GENERAL RESEARCH Volume-Explicit Equation of State for Hard Spheres, Hard Disks, and Mixtures of Hard Spheres Esam Z. Hamad † Chemical Engineering Department, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia A simple volume-explicit equation of state for hard spheres and their mixtures is developed. The equation accurately represents the simulation data. It also predicts the correct volume at high pressures (close packing), unlike the common pressure-explicit equations which predict an unattainable volume. The new equation gives the exact second and third pressure virial coefficients. It predicts higher coefficients within 1.4% (up to the eighth coefficient) compared to 15% for the common pressure-explicit Carnahan-Starling equation. The developed equation can be used as the repulsive part in developing volume-explicit real fluid equations of state or activity coefficient models. An equation for spherical molecules in two dimensions was also developed for use in surface studies. Introduction Equations of state (EOS) are valuable tools in the calculations of thermodynamic properties and phase behavior. EOS that are based on sound statistical thermodynamics have more predictive capabilities com- pared to empirical equations. An important result of the theoretical study of fluids is that repulsive molecular forces are dominant at high densities. This result motivated researches to build EOS around accurate repulsive equations. Examples of such EOS are the works of Chien et al. (1983), Dohrn and Prausnitz (1990), and Tao and Mason (1994). The most commonly used repulsive equation is that of hard spheres due to Carnahan and Starling (1969). This equation is explicit in pressure as most EOS are. In chemical engineering calculations, the pressure and temperature are the usually specified variables. This makes pressure-explicit equations inconvenient to use due to the added numerical procedure to find the correct volume root. It is therefore desirable to have a volume- explicit EOS that expresses the compressibility factor in terms of the independent variables pressure P and temperature T. In addition, the Carnahan-Starling equation does not predict the correct density at which the pressure will diverge. It is the objective of this work to develop a simple volume-explicit equation for hard spheres and their mixtures that overcomes these limita- tions. Such an equation is important for future use in real fluid equations of state and activity coefficient models. An equation for hard spheres on surfaces (hard disks) is also developed for future use for molecules on surfaces. Equations of State The equation of state will be developed based on summation of the pressure virial expansion. The com- pressibility factor, Z ) P/FRT can be expanded as a power series with either density, F, or pressure, P, as the independent variable (Mason and Spurlig, 1969):or where the virial coefficients B i and B′ i are functions of temperature for pure fluids and of temperature and composition for mixtures. They are related to the derivatives of the compressibility factor at zero density or pressure and The two virial coefficients can be related to each other by substituting one expansion in the other and compar- ing coefficients of the same powers. The general rela- tion between the two sets of coefficients has been worked out (Epstein, 1952; Putnam and Kilpatrick, 1953). The explicit expressions for B′ i up to the eighth virial coefficient are given in the Appendix. It is useful to express the virial expansions in dimensionless variables. We define dimensionless den- sity and pressure as and † E-mail: ezhamad@dpc.kfupm.edu.sa. Z ) 1 + ∑ i)2 B i F i-1 (1) Z ) 1 + ∑ i)2 B′ i P i-1 (2) B i ) 1 (i - 1)! lim Ff0 [ ∂ i-1 Z ∂F i-1 ] T (3) B′ i ) 1 (i - 1)! lim Pf0 [ ∂ i-1 Z ∂P i-1] T (4) η ) υF) π 6 σ 3 N A F (5) p ) Pυ RT ) P RT πσ 3 6 N A (6) 4385 Ind. Eng. Chem. Res. 1997, 36, 4385-4390 S0888-5885(96)00763-4 CCC: $14.00 © 1997 American Chemical Society