An asymptotic expression for the critical-region “bird’s beak” isotherm and adjacent isotherms on the vapor–liquid phase diagram of a simple binary mixture

For a binary mixture of a dilute nonvolatile solute in a volatile solvent, an asymptotic expression is derived for isothermal dew-bubble curves in the region just above the solvent critical point. The expression depends only on the solvent coexistence properties and the initial slopes of the continuous critical locus, with no adjustable parameters. It clarifies the mathematical behavior of these curves and shows why, for this situation, classical critical exponents can be used with relatively small error. For supercritical extraction applications, the expression does not apply to solutes with large, complex molecules, since the critical locus with carbon dioxide is usually discontinuous, but it should apply to carbon dioxide+cosolvent mixtures. The formula is in good quantitative agreement with experiment for three simple nonpolar mixtures and for carbon dioxide+acetone, but shows only qualitative agreement for carbon dioxide+ethanol.