Extending the diffuse layer model of surface acidity behaviour: III. Estimating bound site activity coefficients

Abstract

Although detailed thermodynamic analyses of the 2-pK diffuse layer surface complexation model generally specify bound site activity coefficients for the purpose of accounting for those non-ideal excess free energies contributing to bound site electrochemical potentials, in application these terms are ignored based on one or more of the following assumptions: (1) bound site activity coefficients cancel out in the mass action quotients; (2) bound sites display ideal behaviour; and/or (3) these energies are already included in the exponential Boltzmann terms. In this work it is demonstrated that the bound site charging energy terms discussed in the two previous papers in this series have both conceptual and computational analogies to the charging energy contribution to the activity coefficients obtained from the Debye–Huckel Limiting Law. On high charge density colloidal particles at constant counterion condensation (τ), these charging energies can be related to the surface potential (ψ) by: ΔGcharging = (1 – τ)Fψ (where F is the Faraday constant). If one assumes a maximum practical accuracy of ± 10% in experimental estimates of ψ, then it is suggested that charging energies are likely to be experimentally indiscernible under conditions where τ > 0.9. These findings support the historical practice of ignoring bound site activity coefficients with the 2-pK diffuse layer surface complexation model in the following situations: for spherical particles with a radius ≥ 0.1 μm at ionic strengths ≥ 0.001 M (1 : 1), and for spherical particles with a radius >0.01 μm at an ionic strength >0.1 M (1 : 1). In contrast, charging energies (and non-ideal behaviour) are predicted to be significant at all charge densities and ionic strengths for spherical particles with a radius of 0.001 μm.