Shape and curvature of surface tension isotherms for liquid mixtures

Abstract

The different shapes of surface tension isotherms consistent with Butler’s ideal model for the planar surface phase of binary liquid mixtures are comprehensively worked out for the first time. Using the ratio of pure constituent molar surface areas and a parameter named surface factor, which depends only on pure-constituent properties, the Butler equation is expressed and solved as a function of these two independent parameters. The geometrical features for the plots of surface tension against the mole fraction of component with lower surface tension are established for every possible combination of those two parameters. Three main shape types are found, namely: concave up (positive curvature); concave down passing through an inflection point followed by concave up; and concave down (negative curvature). The range of parameter values leading to each shape type is obtained and shown graphically. It is demonstrated that Butler’s ideal model excludes the eventuality of a system presenting a shape with initial positive curvature, then an inflection point and continuing with negative curvature. All the above shapes will appear as a mirror image in the representation of surface pressure against mole fraction. Six organic–organic systems, including two ionic liquid solutions, taken from the literature are examined. It is shown that actual Gibbs adsorption isotherm shapes arise from the sum of ideal surface tension deviations with excess surface tensions, the latter being related to the equilibrium surface/bulk ratio of activity coefficients. It is also shown that this ratio is larger for the component with the greatest molar surface area. From the comparison of surface and bulk activity coefficients at the same chemical composition interesting insights into the surface phase are drawn.