Chemistry Reference
In-Depth Information
1.3.2 c
losed
B
inary
s
ysTems
Binary systems have been the most common system of interest when applying FST.
Most investigations of closed binary systems have involved the determination and
examination of the three KBIs, possibly followed by some description of local compo-
sition or preferential solvation (see Section 1.3.4), all of which is composition depen-
dent. The equations provided in the previous section are more commonly expressed
and simplified by the definition of two additional variables (Ben-Naim 1977),
(
)
++−
(
)
η
== +−
M
ρ
1
NN
ρ
1
NN
12
1
22
12
2
11
21
(1.66)
(
)
(
)
NNN
ζ
==+
NN
1
N
1
1
+
2
1
22
12 1
If we consider a binary mixture of a solvent (1) and a solute (2), then FST provides
the following expressions,
∂
∂
βµ
ρ
η
∂
∂
βµ
ρ
η
∂
∂
βµ
ρ
1
2
1
2
2
=
=
=
ln
m
ln
x
ln
1
+−
NN
2
12
2
12
2
22
12
Tp
,
Tp
,
Tp
1
+−
N
N
1
+−
NN
ζ
η
22
12
11
21
2
V
=
V
=
k
BT
κ
=
(1.67)
1
2
η
η
12
12
12
All the quantities on the left-hand side of the expressions are second derivatives of
the Gibbs free energy, while all the quantities on the right-hand side involve second
derivatives of
pV
—the characteristic thermodynamic potential of the grand canoni-
cal ensemble. Application of the stability conditions for stable (miscible) solutions
indicates that we must have η
12
> 0 and ζ
2
> 0 (Prigogine and Dufay 1954).
The above expressions can be manipulated further to provide relationships for the
various activity coefficients,
(
)
+
(
)
∂
∂
ln
ln
γ
ρ
NN
−
ρ
NN
−
2
2
1
22
12
2
11
21
=−
x
η
12
Tp
,
(
)
++
(
)
∂
∂
ln
ln
γ
m
ρ
NN
−
ρ
2
1
NNN
−
2
1
22
12
11
21
=−
(1.68)
m
η
2
12
Tp
,
c
∂
∂
ln
ln
γ
ρ
NN
−
2
22
12
=−
1
++
NN
22
2
12
Tp
,
which are often of more practical interest. It is well known that the activity coef-
ficient provides an indication of deviations from ideal behavior. Indeed, for dilute
solute standard states on the molarity scale, a positive deviation from unity suggests
a dominance of favorable solute-solvent interactions (
G
12
>
G
22
), while negative
