Chemistry Reference
In-Depth Information
ln
∂
∂
Z
Y
Y
1
=
N
≈
f
+
...
2
2
βµ
2
0
pN
,,
β
1
(12.8)
∂
∂
ln
Z
p
=−
VV
≈+
...
0
β
β,,
fN
21
where the approximation is good for dilute solutions. Considering the volume in
terms of the relative activity
a
2
= f
2
V
1
/V
0
provides,
VaV
a
++
++
...
...
0
2
1
V
=
(12.9)
1
2
Here, we find
V
1
as the partial molar volume. We also find the osmotic coefficient in
this scheme as
1
Z
Y
ϕ=
ln
(12.10)
N
2
0
where
Y
0
represents the pure solvent system where as
Y
1
is a system with one solute
and in
Y
2
the two solutes may interact. In general, if
Y
0
is for pure water we may either
calculate the ratios
Y
N
/ Y
0
(see Section 12.3, “Simulation Results” below) or use them
as parameters to be fit to experiment and thus solve for
f
2
.
Thus, to second order,
Y
Y
Y
Y
1
0
2
0
2
N
=
f
+
2
f
+ ...
(12.11)
2
2
2
and all that is left is to solve a quadratic for
f
2
. In this picture, we may think of
Y
N
/
Y
0
as a measure of how readily the solvent will accept
N
2
solutes. Notice -
k
B
T
ln (
Y
1
/
Y
0
)
is just the Widom insertion chemical potential (Widom 1963). This formula yields
essentially quantitative accuracy when fit to various nonionic cosolvents in water
(see Figure 12.3).
Now let us consider the chemical potential up to terms of only first order. Then
we have
c
Vc
o
3
µµ
3
=+
kT
ln
(12.12)
3
1
−
13
where
μ
3
o
is the standard state chemical potential. Using the derivative of the chemi-
cal potential with respect to the concentration of the osmolyte we get
∂
∂
1
V
Vc
3
3
1
13
β
=+
−
(12.13)
c
c
1
3
Tp
,
