Convergence of Fugacity Expansion and Bounds on Molecular Distributions for Mixtures

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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, p=Σbl1,···, lωΠzαlα,where zα is the fugacity of the αth component, α=1,•••, ω. This series is convergent for ∑ α=1ω|zα|≤[exp(1+2Φ/kT)B]−1,where B={max(α,β)} ∫ |exp[−(kT)−1φαβ(r)]−1|drand where the interaction potential φαβ(r) of a pair of particles of Species α and β satisfies ∑ i<j≤sφαiαj(|xi−xj|)≥−sΦfor all α, x, and s. [For a positive interparticle potential, φαβ(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.

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