Gibbsian interpretation of Langmuir, Freundlich and Temkin isotherms for adsorption in solution

Adsorption is a physicochemical phenomenon important in both natural and engineering processes. In the research and practice of adsorption equilibrium, a long-standing challenge is how to reconcile the classical models proposed by Gibbs, Langmuir, Freundlich, and Temkin for interpreting experimentally obtained adsorption isotherms. Here, we show that the Langmuir, Freundlich, and Temkin isotherms can be derived from the Gibbs equation under different conditions for the change of surface energy (a.k.a. surface tension) by adsorption. When the change of surface energy is predominantly controlled by the change of chemical potential with negligible contribution from the change of internal energy and entropy, the Gibbs equation can be integrated to give the Langmuir isotherm. When changes of internal energy and entropy are no longer negligible, the integration of the Gibbs equation gives the Freundlich and Temkin equations, according to a change of surface energy either independent or linearly dependent on the adsorption capacity. These results indicate that the classical models share the common mechanism of phase equilibrium described by the Gibbsian thermodynamics, thereby providing novel insights for their application.