Chemistry Reference
In-Depth Information
1
1
++ +
NN NN
−
NN
11
22
11
22
12
21
kT
κ
=
NN
(1.52)
BT
(
)
+
(
)
ρ
+−
NN
ρ
2
1 +−
1
22
12
11
21
which is the FST expression for the isothermal compressibility of a binary solution.
This completes the application of FST to binary solutions. Equation 1.49 through
Equation 1.50, and Equations 1.51 and 1.52 involve three thermodynamic properties
of a binary mixture expressed in terms of composition and three KBIs (
G
11
,
G
22
,
G
12
=
G
21
).
1.2.3 g
eneral
m
aTrix
F
ormulaTion
oF
F
lucTuaTion
T
heory
The number of simultaneous equations provided by Equation 1.43 increases as the
number of components increases. It then becomes more efficient to express the results
in terms of matrices. Again, there are many equivalent matrix formulations that
have been presented (Kirkwood and Buff 1951; O'Connell 1971b; Ben-Naim 2006;
Nichols, Moore, and Wheeler 2009). Here, we present one of the simplest. A general
formulation is easiest starting from the first expression in Equation 1.44. Writing
the number fluctuations in matrix form for an
n
c
component system where we also
include the number densities (GD expression at constant
T
) in the first row provides,
ρ
ρ
ρ
1
2
n
c
N
−+
(
1
N
)
1
+−
NN
NN
−
21
11
22
12
2
n
1
n
c
c
M
=
(1.53)
N
−+
(
1
N
)
NN
−
1
+
NN
−
n
1
11
n
2
12
nn
1
n
c
c
cc
c
By taking derivatives of the GD expression with respect to pressure with all
molalities (and
T
) constant, together with a series of derivatives of Equation 1.44 with
respect to each species molality at constant
p
(and
T
), the results can be expressed
in the general form,
V
V
µ
′
µ
′
1
12
1
n
µ
′
µ
′
i
1
ij
M
M
∂
∂
βµ
ln
M
M
2
22
2
n
−
1
i
=
M
V
=
µ
′
=
=
i
ij
>
1
m
j
Tp
,
,, {
m
′
V
µ
′
µ
′
n
n
2
nn
(1.54)
that is, the expressions or values for the partial molar volumes and activity derivatives
are just the elements of the inverse of the
M
matrix, which can also be expressed in
terms of cofactors of the
M
matrix. A simple expression for the compressibility is then
k
B
κ=
NN
M
(1.55)
