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150
3.4.2. Calculating
the osmotic pressure and A
2
We wish to calculate the osmotic pressure and A
2
, as well as the distribution
of ions in this situation. The components to consider are:
•
Polymer (P
z+
) (using chitosan as example)
•
Dissociated counterions (z X
-
)
•
Added salt (co- and counterions, B
+
og X
-
)
The criteria for equilibrium across the membrane are:
1.
Same chemical potential on both sides for diffusible species (water and
salt, not polymer)
2.
Electroneutrality on both sides
Criterium 1 leads to:
Water:
μ
1
(α) = μ
1
(β) = μ
1
0
(10.108)
Salt:
μ
BX
(α) = μ
BX
(β)
(10.109)
The chemical potential of a salt may be approximated by:
μ
BX
= μ
BX
0
+ RT ln C
B
C
X
(molar concentrations)
[strictly: μ
BX
0
+ RT ln (γ
±
2
C
B
C
X
)]
(10.110/111)
We assume that γ
±
2
is the same on both sides (and cancels). We further
assume μ
BX
0
is the same (cancels too). This leads to the equation:
C
B
C
X
(α) = C
B
C
X
(β)
(10.114)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ - + - + - +-
+ - + - + - + -
- + - ++ - + - +
+ - + -
+ - + - + -
+ - + -
+ - + - + -
+ - + - + - + + - + - + - + -