Chemistry Reference
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where
j
stands for:
4
pN
A
b
2
l
B
z
CI
n
CI
þ
z
COI
n
COI
j
¼
(91)
and
f
is a “short-range” parameter. Hao and Harvey did not use the concept of a
volume
V
p
where the condensed counterions are located, but introduced a binding
constant to describe the counterion condensation phenomenon. They expressed that
binding constant using an adjustable, dimensionless (positive) parameter
f
. For
f
1, (
90
) reduces to Manning's approximation (
y
z
¼ y
ð
0
Þ
¼
), whereas for
f
6¼
1
z
the correction term on the right-hand side of (
90
) does not vanish.
5.4 NRTL Model of Nagvekar and Danner
Nagvekar and Danner [
101
] tried to overcome the limitations of the theoretical
expression by combining Manning's result for highly diluted aqueous solutions of a
polyion with the semiempirical electrolyte NRTL (nonrandom two liquid) equa-
tion of Chen and Evans [
109
]. Their expression for the Gibbs energy of an aqueous
solution of a polyion consists of three parts. The first part describes the ideal
mixture, the two other parts describe the excess Gibbs energy
G
E
, which results
from short-range (superscript SR) as well as from long-range (superscript LR)
electrostatic interactions:
X
G
E;
SR
G
E;
LR
G
¼
n
j
m
j;
id
:
mix
:
þ
þ
:
(92)
all components j
The chemical potential of a component
j
in an ideal mixture
m
j;
id
:
mix
:
is defined on
the mole fraction scale using the unsymmetrical convention, i.e., the reference state
for the solvent (water) is the pure liquid solvent. For any solute species, the
reference state is a hypothetical pure liquid where the species experience interac-
tions as if at infinite dilution in water.
As the activity coefficient
g
ðx
j
of component
j
is:
G
E
@
@
RT
ln
g
ðxÞ
j
¼
j;p;T
;
(93)
n
j
n
k6
g
ðxÞ
j
is a product of a short-range and a long-range contribution:
g
ðxÞ
j
SR
;ð
x
Þ
j
LR
;ð
x
Þ
j
¼ g
g
:
(94)
Danner et al. express the short-range contribution using a modification of the
electrolyte-NRTL equation of Chen and Evans [
109
] and take the long-range
contribution from Manning's model (for
the case of
infinite dilution of a
