Chemistry Reference
In-Depth Information
As usual, the TCFI expressions are more complex because they involve matrix
inverses of the second derivatives. Many applications use an EOS model that is cubic
in molar volume, such as the generalized van der Waals form, with pure component
parameters that depend upon composition according to mixing rules that employ
combining rules with unlike interaction parameters of the solution components. An
example is the original van der Waals EOS,
(
)
{}
RT
vbx
ax
v
p
=
)
−
(9.69)
(
{}
2
−
The Helmholtz energy, when written in the form for taking the derivatives of
Equation 9.68 is
2
(
)
=
(
)
,,
{}
{}
ATVN
RT
r
Na N
2
N
VNbN
N
ln
−
(9.70)
(
)
{}
−
RTV
The derivatives can be taken for any set of mixing and combining rules. The well-
known 1-fluid mixing rules are
n
n
c
c
∑
∑
(
)
=
{}
2
Na N
N Na
ii
j
lij
ii
=
1
j
=
1
n
n
c
c
∑
∑
(9.71)
NNb
ii
j
lij
(
)
=
{}
ii
=
1
j
=
1
Nb
N
n
c
∑
N
ii
ii
=
1
and, among the combining rule options, a common choice is
1
2
1
=
(
)
(
)
aaa
−
k
lij
ii
jj
lij
(9.72)
1
2
(
)
(
)
b
=
b
+
b
1
−
l
lij
ii
jj
lij
where
k
lij
and
l
lij
are binary parameters which are zero for pure components (
i
=
j
). For
these choices, Equation 9.68 becomes
2
v
vb
a
kTv
2
ii
1
−=
−
C
−
(9.73)
ii
ii
B
