Ion strength limit of computed excess functions based on the linearized Poisson-Boltzmann equation

The linearized Poisson–Boltzmann (L‐PB) equation is examined for its κ‐range of validity (κ, Debye reciprocal length). This is done for the Debye–Hückel (DH) theory, i.e., using a single ion size, and for the SiS treatment (D. Fraenkel, Mol. Phys. 2010, 108, 1435), which extends the DH theory to the case of ion‐size dissimilarity (therefore dubbed DH–SiS). The linearization of the PB equation has been claimed responsible for the DH theory's failure to fit with experiment at > 0.1 m; but DH–SiS fits with data of the mean ionic activity coefficient, γ± (molal), against m, even at m > 1 (κ > 0.33 Å−1). The SiS expressions combine the overall extra‐electrostatic potential energy of the smaller ion, as central ion—Ψa>b(κ), with that of the larger ion, as central ion—Ψb>a(κ); a and b are, respectively, the counterion and co‐ion distances of closest approach. Ψa>b and Ψb>a are derived from the L‐PB equation, which appears to conflict with their being effective up to moderate electrolyte concentrations (≈1 m). However, the L‐PB equation can be valid up to κ ≥ 1.3 Å−1 if one abandons the 1/κ criterion for its effectiveness and, instead, use, as criterion, the mean‐field electrostatic interaction potential of the central ion with its ion cloud, at a radial distance dividing the cloud charge into two equal parts. The DH theory's failure is, thus, not because of using the L‐PB equation; the lethal approximation is assigning a single size to the positive and negative ions. © 2015 Wiley Periodicals, Inc. A new criterion for the goodness of the linearization of the Poisson–Boltzmann (PB) equation in electrolyte theories—replacing the traditional one (i.e., the “thickness of the ion cloud,” 1/κ)—is the average value of the potential energy of electrostatic ionic interaction, Φave, at radial distance r from the origin, dividing the ion cloud into two parts of equal charge; based on the value of this “Φ1/2” at “r1/2”, the linearized PB equation is shown effective for aqueous electrolyte solutions at 25°C up to at least κ = 1.3 Å−1.