GENERAL RESEARCH
Prediction of Osmotic and Activity Coefficients Using a Modified
Pitzer Equation for Multicomponent Strong Electrolyte Systems at
298 K
Fernando Pe´rez-Villasen˜or and Gustavo A. Iglesias-Silva*
Departamento de Ingenierı´a Quı´mica, Instituto Tecnolo´gico de Celaya, Celaya Gto. C.P. 38010, Mexico
Kenneth R. Hall
Chemical Engineering Department, Texas A&M University, College Station, Texas 77843
We have predicted the osmotic and activity coefficients of strong electrolyte solutions using a
modification of the Pitzer equation. The modified equation can be used for multicomponent
aqueous solutions by applying a mixing rule at the Debye-Hu¨ckel term. We have found that
the modification of the Pitzer equation retains the accuracy of the original equation without
using any characteristic parameters evaluated from the experimental data. The new equation
is predictive and simpler than the original Pitzer equation.
Introduction
The design and operation of industrial processes that
involve electrolyte solutions require knowledge of rigor-
ous models or experimental data to represent the
nonideality of the mixtures. Obviously, the development
of a model is the most economical solution. Loehe and
Donohue1 mention that many theories and empirical
correlations exist that represent the behavior of a solute
in a solvent. Among the most common models are those
proposed by Meissner and Tester,2 Pitzer,3 Chen et al.,4
Haghtalab and Vera,5 Jaretum and Aly,6 and Zhao et
al.7 For multicomponent systems, the problem is more
complex because the models sometimes require, in
addition to the solute-solvent parameters, character-
istic parameters evaluated from experimental measure-
ments. For example, the model developed by Chen and
Evans8 requires parameters that account for the solute-
solute interaction and the extension of the Pitzer
equation9 for multicomponent mixtures needs param-
eters that account for the binary interactions between
ions with charges of the same kind and parameters that
account for the ternary interactions among two ions of
the same charge and one of opposite charge.
Recently, Perez-Villasen˜or et al.10 modified the Pitzer
equation by considering the apparent second virial
coefficient to be independent of the ionic strength and
eliminating the paramater R. They also considered the
Debye-Hu¨ckel (DH) b parameter to be a characteristic
parameter for each solute-solvent system. With these
modifications, the modified Pitzer model became simpler
and more accurate for aqueous solutions.
In this work, we extend the modified Pitzer model10
to include multicomponent mixtures and to predict the
osmotic and activity coefficients for 21 systems at 298.15
K. In the new model, we do not require characteristic
parameters that account for binary and ternary interac-
tions. We compare our results to those of the original
Pitzer model,9 whose interaction parameters are calcu-
lated using the procedure by Pitzer and Kim.9
Modified Pitzer Equation for Mixtures
For multicomponent systems, the excess Gibbs energy
of the Pitzer model9 is
where
for a single electrolyte MvM
zM XvX
zX and (∑mz) ) ∑cmczc )
∑amajzaj. The last two terms include the differences
between the apparent second and third virial terms for
unlike ions of the same sign (these quantities are small).
Using the above equation, the expression for the osmotic
coefficient is
* Corresponding author. Phone: 011 52 (461) 611 7575.
Fax: 011 52 (461) 611 7744. E-mail: gais@iqcelaya.itc.mx.
Gex
wwRT
) f(I) + ∑
c
∑
a
mcma[Bca + (∑mz)Cca] +
∑
c
∑
<c′
mcmc′[2ıcc′ + ∑
a
maªcc′a] + ∑
a
∑
<a′
mama′[2ıaa′ +
∑
c
mcªcaa′] (1)
f(I) ) -
4AI
b
ln(1 + bI1/2) (2)
BMX ) âMX
(0) +
2âMX
(1)
R2I
[1 - (1 + RI1/2) exp(-RI1/2)] (3)
1087Ind. Eng. Chem. Res. 2003, 42, 1087-1092
10.1021/ie0204787 CCC: $25.00 © 2003 American Chemical Society
Published on Web 01/30/2003

where f  ) (1/2)[f ′ - (f/I)] and c and c′ are indices
covering all cations while a and a′ cover all anions. For
a single electrolyte MvM
zM XvX
zX, the expression for the
mean activity coefficient is
In the simplified model, the parameter âMX
(1) is zero;
therefore, the derivatives of BMX with respect to the ionic
strength B′MX are also zero. In this work we consider
the terms ıij, their derivatives with respect to the ionic
strength, and the terms ªijk to be negligible. With these
considerations, the excess Gibbs energy becomes
where F(I) is the DH contribution. Thus, for the osmotic
coefficient, we have
while the mean activity coefficient becomes
where F ) (1/2)[F′ - F/I] and FMX
ç ) (1/2)F′. Obvi-
ously, eqs 7 and 8 are simpler than the original Pitzer
equation. Because the parameter bi is now a character-
istic parameter, eq 2 is no longer valid for multicompo-
nent mixtures. We need a mixing rule for the DH term
and we use the concept of an ionic strength fraction11-16
given by
where the ionic strength of the k electrolyte in solution
is
Here, the index k indicates that for ionic strength we
consider only the associated molalities if the electrolyte
were alone in solution. Therefore,
where q is the total number of electrolytes in solution.
Then, the equivalent expression for the DH term for
multicomponent systems becomes
where ns is the number of binary electrolyte solutions.
A similar equation can be obtained with the treatment
given by Kusik and Meissner.17 Obtaining the deriva-
tives with respect to the ionic strength, we have
and for any electrolyte p in solution with ns dissolved
electrolytes
In this work, we have considered ternary systems,
that is, two electrolytes with a solvent (water). Then,
we can have three different types of systems because
the solvent is always the same: with a common cation
(MX-MY), with a common anion (MX-NX), and with-
out common cations and anions (MX-NY). Table 1
contains these systems. When we have a solution with
ns electrolytes and they have a common ion, then
 - 1 ) -
1
RT∑
i
mi
(@Gex@ww) ) (∑i mi)-1{2If  +
2∑
c
∑
a
mcma[Bca + ∑mz(jzczaj)1/2Cca ] + ∑c ∑c′ mcmc′[ıcc′ +
Iı′cc′ + ∑
a
maªcc′a] + ∑
a
∑
a′
mama′[ıaa′ + Iı′aa +
∑
c
mcªaa′c]} (4)
ln çMX
( ) vMX
-1(vM ln çM + vX ln çX) ) jzMzXjf
ç +
2vM
vMX
∑
a
ma[BMa + (∑mz)CMa + vXvMıXa] +
2vX
vMX
∑
c
mc[BcX + (∑mz)CcX + vMvXıcX] +
∑
c
∑
a
mcma{jzMzXjB′ca + vMX-1(2vMzMCca + vMªMca +
vXªcaX)} +
1
2
∑
c
∑
c′
mcmc′[ vXvMXªcc′X + jzMzXjıcc′] +
1
2
∑
a
∑
a′
mama′[ vMvMXªMaa′ + jzMzXjıaa′] (5)
Gex
wwRT
) F(I) + ∑
c
∑
a
mcma[Bca + (∑mz)Cca] (6)
 - 1 ) (∑
i
mi)
-1{2IF + 2∑c ∑a mcma[Bca +
∑mz
(zcza)
1/2
Cca
 ]} (7)
ln çMX
( ) FMX
ç +
2vM
vMX
∑
a
ma[BMa + (∑mz)CMa] +
2vX
vMX
∑
c
mc[BcX + (∑mz)CcX] + ∑
c
∑
a
mcma
2vMzMCca
vMX
(8)
(Gex/RT)DH ) wwF(I) (9)
yk ) Ik/I (10)
Ik )
1
2
∑
j)1
r
mj
(k) zj
2 (11)
I ) ∑
j)1
q
Ij (12)
F(I) ) ∑
k)1
ns
ykfk(I) ) -4A∑
k)1
ns
[ykI ln(1 + bkI
1/2)] (13)
F ) - ∑
k)1
ns ( AykI1/21 + bkI1/2) (14)
Fp
ç ) -Ajzp
+ zp
-
j( 2bp ln(1 + bpI1/2) + I1/2∑k)1ns yk(1 + bkI1/2))
(15)
Fp
ç ) -A{jzp+ zp-j∑k)1q ykI1/2(1 + bkI1/2) + 2vp[(zpnc)2vpncbp ln(1 +
bpI
1/2) + (zp
cm)2vp
cm∑
k)1
q mcm
p
mcmbk
ln(1 + bkI
1/2)]} (16)
1088 Ind. Eng. Chem. Res., Vol. 42, No. 5, 2003