Conditional apparent free-energy metrics for comparing metal binding at heterogeneous interfaces

Metal binding at heterogeneous interfaces is frequently described by adsorption capacities, isotherm parameters, or conditional binding constants. These quantities characterize uptake or apparent affinity, but changes in fitted constants do not by themselves establish intrinsic cooperative binding. This paper introduces a conditional apparent free-energy difference, $$\:{\varDelta\:G}_{app}\:=\:-RT\:ln({K}_{comp}/{K}_{ref})$$ , for expressing changes between operationally comparable binding constants on a common energetic scale. The metric is conditional because the reported constants incorporate the effects of pH, ionic strength, temperature, surface composition, polymer aging, organic matter composition, metal speciation, and the fitting procedure. Its application requires that the compared constants describe essentially the same binding process and be obtained under genuinely comparable conditions. The approach is illustrated by reanalyzing published datasets for Pb(II) binding in humic acid–polystyrene microplastic systems and Cu(II) binding by microplastic-derived dissolved organic matter. The Pb(II) comparison indicates a change in net system-level apparent affinity involving multiple binding pathways, whereas the Cu(II) comparison quantifies an ultraviolet-aging-induced change in apparent affinity. The proposed metric provides a quantitative framework for screening and comparing apparent-affinity changes and, when supported by independent mechanistic evidence, for evaluating non-additive or cooperative binding.