Chemistry Reference
In-Depth Information
method of Kirkwood and Buff (KB) (1951). For the former, one requires correlations
or potentials of mean force at infinite dilution, and then one typically uses a cluster
theory to obtain the thermodynamics at the desired finite concentration (Friedman
1962; Rasaiah and Friedman 1968; Pettitt and Rossky 1986). With the latter case, the
density-density fluctuation correlations are required at the desired concentration(s).
Both have strengths and difficulties in the multicomponent solution context but KB
is more in keeping with the theme of this volume and we restrict our attention to that
statistical thermodynamic framework.
We start with the change in Helmholtz free energy per particle,
df
=
d
(
F/ N
),
which can be written for a three-component mixture of solvent (1), protein (2), and
other (3) as
(
)
(
)
df
=−
pdv
−+−
sdT
µµ
dx
+−
µµ
dx
3
(12.1)
2
1
2
3
1
Here, the Helmholtz free energy
F
, volume, and entropy divided by the total number
of particles
N
are represented with lower case
f, v,
and
s
, respectively. We shall con-
sider the fluctuations in number of each species and its effect on the thermodynamic
potential. We then partition the system into two subsystems,
a
and
b
, each defined
by a set of variables, [
N
1
a
, N
2
a
, N
3
a
, V
a
, T
] and [
N
1
b
, N
2
b
, N
3
b
, V
b
, T
]. We now apply
a small perturbation away from equilibrium and the state variables correspondingly
readjust to new values [
N
1
a
+
δ
N
1
a
,
N
2
a
+
δ
N
2
a
, N
3
a
+
δ
N
3
a
, V
a
, T
] and [
N
1
b
+
δ
N
1
b
,
N
2
b
+
δ
N
2
b
, N
3
b
+
δ
N
3
b
, V
b
, T
]. One can now write the total change in the free energy,
δ
F
, for this process as
(
)
(
)
++
(
)
(
)
−+
(
)
a
a
a
b
b
b
a
b
δ
FN Nf
=+
δ
+
δ
f
NNf
δ
+
δ
f
NNf
(12.2)
If we substitute the Taylor expansion of
f
a
and
f
b
with respect to
v, x
2
, x
3
,
and apply
the conservation relations,
a
b
a
b
a
b
a
b
δ
NN NN NN VV
+=+=+=+=
δ
δ
δ
δ
δ
δ
δ
0
(12.3)
1
1
2
2
3
3
the irst-order terms cancel and we find the variation in
F
to second order,
NN N
NN
b
+
+
δ
δ
b
∂
∂
2
f
∂
∂
2
f
∂
∂
2
f
(
)
+
(
)
+
(
)
+
2
2
2
b
b
b
δ
F
≅
δ
v
δ
x
δ
x
2
3
a
a
2
2
2
2
v
x
x
2
3
Tx x
,
,
Tvx
,,
T
,
vvx
,
2
23
3
2
2
∂
∂∂
f
vx
∂
∂∂
f
vx
(
)(
)
+
(
)(
)
+
b
b
b
b
2
δδ
vx
2
δδ
vx
2
3
2
3
Tx
,
Tx
,
3
2
)
∂
∂∂
f
xx
2
(
)(
b
b
2
δ
x
δ
x
(12.4)
2
3
23
Tv
,
