Chemistry Reference
In-Depth Information
5.2.3 Contribution from Free Counterions
The aqueous phase contains “free” (or “dissolved”) counterions. These ions are
either dissociated from the polyelectrolyte or result from the dissolution of the salt
S
. Their contribution to the Gibbs energy of the solution is:
ref
G
freeCI
¼
n
freeCI
m
freeCI
þ D
G
freeCI
;
(74)
h
i
ln
c
freeCI
g
ðcÞ
CI
n
o
n
ðpÞ
n
ðsÞ
freeCI
D
G
freeCI
=ð
RT
Þ¼
freeCI
þ
:
(75)
The mole number
n
ðpÞ
freeCI
of the counterions that originate from the polyelec-
trolyte is:
n
p
nð
1
z
CI
y
z
Þ
n
ðpÞ
freeCI
¼
z
p
(76)
z
CI
and the mole number of the same counterionic species from the added salt is:
n
ðsÞ
freeCI
¼
n
s
n
CI
;
(77)
where
n
s
and
n
CI
(either
n
M
or
n
X
) are the mole number of the dissolved salt
S
and
the stochiometric coefficient of the counterion of
S
, respectively. The molarity
c
freeCI
of the counterions in the aqueous solution surrounding the polyelectrolyte is:
n
ðpÞ
freeCI
n
ðsÞ
freeCI
c
freeCI
¼ð
þ
Þ=
V
(78)
or:
z
p
nð
1
z
CI
y
z
Þ
c
freeCI
¼
c
p
þ n
CI
c
s
:
(79)
z
CI
Therefore:
z
p
z
CI
D
G
freeCI
=ð
RT
Þ¼ n
CI
n
s
þ
ð
1
z
CI
y
z
Þnn
p
z
p
n
z
CI
g
ðcÞ
CI
ln
n
CI
c
s
þ
ð
1
z
CI
y
z
Þ
c
p
:
(80)
g
ðcÞ
CI
is the activity coefficient of the counterions in the aqueous solution of ionic
strength
I
S
(on molarity scale) [cf. (
62
)]. That activity coefficient might be set to
unity or be approximated by the Debye H¨ckel theory.
