The Wong–Sandler mixing rule is a thermodynamic mixing rule used for vapor–liquid equilibrium and liquid-liquid equilibrium calculations.[1]
Summary
The first boundary condition is
![{\displaystyle b-{\frac {a}{RT}}=\sum _{i}\sum _{j}x_{i}x_{j}\left(b_{ij}-{\frac {a_{ij}}{RT}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d32b68101d23680fb47ccd017c1c4779e369a12)
which constrains the sum of a and b. The second equation is
![{\displaystyle {\underline {A}}_{EOS}^{ex}(T,P\to \infty ,{\underline {x}})={\underline {A}}_{\gamma }^{ex}(T,P\to \infty ,{\underline {x}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6daea9b6ac2993804ddca1cb47e46f75a2ba946b)
with the notable limit as
(and
) of
![{\displaystyle {\underline {A}}_{EOS}^{ex}=C^{*}\left({\frac {a}{b}}-\sum x_{i}{\frac {a_{i}}{b_{i}}}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a4683c8e2f8d172f84a7f6f11bceb567f383eeb)
The mixing rules become
![{\displaystyle {\frac {a}{RT}}=Q{\frac {D}{1-D}},\quad b={\frac {Q}{1-D}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6221cd677034f347e8eb62076057a51acac04c6f)
![{\displaystyle Q=\sum _{i}\sum _{j}x_{i}x_{j}\left(b_{ij}-{\frac {a_{ij}}{RT}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47f9e27dbf2f7df1baa1358de34565563fc04197)
![{\displaystyle D=\sum _{i}x_{i}{\frac {a_{i}}{b_{i}RT}}+{\frac {{\underline {G}}_{\gamma }^{ex}(T,P,{\underline {x}})}{C^{*}RT}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/489348c5b95fe3ba0d0ee029f6c636366019481b)
The cross term still must be specified by a combining rule, either
![{\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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57075b9bc9a2b0eb3626e2afc917d99adedcafc5)
or
![{\displaystyle b_{ij}-{\frac {a_{ij}}{RT}}={\frac {1}{2}}(b_{ii}+b_{jj})-{\frac {\sqrt {a_{ii}a_{jj}}}{RT}}(1-k_{ij}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5495c3f5aa5173b7797cf0a69f971eb903f2eae)
See also
- Vapor–liquid equilibrium
- Equation of state
References
- ^ 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.