Chemistry Reference
In-Depth Information
a
ð
0
Þ
i
L
and
a
ð
1
Þ
L
denote binary interaction parameters between species (groups)
i
and
L
. These interaction parameters are symmetric, i.e.,
a
ð
0
Þ
i
;
i
;
a
ð
0
Þ
L
i
and
a
ð
1
Þ
¼
¼
;
L
;
i
;
L
a
ð
1
Þ
L
i
. They form a set of adjustable model parameters. The degree of dissociation
a
is
calculated assuming chemical equilibrium between monomers
D
and its dissocia-
tion products
F
and CI in state 5:
;
vdW
CI
vdW
F
m
CI
m
F
m
D
m
G
CI
G
F
na
2
m
p
m
G
G
el
el
K
¼
G
D
¼
G
CI
G
F
;
(170)
1
k
a
vdW
D
G
where all molalities are those in state 5. Chemical reaction equilibrium constant
K
is
one of the adjustable parameters of the model.
When there is also an additional single 1:1 salt
MX
in the aqueous solution, (
162
)
(for the ionic strength) and (
163
) (for the van der Waals contribution to the group
activity coefficient) have to be extended; the sums must also include the ions
M
and
X
. The extension requires the relative surface ratios for
M
and
X
:
Y
M
Y
w
¼
m
MX
q
M
m
q
w
;
M
w
(171)
Y
X
Y
w
¼
m
MX
q
X
m
q
w
:
M
w
(172)
Furthermore, as well as the chemical potential of the polyelectrolyte, the chemi-
cal potential of
MX
is also required (for the calculation of the activity of water, see
below). That chemical potential is given by the sum of the chemical potentials of
cations
M
and anions
X
:
m
M
G
M
m
m
X
G
X
m
ref
M
ref
X
m
MX
¼ m
M
þ m
X
¼ m
þ m
þ
RT
ln
:
(173)
Finally, the activity of water
a
W
is calculated from the chemical potentials of the
solutes (either a single polyelectrolyte or a binary solute mixture of a polyelectro-
lyte and a low molecular weight salt) by applying the Gibbs-Duhem equation:
M
w
X
i
6¼w
m
i
m
o
ðm
w
m
w
pure liquid
dm
w
¼
d
Þ¼
dm
i
:
(174)
Integration at constant temperature for an aqueous solution containing a poly-
electrolyte
P
and a salt
MX
results in:
M
w
ð
mix
water
M
w
ð
mix
water
m
p
m
o
dm
p
m
MX
m
o
Dm
w
¼
RT
ln
a
w
¼
dm
MX
:
(175)
