Chemistry Reference
In-Depth Information
In this chapter, the KB theory of solutions is applied to the study of solubility in
various systems, the emphasis being on multicomponent (higher than binary) sol-
vents. For these systems, only fluctuation theory can provide useful results, and no
traditional thermodynamic approach is available. In addition, the fluctuation theory
is applied to issues involving salting-in or salting-out effects.
10.2 DERIVATIVES OF ACTIVITY COEFFICIENTS VIA
THE KIRKWOOD-BUFF THEORY OF SOLUTIONS
On the basis of fluctuation theory, the following expression for the derivative of the
activity coefficient of a solute (γ
2,
t
) in a solvent (1)-solute (2)-cosolvent (3) mixture
could be derived (Ruckenstein and Shulgin 2001b), and which is valid for any types
of solutes and cosolvents,
∂
ln
γ
2
,
t
=
t
∂
x
3
t
Tpx
,,
2
(10.1)
(
)
[
]
+− −++
[
]
(
)
cc ccG
++
++−−
GGGc
GGGG
1
2
3111
23
12
13
3
12
33
13
23
−
cc
+
2
+++
ccc
∆
+
c c
∆
+
c c
∆
+
c cc
∆
1
3
1
2 2
1
3 3
2
3 3
1
23 123
where
x
i
t
is the mole fraction of component
i
in the ternary mixture,
c
k
is the bulk
molecular concentration of component
k
in the ternary mixture of 1-2-3, and
G
αβ
is
the usual Kirkwood-Buff integral (KBI) given by,
∞
∫
2
G
=
4
π
[
g
( )]
r
−
1
r dr
(10.2)
αβ
αβ
0
where
g
αβ
is the radial distribution function between species α and β,
r
is the distance
between the centers of molecules α and β, and Δ
αβ
and Δ
123
are defined as follows,
∆
αβ
=+−
GG G
2,
αβ
≠
(10.3)
αα
ββ
αβ
and
∆
123
=
GG
+
GG
+
GG
+
2
GG
+
2
GG
+
2
GGG
−
11
22
11
33
22
33
12
13
12
23
13
23
(10.4)
2
2
2
−−−−
GGGGGG
2
−
2
GGG
−
12
13
23
11
23
22
13
33
12
The factors in the square brackets in the numerator of Equation 10.1 and Δ
123
can
be expressed in terms of Δ
αβ
as follows,
+−−=
+−
∆∆∆
13
23
12
GGGG
12
(10.5)
33
13
23
2
