## Abstract

We examine in detail the activity coefficient of higher-charge electrolytes, which, in dilute solutions, can display negative deviations from the Debye–Hückel limiting law instead of the usual positive deviations typical of lower-charge electrolytes. This fact is of considerable relevance for scientists concerned with extrapolation to infinite dilution of thermodynamic and kinetic quantities. It is shown that this “strange” behavior originates merely from the electrostatic interactions between each ion and all other ions, with no necessity of hypothesizing the presence of chemical association; these negative deviations, indeed, are predicted even at the level of the “primitive model” (ions assumed as charged, unpolarizable, rigid spheres inside an unstructured, isotropic, dielectric fluid). Three different approximations for the behavior of the primitive model of low-charge and high-charge electrolytes are tested, in addition to the Debye–Hückel theory; i.e. IPBE (a numerical accurate integration of the Poisson–Boltzmann equation), the Mayer theory of the electrolytes in the so-called DHLL + B2 approximation, and the Bjerrum theory. In the Supporting Information, the fundamentals of the respective algorithms are reported, and the effects produced by the differences of size between cations and anions, are also examined.

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## Notes

DHLL + B2, unlike IPBE and DH, applies consistently also to unrestricted PM (as many distances of closest approach

*a*_{ij}between the centers of two ions*i*and*j*, as the possible combinations of*i*and*j*). The special emphasis on the RPM restriction is to permit comparison with DH and IPBE, which do not admit the use of different*a*_{ij}values. However, as shown in Sect. 4 of the S.I., for solutions of single electrolytes the postulation of different*a*_{ij}values rather than one single*a*has a negligible effect as long as*a*is taken equal to*a*_{+-}, independently of the values of*a*_{++}and*a*_{–}. This observation indicates that for single salts in dilute solutions there exists no perceptible difference between PM and RPM.Mayer's theory is not necessarily limited to PM, it can be extended freely to more realistic descriptions of the short-range interactions. Yet, at the concentrations imposed by the drastic truncation of the cluster series, only suitable in very dilute solutions, any improvement at the level of short range interactions is as effective, as lightening the buttons of the shirt for a man of over 100 kg to make easier his walking.

In the Bjerrum theory,

*d*is a univocal function of charges of ions, temperature, and dielectric constant of the solvent. In water at 298.15 K, Bjerrum values of*d*are: 0.358 nm for 1–1, 0.715 nm for 1–2, 1.073 nm for 1–3, 1.431 nm for 1–4 and 2–2, 2.146 nm for 2–3, 2.861 nm for 2–4, and 3.219 nm for 3–3 electrolytes. When*d*<*a*, no*ion pairs*exist and Bjerrum’s theory exactly coincides with the DH theory.In the present paper the mean activity coefficients are always denoted as

*γ*_{±}. Indeed, the differences between activity coefficients*γ*_{±}(molal scale),*γ*^{'}_{±}(molar scale), and*f*_{±}(rational scale = mole fraction scale) vanish at the dilution levels required for DH, IPBE, DHLL + B2 and BT to agree with the RPM. We selected*γ*_{±}for better consistence with the experimental determinations of activity coefficients, usually performed in solutions of know molality*m*, rather than molarity*C*. Furthermore,*C*and*m*are nearly proportional at high dilutions (*C*≈*m**d*° with*d°*= solvent density), and nearly equal when the solvent is water. Therefore, although the natural variable of ionic interaction theories is*C*rather than*m*, at high dilution the results obtained in function of molar concentrationsor molar ionic strengths apply also to the molal concentrations, or molal ionic strengths. For*C*and*m*we intend generally the dimensionless quantities*C*/(*C*° = 1) and*m*/(*m*° = 1).The single value of

*a*of the RPM is equivalent to the distance of closest approach of cation-to-anion in the PM. Computations applied to PM prove, indeed, that in dilute solutions the effects of the distances of closest approach cation–cation and anion–anion are practically unimportant (see S.I., Sect. 4).Erratum in the S.I. of [13]: page SI 5, line 7, substitute "Mg

^{2+}, ds^{2–}, Na^{+}and Cl^{−}" for "Mg^{2+}, ds^{2–}, K^{+}and Cl^{–}".

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## Acknowledgements

The author is indebted to Chris. Outhwaite and L. Bari Bhuiyan for providing theSPB and MPB data.

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Francesco Malatesta—Retired member, University of Pisa.

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(1) Numerical accurate integration of the Poisson–Boltzmann equation (IPBE); (2) the Mayer theory in the DHLL+B2 approximation; (3) the Bjerrum theory of the "ion pairs"; (4) differences between RPM and PM for single electrolytes; (5) electrolytes 1-2 or 2-1, 1-3 or 3-1, and 1-4 or 4-1; (6) electrolytes 2-3 or 3-2, and 3-3; (7) On Fraenkel's reply to «Comment on 'Negative Deviations from the Debye−Hückel Limiting Law for High-Charge Polyvalent Electrolytes: Are They Real?'»; (8) the source program for DHLL+B2 approximation of Mayer theory. Supplementary file1 (PDF 1053 KB)

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Malatesta, F. The Activity Coefficients of High-Charge Electrolytes in Aqueous Dilute Solutions.
*J Solution Chem* **49**, 1536–1551 (2020). https://doi.org/10.1007/s10953-020-01041-8

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DOI: https://doi.org/10.1007/s10953-020-01041-8