Statistical Physics Approaches to Adsorption Isotherm Models: A Review

The present review critically examines the application of statistical physics to the development and interpretation of adsorption isotherm models, with particular emphasis on systems relevant to wastewater treatment. Rather than describing adsorption solely through empirical uptake curves, statistical physics provides a molecular-level framework that connects equilibrium adsorption data to descriptors of steric effects, energetic heterogeneity, and thermodynamic behavior. The models reviewed encompass monolayer, multilayer, single-solute, competitive, binary, ternary, and quaternary adsorption configurations. More than fifteen statistical-physics-based formulations are discussed, with particular emphasis on their mathematical expressions and the physicochemical significance of their fitted parameters. Thermodynamic quantities, including adsorption entropy, free enthalpy, internal energy, and adsorption-energy distributions, are considered as complementary tools for improving the mechanistic interpretation of adsorption processes. The review further examines the application of these models to pollutants such as dyes, pharmaceuticals, and heavy metals at adsorbent interfaces. In contrast to classical isotherm equations such as the Langmuir and Freundlich equations, statistical physics models provide access to molecular descriptors, including the aggregation number (n), receptor-site density (NM), adsorption energy, and layer-formation tendency, thereby offering deeper insight into adsorption mechanisms. Models derived from the grand canonical partition function can further account for adsorbate orientation, anchoring modes, interaction types, physisorption or chemisorption, and the number of molecules retained per adsorption site. In addition, they enable the macroscopic evaluation of adsorption through internal energy, entropy, and Gibbs free energy. Overall, this review highlights the potential of statistical physics as a powerful framework for bridging macroscopic adsorption measurements with molecular-level mechanisms and for providing a more comprehensive interpretation of pollutant removal at adsorbent interfaces.