A stable tensorial approach for inversion of adsorption potential energy from isotherm data

Underlying the adsorption mechanism is not a simple task due to the complexity of the process. In this context, statistical mechanics provides a versatile approach with relatively low computational cost for describing adsorption phenomena. An alternative strategy consists of obtaining the adsorption potential energy from experimental data through the solution of an inverse problem. In this work, the adsorption potential energy was determined directly from adsorption isotherms using experimental datasets for both aqueous- and gas-phase adsorption systems. The proposed methodology was compared with the traditional two-step procedure, in which the second adsorption virial coefficient is first determined and subsequently used for inversion of the potential energy. Due to the ill-posed nature of the problem, the conventional approach exhibits significant instability and error propagation. In contrast, the direct inversion method provides excellent results, achieving values of $R^2\sim 0.99$ .

A methodology grounded in statistical physics was developed to obtain the adsorption potential energy directly from isotherm data. The proposed approach formulates the problem as a tensor contraction, eliminating the need for the traditional two-step procedure involving the intermediate determination of the second adsorption virial coefficient.