The Activity Coefficients of High-Charge Electrolytes in Aqueous Dilute Solutions
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The Activity Coefficients of High‑Charge Electrolytes in Aqueous Dilute Solutions Francesco Malatesta1 Received: 26 March 2020 / Accepted: 2 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 socalled 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. Keywords Activity coefficient · Electrolyte solutions · Debye–Hückel theory · Bjerrum theory · Mayer theory
1 Introduction In 2018, Fraenkel published a paper [1] that reexamined the activity coefficients of several high-charge electrolytes using his DH-SiS theory [2]. He reached the conclusion that the negative deviations from the Debye–Hückel limiting slope (LL), which have been observed for many years in dilute solutions (ionic strengths, I 2 Si Ii/2 to be negligible, one expects a quasi-linear trend of ln γ’± – LL plotted vs. I, with intercept const (of course, this supposition is suspect in the case of negative deviations from the LL). In reality, the method is problematic also in absence of negative deviations, since the omitted residual terms often generate trends whose slope varies continuously and does not permit linear extrapolation of const. Figure 5 shows the situation observed with Mn(ClO4)2, whose relative activity coefficients have been measured with high precision down to m = 5.739 × 10–5 mol·kg–1 using a cell with permselective liquid membranes [7]. The reference solution was 9.183 × 10–4 mol·kg–1; therefore, in this case const means the value 5
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.
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