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140
Δ
H
mix
=
0
Δ
S
mix
=
−
k
N
i
ln
X
i
=
−
R
∑
n
i
ln
X
i
∑
⇒
Δ
G
mix
=
−
T
Δ
S
mix
=
RT
n
i
ln
X
i
∑
<
0
⇒
Mixing is energetically favoured in an ideal solution
Interesting consequence:
A low molecular weight compound is generally more soluble than a high
molecular weight compound because n
i
and X
i
are higher for smaller
molecules when we consider a fixed amount (e.g. 1 gram)
3.2.5. Chemical potential of a
simple
two-‐component
system
Let us consider a simple two-component system (water and biopolymer)
X
1
+
X
2
=
1
(Sum of mole fractions)
Since we have a dilute solution (X
2
<<1) , we can simplify and obtain:
X
2
1
⇒
ln(1
−
X
2
)
=
−
X
2
−
1
2
X
2
2
−
1
3
X
2
3
−
...... (
Eq
.10.46)
The mole fraction (X) is a useful parameter in simplifying the equations, but
normally we use the concentration given as g/ml (or similar). However, X
2
and
c
2
are related through V
1
0
and M
2
:
0
2 1
2
2
( .10.54)
c V
X
Eq
M
=
Combining these equations yields, for an ideal solution, the following
equation: