Chemistry Reference
In-Depth Information
γ
i
G
E
m
μ
ij
ρ
κ
T
V
m
V
i
G
ij
FIGURE 1.2
The Kirkwood-Buff inversion approach for obtaining KBIs from the avail-
able experimental data. See Prolegomenon for symbol definitions. (Ploetz, E. A., and Smith,
P. E., 2011, Local Fluctuations in Solution Mixtures,
Journal of Chemical Physics
, 4, 135.
With permission of the American Institute of Physics.)
The above equations can then be expressed in matrix form,
*
*
*
*
1
+
N
N
1
+
µ
µ
1
−
ρ
V
−
ρ
V
11
n
1
11
n
1
11
1
n
=
N
1
+
N
*
*
*
*
µ
1
+
µ
−ρ
V
1
−
ρ
V
1
n
nn
1
n
nn
n
1
n
n
(1.61)
which provides a simple matrix expression for the elements of the transposed
NN
matrix (and thereby the KBIs) in terms of a matrix containing just volumes and
a matrix containing just chemical potentials, both of which can be expressed in
terms of experimental data.
Alternatively, one can simply use the matrix relation between the
A
and
B
matrices,
and a thermodynamic relation between the isochoric and isobaric chemical potential
derivatives to provide the elements of the
A
matrix. The relevant expressions are
ij
A
A
B
=
ij
(1.62)
∂
∂
βµ
VV
kT
i
ij
AV
N
=
+
ij
κ
j
B
T
Tp N
,,{}
′
Explicit expressions for up to four components have also been provided (Kang
and Smith 2008). The overall inversion approach is summarized schematically in
Figure 1.2.
1.2.5 s
ummary
oF
F
undamenTal
r
elaTions
At this point it is worth pausing and reminding ourselves what we have accomplished
here. The above expressions correspond to thermodynamic properties of solutions
that are amenable to experiment. The expressions involve fluctuating quantities
for an equivalent open system in which the average particle numbers are equal to
the fixed number of particles used in the experiment. The same argument holds
for the chemical potentials, except in the opposite direction, and the volume and
the pressure, that is, we are invoking the equivalence of ensembles approach. In
