Chemistry Reference
In-Depth Information
The chemical potential derivative in Equation 4.13 is defined as
∂
∂
µ
i
µ
′ =
(4.14)
ij
N
j
Tp N
,,{}
′
The equations that are commonly used to represent experimental data of
Z
E
(
Z
E
=
G
E
,
V
E
) and μ
i
are expressed as a function of
x
i
, whereas in Equation 4.14 derivatives
with respect to
N
i
are required. We need therefore to express them in a function of
x
i
. Taking into account the definition of the excess partial molar quantity,
Z
i
E
, as a
function of
N
i
, the relationship between
x
i
and
N
i
, the differentials of
Z
E
=
f
(
x
1
,
x
2
)
with respect to
x
i
and of
x
i
with respect to
N
i
, and applying the treatment to one mole
of mixture, after some substitutions and rearrangements, the diagonal elements μ
ii
can be expressed in a function of
x
i
and of four derivatives of the chemical potential
of components 1 and 2,
∂
∂
µ
∂
∂
µ
(
)
1
1
µ
′
=−
1
x
−
x
(4.15)
11
1
2
x
x
1
2
Tpx
,,
Tpx
,,
1
2
∂
∂
µ
∂
∂
µ
(
)
2
2
µ
′
=−
1
x
−
x
(4.16)
22
2
1
x
x
2
1
Tpx
,,
Tpx
,,
2
1
1
∂
∂
µ
∂
∂
µ
∂
∂
µ
∂
µ
2
1
1
2
2
2
µ
′
=
x
+
xx
+
+
x
(4.17)
33
12
x
x
x
x
∂
x
3
1
2
1
2
Tpx
,,
Tpx
,,
T px
,,
2
1
2
Tpx
,,
1
The remaining three off-diagonal μ
ij
(
μ
is a symmetrical matrix) can be expressed in
terms of the μ
ii
thanks to the Gibbs-Duhem relationship,
2
2
2
x
µ
′
−
x
µ
′ −
x
µ
′
k k
i
ii
j
jj
µ
′ =
(4.18)
ij
2
xx
ji
The derivatives in Equation 4.15 through Equation 4.17 are easily obtained from the
expression of the chemical potential on the rational scale,
o
µµ
=+
RT
ln
γ
x
(4.19)
i
i
ii
∂
∂
βµ
=+
∂
∂
δ
ln
γ
i
ij
i
(4.20)
x
x
x
j
j
j
Tp x
, {}
′
T
px
′
,{ }
