The Wong–Sandler mixing rule is a thermodynamic mixing rule used for vapor–liquid equilibrium and liquid-liquid equilibrium calculations.
Summary 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 } (and V _ i → b , {\displaystyle {\underline {V}}_{i}\to b,} V _ m i x → b {\displaystyle {\underline {V}}_{mix}\to b} ) of
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}).}
See also Vapor–liquid equilibrium Equation of state
References