Chemistry Reference
In-Depth Information
This can, however, cause problems if the solvent concentration disappears. A similar
differential for changes in the mole fractions can also be developed by noting that
d
ln
x
i
=
d
lnρ
i
-
k
x
k
d
ln ρ
k
and therefore,
∑∑
dx
ln
=
δ
+
N
−
x
(
δ
+
Nd
)
βµ
(1.45)
i
ij
ij
k j
kj
j
j
k
The three previous differentials are valid under isothermal conditions. Taking
the derivative of Equation 1.43 with respect to
p
with composition and
T
constant
one finds,
∑
(
kT
κ
=
δ
+
NV
)
(1.46)
BT
ij
ij
j
j
for any
i
species. Alternatively, taking derivatives with respect to composition with
p
and
T
constant provides,
∂
∂
βµ
∑
j
δρ
−=
V
(
δ
+
N
)
(1.47)
ik
ik
ij
ij
ln
N
k
Tp N
,,{}
′
j
The last two expressions involve a mixture of fluctuating quantities and thermody-
namic properties. The thermodynamic properties on the right-hand side will ulti-
mately be expressed in terms of fluctuating quantities. However, the above mixed
expressions are often extremely useful in themselves.
If desired, the relationships in Equation 1.43 through Equation 1.45 can be further
manipulated by eliminating one of the chemical potentials (usually the primary sol-
vent) using the GD expression at constant
T
and
p
. The results for components other
than the solvent are then given by
∑
d
ln
ρ
=
(
δ
+
N
−
mN d
)
βµ
i
ij
ij
j
i
1
j
j
>
1
(1.48)
∑
Nd
ij
dm
ln
=
(
δ
+
) βµ
i
ij
j
j
>
1
+
where
NNmNNN
ij
1
11 1 1
has been used as a shorthand (Hall 1971),
and represents essentially the matrix elements used in the original paper (Kirkwood
and Buff 1951). The expressions are now only valid under constant
T
and
p
condi-
tions. The mole fraction differential is significantly more complicated and is rarely
used (Smith and Mazo 2008).
=+ +−−
(
)
ij
j
i
j
