Direct calculation of the equilibrium composition for multi-component Langmuir isotherms in batch adsorption

The extended Langmuir isotherm is commonly used to describe multi-component adsorption. In batch adsorption, the equilibrium composition can be computed by solving simultaneously the Langmuir equations and the material balance equations numerically. We demonstrate that for given values of the initial fluid-phase concentrations, the adsorbed phase composition of all components can be expressed in terms of the fluid-phase concentration of a reference species. As a result, the adsorbed phase composition is easily derived by combining the isotherm expression for the reference species with the corresponding material balance. We derive a general graphical representation of the underlying mathematics valid for N components along with the analytic solution for two-component systems.