Equations of state for hard sphere mixtures

The following are equations of state for mixtures of hard spheres.

Mansoori, Carnahan, Starling, and Leland

The Mansoori, Carnahan, Starling, and Leland equation of state is given by (Ref. 1 Eq. 7):

$Z={\frac {(1+\xi +\xi ^{2})-3\xi (y_{1}+y_{2}\xi )-\xi ^{3}y_{3}}{(1-\xi )^{3}}}$

where

$\xi =\sum _{i=1}^{m}{\frac {\pi }{6}}\rho \sigma _{i}^{3}x_{i}$

where $m$ is the number of components, $\sigma _{i}$ is the diameter of the $i$th component, and $x_{i}$ is the mole fraction, such that $\sum _{i=1}^{m}x_{i}=1$.

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y_{1}=\sum _{j>i=1}^{m}\Delta _{ij}{\frac {\sigma _{i}+\sigma _{j}}{\sqrt {\sigma _{i}\sigma _{j}}}}}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y_{2}=\sum _{j>i=1}^{m}\Delta _{ij}\sum _{k=1}^{m}\left({\frac {\xi _{k}}{\xi }}\right){\frac {\sqrt {\sigma _{i}\sigma _{j}}}{\sigma _{k}}}}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y_{3}=\left[\sum _{i=1}^{m}\left({\frac {\xi _{i}}{\xi }}\right)^{2/3}x_{i}^{1/3}\right]^{3}}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Delta _{ij}={\frac {\sqrt {\xi _{i}\xi _{j}}}{\xi }}{\frac {(\sigma _{i}-\sigma _{j})^{2}}{\sigma _{i}\sigma _{j}}}{\sqrt {x_{i}x_{j}}}}

Santos, Yuste and López De Haro

Ref. 2

Hansen-Goos and Roth

Ref. 3 Based on the Carnahan-Starling equation of state

References

1. G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W. Leland, Jr. "Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres", Journal of Chemical Physics 54 pp. 1523-1525 (1971)
2. Andrés Santos; Santos Bravo Yuste; Mariano López De Haro "Equation of state of a multicomponent d-dimensional hard-sphere fluid", Molecular Physics 96 pp. 1-5 (1999)
3. Hendrik Hansen-Goos and Roland Roth "A new generalization of the Carnahan-Starling equation of state to additive mixtures of hard spheres", Journal of Chemical Physics 124 154506 (2006)