Wong–Sandler mixing rule

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

$b-{\frac {a}{RT}}=\sum _{i}\sum _{j}x_{i}x_{j}\left(b_{ij}-{\frac {a_{ij}}{RT}}\right)$

which constrains the sum of a and b. The second equation is

${\underline {A}}_{EOS}^{ex}(T,P\to \infty ,{\underline {x}})={\underline {A}}_{\gamma }^{ex}(T,P\to \infty ,{\underline {x}})$

with the notable limit as $P\to \infty$ (and ${\underline {V}}_{i}\to b,$ ${\underline {V}}_{mix}\to b$) of

${\underline {A}}_{EOS}^{ex}=C^{*}\left({\frac {a}{b}}-\sum x_{i}{\frac {a_{i}}{b_{i}}}\right).$

The mixing rules become

${\frac {a}{RT}}=Q{\frac {D}{1-D}},\quad b={\frac {Q}{1-D}}$

$Q=\sum _{i}\sum _{j}x_{i}x_{j}\left(b_{ij}-{\frac {a_{ij}}{RT}}\right)$

$D=\sum _{i}x_{i}{\frac {a_{i}}{b_{i}RT}}+{\frac {{\underline {G}}_{\gamma }^{ex}(T,P,{\underline {x}})}{C^{*}RT}}$

The cross term still must be specified by a combining rule, either

$b_{ij}-{\frac {a_{ij}}{RT}}={\sqrt {\left(b_{ii}-{\frac {a_{ii}}{RT}}\right)\left(b_{jj}-{\frac {a_{jj}}{RT}}\right)}}(1-k_{ij})$

or

$b_{ij}-{\frac {a_{ij}}{RT}}={\frac {1}{2}}(b_{ii}+b_{jj})-{\frac {\sqrt {a_{ii}a_{jj}}}{RT}}(1-k_{ij}).$

See also

Vapor–liquid equilibrium

Equation of state