Chemistry Reference
In-Depth Information
with
A
=+
1
ρ (
G
+−−
GGG
)
(1.76)
i
i
ii
jk
ij
ik
where
i
,
j
, and
k
can be 1, 2, or 3 and therefore,
ηρ
=+
AA
ρ
(1.77)
ij
i
j
ji
which is a generalization of Equation 1.66. The following expressions are then
obtained for ternary systems (Smith 2006a):
∂
∂
βµ
ρ
η
A
∂
∂
βµ
ρ
η
A
∂
∂
βµ
η
η
1
32
2
31
3
12
=−
=−
=
ln
m
ln
m
ln
m
3
123
3
123
3
123
Tpm
,,
Tpm
,,
T pm
,,
2
2
2
(
)
−−
(
)
1
+−
NN
1
+−
NN
(
NN
)(
NN
−
)
22
12
33
13
32
12
23
13
V
=
1
η
123
ζ
η
3
123
k
BT
κ
=
(1.78)
Other derivatives and partial molar volumes can be obtained by a simple index change.
When the solute appears at infinite dilution the following limiting expressions apply:
∞
∞
m
∞
∞
∂
∂
βµ
∂
∂
ln
ln
γ
ρρ
(
GG
−
)
∂
βµ
2
2
3
13
23
21
2
=
=−
−
φ
=
1
3
ln
m
m
η
∂
ln
m
3
13
2
Tpm
,,
T ppm
,
,
Tpm
,,
2
2
3
∞
∂
ln
γ
m
ρ
AA
(
)
−
2
113
∞
∞
∞
=−
ρ
GGGG
+−−
(1.79)
1 2
31
21
23
∂
m
η
2
13
Tpm
3
,,
(
)
+−
(
)
−−
∞
∞
∞
1
+
NNN
−
1
NN
(
NNNN
)(
−
)
11
21
33
13
31
21
13
23
∞
∞
∞
V
=
=
kT
κ
−
NV
−
NV
2
BT
21
1
233
η
13
and all other expressions reduce to those of the corresponding binary mixture of
1 and 3.
We do not provide expressions for the KBIs in a general form. This is a result of
the many equivalent forms for the expressions that can be obtained when the chemi-
cal potential derivatives are interchanged using the expressions provided by the GD
relationship. However, a useful set of expressions is (Smith 2008)
(
)
(
)
µ
′
VV
−
µ
′
φµ
+
′
V
−
µ
32
′
V
φ
33
2
233
2
223
23
1
+=
Nk T
ρκρ
+
11
1
BT
1
µµ
′
′
−
µµ
′
′
22
33
23
32
(1.80)
(
)
(
)
µ
′
VV
−
µφµ
′
+
′
VV
−
µ
′
φ
1
333
33
12
32
1
123
3
Nk T
BT
=
ρκρ
+
12
2
2
µµ
′
′ −
µµµ
12
′
′
32
13
33
where other KBIs are provided by a suitable index change.
