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):

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

where

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

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

${\displaystyle y_{1}=\sum _{j>i=1}^{m}\Delta _{ij}{\frac {\sigma _{i}+\sigma _{j}}{\sqrt {\sigma _{i}\sigma _{j}}}}}$
${\displaystyle y_{2}=\sum _{j>i=1}^{m}\Delta _{ij}\sum _{k=1}^{m}\left({\frac {\xi _{k}}{\xi }}\right){\frac {\sqrt {\sigma _{i}\sigma _{j}}}{\sigma _{k}}}}$
${\displaystyle y_{3}=\left[\sum _{i=1}^{m}\left({\frac {\xi _{i}}{\xi }}\right)^{2/3}x_{i}^{1/3}\right]^{3}}$
${\displaystyle \Delta _{ij}={\frac {\sqrt {\xi _{i}\xi _{j}}}{\xi }}{\frac {(\sigma _{i}-\sigma _{j})^{2}}{\sigma _{i}\sigma _{j}}}{\sqrt {x_{i}x_{j}}}}$

Ref. 2

Hansen-Goos and Roth

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