The Temperature Dependence of the Langmuir Adsorption Model for a Single-Site Metal-Organic Framework

The single-site Langmuir adsorption model, also known as the Langmuir isotherm equation, is one of the simplest possible descriptions of adsorption phenomena and yet finds widespread applicability across a range of disciplines. In its simplest form, it is deployed to treat adsorption equilibria at constant temperature (i.e., along isotherms); however, at the heart of its derivation is a more general class of models that each incorporates an explicit temperature dependence, subject to assumptions about the spatial/translational degrees of freedom of the adsorbed species. In this work, measurements of the temperature dependence of supercritical adsorption of H₂ on a single-site metal-organic framework (MOF) are presented and fitted using a range of Langmuir models with distinct treatments of degrees of freedom in the adsorbed phase. Surprisingly, all of the models can be used to adequately represent the measured data (to within 0.0003 mmol g^-1 per point), despite yielding significantly different values for binding energy and the temperature dependence of the isosteric enthalpy of adsorption (i.e., the isosteric heat, q_st). However, a critical finding of this work is that the mean-temperature isosteric enthalpy of adsorption remains consistent across all models within experimental error (±0.1% or <0.1 kJ mol^-1), highlighting its reliability for evaluating adsorption thermodynamics.