Chemistry Reference
In-Depth Information
+−−=
+−
∆∆∆
12
13
23
GGGG
11
(10.6)
23
12
13
2
and
2
2
2
(
∆
)
+
(
∆
)
+
(
∆
)
−
2
∆
∆∆∆
−
2
−
2
∆∆
12
13
23
12
13
12
23
13
23
∆
=−
(10.7)
123
4
The insertion of Equation 10.5 through Equation 10.7 into Equation 10.1 provides
expressions for the derivatives
∂
ln
γ
2
,t
t
∂
x
3
t
Tpx
,,
2
in terms of Δ
αβ
and concentrations.
It should be noted that Δ
ij
is a measure of nonideality (Ben-Naim 1977) of the
binary mixture α-β, because for an ideal mixture Δ
αβ
= 0. For a ternary mixture
1-2-3, Δ
123
also provides a measure of nonideality. Indeed, inserting
G
αβ
id
for an ideal
mixture (Ruckenstein and Shulgin 2001a) into the expression of Δ
123
, one obtains
that for an ideal ternary mixture Δ
123
= 0.
At infinite dilution of component 2, Equation 10.1 becomes,
(
)
(
)
(
)
(
(
)
)
0
0
0
0
0
0
cc
+
ccc
+
∆∆
−
+−
cc
∆
∂
ln
γ
1
3
1
3
12
23
t
1
3
13
t
x
=
0
x
=
0
2
,
t
lim
=−
2
2
(10.8)
t
∂
x
t
++
(
)
0
0
0
0
x
→
0
2
cccc
∆
3
2
t
Tpx
,,
1
3
1
3
13
t
x
=
0
2
2
where
c
1
0
and
c
0
represent the bulk molecular concentrations of components 1 and 3
in the solute-free binary solvent 1-3. In addition to Equation 10.8, one can write
for the derivative of the activity coefficient in a binary mixture with respect to the
mole fractions the expression,
0
0
∂
ln
γ
c
cx
∆
1
3
13
γ
≡
=−
(10.9)
11
0
0
0
∂
x
1
+
∆
1
1
3
13
pT
,
where
x
0
and γ
1
0
are the mole fraction of component 3 and the activity coefficient of
component 1 in the solute-free binary solvent of 1-3.
By combining Equations 10.8 and 10.9, one obtains an expression for the deriv-
ative of the activity coefficient of an infinitely dilute solute with respect to the
cosolvent mole fraction in terms of characteristics of the solute-free binary solvent
(γ
11
,
c
1
0
, and
c
0
) and the parameters Δ
12
and Δ
23
, which characterize the interactions
of an infinitely dilute solute with the components of the mixed solvent. Even though
Equation 10.8 constitutes a formal statistical thermodynamics relation in which all
