Critical lines and phase equilibria in binary van der Waals mixtures

Abstract

The van der Waals equation of state is used to determine phase diagrams for a wide variety of binary fluid mixtures. The locus of the critical line in pressure-temperature-composition space is determined exactly by solving a set of equations with the aid of a computer. The van der Waals constants a$_{\text{m}}$ and b$_{\text{m}}$ for the mixture depend quadratically and linearly upon the mole fractions x$_{\text{i}}$: a$_{\text{m}}$ = $\sum_{\text{i}}\sum_{\text{j}}$ x$_{\text{i}}$ x$_{\text{j}}$a$_{\text{ij}}$ and b$_{\text{m}}$ = $\sum_{\text{i}}$x$_{\text{i}}$b$_{\text{ii}}$. Mixtures are characterized by three non-dimensional parameters: $\xi $ = (b$_{22}$-b$_{11}$)/(b$_{11}$+b$_{22}$), $\zeta $ = (a$_{22}$b$_{22}^{-2}$-a$_{11}$b$_{11}^{-2}$)/(a$_{11}$b$_{11}^{-2}$+a$_{22}$b$_{22}^{-2}$) and $\Lambda $ = (a$_{11}$b$_{11}^{-2}$-2a$_{12}$/b$_{11}$b$_{22}$+a$_{22}$b$_{22}^{-2}$)/(a$_{11}$b$_{11}^{-2}$+a$_{22}$b$_{22}^{-2}$). The parameter $\Lambda $ can be related to the low-temperature enthalpy of mixing and the parameter $\zeta $ to the difference between the gas-liquid critical pressures of the pure fluids. Most of the calculations are for molecules of equal size ($\xi $ = 0), but calculations for a size ratio of two ($\xi $ = $\frac{1}{3}$) are also reported. Nine characteristic types of critical lines are distinguished and these correspond to nine separate regions on a $\Lambda $, $\zeta $-diagram. Isobaric temperature-composition diagrams and pressure-temperature projections are given for one example from each region to illustrate the possible types of phase equilibrium. Special attention is given to the details of lower critical solution temperature behaviour (type IV) such as is found in the system methane + n-hexane, to tricritical points (symmetrical and unsymmetrical), to azeotropy, and to the possibility of double azeotropy. The phase diagrams calculated from the van der Waals equation seem to account, at least qualitatively, for all but one of the varieties of phase equilibria found in binary fluid mixtures: the low-temperature lower critical solution points in some highly structured aqueous solutions of alcohols and amines.