Physico‑chemical description at nanoscale of the adsorbed phase of phenol onto coal fly ash grain by applying statistical physics treatment

This study applies the formalism of statistical physics to elucidate the molecular adsorption mechanism of phenol onto coal fly ash spherical grains (CFAPsg) and coal gangue spherical grains (CFAGsg). Experimental adsorption isotherms were interpreted using statistical physics models to identify the most appropriate adsorption scenario and to determine the steric, energetic, and thermodynamic characteristics governing the adsorption process. Error analysis demonstrated that the adsorption equilibrium is best described by a two-layer model involving two adsorption energies, providing a realistic representation of the adsorbed phase. The steric investigation revealed that the number of adsorbed molecules per active site (δ) ranges from 1.22 to 2.75, indicating a cooperative adsorption mechanism in which phenol molecules adopt a predominantly perpendicular orientation with the possibility of molecular aggregation. The calculated adsorption energies lie within 14.934–25.113 kJ mol−1, confirming that phenol retention is mainly governed by physical interactions. Thermodynamic analysis further revealed a pronounced dependence of the configurational entropy on both temperature and phenol concentration, with a maximum at intermediate concentrations corresponding to the progressive occupation of active sites before surface saturation. Moreover, topographic analysis showed that the energies are distributed within 9200–30,085 J mol-1 for the Phenol–CFAGsg system and 7500–30,000 J mol-1 for the Phenol–CFAPsg system.