Perturbation and Variational Approaches to Equilibrium Thermodynamics of Gases, Liquids, and Phase Transitions

TLDR: Perturbation and variational approaches to equation of state and other equilibrium thermodynamic properties of simple liquids and phase transitions, vapor-liquid and liquid-solid, are reviewed and discussed in this article.

Abstract: Perturbation and variational approaches to equation of state and other equilibrium thermodynamic properties of simple liquids and phase transitions, vapor-liquid and liquid-solid, are reviewed and discussed. count for the lack of convergence of the virial equation for the liquid state. Probably the most widely used equation of state today, indeed the standard of reference often used when proposing new equations of state, is the Benedict-Webb-Rubin equation. Both the form of this equation and the recipes for estimating mixture behavior from known behavior of its components are based upon virial expansion. While the virial expansion has been valuable in treating gases and gaseous mixtures, its failure to converge in the liquid state forces one to look for a more complete description, either empirically or theoreticaliy. The theoretical problem has been approached in several ways, and these are reviewed briefly below. The main emphases of this review are perturbation and variational approaches to evaluate the partition function, hence thermodynamic properties and equation of state, because these approaches show considerable promise for practical application. As a point of reference to the reader, this review is based largely on references available to the authors as of June 1969. Interpretive Approaches (2) Interpretive approaches to the equation of state start from an approximate description of the structure of the system. These approaches are called “lattice” theories because the structure customarily is assumed to resemble the regular lattice structure of crystalline solids. Assumptions regarding structure must be guided both by physical reality and by the ability to calculate the partition function resulting from the assumed structure. For crystalline solids, these requirements are harmonious because solids are known to have regular structures which are disturbed only slightly by thermal motions and such regular, more or less static, structures lead to a partition function which can be evaluated. Liquids, on the other hand, offer a serious challenge to the viability of interpretive approaches because their structure, if it can be called that, is continually changing and can be visualized only on an instantaneous basis. T o account for physical reality in liquids, a static average over the instantaneous structure suggests itself and such an approach may lead to a satisfactory “lattice” theory for liquids. This is true only if one can solve the mathematical problems in the form of very difficult and complicated combinatorial versions (17), which is still a difficult task. Predictive Approaches (58) Predictive approaches place emphasis, initially, on the process by which the intermolecular forces determine the structure, in the hope that a correct mathematical description of this process will lead to equations whose solutions describe the actual structure. Theories of this class are called “distribution function” theories because the equations involve distribution functions specifying the probability of finding sets of molecules in particular configurations. The three theories, customarily called Born-Green (BG) (12), hypernetted chain (HNC), and PercusYevick (PY) (52), of which any discussion of the theory of fluids must take account, had very different origins, and each requires different initial assumptions to get tractable results. The first, BG, which was proposed in 1936, requires the direct assumption that the triplet distribution function can be expressed as a product of pair distribution function for the three pairs involved, that is, the superposition approximation. Although the HNC and PY theories bypass direct assumptions regarding the triplet distribution function in their initial formulation, attempts to improve these theories ultimately lead to approximation or calculation of the triplet distribution function. The PY theory, while capable of producing an analytic and approximately correct equation of state for hard spheres (62, 65), must be improved before it is equally successful for more realistic systems. Rice and coworkers (67) have replaced the superposition approximation in the BG theory with a more adequate approximation, but extensive calculations have demonstrated that none of the predictive theories are capable of predicting the properties of simple liquids and phase transitions at the present time. The triplet distribution function, complicated and unknown, stands in the way of successful predictive approaches to the equation of state and other equilibrium properties of dense gases and liquids. Perturbation and Variational Approaches Variational and perturbation approaches, the subjects of this review, are neither predictive nor interpretive theories. They are mathematical * means of expanding the configurational partition function of an original system around a relatively simple reference system whose properties are known. The reference system can be: a real substance whose properties are known (55, 69) or can be a hypothetical system, such as hard spheres, as long as relationships are available to describe its behavior. This review is concerned only with the second approach where properties of the reference system can be calculated from a statistical mechanic theory. That is, by defining a particular intermolecular potential function for the reference system, one must be able to calculate other properties of the reference system such as Helmholtz free energy, compressibility, and the radial distribution function. The reference system should resemble the original system as close as possible to speed the convergence of the power series of the partition function of the original system, and could be the result of a predictive or an interpretive theory. The equation of state for a gas at very high temperatures is determined largely by the repulsive forces acting between molecules, that is, a real gas at very high temperatures behaves much like an assembly of hard spheres. Thus, it is reasonable to expect that the equation of state a t somewhat lower temperatures could be obtained by treating the attractive forces as perturbations about the repulsive forces. If the perturbational approach is valid, the equation of state for gases at moderate temperatures will be of the following form: VOL. 6 2 NO. 8 A U G U S T 1 9 7 0 13 Perturbation and Variational Approaches to Equilibrium Thermodynamics of Gases, Liquids, and Phase Transitions