Chemistry Reference
In-Depth Information
∂
N
∂
N
V
∑
∂
∂
N
i
i
i
dN
=
d
β
+
dV
+
d
βµ
(1.41)
i
j
∂
β
∂
βµ
j
ββµ
,{
}
V
,{
βµ
}
β µ
,,{}
V
′
j
valid for each
i
species and any number of solution components. All the partial deriva-
tives can be expressed in terms of ensemble averages in the grand canonical ensemble.
Using the expressions provided in Equations 1.32, the above differential simplifies to
N
V
+
∑
i
dN
=−
δδ
NEd
β
+
dV
δ
NN d
δ
µ
(1.42)
i
i
i
j
j
j
Derivatives of the above differential can be used to develop expressions valid
for a variety of ensembles. Some applications where changes in
T
are important are
provided in the literature (Buff and Brout 1955; Debenedetti 1988; Jiao and Smith
2011; Ploetz and Smith 2011b). However, the vast majority of applications of FST
are restricted to isothermal systems. Hence, the first term on the right-hand side of
Equation 1.42 disappears and the above expression can be rewritten, after dividing
throughout by ⟨
N
i
⟩ and rearranging as
∑
d
ln
ρ
=
(
δ
+
N
)
d
βµ
(1.43)
i
ij
ij
j
j
This set of simultaneous equations, coupled with the Gibbs-Duhem equation at con-
stant
T
, are all we need to derive the most common forms of the basic theory pro-
viding a range of relationships between fluctuating quantities and thermodynamic
properties for multicomponent systems.
In the original Kirkwood and Buff paper (Kirkwood and Buff 1951), the start-
ing point for their derivation is the
A
matrix (see also the Prolegomenon) which
has elements of the form
V
βµ /
. When used in combination with the
B
matrix, with elements
B
ij
given by Equation 1.38, one finds the matrix relationship
I
=
A B
, which is obtained directly from Equation 1.43 by taking derivatives with
respect to particle numbers with volume and
T
constant. Others have adopted similar
approaches (Ben-Naim 2006). This is usually followed by a series of thermodynamic
transformations to convert the isochoric chemical potential derivatives to provide
the more common and useful isobaric expressions. Here, we wish to eliminate the
majority of these transformations. In fact, the results for a small number of compo-
nents can be obtained directly from Equation 1.43, as we shall see in the next section.
Before leaving this section, we note that Equation 1.43 also provides several addi-
tional relationships that are of general use and/or might be more convenient for spe-
cific applications (Smith and Mazo 2008). An expression for changes in the species
molality (
m
i
= ρ
i
/ρ
1
) can be obtained by subtracting the differential for the primary
solvent (
i
= 1) to give
(
∂∂
N
)
,,{}
i
j
TV N
′
∑
δ
dm
ln
=
(
+
N
−−
δ
N
)
d
βµ
(1.44)
i
ij
ij
1
j
1
j
j
j
