Effect of Solvent Permittivity on the Thermodynamic Behavior of HCl Solutions: Analysis Using the Smaller-Ion Shell Model of Strong Electrolytes

Reply to "Comment on 'Negative Deviations from the Debye-Hückel Limiting Law for High-Charge Polyvalent Electrolytes: Are They Real?' "

In their "Comment on 'Negative Deviations from the Debye-Hückel Limiting Law for High-Charge Polyvalent Electrolytes: Are They Real?' " Biver and Malatesta, citing my respective paper, argue that my analysis of experimental data of their group is wrong and so is my conclusion that negative deviations observed by them are not physically real. Here I rebut their arguments and explain why their rejection of my work is unjustified, and why those authors do not in any way prove me wrong. The core of my study and conclusion remains intact: Negative deviations in the case of high-charge electrolytes with | z₊ z_-| > 3 do not always happen, and so far, such observed deviations are not convincingly supported by theory.

Negative Deviations from the Debye-Hückel Limiting Law for High-Charge Polyvalent Electrolytes: Are They Real?

In the past few decades, electromotive force (emf, E) measurements using improved electrochemical cells have afforded the derivation of mean ionic activity coefficients (γ_±'s) of very dilute solutions of binary electrolytes, within the 10^-6-10^-3 molar range (Malatesta et al., J. Solution Chem. 1994, 23, 11 ). Such measurements are especially important for highly charged electrolytes whose behavior at very low concentration is not yet fully understood, and whose γ_± values, derived from E data, have been claimed by Malatesta and co-workers to exhibit "negative deviations" from the Debye-Hückel (DH) limiting law. Here I examine electrolytes studied by the Malatesta group, which belong to the 3-1, 1-3, 2-3, and 3-3 valence families, and analyze their E and γ_± data using the Smaller-ion Shell (SiS) theoretical treatment ("DH-SiS") of strong electrolyte solutions (Fraenkel, J. Chem. Theory Comput. 2015, 11, 178 and 193 ). The DH-SiS physical model incorporates all three ion-size parameters of a binary ionic solution as the "true" ion-ion collision distances, and leads to an improved DH-like electrostatic theory of ionic activity. Correcting Malatesta's data by better extrapolation of E to zero concentration results in a more accurate "standard potential", E°; this affords improved γ_± values that (1) fit well with the DH-SiS equations and (2) agree with the DH limiting law.

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.

The effect of concentration- and temperature-dependent dielectric constant on the activity coefficient of NaCl electrolyte solutions

Our implicit-solvent model for the estimation of the excess chemical potential (or, equivalently, the activity coefficient) of electrolytes is based on using a dielectric constant that depends on the thermodynamic state, namely, the temperature and concentration of the electrolyte, ε(c, T). As a consequence, the excess chemical potential is split into two terms corresponding to ion-ion (II) and ion-water (IW) interactions. The II term is obtained from computer simulation using the Primitive Model of electrolytes, while the IW term is estimated from the Born treatment. In our previous work [J. Vincze, M. Valiskó, and D. Boda, "The nonmonotonic concentration dependence of the mean activity coefficient of electrolytes is a result of a balance between solvation and ion-ion correlations," J. Chem. Phys. 133, 154507 (2010)], we showed that the nonmonotonic concentration dependence of the activity coefficient can be reproduced qualitatively with this II+IW model without using any adjustable parameter. The Pauling radii were used in the calculation of the II term, while experimental solvation free energies were used in the calculation of the IW term. In this work, we analyze the effect of the parameters (dielectric constant, ionic radii, solvation free energy) on the concentration and temperature dependence of the mean activity coefficient of NaCl. We conclude that the II+IW model can explain the experimental behavior using a concentration-dependent dielectric constant and that we do not need the artificial concept of "solvated ionic radius" assumed by earlier studies.

Theoretical analysis of aqueous solutions of mixed strong electrolytes by a smaller-ion shell electrostatic model

In spite of the great importance of mixed electrolytes in science and technology, no compelling theoretical explanation has been offered yet for the thermodynamic behavior of such systems, such as their deviation from ideality and the variation of their excess functions with ionic composition and concentration. Using the newly introduced Smaller-ion Shell treatment - an extension of the Debye-Hückel theory to ions of dissimilar size (hence DH-SiS) - simple analytic mathematical expressions can be derived for the mean and single-ion activity coefficients of binary electrolyte components of ternary ionic systems. Such expressions are based on modifying the parallel DH-SiS equations for pure binary ionic systems, by adding to the three ion-size parameters - a (of counterions), b+ (of positive coions), and b- (of negative coions) - a fourth parameter. For the (+ + -) system, this is "b++," the contact distance between non-coion cations. b++ is derived from fits with experiment and, like the other b's, is constant at varying ion concentration and combination. Four case studies are presented: (1) HCl-NaCl-H2O, (2) HCl-NH4Cl-H2O, (3) (0.01 M HX)-MX-H2O with X = Cl, Br, and with M = Li, Na, K, Cs, and (4) HCl-MCln-H2O with n = 2, M = Sr, Ba; and n = 3, M = Al, Ce. In all cases, theory is fully consistent with experiment when using a of the measured binary electrolyte as the sole fitting parameter. DH-SiS is thus shown to explain known "mysteries" in the behavior of ternary electrolytes, including Harned rule, and to adequately predict the pH of acid solutions in which ionized salts are present at different concentrations.

Electrolytic nature of aqueous sulfuric acid. 1. Activity

According to the literature, when H(2)SO(4) dissolves in water, (1) it retains its molecular formula and tetrahedral structure of two O atoms and two OH groups bonded to a central S atom, and (2) it ionizes partially, as a 1-1 electrolyte, to H(+) (H(3)O(+)) and HSO(4)(-); the latter ion further dissociates at low concentrations (<0.1 M) to H(+) and SO(4)(2-). Using the Debye-Hückel (DH) limiting law at very low concentration, and the smaller-ion shell (SiS) model of strong electrolyte solutions-an extension of the DH model for ion size dissimilarity-up to moderate concentration, I examine the theory-experiment fit of the mean ionic activity coefficient (γ(±)) of the acid as a function of concentration (at 0 to ∼6 m) and of temperature (at 0-60 °C). The fit is impossible if H(2)SO(4) in water is assumed to be a 1-1 or 1-2 electrolyte, but is excellent when the acid is treated instead as a strong 1-3 electrolyte; that is, aqueous sulfuric acid behaves as a fully dissociated H(3)A acid. At 25 °C, the SiS best fit is achieved with the H(+) diameter being 1.16 Å (as obtained for strong mineral 1-1 protonic acids) and with the A(3-) ionic diameter being 5.77 Å. On the basis of the present study, H(2)SO(4) in water may be H(4)SO(5) (dubbed "sulfoxuric", or parasulfuric acid) completely ionized to 3H(+) and the ("bisulfoxate", or parabisulfate) anion HSO(5)(3-). The calculated standard potential of a newly proposed half-cell reaction, H(2) + HSO(5)(3-) ↔ H(+) + SO(4)(2-) + H(2)O + 2e(-), at 25 °C, is -1.0933 V.

Single-ion activity: experiment versus theory

The almost century-old dispute over the validity of the experimentally derived activity of a single ion, a(i), is still unsettled; current interest in this issue is nourished by recent progress in electrochemical cell measurements using ion-specific electrodes (ISEs) and advanced liquid junctions. Ionic solution theories usually give expressions for a(i) values of the positive and negative ions, that is, the respective a(+) and a(-), and combine these expressions to compute the mean ionic activity, a(±), that is indisputably a thermodynamically valid property readily derivable from experiment. Adjusting ion-size parameters optimizes theory's fit with experiment for a(±) through "optimizing" a(+) and a(-). Here I show that theoretical a(i) values thus obtained from the smaller-ion shell treatment of strong electrolyte solutions [Fraenkel, Mol. Phys. 2010, 108, 1435] agree with a(i) values estimated from experiment; however, theoretical a(i) values derived from the primitive model, the basis of most modern ionic theories, do not agree with experiment.