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The thermodynamic stability of the hard-sphere gas has been examined, using the formalism of scaled particle theory [SPT], and by applying explicitly the conditions of stability required by both the second and third laws of thermodynamics. The temperature and volume limits to the validity of SPT have also been examined. It is demonstrated that scaled particle theory predicts absolute limits to the stability of the fluid phase of the hard-sphere system, at all temperatures within its range of validity. Because scaled particle theory describes fluids equally well as dilute gases or dense liquids, the limits set upon the system stability by SPT must represent limits for the existence of the fluid phase and transition to the solid. The reduced density at the stability limits determined by SPT is shown to agree exactly with those of that estimated for the Alder-Wainwright, supercritical, high-density gas-solid phase transition in a hard-sphere system, at a specific temperature, and closely over a range of more than 1,000K. The temperature dependence of the gas-solid phase stability limits has been examined over the range 0.01K-10,000K. It is further shown that SPT describes correctly the variation of the entropy of a hard-core fluid at low temperatures, requiring its entropy to vanish as T goes to zero by undergoing a gas-solid phase transition at finite temperature and all pressures.

A thermodynamic argument has been developed which relates the chirality of the constituents of a mixture of enantiomers to the system excess volume, and thereby to its Gibbs free enthalpy. A specific connection is shown between the excess volume and the statistical mechanical partition function. The Kihara-Steiner equations, which describe the geometry of convex hard bodies, have been extended to include also chiral hard bodies. These results have been incorporated into an extension of the Pavlicek-Nezbeda-Boubik equation of state for convex, aspherical, hard-body systems. The Gibbs free enthalpy has been calculated, both for single-component and racemic mixtures, for a wide variety of hard-body systems of diverse volumes and degrees of asphericity, prolateness, and chirality. The results show that a system of chiral enantiomers can evolve to an unbalanced, scalemic mixture, which must manifest optical activity, in many circumstances of density, particle volume, asphericity, and degree of chirality. The real chiral molecules fluorochloroiodomethane, CHFClI, and 4-vinylcyclohexene, C8H12, have been investigated by Monte Carlo simulation, and observed to manifest, both, positive excess volumes (in their racemic mixtures) which increase with pressure, and thereby the racemic-scalemic transition to unbalanced distributions of enantiomers. The racemic-scalemic transition, responsible for the evolution of an optically active fluid, is shown to be one particular case of the general, complex phase behavior characteristic of "closely-similar" molecules (either chiral or achiral) at high pressures.

The spontaneous genesis of hydrocarbons which comprise natural petroleum have been analyzed by chemical thermodynamic stability theory. The constraints imposed upon chemical evolution by the second law of thermodynamics are briefly reviewed; and the effective prohibition of transformation, in the regime of temperatures and pressures characteristic of the near-surface crust of the Earth, of biological molecules into hydrocarbon molecules heavier than methane is recognized. A general, first-principles equation of state has been developed by extending scaled particle theory (SPT) and by using the technique of the factored partition function of the Simplified Perturbed Hard Chain Theory (SPHCT). The chemical potentials, and the respective thermodynamic Affinity, have been calculated for typical components of the hydrogen-carbon (H-C) system over a range pressures between 1-100 kbar, and at temperatures consistent with those of the depths of the Earth at such pressures. The theoretical analyses establish that the normal alkanes, the homologous hydrocarbon group of lowest chemical potential, evolve only at pressures greater than approximately thirty kbar, excepting only the lightest, methane. The pressure of thirty kbar corresponds to depths of approximately 100 km. Special high-pressure apparatus has been designed which permits investigations at pressures to 50 kbar and temperatures to 2000 K, and which also allows rapid cooling while maintaining high pressures. The high-pressure genesis of petroleum hydrocarbons has been demonstrated using only the solid reagents iron oxide, FeO, and marble, CaCO3, 99.9% pure and wet with triple-distilled water.

A general equation of state for the hard-body reference system of real fluid has been developed from first principles, statistical mechanical arguments using metric differential geometry to describe the "available volume," V0, and its determining surface, S0, of a hard-body fluid. The rigorous, exact results of scaled particle theory of Reiss et al., which themselves obtain from statistical geometry, have been applied following the extension of Boublik et al. for hard bodies of non-spherical shape. The geometric description of the hard-body system can be used with the Boublik equation of state at low and modest densities. At high densities, this geometric description specifies the procedure to specify V0 and S0, notwithstanding that both become multiply-connected.

Microcanonical thermodynamics allows the application of statistical mechanics both to finite and even small systems and also to the largest, self-gravitating ones. However, one must reconsider the fundamental principles of statistical mechanics especially its key quantity, entropy. Whereas in conventional thermostatistics, the homogeneity and extensivity of the system and the concavity of its entropy are central conditions, these fail for the systems considered here. For example, at phase separation, the entropy, S(E), is necessarily convex to make exp[S(E)-E/T] bimodal in E. Particularly, as inhomogeneities and surface effects cannot be scaled away, one must be careful with the standard arguments of splitting a system into two subsystems, or bringing two systems into thermal contact with energy or particle exchange. Not only the volume part of the entropy must be considered. As will be shown here, when removing constraints in regions of a negative heat capacity, the system may even relax under a flow of heat (energy) against a temperature slope. Thus the Clausius formulation of the second law: ``Heat always flows from hot to cold'', can be violated. Temperature is not a necessary or fundamental control parameter of thermostatistics. However, the second law is still satisfied and the total Boltzmann entropy increases. In the final sections of this paper, the general microscopic mechanism leading to condensation and to the convexity of the microcanonical entropy at phase separation is sketched. Also the microscopic conditions for the existence (or non-existence) of a critical end-point of the phase-separation are discussed. This is explained for the liquid-gas and the solid-liquid transition., Comment: 23 pages, 2 figures, Accepted for publication in the Journal of Chemical Physics