Conductivity of concentrated salt solutions

TL;DR

The paper tackles the century-old puzzle of predicting ionic conductivity in concentrated salt solutions, where dilute-solution theories fail. It introduces a minimal macroscopic model combining a mean-field electrostatic description with colloid electrophoresis, treating ions as finite-radius centers with Debye screening clouds. A compact analytical relation for the relative conductivity $K/K_0$ is derived, depending only on the harmonic mean radius $R_h$ and concentration via a universal scaling with $R_h\sqrt{c_\infty}$, and it aligns with experimental data for univalent salts up to several mol/L. The study concludes that conductivity can be interpreted without invoking ion pairing, with relaxation corrections important mainly at very low dilution and negligible at higher concentrations, and highlights potential extensions to multivalent salts and temperature effects.

Abstract

The conductivity of concentrated salt solutions has posed a real puzzle for theories of electrolytes. Despite a quantitative understanding of dilute solutions, an analytical theory for concentrated ones remains a challenge for almost a century, although a number of parameters and effects incorporated into theories increases with time. Here we show that the conductivity of univalent salt solutions can be perfectly interpreted using a simplest model that relies on a modified mean-field description of electrostatic interactions and on a classical approach to calculating colloid electrophoresis. We derive a compact equation, which predicts that the ratio of conductivity to that at an infinite dilution is the same for all salt and depends only on a product of the harmonic mean of ion hydrodynamic radii and the square root of concentration. Our equation fits very well the data for inorganic salts (up to a few mol/l), although at a very high dilution the relaxation correction seems necessary.

Figures (7)

• Figure 1: Sketch of the charge distribution in the (a) real electrolyte solution, (b) Debye-Hückel classical model of finite size ions with the charges located at their center, and (c) model of ions with neutral interior, but constant surface charge densities.
• Figure 2: Sketch of the cation and anion with hydrodynamic radii $R_+$ and $R_-$ in the bulk electrolyte solution characterized by $\lambda_D$. The cation propels with the speed $V_+$ in the direction of the electric field $E$. Its zeta potential $\zeta_+$ (or dimensionless mobility) is positive. The anion of negative $\zeta_-$ migrates against the field with the velocity $V_-$.
• Figure 3: Function $\mathcal{F_{\pm}}/(1+\rho_{\pm})$ vs $\rho_{\pm}$ with $\mathcal{F_{\pm}}$ calculated from Eqs. \ref{['eq:henry']} (solid curve), \ref{['eq:henry_lin']} (dashed curve), and $\mathcal{F_{\pm}} = 2/3$ (dash-dotted curve). Dotted curve shows the electrophoretic term derived in avni.y:2022.
• Figure 4: Zeta potentials $\zeta_{\pm}$vs.$c_{\infty}$ computed for K$^{+}$, Li$^{+}$, Cl$^{-}$, I$^{-}$ (solid curves from top to bottom). Dashed lines are obtained using Eq. \ref{['eq:zeta_ion_max']}. Symbols show calculations from Eq. \ref{['eq:zeta_ion+2']}.
• Figure 5: $K/K_{0}$ as a function of $c_{\infty}$ for KBr, NaCl and LiI aqueous solution calculated from Eqs. \ref{['eq:MM0']}, \ref{['eq:K_appr']} and \ref{['eq:onsager']} (solid, dash-dotted and dashed curves). Open and filled circles indicate experimental data from vanysek.r:2000 and miller.dg:1966, open and filled squares show data from dobos.d:1975 and lobo.vmm:1984.