Perturbation Theory and the Equation of State of Mixtures of Hard Spheres

The equation of state of a mixture of hard spheres of diameter ${\sigma }_{11}$ and ${\sigma }_{22}$ is evaluated by expanding the free energy of the mixture to second order in powers of the differences ${\sigma }_{\mathrm{\alpha \beta }}\mathrm{\hspace{0.17em}-\hspace{0.17em}\sigma }$ , where ${\sigma }_{\mathrm{\alpha \beta }}\mathrm{\hspace{0.17em}=\hspace{0.17em}}\frac{1}{2}{\mathrm{(\sigma }}_{\mathrm{\alpha \alpha }}{\mathrm{\hspace{0.17em}+\hspace{0.17em}\sigma }}_{\mathrm{\beta \beta }})$ and $\sigma$ is the diameter of the unperturbed hard spheres. The parameter $\sigma$ is chosen to make the sum of the first‐order terms zero. The second‐order terms are evaluated using the superposition approximation, and the equation of state is determined by analytic differentiation. The agreement of this perturbation calculation with quasiexperimental Monte Carlo and molecular dynamics results is excellent, even for quite large values of ${\mathrm{R\hspace{0.17em}=\hspace{0.17em}\sigma }}_{22}{\mathrm{\hspace{0.17em}/\hspace{0.17em}\sigma }}_{11}$ , but becomes less satisfactory as $R$ becomes very large.