The first boundary condition is
b
−
a
R
T
=
∑
i
∑
j
x
i
x
j
(
b
i
j
−
a
i
j
R
T
)
{\displaystyle b-{\frac {a}{RT}}=\sum _{i}\sum _{j}x_{i}x_{j}\left(b_{ij}-{\frac {a_{ij}}{RT}}\right)}
which constrains the sum of a and b . The second equation is
A
_
E
O
S
e
x
(
T
,
P
→
∞
,
x
_
)
=
A
_
γ
e
x
(
T
,
P
→
∞
,
x
_
)
{\displaystyle {\underline {A}}_{EOS}^{ex}(T,P\to \infty ,{\underline {x}})={\underline {A}}_{\gamma }^{ex}(T,P\to \infty ,{\underline {x}})}
with the notable limit as
P
→
∞
{\displaystyle P\to \infty }
V
_
i
→
b
,
{\displaystyle {\underline {V}}_{i}\to b,}
V
_
m
i
x
→
b
{\displaystyle {\underline {V}}_{mix}\to b}
A
_
E
O
S
e
x
=
C
∗
(
a
b
−
∑
x
i
a
i
b
i
)
.
{\displaystyle {\underline {A}}_{EOS}^{ex}=C^{*}\left({\frac {a}{b}}-\sum x_{i}{\frac {a_{i}}{b_{i}}}\right).}
The mixing rules become
a
R
T
=
Q
D
1
−
D
,
b
=
Q
1
−
D
{\displaystyle {\frac {a}{RT}}=Q{\frac {D}{1-D}},\quad b={\frac {Q}{1-D}}}
Q
=
∑
i
∑
j
x
i
x
j
(
b
i
j
−
a
i
j
R
T
)
{\displaystyle Q=\sum _{i}\sum _{j}x_{i}x_{j}\left(b_{ij}-{\frac {a_{ij}}{RT}}\right)}
D
=
∑
i
x
i
a
i
b
i
R
T
+
G
_
γ
e
x
(
T
,
P
,
x
_
)
C
∗
R
T
{\displaystyle D=\sum _{i}x_{i}{\frac {a_{i}}{b_{i}RT}}+{\frac {{\underline {G}}_{\gamma }^{ex}(T,P,{\underline {x}})}{C^{*}RT}}}
The cross term still must be specified by a combining rule, either
b
i
j
−
a
i
j
R
T
=
(
b
i
i
−
a
i
i
R
T
)
(
b
j
j
−
a
j
j
R
T
)
(
1
−
k
i
j
)
{\displaystyle b_{ij}-{\frac {a_{ij}}{RT}}={\sqrt {\left(b_{ii}-{\frac {a_{ii}}{RT}}\right)\left(b_{jj}-{\frac {a_{jj}}{RT}}\right)}}(1-k_{ij})}
or
b
i
j
−
a
i
j
R
T
=
1
2
(
b
i
i
+
b
j
j
)
−
a
i
i
a
j
j
R
T
(
1
−
k
i
j
)
.
{\displaystyle b_{ij}-{\frac {a_{ij}}{RT}}={\frac {1}{2}}(b_{ii}+b_{jj})-{\frac {\sqrt {a_{ii}a_{jj}}}{RT}}(1-k_{ij}).}
^ Wong, D. S. H. & Sandler, S. I. (1992). "A theoretically correct mixing rule for cubic equations of state". AIChE Journal . 38 (5): 671–680. doi :10.1002/aic.690380505 .
