Rigorous statistical thermodynamics of a model binary solution: One-dimensional soft rods with hard rods

Abstract

From canonical ensemble theory, the thermodynamic functions A, F (the one-dimensional pressure), μ, S, G, and H are obtained in the thermodynamic limit for a system of soft rods (two equal- mass points between which there is a potential proportional to separation) and for a system of hard rods of the same total mass and arbitrary length σ, on a circle. Introduction of a tangent “interchange loop” allows the evaluation of the partition function of an arbitrary mixture of physically indistinguishable soft rods with physically indistinguishable hard rods. All circular sequences of the two types of particles give the same previously evaluated integral; the number of sequences is related to the number of different linear arrangements in a way exact in the thermodynamic limit. The same thermodynamic functions are evaluated for the mixture. The fugacities reveal that the two non-ideal components surprisingly form a perfect solution with each other, regardless of the volume fraction, often invoked in three-dimensional solution theory.