Chemistry Reference
In-Depth Information
of
G
E;
el
p
G
cond
:
CI
, Manning assumed that the condensation process
reaches an equilibrium when the sum (
G
E;
el
p
and a decrease of
D
+
D
G
cond
:
CI
) reaches a minimum:
0
1
G
E;
el
p
@
þ D
G
cond
:
CI
@
A
¼
0
:
(68)
@y
z
This equation is evaluated for the aqueous solution of the single polyion, also
assuming that the influence of
y
z
on the ionic strength can be neglected. The molar
volume
u
p
is calculated from:
!
þ
y
z
z
p
u
p
c
CI
2
z
CI
1
þ
ln
z
p
l
1
ð
z
CI
y
z
Þ
ln 1
½
exp
ðkb
Þ
¼
0
:
(69)
For low concentrations of the polyelectrolyte (i.e., when
c
CI
!
0) that equation
can only be fulfilled when:
1
z
CI
z
p
z
CI
l
y
ð
0
Þ
z
¼
c
CI
!
0
y
z
¼
lim
1
:
(70)
As
y
ð
0
z
z
CI
is the ratio of the number of actual charges on the polyelectrolyte
backbone to the maximum number of charges on that backbone,
i.e.,
(
y
ð
0
z
z
CI
)
1, then
y
ð
0
z
z
CI
is positive as long as
z
p
=ð
0
z
CI
lÞ <
1. The poly-
y
ð
0
z
z
CI
=0.
electrolyte is completely dissociated when
z
p
=ð
z
CI
lÞ¼
1,
i.e.,
y
ð
0
z
cannot be negative even if
z
p
=ð
z
CI
lÞ >
1. Therefore, two cases have to be
distinguished:
Case A:
1
z
CI
z
p
z
CI
l
y
ð
0
Þ
z
z
p
=ð
z
CI
lÞ
1
;
¼
1
:
(71)
Case B:
y
ð
0
Þ
z
z
p
=ð
z
CI
lÞ >
1
;
¼
0
:
(72)
u
p
is calculated from (
69
) by replacing
y
z
by
y
ð
0
Þ
which is taken from (
71
). As
for the case when
y
ð
0
z
= 0, no counterion condensation occurs and therefore there is
no contribution from condensed counterions to the Gibbs energy.
The result for the molar volume
u
p
is:
;
z
z
CI
l
z
p
u
p
¼
4
pN
A
z
p
b
3
ð
1
Þ
exp
ð
1
Þ:
(73)
