Physics: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

$b-\frac{a}{RT}=\sum _i\sum _j{x}_i{x}_j(b_{ij}-\frac{a_{ij}}{RT})$

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

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

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

${\underset{\_}{A}}_{EOS}^{ex}=C^{*}(\frac{a}{b}-\sum x_i\frac{a_i}{b_i}).$

The mixing rules become

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

$Q=\sum _i\sum _j{x}_i{x}_j(b_{ij}-\frac{a_{ij}}{RT})$

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

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

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

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