Chemistry Reference
In-Depth Information
and
⊗
denotes either the pure component, or infinite dilution condition. Obviously,
the 15 expansion coefficients are not all independent, since they are connected by
two thermodynamic constraints, that is, they must satisfy simultaneously the Gibbs-
Duhem (GD) equation and the exactness (Maxwell relations, MR) of the mixture's
residual Gibbs free-energy function
G
r
(
p
,
T
,
N
1
,
N
2
,
N
3
) (O'Connell and Haile 2005).
By following the procedure described in detail elsewhere (Chialvo et al. 2008),
we arrive at the desired expressions,
o
2
2
ln
φ
(,) n
xx
=
φ
+
05
.
kx
+
05
.
kx
+
kx
x
123
1
22
2
33
3
23
2
3
∞
2
2
ln
φ
(,) n
xx
=
φ
−
kx
−
kx
+
05
.
kx
+
005
33
.
kx
+
kxx
(8.38)
223
2
22
2
233
22
2323
ln
φ
(,) n
xx
=
φ
∞
−
kx
−
k
x
+
05
.
k
x
2
+
05
.
k
xkxx
2
+
323
3
23
2
33
3
222
33
3
2323
where the coefficients
k
ij
(
p
,
T
) can be interpreted in terms of the infinite dilution
limiting slopes, that is,
(
)
∞
kpT
(, )
≡− ∂
ln
φ
∂
x
(8.39)
ij
i
j
pT x
k
,,
whose microscopic interpretation will be given below in Section 8.3.2. A close
inspection of Equation 8.38 allows us to recast the solute properties in a more com-
pact form, that is,
∞
o
ln
φ
(,) n
xx
=
φ
−
kx
−
kx
+
ln(
φ
xx
,)
φ
223
2
22
2
233
123
1
(8.40)
∞
φ
o
ln
φ
(,) n
xx
=
φ
−
kx
−
kx
+
ln(
φ
x
,,
x
3
)
323
3
23
2
333
12
where we have separated the linear from the quadratic composition dependence,
a feature that will become handy in our discussion (Section 8.3.4) of the valid-
ity of current irst-order truncated expansions. According to Equation 8.40, the
composition dependence of the residual Gibbs free energy of a dilute ternary
system becomes
r
r
gpTx xGpT NN NNNN
(, ,,)
=
(
,, ,
,
)(
+
+
)
23
123
1
2
3
(
)
+
(
)
o
∞
o
∞
o
=
kT
ln
φ
+
x
ln
φ
φ
x
φ
φ
−
(8.41)
B
1
2
2
1
33 1
2
2
05
.
kx
−
05
.
kx
−
kxx
22
2
33
3
2323
Because Equation 8.38 through Equation 8.40 are thermodynamically consis-
tent, the corresponding pressure and temperature derivatives, that is, the partial
molecular volumetric, enthalpic, and entropic counterparts will automatically sat-
isfy the thermodynamic constraints discussed above. For example, by recalling that
