Convergence of Fugacity Expansion and Bounds on Molecular Distributions for Mixtures

Abstract

We have generalized the methods developed recently by Groeneveld and Penrose for a one‐component system to obtain a lower bound on the domain of convergence of the Mayer fugacity expansion of the pressure, $${\mathrm{p=\Sigma b}}_{l_1}{\mathrm{,···,}}_{l_{\omega }}{\mathrm{\Pi z}}_{\alpha }^{l_{\alpha }},$$ where z_α is the fugacity of the αth component, α=1,•••, ω. This series is convergent for $${\displaystyle \sum _{\text{α=1}}^{\omega }}{\mathrm{|z}}_{\alpha }\mathrm{|\le [}\exp {\text{(1+2Φ/kT)B]}}^{\text{−1}},$$ where $$\mathrm{B=\{}\max \mathrm{(\alpha ,\beta )\}}{\displaystyle \int }|\exp {\mathrm{[-(kT)}}^{\text{−1}}{\phi }_{\mathrm{\alpha \beta }}\text{(r)]−1|d}\mathbf{r}$$ and where the interaction potential φ_αβ(r) of a pair of particles of Species α and β satisfies $${\displaystyle \sum _{\mathrm{i<j\le s}}}{\phi }_{{\alpha }_i{\alpha }_j}{\mathrm{(|x}}_i{\mathrm{-x}}_j\mathrm{|)\ge -s\Phi }$$ for all α, x, and s. [For a positive interparticle potential, φ_αβ(r)≥0, Φ=0.] Consequently the system remains in a single phase in this region. We have also generalized the inequalities of Lieb, Penrose, Lebowitz, and Percus to this case. For positive potentials upper (lower) bounds are gotten for the pressure and the distribution functions by expanding in a Taylor series up to terms of even (odd) total order in the fugacities.