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152
3.4.4. A
2
: The
ideal Donnan
term
The redistribution of salt contributes to A
2
(in addition to the excluded volume
term), and may often dominate completely. This can be estimated with some
simplifications.
Assuming we work in dilute solutions so that A
2
does not contribute (1/M >>
A
2
c) For each species we have:
Π =
RT c
M
=
RTC
(Note C is here the molar concentration)
For the macromolecule, there is only a contribution to
Π
on the
α
side. For the
salt, there is a difference in molar concentration between the
α
-side and the
β
-
side. Therefore, their contribution to the osmotic pressure on the
α
-side is:
Π
B
=
RT C
B
(
α
)
−
C
B
(
β
)
[
]
Π
X
=
RT C
X
(
α
)
−
C
X
(
β
)
[
]
The total osmotic pressure on the
α
-side is the sum of each term:
Π = Π
I
I
∑
= Π
P
+
Π
B
+
Π
X
=
RT C
P
+
C
B
(
α
)
−
C
B
(
β
)
[
]
+
C
X
(
α
)
−
C
X
(
β
)
[
]
{
}
(10.128)
Assuming a large excess of salt (C
BX
) is added, and that the difference in salt
concentration is small on the
α
and
β
sides, the equation simplifies (after a
few intermediate calculations, see textbook) to:
Π =
RT C
p
+
z
2
C
p
2
4
C
BX
⎡
⎣
⎢
⎤
⎦
⎥ =
RT c
2
M
2
+
z
2
c
2
2
4
M
2
2
C
BX
⎡
⎣
⎢
⎤
⎦
⎥
or
Π
c
2
=
RT
1
M
2
+
z
2
4
M
2
2
C
BX
c
2
⎡
⎣
⎢
⎤
⎦
⎥
Hence,