Prog. Theor. Phys. Vol. 60 No. 6 (1978) pp. 1653-1668

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Generalized Theory of Condensing Systems. VII

— An Imperfect Gas Obeying the Perfect-Gas Law in the Gaseous State —

Kazuyosi Ikeda

Department of Applied Physics, Faculty of Engineering, Osaka University, Suita 565

(Received December 14, 1977)

Abstract:

This paper theoretically discusses an imperfect gas which obeys the perfect-gas law in the gaseous state and begins to condense at a certain specific volume. The gas is defined by the simple expressions for the cluster integrals: b₁ ≡1, b_l = a^-l/V (for l ≥2), where l is the number of particles composing a cluster integral (i.e., the size of a cluster), V is the volume of the gas, and a is independent of l and V but may depend on the temperature T. In the limit N(=number of particles in the gas)→∞ with v = V/N fixed, the Helmholtz free energy f per particle, the pressure p and the activity z are obtained from the partition function expressed in terms of the cluster integrals. Not all parts of the theory of systems with volume-dependent cluster integrals, which has been developed in the author's previous papers, are applicable to this gas. It is rigorously proved that, for 0< v <a^-1, the p-v isotherm is horizontal, z has a constant value a, and at equilibrium a “huge” (i.e., macroscopic-sized) cluster, representing the liquid phase, coexists with a set of clusters of size one, representing the saturated vapour, and that, for v ≥a^-1, the equation p v = kT holds and at equilibrium the gas contains only clusters of size one. Thus it is deduced that the gas condenses at the “non-analytical” singularity z=a. The uniform convergence (for v) of the thermodynamic functions is discussed. A remark is made on a similar problem in the theory of distribution of zeros of the grand partition function.

URL : http://ptp.ipap.jp/link?PTP/60/1653/
DOI : 10.1143/PTP.60.1653

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Citing Article(s) :

Progress of Theoretical Physics Vol. 65 No. 5 (1981) pp. 1542-1564 :
Statistical Mechanics of One-Dimensional Systems. I

Kazuyosi Ikeda and Takehiko Takano
Progress of Theoretical Physics Vol. 71 No. 4 (1984) pp. 689-706 :
Phase Transitions of Lattice Gases

Kunisuke Nisizima and Kazuyosi Ikeda