Further Development of Scaled Particle Theory of Rigid Sphere Fluids

Abstract

A new approach to the scaled particle theory of fluids is developed which is based on detailed consideration of the reversible work required to create a fixed spherical cavity of radius $r$ in a rigid sphere fluid. The new results depend only on the mechanical concept of work and the classical virial theorem. A cycle greatly facilitates the necessary detailed analysis. A new integral equation for $\mathrm{G(r)}$ , the primary function in scaled particle theory, culminates the new results. Approximation schemes may be employed relatively easily in this integral equation. The series approximation $$\mathrm{G(r)\hspace{0.17em}=\hspace{0.17em}}{\displaystyle \sum _{\text{n\hspace{0.17em}=\hspace{0.17em}0}}^{\infty }}{G}_n{\mathrm{(r}}^{\text{−1}}{)}^n,$$ which is asymptotically correct as $\mathrm{r\hspace{0.17em}\to \hspace{0.17em}\infty }$ , is discussed. A new set of exact conditions on the coefficients $G_n$ in this asymptotic expansion, the most striking of which is $G_3\text{\hspace{0.17em}≡\hspace{0.17em}0}$ in three dimensions, is derived. These exact conditions relate to surface properties. The infinity and integral exact conditions on $\mathrm{G(r)}$ are shown to be truly independent. The jump condition on the second derivative of $\mathrm{G(r)}$ is incorporated successfully for the first time with other exact conditions in the derivation of two new expressions for $\mathrm{G(a)}$ , both of which give rise to equations of state. One of these expressions yields the best virial coefficients thus far obtained for a rigid sphere system from any analytic equation of state not based on cluster expansions. The new approach and results are easily generalized to arbitrary dimension. Future work will center on attempts to extend the new integral equation for $\mathrm{G(r)}$ , on possible extensions of the present method to real fluids, and on studies of the nature of the asymptotic series approximation scheme.