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138
Please note that semidilute and concentrated solutions are technically and
biologically important (taught in detail in higher courses like ‘Biomaterials’)
3.2.4. The general
thermodynamic equation
for dilute
solutions:
The starting point for further thermodynamic calculations is the following
equation:
µ
1
−
µ
1
0
=
−
RTV
1
0
c
2
1
M
+
A
2
c
2
+
A
3
c
2
3
+
⋅ ⋅ ⋅
⎛
⎝⎜
⎞
⎠⎟
(Textbook Eq. 10.60)
µ
i
=
∂
G
∂
n
i
⎛
⎝⎜
⎞
⎠⎟
T
,
P
,
n j
≠
i
Symbols:
µ
1
= Chemical potential of solvent (water/buffer in our case) in the presence of
biopolymer (solute)
µ
1
0
= Chemical potential of pure solvent
V
1
0
= Molar volume of pure solvent (0.018 l/mol for water at RT)
A
2
= Second virial coefficient
c
2
= concentration of biopolymer (solute) (g/ml)
Arriving at this equation involves a couple of assumptions and calculation
steps. It is convenient to start with a thermodynamically
ideal
solution. An
ideal solution is formed from its components without any change in enthalpy
(∆H
mix
= 0), and the change in entropy (∆S
mix
) equals the statistical mixing
term, which will be calculated below. The ideal solution serves as a basis of
comparison when dealing with real (non-ideal) solutions.
Calculation of ∆S
mix
for an ideal solution starts with Boltzmann’s famous
formula, which assumes all components are of the same size (certainly not
true for macromolecules):