Wong–Sandler mixing rule explained

The Wong–Sandler mixing rule is a thermodynamic mixing rule used for vapor–liquid equilibrium and liquid-liquid equilibrium calculations.^[1] __TOC__

Summary

The first boundary condition is

b - (a)/(RT) = \sum _{i} \sum _{j} x _{i} x _{j} \left(b _{ij} - (a _{ij})/(RT) \right)

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

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

with the notable limit as

P\to inf ty (and \underline{V} _{i\to} b, \underline{V} _{mix} \to b ) of\underline A ^{ex} _{EOS} = C ^{*} \left( (a)/(b) - \sum x _{i} (a _{i})/(b _{i}) \right).

The mixing rules become

(a)/(RT) = Q (D)/(1- D) , b = (Q)/(1- D) Q = \sum _{i} \sum _{j} x _{i} x _{j} \left( b _{ij} - (a _{ij})/(RT) \right) D = \sum _{i} x _{i} (a _{i})/(b _{i} RT) + (\underline{G)/(^{ex}) _{\gamma(T,} P, \underline x) }{C ^{*} RT}

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

b _{ij} - (a _{ij})/(RT) = \sqrt{\left(b _{ii} - (a _{ii})/(RT) \right)\left(b _{jj} - (a _{jj})/(RT) \right)} ( 1 - k _{ij} )

or

b _{ij} - (a _{ij})/(RT) = (1)/(2) (b _{ii} + b _{jj} ) - (\sqrt{a _{ii} a _{jj})/() }(1 - k_).

See also

Vapor–liquid equilibrium

Equation of state

Notes and References

1. Wong, D. S. H. . Sandler, S. I. . amp . A theoretically correct mixing rule for cubic equations of state . AIChE Journal . 1992 . 38 . 5 . 671–680 . 10.1002/aic.690380505 .