Prog. Theor. Phys. Vol. 60 No. 6 (1978) pp. 1653-1668
Generalized Theory of Condensing Systems. VII
— An Imperfect Gas Obeying the Perfect-Gas Law in the Gaseous State
Department of Applied Physics, Faculty of Engineering, Osaka University, Suita 565
(Received December 14, 1977)
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, bl = 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.
DOI : 10.1143/PTP.60.1653
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