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; andyor (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 (t), these charging energies can be related to the surface potential (C) by: DG charging ¼ ð1 À tÞFC (where F is the Faraday constant). If one assumes a maximum practical accuracy of AE 10% in experimental estimates of C, then it is suggested that charging energies are likely to be experimentally indiscernible under conditions where t ! 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 mm at ionic strengths ! 0.001 M (1 : 1), and for spherical particles with a radius 40.01 mm at an ionic strength 40.1 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 mm.

Extending the diffuse layer model of surface acidity constant behaviour: IV. Diffuse layer charge/potential relationships

“…If one accepts + 10% as a reasonable figure of merit for estimating interfacial potentials (Loux, 2009), then the relative degree of agreement among the non-surficial potential estimates depicted in Figure 4 suggests that any of the procedures would be reasonable for describing these phenomena at charge densities below 0.125 C m À 2 . Furthermore, as the EB potential estimate may be too high at a charge density of 0.025 C m À 2 (the expression dTDSyjzjen excess for estimating the partial molar entropy of mixing is most valid at high charge densities), the % CV for all of the other non-surficial data falls below 10% at charge densities equal to or less than 0.125 C m À 2 .…”

“…Alternatively, these estimates do not account for counterelectrolyte ion ''site binding'' (e.g. see references cited in Loux, 2009) which in turn could tend to reduce the associated interfacial potential estimates.…”

“…In general the EB iteratively finds conditions whereby C & TDSyjzjen excess (where n excess is the excess number of electrolyte counterions in the surface volume). From arguments presented by Loux (2009), potentials estimated with the EB are likely to be most accurate under conditions either where t ! 0:9 (i.e.…”

“…When jeCj 5 kT and in the absence of specific ion adsorption, there is generally minimal counterion condensation andyor ion pair formation between bound sites and electrolyte ions and the vast majority of charge neutralization for a given charged particle occurs in the diffuse layer distant from the particle surface. In historical Debye Huckel theory (where jeCj is assumed to be much less than kT), the diffuse layer potential contribution to the total particle surface potential is presumed to arise from a hollow charged spherical envelope that is 1yk distant from the particle surface [see discussions in Hunter (1987) and Loux (2009)]. Computationally, when Figure 1 Comparison of diffuse layer potential estimates for particles with a charge density of 0.1 C m À 2 in a room temperature aqueous medium with an ionic strength of 0.001 M (1 : 1).…”

“…In agreement with observations published elsewhere in the literature, the authors conclude that electrostatic and hydration energies are likely to dominate those variable excess free energy terms moderating mobile ion electrochemical potentials at the solidywater interface. Loux (2006) and Loux (2009) further examined possible contributions to acidity behaviour resulting from limited counterion condensation and concluded that charging energiesenergies that moderate interfacial bound site electrochemical potentials -are likely to be indiscernible in experimental data obtained from systems where the ionic strength is ! 0:001 M and the particle radius !…”