Fluid Phase Equilibria 241 (2006) 25–30 An algebraic formulation for an equal area rule to determine phase compositions in simple reactive systems Gustavo A. Iglesias-Silva a,∗ , Adri´ an Bonilla-Petriciolet a , Kenneth R. Hall b a Departamento de Ingenier´ ıa Qu´ ımica, Instituto Tecnol´ ogico de Celaya, Celaya, Gto. CP 38010, Mexico b Chemical Engineering Department, Texas A&M University, College Station, TX 77843-3122, USA Available online 7 February 2006 Abstract Equal area rules are fundamental constructs in thermodynamics for determining conditions of equilibrium. The Maxwell equal area rule (MEAR) provides saturated volumes when the two-phase isotherm/isobar cuts the van der Waals loops such that the areas above and below are equal. Another equal area rule exists for determining the phase compositions in a binary mixture when plotting the derivative of the total Gibbs energy against the composition. In this work, we show that in the determination of phase compositions in reactive systems, an equal area rule exists under the correct transformation of the mole fractions. © 2006 Elsevier B.V. All rights reserved. Keywords: Chemical equilibrium; Phase equilibrium; Equal areas; Compositions; Reactive systems John M. Prausnitz One of us (KRH) had the singular good fortune to have John as his instructor in graduate thermodynamics at Berkeley and to have benefited form some discussions with him. At that time, I had not really settled upon a research area in which I was pas- sionate, but John changed that and instilled an excitement in me for the field that has lasted 43 years. Although for personal reasons, I could not remain at Berkeley and study under his tute- lage, I have always felt that John was available for consultation and advice throughout my career. I have benefited enormously from this relationship, and I am certain that everyone has who has come into contact with this exceptional person feels the same. From John, I learned to think differently from the crowd. I would like to think that he enjoys reading this paper that extends a common thermodynamics construct to chemical equilibrium. Even an experimentalist dabbles in theory sometimes. 1. Introduction Two equal area rules have appeared in the literature for determining equilibrium conditions. The classical one is the ∗ Corresponding author. E-mail address: gais@iqcelaya.itc.mx (G.A. Iglesias-Silva). Maxwell equal area rule (MEAR). This construct appears in most thermodynamics textbooks. It locates the equilibrium vol- umes for pure components from an equation of state (EOS) by equalizing the areas formed by the van der Waals loops of an isotherm. More recently, Eubank and Hall [1] proved the existence of another equal area rule that enabled finding the equilibrium compositions for binary mixtures at constant temperature and pressure. They plotted the mole fraction derivative of the total Gibbs energy against composition to find the equilibrium com- positions using an equal area rule. Shyu et al. [2] used the equal area method to calculate the compositions of two or three phases in equilibrium for ternary systems. Hanif et al. [3,5] discussed MEAR in connection with pure component phase equilibrium using an EOS. Then, Shyu et al. [4] proposed the maximum partial area rule for applying the equal area rule to phase equi- librium calculations. Later, Hanif et al. [3,5] extended the use of the equal area rule for calculating multi-phase equilibrium problems. Finally, Iglesias-Silva et al. [6] collected all the equal area techniques into an efficient algebraic method. Recently, Ung and Doherty [7,8] used transformed mole fractions to solve the problem of phase equilibrium in reactive systems. Using this transform changed the constrained problem into an uncon- strained problem. In this work, we have used the transformed mole fractions and show that an equal area construct can determine the phase 0378-3812/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2005.12.035