Chemistry Reference
In-Depth Information
5.2.1 Contribution from the Polymer
Manning assumed that there are only contributions from electrostatic interactions.
He approximated these contributions from the cylindrical version of the Debye-
H¨ckel theory:
ref
G
E;
el
p
G
p
¼
n
p
m
p
þ
(57)
with
2
G
E;
el
p
=ð
RT
Þ¼
n
p
n
1
ð
z
CI
y
z
Þ
l
ln 1
½
exp
ðkb
Þ
:
(58)
n
p
and
n
are the number of moles of the polyion and the number of dissociable
repeating units in that polyion, respectively (i.e.,
n
p
n
is the total mole number of
dissociable electrolytic groups in the backbone).
z
CI
is the (absolute) valency of the
counterions of the polyelectrolyte.
z
CI
y
z
is the ratio of charges carried by those
counterions that are “condensed” to the backbone to the maximum number of
charges on that backbone. That ratio is also called the “neutralization fraction of
the polyion”. Thus 1
is the ratio of the actual number of charges
q
to the
maximum number
q
max
of charges on the backbone of the polyelectrolyte.
q
q
max
¼
ð
z
CI
y
z
Þ
ð
1
z
CI
y
z
Þ:
(59)
Consequently, the number of moles of dissociated repeating units,
n
p;
diss
is:
n
p;
diss
¼
n
p
n
1
ð
z
CI
y
z
Þ:
(60)
l
is the charge density parameter [cf. (30)]. When the charge density is small (i.e.,
the distance
b
between two dissociable groups is large so that
l <
1) the poly-
electrolyte is completely dissociated. Thus, the first part on the right-hand side of
(
58
) [i.e.,
n
p
n
1
2
l
] is the number of moles of dissociated polymer groups
times the charge density parameter. Parameter
k
is the inverse of the radius of the
ionic cloud in the aqueous solution, as introduced in the Debye-H¨ckel theory:
ð
z
CI
y
z
Þ
e
2
ee
0
kT
I
S
¼
2
k
¼
2
N
A
8
pN
A
I
S
l
B
;
(61)
where
I
S
is the ionic strength of the aqueous solution on the molarity scale.
When a single polyelectrolyte and a single low molecular weight salt
M
n
M
X
n
X
are
dissolved in water, that ionic strength is:
c
s
þ
;
1
2
n
M
z
2
M
þ n
X
z
X
I
S
¼
z
CI
z
p
1
ð
z
CI
y
z
Þnc
p
(62)
