
W.. potential of mean force.
X. = N/N = mole fraction of species i.
X = mole fraction of component a on a
Z = dimensionless parameter in exponential integral.
Z. = valence of ion i.
a = Euler's constant, empirical universal constant
for ion-solvent correlations.
i = activity coefficient of species i.
a = activity coefficient of component a.
6.. = Kroniker delta.
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.
S = -6 p. a. = reduced density.
K 6. 1 1
S = P., osmotic pressure.
p density of all species.
p = vector of species densities.
P = = density of species i.
po = density of component a.
.ijk = spatial integral of .ijk"
Aijk = spatial integral of Ai .j
ijk = microscopic three-body coefficient.
Aijk = ijk ijk = difference of microscopic three-
= orientation dependence of dipole-dipole
Q = f dw. = integral over orientation coordinates.
. = angular orientation coordinates of i.
1 n 2
= Z v. Z..
Y 2 i= y 1
F = Final.
FLL = Friedman's limiting law.
HNC = hypernetted chain.
HS = hard sphere.
KB = Kirkwood-Buff.
LR = long range or field type correlations or
MM = McMillan-Mayer.
P = Pure component.
PY = Percus-Yevick.
< > = integration over orientation.
= Three body.
= infinite dilution in salt.
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
Heriberto Cabezas, Jr.
Chairman: Dr. J.P. O'Connell
Major Department: Chemical Engineering
Fluctuation solution theory relates derivatives of the
thermodynamic properties to spatial integrals of the direct
correlation functions. This formalism has been used as the
basis for a model of aqueous strong electrolyte solutions
which gives both volumetric properties and activities.
The main thrust of the work has been the construction
of a microscopic model for the direct correlation func-
tions. This model contains the correlations due to the hard
core repulsion, long range field interactions, and short
range forces. The hard core correlations are modelled with
a hard sphere expression derived from the Percus-Yevick
theory. The long range field correlations are accounted for
by using asymptotic potentials of mean force and the hyper-
netted chain equation. The short range correlations which
include hydration and hydrogen bonding are modelled with a
density expansion of the direct correlation function. The
model requires six parameters for each ion and two for
water. The ionic parameters are valid for all solutions
and those for water are universal.
The model has been used to calculate derivative prop-
erties for six 1:1 electrolytes in water at 25C, 1 ATM. The
calculated properties have been compared to experimentally
determined values in order to confirm the adequacy of the
Aqueous electrolytes are present in many natural and
artificial chemical systems. For example, the chemical
processes of life occur in an aqueous electrolyte medium.
All natural waters contain salts in concentrations ranging
from very low for fresh water to near saturation for geo-
thermal brines. Industrially, electrolytes are used in
azeotropic distillation, electrical storage batteries and
fuel cells, liquid-liquid separations, drilling muds, and
many other processes. Since a quantitative description
of the properties of these systems is required for under-
standing, design, and simulation, the ability to predict
and correlate the solution properties of electrolytes is
both scientifically and technologically important.
In attempting to fill this need, many models of aqueous
salt solutions have been developed. Essentially all describe
only activities of the components but ignore the volumetric
properties. Several extensive reviews of electrolyte solu-
tion models are available in the literature (Pytkowicz,
1979; Mauer, 1983; Renon, 1981). To be concise, the various
models have been classified here into three general cate-
gories and a few examples of each briefly discussed. First,
there are models based on relatively rigorous statistical
mechanical results which can be called "theoretical."
Second, there are those composed of a mixture of rigorous
theory and empirical corrections which can be named "semi-
empirical." Third, there are those models which directly
correlate experimental data and are thus termed "empirical."
Neither this classification nor the following list pretends
to be either unique or all-inclusive.
Among the "theoretical" models, the earliest and still
the most widely accepted is the theory of Debye and Huckel
(1923) which gives the rigorous relation at very low salt
concentration (the limiting law) for salt activity coeffi-
cients but fails at higher salt concentration. This theory
has been amply treated in the literature (Davidson, 1962;
Harned and Owen, 1958). The Debye-Huckel theory considers
an electrolyte solution as a collection of charged hard
spherical ions embedded in a dielectric solvent which is
continuous and devoid of structure. This is the physical
picture generally called the "Primitive Model." The correct
formalism for the application of modern statistical mechani-
cal techniques to the "Primitive Model" is given by the
McMillan-Mayer theory (1945). A major method developed
for this formalism is a resumed hypernetted chain approxima-
tion to the direct correlation function. This, together
with the Ornstein-Zernike equation (1914), forms a solvable
integral equation for the primitive model ion-ion
distribution function which has been used to calculate
the properties of electrolytes up to 1 M salt concentration
(Rasaiah and Friedman, 1968; Friedman and Ramanathan, 1970;
Rasaiah, 1969). This method requires tedious numerical
calculations to obtain the properties. A simpler and more
generalizable approach is the Mean Spherical Approximation
(MSA) which has been applied to both primitive (Blum, 1980;
Triolo, Grigera, and Blum, 1976; Watanasiri, Brule, and
Lee, 1982) and nonprimitive (Vericat and Blum, 1980;
Perez-Hernandez and Blum, 1981; Planche and Renon, 1981)
electrolyte models. The MSA method essentially consists
of solving the Ornstein-Zernike (1914) equation for the
distribution functions subject to the boundary conditions
that the total correlation function is minus one inside
the hard core and that the direct correlation function
equals the pair potential outside the hard core. This
is equivalent to the Percus-Yevick method for rigid nonionic
systems (Lebowitz, 1964). The MSA generally gives good
thermodynamic properties if these are calculated from the
"Energy Equation" (Blum, 1980). It does not yield good
correlation functions and further suffers from the need
to numerically solve complex nonlinear relations for the
value of the shielding parameter at each set of conditions.
This last problem grows progressively worse as the sophisti-
cation of the model increases. Due to their complexity none
of the modern "theoretical" models is widely used in
The most successful of the semiempirical models is
that due to Pitzer and coworkers (Pitzer, 1973; Pitzer
and Mayorga, 1973; Pitzer and Mayorga, 1974; Pitzer, 1974;
Pitzer and Silvester, 1976). Model parameters for activity
coefficients have been evaluated for a large number of
aqueous salt solutions, but volumetric properties and multi-
solvent systems have not been treated. To construct the
model, Pitzer adopted the "Primitive Model" and inserted
the Debye-Huckel radial distribution function for ions
into the osmotic virial expansion from the McMillan-Mayer
formalism. This latter is analogous to using the "Pressure
Equation" of statistical mechanics (Pitzer, 1977). The
resulting expression contains the correct limiting law.
He then added empirical second and third virial coefficients
which are salt and solvent specific. Although Pitzer's
model correlates aqueous activity coefficients superbly,
it does not add to the fundamental understanding of these
solutions; further, its extension to multisolvent systems
would pose some serious problems associated with the mixture
dielectric constant as has been recently pointed out (Sander,
Fredenslund, and Rasumussen, 1984). Another semiempirical
approach uses the NRTL model for solutions of nonelectrolytes
(Renon and Prausnitz, 1968) adapted for short range ion
and solvent interactions (Cruz and Renon, 1978; Chen, Britt,
Boston, and Evans, 1979) in nonprimitive models of electro-
lyte solutions. Cruz and Renon separate the Gibbs energy
into three additive terms: an elecrostatic term from the
Debye-Huckel theory, a Debye-McAulay contribution to correct
for the change in solvent dielectric constant due to the
ions, and an NRTL term for all the short range intermolecular
forces. Chen et al. adopted a Debye-Huckel contribution
and an NRTL term for the Gibbs energy but no Debye-McAulay
term. More recently, the UNIQUAC model for nonelectrolytes
has been modified for short range intermolecular forces
in electrolyte solutions (Sander, Fredenslund, and Rasmussen,
1984). The resulting UNIQUAC expression has been added
to an empirically modified Pitzer-Debye-Huckel type electro-
static term to form the complete Gibbs energy model.
Although the two NRTL and the UNIQUAC models correlate
activity coefficient data reasonably well even in multi-
solvent systems, they have to be regarded as mainly
empirical. First, their resolution of the Gibbs energy
into additive contributions from each different kind of
interaction is not rigorous. Second, the problems associated
with the mixture dielectric constant are resolved in an
empirical and somewhat arbitrary fashion. As a result,
such models add little to our understanding of these systems
and may not be reliable for extension and extrapolation.
Of the various empirical methods developed, two have
been chosen to be discussed here because they represent
distinct approaches. First, there is the method of Meissner
(1980) which is a correlation for the salt activity coeffi-
cient in terms of a family of curves that are functions
of the ionic strength and a single parameter which can
be selected from a single data point. This method has
been extended to multicomponent electrolyte solutions and
is useful over a wide range of salt concentration (0.1-20
MOLAL), though it is not very accurate. Second, there
is the method of Hala (1969) which is more conventional
in that it consists of a purely empirical model for the
Gibbs energy of the solution. This method is an excellent
correlational tool, but it is not predictive. It has four
parameters per salt-solvent pair.
The existence of so many models to correlate and predict
the thermodynamic behavior of electrolyte solutions is
indicative of the complexity of these systems and, perhaps,
the relatively poor state of the art.
As examples of the physical complexity of electrolyte
solutions, the composition behavior of the salt activity
coefficient (Figure 1) and of the species (ions and solvents)
density (Figure 2) is presented. Figure 1 shows the large
deviation from ideal solution behavior (y= 1) even at
very low salt concentration for all salts. Second, it
indicates that salts of the same charge type show similar
behavior at low salt concentration but are widely different
at higher salt concentration. In Figure 2, the difference
in the salt composition behavior of the species density
is obvious even for relatively similar salts, i.e., the
solution seems to expand for KBr while it seems to contract
for all other salts. The activity coefficient data were
taken from the compilation by Hamer and Wu (1972). For
NaCl and NaBr the density data of Gibson and Loeffler (1948)
were used. For LiCI, LiBr, and KBr the density data were
taken from the International Critical Tables. For KC1
the density data of Romankiw and Chou (1983) were used.
In the hope of improving the situation for obtaining
properties of solutions, a new model of strong aqueous
electrolyte solutions is presented here. This model has
been carefully constructed so that it overcomes a number
of the deficiencies of previous methods. For example,
this model is simple enough for economical engineering
calculations, yet sufficiently sophisticated to rigorously
include all the different interactions (ion-ion, ion-solvent,
solvent-solvent) and the principal physical effects (electro-
static, hard core repulsion, hydration, etc.) that contribute
to each interaction. The model is also extendable to multi-
salt and multisolvent systems in a straightforward fashion.
Finally, it addresses both activity and volumeric
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
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.
0.5 1.0 1.5 2.0 2.5
Salt Activity Coefficient in Water at
250C, 1 ATM. Data of Hamer and Wu
0.00 0.02 0.04 0.06
0.08 0.10 0.12
Species Density in Aqueous Electrolytes
at 250C, 1 ATM. For data sources see
FLUCTUATION THEORY FOR STRONG ELECTROLYTE SOLUTIONS
There are three general relations among the thermodynamic
properties of a solution and statistical mechanical correlation
functions. The first two are the so-called "Energy Equation"
and "Pressure Equation" which are obtained from the canonical
ensemble with the assumption of pairwise additivity of inter-
molecular forces. These equations relate the configurational
internal energy and the pressure respectively to spatial
integrals involving the intermolecular pair potential and
the radial distribution function (Reed and Gubbins, 1973;
McQuarrie, 1976). The third relation is the so-called "Com-
pressibility Equation" which is derived in the grand canonical
ensemble without the need to assume pairwise additivity of
intermolecular forces. This equation relates concentration
derivatives of the chemical potential to spatial integrals
of the total correlation function (Kirkwood and Buff, 1951)
and to spatial integrals of the direct correlation function
(O'Connell, 1971; O'Connell, 1981). This last method is
generally known as Fluctuation Solution Theory.
Fluctuation solution theory has been applied to the
case of a general reacting system (Perry, 1980; Perry and
O'Connell, 1984), and the formalism has also been adapted
to treat strong electrolyte solutions which are considered
as systems where the reaction has gone to completion (Perry,
Cabezas, and O'Connell, 1985). The main body of this chapter
consists of a derivation of the general fluctuation solution
theory. Although the final results are identical to those
previously obtained by Perry (1980), the development is
more intuitive and mathematically simpler, though less
general. The remainder of the chapter illustrates the
calculation of direct correlation function integrals (DCFI)
from solution properties and sets theoretically rigorous
infinite dilution limits on the DCFI's.
Thermodynamic Property Derivatives and Direct
Correlation Function Integrals
A general multicomponent electrolyte solution, contain-
ing n species (ions and solvents) formed from no components
(salts and solvents) by the dissociation of the salts into
ions, is not composed of truly independent species due
to the stoichiometric relations among ions originating
from the same salt. It is, therefore, not possible to
change the number of ions of one kind independently of
all the other ions. However, the independence of ions
has been assumed traditionally for theoretical derivations,
and it will lead us to the correct results by a relatively
simple mathematical route. Thus, with the assumption that
any two species i and j are independent of all other species,
Fluctuation Solution Theory gives the following well known
result (O'Connell, 1971; O'Connell, 1981):
1 ^i 6.. c.
RT 9N. N. N
where = the chemical potential per mole of
N. = the number of moles of species i.
N = the total number of moles of all species.
6.. = the Kroniker delta.
C. = 4Tp J r dr = spatial integral of
the direct correlation function.
p = = molecular density of all species.
The microscopic direct correlation function
is an angle averaged direct correlation function defined
ij> c1 dw. dw. (2-2)
'J j2 ij 1 J
where Q2 = dw. d .
In order to arrive at the first and simplest of the
desired relations, we define the activity coefficient for
species i on the mole fraction scale as
(T,P) = MP(T) + RT In X.Y.(T,P) (2-3)
where = the reference chemical potential.
Xi = N = mole fraction of species i.
Yi = the activity coefficient of species i.
P = the vector of species mole densities.
By differentiating equation (2-3) with respect to
the number of moles of species j, we obtain
ny+ i1 1 (2-4)
3N. N. N
3 T,V,Nk j
which upon insertion in equation (2-1) gives
1ny. 1 C..
1 = N1i (2-5)
and when multiplied by the system volume on both sides
of the equation,
81ny. 1 C..
1 1- (2-6)
p Tp p
where p = n-= molar density of species i.
By performing a sum over all species i and j on equation
S n n lny.
v v- v iaej B Pj
a B i=l j=l j T
n n 1-C..
S1 D1 (2-7)
a i=l j=l
where vi = number of species i in component a.
Sa = total number of species in component a.
By noting the definition of the mean activity
coefficient of a component a,
Iny, = V. Iinyi (2-8)
a v a i-i i
and also assuming that species j is formed from an arbitrary
component 8 so that
Pj = VjB Po (2-9)
one then arrives at the first relation
1 n n 1-C .
i=1 j=l a6
which upon identification of
n n 1-C..
1 C = (2-11)
i=l j=l a3S
assumes the simpler form
PV p= v (1-C ) (2-12)
Equation (2-12) relates the DCFI to the derivative
of the activity coefficient of any component a (salt or
solvent) with respect to the molar density of any component
3 (salt or solvent) at constant volume, temperature, and
mole number of all components other than 8.
Because most experiments and many practical calculations
are performed at constant pressure rather than constant
volume, it is of interest to derive a relation between
the activity coefficient derivatives at constant pressure
and direct correlation function integrals. First, a change
of variables is executed.
3 T,V,Nkj 3 T,P,Nk/j
yi N .P (2-13)
and the following identifications are made,
p = V (2-14)
P T,N = r (2-15)
aN. @N. VK
3 TV'Nkj V TPNkj
Then equations (2-1), (2-14), and (2-15) are inserted into
equation (2-13) to obtain
lli 6. C.. V.V.
1 = -1 13 1 3 (2-16)
RT 3N. N. N VKTRT
3 T,P,Nk 1 T
where = partial molar volume
volume of species i.
V = the system volume.
KT V P = isothermal compressibility.
To develop a relation in terms of activity coefficients,
the chemical potential is written as in equation (2-3)
and the proper constant pressure derivative is taken.
1 i lny
RT aN. = Nj +
3 TP,Nk.j TP,Nkj
13 1 (2-17)
From equations (2-16) and (2-17),
81ny. 1 -e. V.V.
1 = L- 1 3
aN N VK RT (2-18)
By multiplying equation (2-18) by the system volume
and rearranging, one finds
N alny. 1 C V. V.
pKTRT aN pKTRT TRT K TRT (2-19)
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
1 1 n n
1 v. (1-C..) -
KT a 8 i=l j=1
12 1. v V V. (2-20)
(K RT)2 V v i=l j=l a j8 i j
T a B
Equation (2-20) must be simplified so that all the properties
appear as component rather than species quantities. This
is done with the aid of equations (2-8), (2-11), and the
assumption that species j is formed from an arbitrary
component 8 so
Nj = vjNo (2-21)
Additionally, the partial molar volume of a component a
or B is expressed as a sum of the species partial molar
-V 1 aV
V- N- N (2-22)
DN NiB No
1 iS Of3
Vo8 N =i i Vi (2-23)
Equation (2-20) is now transformed to
PK RT 8N
v V (1-C ) V V (2-24)
a s aS oa oS (2-24)
PKTRT K RT KTRT
To make further progress, the relationship of the
bulk modulus of the solution (p TRT) and the group, -T,
to the direct correlation function integrals must be found.
First, the compressibility equation is derived from the
basic fluctuation theory result of equation (2-1) starting
with the Gibbs-Duhem equation for an isothermal but
N N.dP. = VdP (2-25)
i=l 1 1
Upon differentiation of equation (2-25) with respect
to the mole number of an arbitrary component j,
SN. = V P
i=1 i T'V j aN
T,V,N dj T,V,N kj
and by insertion of the equation (2-1),
RT Z (6 X.C..) = V
i=l ij 13 N.
1 P 1- 1 x. C.
RT ap T Fk 1 13
RT j i=
Equation (2-28) is the general multicomponent compres-
sibility equation expressed in terms of species quantities.
This relation is now transformed to one in terms of com-
ponents by performing a summation over species j and use
of equations (2-9) and (2-29).
X = 1 Vi X o
1 n ap
RT X Vj
= 1 X.v C.
i=1 j= 1
o n n
1 XI v. Xi v .' C..
@=1 i=1 j=1
which by use of equation (2-11) becomes
1 @P 1 = v Xo Ca (2-32)
RT Dp oa 0 0
Equation (2-32) is the multicomponent compressibility
equation expressed in terms of components. The density
derivative of pressure is related to the partial molar
-V P aV o (2-33)
ap V 8N
o T,N oa T
T,p o T,P,N
which when inserted in equation (2-32) gives one of the
oa v v X (1-C ) (2-34)
KTRT a O~ X
In order to relate the bulk modulus to direct correla-
tion function integrals, the total volume is related to
the partial molar volumes.
V= ON = V No (2-35)
VN oa oa oa
a=l oa T,P,N a=1
Dividing equation (2-35) by the mole number of species
1 V n
S= X (2-36)
p N oa oca
and when applied to equation (2-34)
RT= 1 VX X (1-Ca)
OK RT l a 6 oa O (
T a=1 8=1
which is the second necessary relation.
Substitution of equation (2-34) and equation (2-37)
into equation (2-24) gives
SV vX X [(l-C )(l-C ) (1-C )(1-C )]
y=l 6=1 Y 6X y 06 y6 ay a5
which is further transformed by substituting for the bulk
Nv ny a
v v I vI v v X Y6[(1-C Y)(1-C a)-(1-C Y)(1-C6 )]
y=l 6=1 o 06
S V X oyX 06(1-C y6
y=l 6=1 (2-39)
Equation (2-39) relates changes in the activity coefficient
of any component a with changes in the mole number of any
component 8 where the process occurs at constant pressure
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
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.
PV D v v (1-C ) (2-12)
TRT- V V X (1-C ) (2-34)
KTRT a =1 o aB
SP/RT 1 o o
~ P/RT 1 v0 vX X (1-C- 1 (2-37)
P T,N TRT a=l =iTxaB=1
a 8 6=1 Y 6 oy 0
[(l-Cy )(1-Ca ) (1-Cay)(l-C68)]
N N v. v. (1-C..)
S-C = ia jB 1
i=l j=l a Y
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
oa = (1-C )0
v K RT al
S (1-C11 )
P K RT
1 No IT,P,N
V 3N 0
3 ~o T,P,N
-(1 V K 1KRT
(1-Cal) (L-C )
where (1-C ) = (1-C )
and where equations (2-42) represent a constant pressure
limit on the DCFI. A corresponding constant volume limit
can be obtained from equation (2-12).
For a binary system consisting of one solvent (1)
and one salt (2), the fluctuation relations become
n2 22 (2-43)
ap02 T 01
o2 = (1-C ) + V2X (1-C ) (2-44)
SRT = 2Xol 12 2 o2 22
aP/RT 1 2
p pKRT ol (1-C )+
3P T,N PKTRT 01 11
2olXo2(-12) + (-22 (2-45)
pK RT 8No2
2X [(1-C 1 )(1-C22) (1-C12)2] (2-46)
2 ol 11 22 12
v+2 (1-C 1+) + V-2 (1-C 1-)
1 -C12 = (2-47)
v (1-C )+2+2v (1-C )+ (1-C__ )
+2 ++ +2-2 +- -2
and the respective infinite dilution limits are
V 2Kl1RT = (C21
P p K RT
Po (V )
22 V2 K RT
aN o l
(1-C ) (1-C12
The significance of equations (2-51) can be further
understood by realizing that any correct model for the
activity coefficient of an electrolyte must approach the
Debye-Huckel Limiting Law at very low salt concentration.
Thus, the mean activity coefficient of a salt on the mole
fraction scale is given by this law as
1 n 2 1/2
lny = S ( Z) (2-52)
ny2 i=l 2
where S =( 332T )
S Dk T
Debye-Huckel limiting law coefficient.
e = the electronic charge.
D1 = pure solvent dielectric constant.
k = Boltzmann's constant.
T = temperature.
Z. = valence of ion i.
I = ZiP = ionic strength.
and when the proper derivative is taken,
N P -1/2
0ol 2 y ol
2 3N2 4v2
n n 2 2
v1 1 i2 Ziz2 (2-53)
Insertion of equation (2-53) into equation (2-51a)
Sp I n n
Y ol- 2 +
2 i2j22 1
4v2 i=1 j=1
Po (V O)
21 02 = (1-C2) (2-54)
which approaches negative infinity as the salt concentration
In order to construct a model capable of correlating
and predicting the solution properties of electrolytes,
it is helpful to calculate the experimental behavior of
the DCFI's from solution properties. To that purpose,
equations (2-43), (2-44), and (2-45) have been inverted
so that the three DCFI's can be calculated from
1 C11 = [1 X V p] +
11 2 OKRTo2 o2
X 2 iny2
ol V N 2 (2-55)
2 2 DN
1-C =1[-X V p1-
12 V2XolKTRT -o2 o2
Xo2 N n 2 (2-56)
0ol o2 T
p(Vo ) N in 2
1 C = -- (2-57)
22 2 vK aN
2 T T,P,Nol
Figures 3-5 show the results of equations (2-55) and
(2-57) for six different salts at 1 ATM and 25C. The
compressibility data used were those of Gibson and Loeffler
(1948) for NaCL and NaBR. For LiCL, LiBR, KCL and KBR
the compressibilities of Allam (1963) were used. The
activity coefficient data were taken from the compilation by
Hamer and Wu (1972). The density data of Gibson and Loeffler
(1948) were again used for NaCL and NaBR. For LiCL, LiBR,
and KBR the density data were taken from the International
Critical Tables. The newer density data of Romankiw and
Chou (1983) were used for KCL. The pure water data were
those of Fine and Millero (1973). The infinite dilution
partial molar volumes were also from Millero (1972).
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.
0.00 05 02
0.04 0.06 0.08 0.10
Figure 3. Salt (2)-Salt (2) DCFI in Aqueous
Electrolyte Solutions at 250C, 1 ATM.
For data sources see text.
0.04 0.06 0.08 0.10 0.12
Figure 4. Salt (2)-Water (1) DCFI in Aqueous
Electrolyte Solutions at 250C, 1 ATM.
For data sources see text.
0.00 0.02 0.04 0.06 0.08 0.10 0.12
Figure 5. Water (1)-Water (1) DCFI in Aqueous
Electrolyte Solutions at 250C, 1 ATM.
For data sources see text.
A MODEL FOR DIRECT CORRELATION FUNCTION INTEGRALS
IN STRONG ELECTROLYTE SOLUTIONS
In order for the formalism introduced in the previous
chapter to be of practical value, a model to express direct
correlation function integrals in terms of measurable
quantities (p, T, x) must be constructed. The present
chapter describes such a model. First, a general physical
picture of electrolyte solutions and its relation to micro-
scopic direct correlation functions is discussed. Second,
a rigorous statistical mechanical basis is laid for the
microscopic direct correlation functions and their spatial
integrals. Third, equations are given for each type of
pair correlations in the system (ion-ioin, ion-solvent,
solvent-solvent). Lastly, a summary is presented of the
model parameters and estimated sensitivity of results to
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
solvent molecules. In the absence of a complete understand-
ing of all these forces, models use simpler or, at least
tractable, interactions which may have the essential charac-
teristics of the real forces. In addition, some semiempiri-
cal terms are used to account for those interactions that
cannot be simply approximated.
Thus the interactions between the ions at long distances
are modeled as those of charges in a dielectric medium
containing a diffuse atmosphere of charges. At very short
range, however, the dominant interaction becomes a hard
sphere-like repulsion. There exist rigorous statistical
mechanical methods to treat these two types of interactions,
but these two are not adequate to correlate and predict
the solution behavior with sufficient accuracy. Interactions
that are important at intermediate ion-ion ranges must
be incorporated. Unfortunately, these intermediate range
forces cannot be simplistically approximated because they
involve strong many-body effects such as dielectric satura-
tion, ion-pairing, polarization, etc., which are not well
understood. In the present model the ionic and hard sphere
interactions are treated theoretically while the rest are
included in a semiempirical fashion.
The interactions between ions and solvent molecules
at large separation can be treated as those of charges and
multipoles in a dielectric medium containing an ionic
atmosphere. In general, quadrupoles and higher order
multipoles are not included, because their contribution
is expected to be numerically insignificant in an aqueous
system. The short range interactions are treated as hard
sphere repulsion. Intermediate range forces for the ion-
solvent case are very important because they include solva-
tion which makes a larger contribution than the long range
charge-multipole forces. Solvation of the ion by the solvent
is intimately related to the partial molar volume of the
salt and must be incorporated if there is to be any hope
of correlating and predicting the volumetric behavior of
the solution. As for ions, the long and short range intera-
tions are treated theoretically while the intermediate
range forces are incorporated semiempirically.
The forces between solvent molecules at long range
can be considered to be those of dipoles in a dielectric
medium which has an ionic atmosphere. Higher order multi-
poles may again be neglected because their contributions
are less important and can be covered in other ways. The
short range forces are again treated as hard sphere repul-
sions. The intermediate range interactions for the solvent-
solvent case are dominated by association type forces such
as hydrogen bonding which make a larger contribution than
the long range dipole-dipole term. As above, the long
and short range interactions are treated theoretically
while the effects of the intermediate range forces are
In summary, there are three distinct classes of inter-
action: ion-ion, ion-solvent, solvent-solvent. Each class
has unique contributions from long-range, field-type forces,
short-range, repulsive forces, and intermediate range forces.
Traditionally, models have been written for the excess
Gibbs or Helmholtz energy of the system by adding contribu-
tions from some of the above forces in an ad hoc and, gener-
ally, nonrigorous fashion. The fact that free energy
contributions do not naturally separate into the types
of forces and that experimental values for each cannot
be separately determined has caused many of these models
to be complex and/or inconsistent. Further, they do not
yield volumetric properties along with the activities.
Within the framework of Fluctuation Solution Theory,
the contributions of the pair correlations to the thermo-
dynamic properties can be rigorously added. Thus, there
are terms from the salt-salt, salt-solvent, and solvent-
solvent DCFI's, as shown in Chapter 2. Further, the experi-
mental behavior of each of the three DCFI types can be
separately calculated from solution data as seen in the
previous chapter. It is then possible to construct separate
and accurate models for each one of the DCFI's. These
models can later be manipulated to yield thermodynamic
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,
c..(T,p,r ,r ., .) = g. 1 Zn g + B.. (3-1)
13 -j -1- n g ij kT 13
where gij = e = radial distribution function.
W = potential of mean force.
ui. = pair potential.
B.. = sum of all bridge diagrams also known
as elementary clusters.
Although equation (3-1) is an exact expression for
the direct correlation function, it is of little practical
value because the bridge diagrams cannot be summed analyti-
cally. This series is
B.. f.f f .f.f. dr drd d
Bl 2'! Okp fikfk fZjfifkj dkd wkd
for a system consisting of n species.
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
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
- 1 ng. -
From the definition of the radial distribution
Z.n g..= 11
W.. W .
giJ -1 1 kT ) 2
13kT 72T kT
which on insertion into equation (3-3) gives,
HNC u W.. W..
Cij T 1 3! 1 ) +
ij IT 2! kT 3! kT "
To apply equation (3-6) requires at least approximate
expressions for the potential of mean force in terms of
measurable variables. Such expressions, valid in the limit
of zero salt concentration and large separation between
the two interacting species, are available for ion-ion
interactions from the Debye-Huckel theory and for ion-dipole
and dipole-dipole interactions from more recent work (H'ye
and Stell, 1978; Chan, Mitchell, and Ninham, 1979) which
yields results identical to those of Debye and Huckel for
ionic activities. Thus, the long range direct correlation
function is based on these potentials of mean force, WLR.
Then, our HNC approximation is
I-*o c.. c.. (3-7)
r ij _0_00 3 13
LR 2 LR 3
LR 1 W1 W
cL. + 1 -+ (3-8)
13 kT 2! kT 3! kT
The potentials of mean force, however, are unphysical
inside the hard core of the molecules and must be set equal
W R = 0 r.. < a..
i] 1] ji
W.. = W.. r.. > a..
ij ij] 3 13
where aij = (aii + a..) = distance of closest approach
of species i and j
At contact and inside the core of the molecules, the
direct correlation function is dominated by a very strong
repulsion which is modelled as a hard sphere interaction.
To obtain the appropriate expressions for the hard sphere
direct correlation functions, the Percus-Yevick theory
(1958) was used since it has been shown to give a compres-
sibility equation of state which is in good agreement with
simulation results for hard spheres (Reed and Gubbins,
1973). The Percus-Yevick (PY) microscopic direct correlation
function for hard spheres is zero outside the core. Thus,
c.. = c.. (3-10)
PY-HS HS (1 ijkT
cij = gij (1 e ) (3-11)
u. = 0 r.. > a..
u.. = r.. < a..
1] 1] 13
Although the PY microscopic direct correlation function
is formally used in the development that follows, it was
not actually employed in obtaining the final expressions
for the DCFI's. Rather, the expression for the hard sphere
chemical potential as derived from Percus-Yevick theory
through the compressibility equation was used together
with equation (2-1) to obtain the desired relation (see
Appendix A). Although the more exact Carnahan-Starling
(Carnahan and Starling, 1969; Mansoori, Carnahan, Starling,
and Leland, 1971) expression could have been used, it is
somewhat more complex and relatively little improvement
in accuracy would be expected.
At this point, we have established a viable, albeit
traditional, theory for the behavior of the direct correla-
tion function as r.. m and at r.. < a... However, many
1] I] J1
interactions which are important in aqueous electrolyte
systems such as hydration of ions by water, hydrogen bonding
between water molecules, and ion pairing are strongest
at r.. just outside the core. Further, this is that kind
of interaction for which liquid state theory is not well
developed. Therefore, we attempt here to develop a method
for interpolation of the direct correlation function between
long and short range. Because generally available theory
offers little guidance, the method can at best be semiempiri-
cal. For this purpose, the Rusbrooke-Scoins expansion
of direct correlation function (Reichl, 1980; Croxton,
1975) for species i and j in a system of n kinds of species
is now introduced.
cij(T,p,r,rj ,wi'O.) = fij(T) +
+ Z pk ijk(T) + (3-12)
where (T) = J f. fi fj dr
ijk ij ik jk d-k k
Since equation (3-12) represents the entire direct
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
c.. = c.H + c. + Ac.. (3-13)
] i i] 1]
where cij is defined by equation (3-11), c by equation
Ac.. = c c.. c. (3-14)
13 1] 13 3
which is approximated by the Rushbrooke-Scoins expansion
HS- LR n
Ac.. = (fi.. f f ) + I (p
] i] ij k= k ijk
HS o LR
P H P o L (3-15)
k ijk k ijk (3-15)
o LR LIM
where = r.. P
k ijk 13 k ijk
The series in equation (3-12) is truncated at the
first order term in density to be consistent with the HNC
theory and because inclusion of the more complex higher
order terms was empirically unnecessary.
For the sake of simplicity in notation equation (3-15)
is expressed as
Ac = Af.. + n po LR
Skl k ijk ijk3-16)
where Af.. = f.. fHS fLR
13 i] ij ij
No attempt was made in this work to analytically calcu-
late the coefficients in equation (3-16); rather, their
spatial integrals were fitted to data. The importance
of equation (3-16), however, is in providing a theoretical
framework for describing the properties for a class of molecu-
lar interactions which are not well understood. Thus,
the first term represents the contribution of pairing or
repulsion in the case of ion pairs, solvation in the case
of ion-solvent pairs, and hydrogen bonding in the case
of solvent pairs. The second term represents the effect
of a third body (k) on the direct correlation between species
i and j. If one or two of the three are solvent and the
rest ions, then this term is dominated by hydration. If
all three species are ions, then this term is dominated
by ion association or repulsion. The physical significance
of these terms will be discussed further below.
As pointed out in Chapter 2, solution properties are
related to spatial integrals of the direct correlation
function. In order to relate this model to thermodynamic
properties, equation (3-13) is integrated over angles first
and separated later. Thus
i c. d dW. (3-17)
C (T,p) = 47'p f r2. dr.. (3-18)
S o 0 ij 13
Cj(T,P) = CHS + CLR + C. (3-19)
ij 13 1j 13
where C is obtained directly from the chemical potential
as shown in Appendix A. Thus, CLR is defined by
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
Lastly, AC.j is defined by formally integrating equation
ACij = pAFij(T) + p (kAO(T) pk(T)) (3-21)
k=l ijk ijk
where AF(T) = 47 J r.2 dr.
ij o ] W 3 13
A(T) = 47 f r.2 dr..
ijk o ijk 13
LR > 2
(T) = 47 r. dr.ij
Equations (3-19), (3-20), (3-21), and the expression
for C. from Appendix A are the general forms of the model
for species direct correlation function integrals. To
obtain practical expressions one needs merely to introduce
the appropriate pair potential and potential of mean force
into equation (3-20) and perform the indicated integration
as illustrated in the sections that follow.
Since the coefficients in equation (3-21) are fitted
to data rather than evaluated analytically, it is of
importance to develop mixing rules to reduce the amount
of data necessary to model multicomponent systems. The
aim here is to predict all the coefficients from quantities
associated with no more than two different species so that
only binary or common-ion solution data would be required.
For aqueous electrolytes, the situation can be improved
due to the relative simplicity of ion-ion interactions
which can be generally scaled with the ionic charge (Kusalik
and Patey, 1983). Thus, two and three ion coefficients
are expressed from quantities related to a single ion.
If i, j, and k are ions, then
AF (T) 1 (AFi + AF ) (3-22)
ij 2 ii ji
AQ(T) (A + Aj + AD ) (3-23)
3 iii jjj kkk
L(T) (L + .. + LR ) (3-24)
ijk 3 11i 33 kkk
If one or two of the species i, j, and k are solvents while
the remainder are ions, then the mixing rule must be
expressed from quantities involving each of the species
and water. The reason for this is that ion-solvent inter-
actions cannot possibly be predicted from solvent-solvent
and ion-ion interactions separately. Therefore, if i is
an ion and j a solvent, then
AF..(T) = AF.. (3-25)
If i and j are ions while k is a solvent, then
AO(T) = (A ii + A k) (3-26)
(T) ( + LR (3-27)
i 2 iik jjk
If i is an ion and j and k are solvents, then
AM(T) = A-jk (3-28)
$(T) = D(T) (3-29)
Lastly, if i, j, and k are all solvents, then
AF(T) = AF. (3-30)
AI(T) = Ai. (3-31)
((T) = ijk (3-32)
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.
Second, the extension of equation (3-20) for cLR requires
potentials of mean force applicable to the system. Assuming
all solvents are dipolar requires only knowing the dipole
moment of each of the solvent molecules and the dielectric
constant of the solvent mixture. Neither of these are
expected to present a problem in general. Third, the exten-
sion of equation (3-21) for Ac.. involves a few more coeffi-
cients and slightly different mixing rules for some three
body terms. Thus, while equations (3-22) to (3-27) would
remain the same for all solvents, equations (3-28) and
(3-29) where i is an ion and j, k solvents would be altered
D(T) = (LR. + LR (3-34)
ijk 2 ii3 ikk
which reduce to the previous result only when j and k are
equal. Here, any nonadditive interaction between j and
k has been tacitly ignored because the difference in the
interactions between different solvents is likely to be
less important to direct correlation function integrals
than that from the much stronger ion-solvent interactions.
This assumption is based on previous investigation of
solvent-solvent interactions which are dominated by angle
independent forces (Brelvi, 1973; Mathias, 1978; Telotte,
1985; Campanella, 1983; Gubbins and O'Connell, 1974; Brelvi
and O'Connell, 1975). Finally, equations (3-30), (3-31),
and (3-31) where i, j, and k are solvents would become
AF (T) (AF + AF ) (3-35)
ij 2 ii Fj
AM (T) = (At. + A + ) (3-36)
ijk 111 331 kkk
LR 1 LR LR LR
f(T) = i ii + + k) (3-37)
ijk 3 111i 333 kkk
The above mixing rules for an aqueous system (single
solvent) have been tested against data for a number of
salts and may be regarded as established. The rules for
a multisolvent system, however, have not been tested.
They can only be seen as physically reasonable in the light
of previous experience but still tentative.
The next two sections deal with the application of
the theory developed here to specific interaction in order
to construct practical expressions.
Expression for Salt-Salt DCFI
The salt-salt direct correlation function integral
(C a) can be expressed as a stoichiometric sum of ion-ion
DCFI's (c. ) given by equation (2-11).
n n v. v (1-C..)
1 C = 1 la 18 (2-11)
i=l j=l ae
It is, thus, only necessary to develop general and practical
expressions for the ion-ion DCFI's and insert these into
equation (2-11) to obtain a general expression for the
salt-salt DCFI. The basic model for ion-ion DCFI's is
represented by equation (3-19). The expression for CHS
has been developed in Appendix A and that for AC.. is given
by equation (3-21). This section is then chiefly concerned
with performing the integration in equation (3-20) to
obtain an expression for CLR
The pair potential between two ions is given by
LR z _
u. L3 (3-38)
Here the potential of mean force is approximated by a gener-
alized form given by the Debye-Huckel theory.
Z.e K(a. .-r. ij)
"LR = i e r > a.. (3-39a)
ij kTr.. D(1+Ka..) l]
W. = 0 r < a.. (3-39b)
K2 4ne2 n 2
K DkT zi Pi = Debye-Huckel
D = the dielectric constant of the solvent
or mixture of solvents.
Insertion of equations (3-38) and (3-39) into equation
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
2JrpZ.Z .e 3Ka
i e ] f
3(DkT) (1+Ka..) a..
e 13 dr.. -
e i rj
e dr.. +
The first term of equation (3-40) contains a divergent
integral. However, when it is introduced into equation
(2-11) which relates it to thermodynamic properties, electro-
neutrality makes the coefficients of the integrals sums
to exactly zero.
2 n n
SkTi 1 Z vjZ f r.. dr.. = 0 (3-41)
v V kT i=la i j=1 1
where v. Z. = o
i=l ic 1
The second term of equation (3-40) is integrable and
contains the implications for DCFI's of the Debye-Huckel
limiting and extended laws (see equation 2-54). Then,
27rpZ2 2e 2Ka.. m -2Kr..
i e 13 e dr..
(DkT)2(1+Ka..)2 a. 1J
2 2 -1/2
= 1 'Y (3-42)
where 2e6 1/2
where S = ( 2 )
K = B I1/2
B =K I /2 DkT
1 2 Zi Pi
The third term of equation (3-40) is also integrable
but more complex. The integral is the first order member
of a class of functions known as the exponential integrals.
These cannot be evaluated explicitly but a number of
asymptotic expansions and numerical approximations are
available (see Appendix D). It is convenient to express
the integral in dimensionless form.
Letting X = r/ai. then
= e- dx = E (3Ka..)
1 x 1 l]
where E1(3Ka ij) = the first exponential integral
The third term in equation (3-40) becomes
3 3 2 3a. y12
Z.Z Spe Pe
S(l+a. .B E (3a..B 11/2)
3 1/2 13
which contains the implications for DCFI's to a higher
order limiting law for unsymmetric electrolytes (Friedman,
1962). Because of electroneutrality, this term, when
inserted into equation (2-11), is always very small for
symmetric electrolytes, and it approaches zero as the con-
centration of salt decreases. For unsymmetric electrolytes,
however, the sum over the ionic charges is not small and
this term actually diverges logarithmically as the salt
concentration decreases. To further explore the relation
of (3-44) to Friedman's limiting law and to elucidate the
low salt concentration behavior, the exponential integral
(E1) can be expanded for low values of the ionic strength
E (3ai.B I1/2) = n(3aiB I1/2) a + O(I1/2) (3-45)
where a = 0.5772 = Euler's constant.
This expansion is valid only at extremely low ionic
strength. Equation (3-44) then becomes
ji S2p PE (3aB 11/2-
3 Y olEl(3ai ) Y
3 ol 2 n + + n 3a
+ 0(11/2) (3-46)
where InI diverges as I 0 while a + In 3a..B are all
The contribution of Friedman's limiting law to the
activity coefficient of a salt (a) is
1 n .3 2
V x ia i
FLL 1 _i=l 2
FLLn y1 i= Z S2 IknI (3-47)
a 3 n 2 Y
i=l La 1
and by taking the first derivative with respect to the
mole number of a salt 8 at I o,
FTL 2 P
Sn y Sypo n n 3
2-1L 01 3 ..zz
N 23v v X 3 3
oNo ITPN 3vaB i=l j=la Z
(- An I + -) (3-48)
Rearrangement of equation (2-24) gives
N any pV V0
N -C oa oB (3-49)
v @N 0CaB$ v K TRT
S oBS a c aB
If equations (3-48) and (3-49) are compared, it is
clear that the contribution of Friedman's form of the limit-
ing law to the salt-salt direct correlation function integral
cFLL S ol n n1
CF LLX z z (- Zn I + ) (3-50)
U 3vv ia j i j ]2 2
S 3vaeB i=1 j=1
Comparing equations (2-11) and (3-50) gives the ion-ion
CFLL S2 P 1 + 1) (3-51)
ij 3 y ol 2 2-
Substitution of equation (3-46) in equation (3-40)
gives the expression for the limiting contribution of the
third term in equation (3-40) to the ion-ion DCFI.
SS2 P ( nI + + In 3a.. B) (3-52)
3 y o 2 13
Equations (3-51) and (3-52) have essentially the same
behavior as I 0 since they differ only by a small constant
which is negligible compared to ZnI as I 0. Therefore,
equation (3-44) contains the higher order limiting law.
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 )
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)
The expression for CHS is
HS 1 n n HS
CS 1 V V. V. C (3-54)
a ji=l j=l
Lastly, the expression for ACB is
AcB = v I Via. AF. +
a B i=l j=l aB
n n n
+ v (p A po ) (3-55)
Va i=l j=l k=l a j k ijk kjk
Equations (3-53), (3-54), and (3-55) form the complete
model for the salt-salt DCFI.
S = CHS + CLR + AC (3-56)
CaB CaB aB
Since the limits of DCFI's as salt concentration
approaches zero are well defined, it is advantageous to
use equations (3-53) to (3-55) to model the deviations
from this limit. To this purpose, the infinite dilution
limit of the salt-salt DCFI is now explored. From equation
N n p V
N1 N y = (1-Co) s
S(1T-C ) ola (2-42a)
S N a v vT,P, K RT
S oS T~~Noy a6
it is seen that the constant temperature and pressure limit
has divergent terms associated with the activity coefficient,
a first constant related to the partial molar volume, and
a second constant associated with the activity coefficient
and which is not so well defined. This second constant
is loosely related to a term linear in salt density which
often appears in empirical expressions for the salt activity
coefficient (Guggenheim and Turgen, 1955; Guggenheim and
Stokes, 1969). In the present model the divergent terms
are contained in equation (3-53). The first constant can
be calculated directly from infinite dilution partial molar
volumes and solvent quantities. The second constant must
be fitted to data using terms from equation (3-55) which
have only ion-ion and long range ion-water correlations.
This reflects the fact that triple ion direct correlations
are zero at infinite dilution and any contributing short
range ion-solvent correlations would generally be contained
in the first constant. Thus,
0 LIM LR TB
C XL1 (C aB- A ) (3-57)
a X +91 aB -ca aB
where TB n n n
where VAC = v. v jB ,ijk
SB i=l j=l k=l
p- o_ o
p V V n n
1-c oa oa B p (F
(1-C) KRT v v v (AF
Sa i=l j=1
+ P 1l ) (3-58)
Finally, the general expression for the salt-salt
DCFI model including the infinite dilution limit is
m LR HS HSo'
1-C = (-C ) (C -HS C
aB cB aaB ca6 )
(A AC ) (3-59)
where cHS = LIM HS
where C C
aB Xol l aB
TBC LIM TB
aB X oll aB
Pol n n
= 01 y y v PA
V v I I itX jp o1 ijl
Va i=l j=l
Although equation (3-56) can be used in place of
equation (3-59), it was felt that the latter was more
appropriate for calculations at constant temperature and
pressure. Therefore, equation (3-59) was used in the com-
parisons and correlations in this work. In calculations
where pressure varies, equation (3-56) would be more
convenient since it would eliminate the need to obtain
partial molar volumes as a function of pressure.
For illustrative purposes, equation (3-59) will now
be written for a binary system consisting of a solvent
(1) and a salt (2) which dissociates to formvy cations
and v anions.
n = 1 + v +
V = V +
v2 = + -
22 = 22 22 (C22
- (ACTB ACT )
TB TB- C P
(AC22 AC ) -2
+ 2v+v _(plAl+-
[v (p AD + p AD + p AD ) +
+ 1 1++ + +++ -++
+ p-A(_+ ) +
+ V (PlAl--_ + p+AI+__ + pA___ )] +
(P p ) 2 9
01 [v 2A + 2v v A+ + v2A1 ]
2 + ++ + 1+- 1--
R S -1/2 2 4
cLR= ++ +
22 2 1/2 2
4v (l+a BI )
2 ++ y
2v _Z2Z2 v2 Z
+ ( BI1/22 + (1/2)2
(l+a+ BI ) (l+a BI )
+- -- y
2 6 + y
2v 3a B 1/2
3 3 3a+- B
2+ Z_ -+
E (3a+ B 1/2)
E (3a B 1 /2)
2 6 3a B II/
2V -- y
(l+a B 11/2)3
SPol(Vo2) p 2
1 -C22 [V (aF
SC22 =2 -2 + ++
v2 1RT 2
+ 2v+v_ (AF_
(CHS HSm) 1
22 22 2
+ 2V v_ (CHS
- P l++) +
P LR 2 P LR
- Po (l ) + v (AF Po l )
S1+- -- 01 1--
[v2 (CHS ) +
+ ++ ++
CHS) + v2 (HS HSm)
+- + __
(1 B1/2 3
(l+a BI )
E (3a__ By1/2 )
Expression for Salt-Solvent DCFI
The development in this section parallels that of
the previous one. Thus, a general expression for the
solvent-ion DCFI is derived and then inserted in equation
(2-11) to yield the salt-solvent DCFI relation.
Although any type of interaction can, in principle,
be included, it was assumed here that ion-solvent interac-
tions are dominated by dipole-charge forces at large separa-
tion, and no other interactions were included. The pair
potential for an ion (i) and a dipolar solvent (1) is
LR l e
u = cos 6 (3-65)
where pl = the dipole moment of solvent 1
S= the Eulerian angle between dipole
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.
LR i 1 e il
il 2 (cos 0) e r > ail (3-66a)
Wi = 0 r ail (3-66b)
where a is a universal constant that we have set equal
to 4.4 empirically.
Since equations (3-65) and (3-66) are functions of
orientation, it is necessary to first perform the integration
over angles as indicated in equation (3-20).
= u LR dw dwl (3-67)
il 2 i1 i 1
where dwo = sin e.di.d4.
1 1 1 1
S= f dw. = f sin 6.de. f2 7di = 47
1 0 1 1 0 i
When the integral in equation (3-67) is evaluated, it is
= 0 (3-68)
The second term in equation (3-20) has
LR 2 1 I (W)LR2
<(W ) > (W) dw. dw (3-69)
il Q 2 il 1
After the integral in equation (3-69) is evaluated, it
L2 2 2K(ail-ril)
2 Z ile )e
<(W i) > = T 4 (3-70)
The third term in equation (3-20) contains
LR 3 1 )3
<(W ) > ( dw d
ii 2 il 1
which also equals zero.
<(W ) > = 0
Therefore, for ion-dipole pairs there is only one term
in equation (3-20).
i 2 Tr c 2 2Ka -2Kril
LR 2I 1I ) 2 il e e
il 3 P DkT ) e f -
The integral in equation (3-73) is also an exponential
integral (E2) which is expressed in dimensionless form
C -2Kr il -(2Ka )X
S1 il E (2Kail )
2 e dr = 1 f e dX = ail
2 i1 a.i 1 a
Equation (3-73) then becomes
iR -2T lZ el e E2(2Kail
11 3 DkT ail
Then, the general expression for the salt (a) and
solvent (1) DCFI is
S2a iB I1/2
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
The expression for CHS is given by
HS 1 HS
Cl v Via Cil
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
- p ik)
Again, equations (3-76), (3-77), (3-78), and (3-79)
form the complete general model for the salt-solvent DCFI.
C = CHS + C + AC
al al al al
As previously discussed, it is convenient, particularly
for isobaric calculations, to use the model only for
deviations from infinite dilution. (For nonisobaric calcu-
lations, equation (3-79) would be more appropriate.) The
infinite dilution limit of Cal is given by equation (2-40)
and that of CLR is (see Appendix D)
P2eep 2 n y Z2
LRO LIM LR 2ol ( ea 2n i (3-80)
1C C = ) P 1 a1 (3-80)
al X ol+1 l 3v DkT 1 ail
ol a i=l a
while the infinite dilution limit of C H is formally
HSm LIM HS (3-81)
C1 =X1 Cl (3-81)
Lastly, the infinite dilution limit of AC is
A LIM ol n P LR
al X o1 al i (iFil Pol ill
+ 1 o' pnolA l (3-82)
iV Vi P1 ill
The complete general relation for the salt-solvent
DCFI including the infinite dilution limit is
oc HS HS o LR LRm
1- IT- RT Cl- Cal) (C Cal
(ACal ACal) (3-83)
Finally, equation (3-83) will now be written for a
binary system consisting of solvent (1) and salt (2) with
v+ cations and v_ anions.
V o 02 (HS
21 v K2 RT 21
- C2 ) (C2
HS _HS- 1 HS
21 21 v2 +(C+1
-HS + (HS HSm)
+1 ) -1 -1 )
CLR CLR 2- r ( ) e )
21 21 3v DkT
2 + +
2a +B I1/21/
(pe Y E2 (2a+1B
AC C = [v (AF+
1 C21 + +1
2a 1B BI/2
(pe E2(2a lBI2) p)]
P LR P LR
- i ) + _(AF Pol )]
01 +11 -1 01 -1
(p p ) +
+ ol + p+ p +
V2 + +1 1+1 + P+A++ + -+
+ v_(plA11_ + P+A+ + P_A __ )]
[v+ &1+1 + v _l_ (3-87)
Expression for Solvent-Solvent DCFI
The solvent-solvent direct correlation function integral
has the simplest relation since the solvent does not
dissociate so the species and component integrals are the
As previously noted, any type of interaction can gen-
erally be included in this theory, but it was assumed that
solvent-solvent interactions at large separation are domi-
nated only by dipole-dipole forces. The solvent (1)-solvent
(1) pair potential is
LR 1 1
u 3 1 (3-88)
where ( = 2 cos 611 cos e12
sin 011 sin 012 cos (11 012
611' 11 = Eulerian angles of solvent molecule
612' 12 = Eulerian angles of solvent molecule
The potential of mean force is approximated by a
function inspired by previously mentioned work (Chan,
Mitchell, and Ninham, 1979; Hzye and Stell, 1978).
LR 1 1 e 1
W = kr D
W11 3 D
W I 01
r > al
r < al
Again, equations (3-88) and (3-89) are inserted into
equation (3-20) and the required integration over angles
SLR = L l d11
11 Q 2 11 d11 12
dli= sin 1i deli d li
S= f d6l = f sin 6li dli
f di = 47
The integration of equation (3-90) gives
The second term in equation (3-20) has
LR 2 1 LR 2
<(W ) > f (W ) dw dw
<11 ( 2 (W11 d11 d12
which yields upon evaluation,
LR 2 1 i l11 2
<(W ) > (D -)
11 w 3 DkT
The third term in equation (3-20) has
LR 3 1 LR 3
(W ) > (W ) d
<11 2 11 1 12
which becomes upon integration
<(W )> = o
Thus, for dipolar solvents only one term of equation
(3-20) remains after the angle integration.
R = 2 e2Kal -2Krl
1 3 (DkT e r 4
The integral in equation (3-96) is also an exponential
integral (E4) which can be expressed in dimensionless form.
S11 -(2Ka11 )X E (2Ka )
e 1 e 4 11
d 4 dr l = -- -4 dX = 3
al r11 all 1 X all
Equation (3-96) is then transformed
LR 4 ( 1 (1 1)1
C11 3 DkT 3
E (2a B 11/2)
which is the general expression for the solvent-solvent
Since the solvent does not dissociate, there is no
summation over species in C11. However, AC11 does have
a sum over third bodies.
CFo R ( 9LR
AC11 = pA1 + (Pkllk llk (3-99)
Equations (3-98), (3-99), and (3-100) form the complete
general model for the solvent-solvent DCFI.
C = CHS + + AC1 (3-100)
11 11 11 11
Again, the infinite dilution limit of C11 is introduced
so that for isobaric calculations the model need only account
for deviations from the infinite dilution value. Also,
equation (3-100) would be more practical for nonisobaric
cases. The infinite dilution limit of C11 is the bulk
modulus of the pure solvent given by equation (2-41).
The infinite dilution limit of CLR is given by
LRm LIM cLR 4pol r T D 2
CLR- XLIM CII l ( (3-101)
11 X 01 11 3 fDkT )-0
and that for C S is formally
HS- LIM HS
C =1 Xol1 CI (3-102)
11 X -1 11 (3-102)
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
Finally, the complete general expression for the
solvent-solvent DCFI including the infinite dilution limit
1 HS HS" LR LR"
1 C (Cl l ) (C C )
--11 11 11 11
(ACll ll) (3-104)
Again, the application of equation (3-104) to a binary
system consisting of solvent (1) and a salt (2) with v
cations and V_ anions is shown. However, for the solvent-
solvent DCFI all of the terms except AC11 appear similar
to the general case since they have no summations over
species. Thus, only AC11 is illustrated below.
oo P P LR
AC ACI = (P p )(AF Po +
11 11 01 11 P ll
+ P(P1 111 + P+ A)11 + pA-11) -
(p P)2 A (3-105)
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 (C C R AC ) will now be discussed, the
model parameters will be listed, and the sensitivity of
solution properties to parameter value considered.
The salt-salt DCFI is dominated at very low salt con-
centration by C n which contains the long ranged electro-
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
The salt-solvent DCFI is always dominated by C H with
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
The solvent-solvent DCFI is also dominated by CHS
over the entire range of salt concentration up to about
6M. C11 makes a negligible contribution reflecting the
relative weakness of long range dipole-dipole interactions.
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
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.
AC has AF POaLR i l and A. AC22 includes
1A2 i 01o lil 1il' 1i22
AF. p A and AoP.... This totals to two solvent
11 ol iin, ii, 111
specific parameters if AO11 is neglected and six parameters
for each ion (note that A lii = A.il and AD. = Al )
lil ill 111 ii
three of which involve solvent-ion pairs.
Properties predicted with the model are most sensitive
to the value of the hard sphere diameters because the C
is a very strong function of the diameters. But it is
not as sensitive as is the case with other models. This
is due to the fact that the two body coefficients AF..
are fitted to infinite dilution quantities that include
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.
APPLICATION OF THE MODEL TO AQUEOUS STRONG ELECTROLYTES
In Chapter 2, the formal relations between DCFI's and
thermodynamic properties were introduced. In Chapter 3, a
model expressing the DCFI's in terms of measurable variables
was constructed. In the present chapter we illustrate the
use of the formal relations and the model in the calcula-
tion of thermodynamic properties. We also explore the
scheme used to fit model parameters; further we compare
calculated values to experimental ones for the salt-salt,
salt-solvent, and solvent-solvent DCFI's and for the bulk
modulus, partial molar volume, and salt activity coeffi-
cient. Finally, a discussion of the above results and a few
conclusions are presented.
The use of Fluctuation Theory in general fluid phase
equilibria problems has been treated in detail by O'Connell
(1981). The specific case of liquids containing super-
critical components has been addressed by Mathias and
O'Connell (1981) and Mathias (1978). The present treatment
generally follows these developments, but there are
important differences for the present case of electrolytes.
Calculation of Solution Properties
from the Model
The formal relations between solution properties and
DCFI's are given by equations (2-12), (2-34), (2-37), and
(2-38) for a system consisting of no components, salts (a,B)
and one solvent (1).
ac ap~ ~
= V v (1-C )
OT = Ea Vx (1-Ca
aP/RT o o
PT = E Z v VX X (1-C
Tp a=l =l a oa oB aB
PK TRT a@i
v v V Y v vvX X 0
a y=1 6=1 6 oy 06
[(1-C )(1-C a) (1-Cy )(1-C )]
ae P/RT 1
where = -
ap I PK RT
In order to evaluate the change in solution density
with pressure while the composition and temperature are
constant, one needs to integrate equation (2-37) from a
known reference density (pR) at the reference pressure (pR)
at the temperature and composition (mole fraction) of the
system up to the desired density (p ) at the system pressure
n n F
R o o P (T,P,X)
v V X X (1-C )dp (4-1)
RT E =i =l oa o R R T,N (
p R(T,R ,X)
Equation (4-1) represents an implicit equation for the
unknown density (p ) which can only be solved numerically
with realistic models.
It should be appreciated that equation (4-1) cannot be
applied to an isobaric change because that would imply that
pressure, as well as temperature and composition, were
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
dependent variable, a function of temperature, density, and
P = P (T,p,X) (4-2)
Taking the total differential of pressure gives
dP dp+ dT
ap T,TN Tp,
If the change is isothermal and if we divide by RT,
1- 3- dp
RT 3p T
By making some identifications we obtain
Inserting equations (4-5) and (4-6) into equation (4-4)
dP 1 aP dp +
RTT RT ap T
T a TN
We next insert equations (2-34) and (2-37) into
1 dP = [= V V X (1-C ) dp +
RT T c=l 8=1 a oaoB 0
n n n
+ 0 V X (1-C pdXo (4-8)
a=2 8=1 y=l Y 6 -vX
c G B-uoc
Equation (4-8) permits us to evaluate the change in
solution density with both pressure and composition along an
isotherm. This equation is also applicable to an isobaric
and isothermal process where the solution density changes as
a function of composition only.
To obtain the density (p ) of a given solution at a
known temperature, pressure, and composition (X ), we
isothermally integrate equation (4-8) from the known
reference density (p ) at a system temperature and a conven-
iently chosen reference pressure (P R) and composition (X )
up to the desired density and composition (p and X ). It
is suggested that for aqueous electrolytes the reference
density be chosen to be that of pure saturated water at the
SAT n n P (T,P,X )
P-P o 0 -o
R = I v B SAT X X (1-Ca )) dp +
RT SAT oc oaS
Ra=l =1 Poi (T) T
n n n X
o o o oa pdX
+ I I oy(-C) (4-9)
a=2 8=1 y=l Y oy a6 BXa T
In evaluating the integrals of equation (4-9) each
integral involves variables appearing in other integrals.
To explicitly find pF requires further manipulations
The activity coefficient on the mole fraction scale for
any component (a) can be obtained by integration of equation
(3-12) from the reference molar density (P R) to the molar
density of each component (PF ) at constant temperature.
Y n p 1-C
n v R dP oS (4-10)
R R R p a T
7 8=1 8 Po T
where Po5 = Xo8P
Equation (4-10) is applicable to any isothermal change,
isobaric or not, and the reference state composition where
YaR = 1 need not be that chosen. However, for aqueous
electrolytes it is natural to choose pure saturated liquid
water at the system temperature as was done for equation
P (T,P,X)I n P (T,P,X)
ZnYc = al dol + Z f 1-Ca dp
pSATT T 8=2 o p OT
Equation (4-11) can be used for either isobaric or
In equations (4-9) and (4-11) one can use the DCFI
model represented by equations (3-59), (3-83), and (3-104)
for isobaric integration. But, for nonisobaric integra-
tions with equations (4-1), (4-9), and (4-11) the DCFI model
of equations (3-56), (3-79), and (3-100) will be more
applicable because the pressure behavior of the DCFI
infinite dilution limits, some of which involve salt partial
molar volumes, is not generally available.
The composition behavior of component activity coeffi-
cients on the mole fraction scale at constant temperature
and pressure could also be obtained from equation (2-38)
with composition expressed as mole fractions. Thus, we
express the differential of the activity coefficient of a
component (a) as
dnya = z X dX (4-12)
ST,P B=2 a TPX
eny o 8 ny aN
= 3o o N(4-13)
06 T,P, Xo l T,P,Noa 0 N
OY 8 oy/c OY7c
= N (4-14)
ax 6 -v x
Inserting equations (4-14) and (2-38) into (4-13) and
then putting the resulting expression into equation (4-12)
n n n n
o o PK RT o o
diny = I dX I Z v TV
T,P 6=2 =1 B O 0 Y=l 6=1
XoyXo (1-C 6)(-Ca) (-Cy) (1-C6)] (4-15)
To obtain the activity coefficient, equation (4-15) is
isothermally and isobarically integrated from the reference
to the desired state.
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
Xo Xo[(1-C 6)(1-C ) (1-C )(1-C6) ]
e o o
where = v v X Xo (1-C )
SRT = 8=1 B a o
6 = Kroniker Delta
Equation (4-16) can only be used for isothermal,
isobaric changes and thus either the DCFI model of equations
(3-59), (3-83), and (3-104) or that of equations (3-56),
(3-79), and (3-100) may be used.
Equations (4-1), (4-9), (4-11), and (4-16) express
integration of the DCFI model formally. However, these
cannot be explicitly evaluated because of the multiple
variables involved in the integrals. To actually evaluate
these integrals requires a change of variables as discussed
by Mathias (1979) and O'Connell (1981). Rather than give
their formal equations, we now give the above relations with
explicit expressions for the present DCFI model. Those
parts that are analytically integrable have been evaluated
while simplified integrals are given for the others. The
DCFI model used is that of equations (3-56), (3-79), and
(3-100) which does not contain the DCFI infinite dilution
limits. This form of the model yields simpler expressions
which can be applied to both isobaric and nonisobaric
changes. We start by rewriting equation (4-1) as
_pR pPY-HS(pX) F PY-HSpRX)
RT RT RT
- (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
F F FR R R
p P p -p p p
FF P,F RR P,R
P P P -pp p
where p(t) = I(pR +
X.X. AF.. -
1 3 13
SI X.X X Aj +
j=1 k=l 1 3 k
S z x.x. ,LR
i=1 j=1 ijl
F R o
(P F-P R)t)= I v p (t)
oy oy Y Y
Equation (4-9) can also be changed to
SAT PY-HS F F PY-HS R R
P-P TT) P ( ,X ) P -(p ,X)
1 -o 0
RT RT RT
n n n LR
0 FF RR a C t)
v v (X p -X p ) fp (t)p (t) p dt -
= 8 = oY -oy o o o p(t)p(t)
a=l 8=1 y=1 o
n n n n 1
oo o o 1 1-C (t)
2 ll yl =1 Y o P (t) (t) F (t) dt -
a=2 6=1 Y=I 6=1 0 oy poa(t) a
6 -v p(
rB 8 p(t)
F FFF R RRR
S(X. X p p -X X. pp ) AF. -
j=l 1 3 1 3
F F FFFF R R RRRR
S(X. /X. X pp-X. X. A4. +
j=1 k= i j P P -Xi j X P p p ) A ijk
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
R P,R SAT
S=P = T)
X = 1
X R =0 a 1
X. = v. X
1i L )i Xot
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)
0 FF R R
p o (t) (X p -XR p )
oa p oB O
The expression for PY-HS is given by equation (A-l) and
that for C by equations (3-53), (3-76), and (3-98).
In a similar fashion we transform equation (4-11) to
- n ) (p
PY-HS TpF PY-HS pR
i T, ) pi TT, )
1 C LR(t)
F R) cl d
ol 1o p(t)
1 CLRt) F
- ) a dt + n -
n F F
I .. (PijP
k=l la 3 k
S. (p .p
i cc ol0
Pi = p v iaX
h j ijk
R P,R LR
j- ol )ijl
= v. P
l ia oa
and the other quantities have been defined above.