Chemistry Reference
In-Depth Information
where
NN
is the matrix of δ
ij
+
N
ij
elements. At this point, we have a matrix formu-
lation for the chemical potential derivatives, partial molar volumes, and isothermal
compressibility. This is the theory of Kirkwood and Buff.
A general formulation of KB/FST theory in terms of direct correlation functions
is also possible. Because the DCFIs are related to TCFIs through a matrix inversion,
their general multicomponent formulas do not involve matrix cofactors (O'Connell
1971b). The equivalents of Equations 1.54 and 1.55, which apply over the entire com-
position range are then
∑
∑
(
)
1
ρκ
kT
=
xx
1
−
C
(1.56)
BT
ij
ij
i
=
1
j
=
1
∑
(
)
VkT
κ=
x
1
−
C
(1.57)
i
B T
j
ij
j
=
1
N
kT
∂
∂
βµ
i
i
=−
δ
C
(1.58)
ij
ij
ρκ
N
BT
j
TV N
,,
kj
≠
and
∑∑
1
(
)
(
)
x
1
−
C
x
1
−
C
k
ik
k
jk
N
kT
∂
∂
βµ
i
i
k
=
k
=
1
=−−
δ
C
(1.59)
ij
ij
ρκ
N
∑
∑
BT
j
(
)
TpN
,,
xx
1 −
C
mn
kj
≠
mn
m
=
1
n
=
1
for any number of components.
1.2.4 i
inversion
oF
F
lucTuaTion
T
heory
The KB/FST inversion procedure is the process of obtaining expressions for the par-
ticle number fluctuations or KBIs in terms of experimentally available (isothermal-
isobaric) data. Again, there are multiple approaches to the inversion procedure
(Ben-Naim 1977; O'Connell 1994; Smith 2008). Arguably, the simplest approach
involves the pseudo chemical potential and partial molar volumes (Ben-Naim 2006).
First, we note that combining Equations 1.46 and 1.47 provides
∑
*
*
δρ
−=
V
(
δ
+
N
)(
δ
+
µ
)
ik
ik
ij
ji
kj
kj
j
*
VVkT
=−
κ
(1.60)
k
k
BT
∂
∂
βµ
k
*
=+−
µδφ
kj
kj
j
ln
N
j
pT N
,,{}
′
