Chemistry Reference
In-Depth Information
6
4
2
0
25
0
-25
-50
0
0.5
1
1.5
2
2.5
r
or
R
FIGURE 1.1
A typical center of mass-based pair radial distribution function as a func-
tion of molecule separation,
r
(in nm) (top), and the corresponding Kirkwood-Buff integral
(in cm
3
/mol) as a function of integration distance,
R
(in nm) (bottom).
one encounters in the study of imperfect gases and the McMillan-Mayer theory of
solutions. However, the FST approach does not suffer from convergence limitations
in typical series expansions, and the KBIs are relevant at any solution composition—
not just infinitely dilute solutes in the gas phase or a primary solvent.
Multiplying the KBIs by number densities (concentrations) one obtains an alter-
native picture of the integrals. The quantities,
N
ij
= ρ
j
G
ij
, have been referred to as
excess coordination numbers
and quantify the change in the number of
j
molecules
observed in an open volume of solution on introduction of a central
i
molecule from
that observed for the same volume of bulk solution in the absence of the
i
molecule
(Newman 1994).
Multiple combinations of KBIs and number densities have appeared in the litera-
ture. Some of the more common variations include,
δδ
NN
V
(
)
=
i
j
BB
==
ρδ
+
ρ
G
ij
ji
i
ij
j
ij
(1.38)
δδ
N
i
NN
i
j
δ
+=
N
ij
ij
The most convenient form often depends upon the application and/or the derivation
used. In many applications, total correlation function integrals (TCFIs) are often
used in place of the KBIs (O'Connell 1971b). These are typically defined and denoted
as
HGxN
ij
==ρ
1
, but care should always be taken to ensure other closely related
definitions are not being used.
Finally, we note that particle number fluctuations increase, and many properties
diverge, as one approaches a critical point. O'Connell and coworkers have shown
that one can still apply FST under these circumstances by using integrals over direct
ij
j
ij
