Cube-root organization of electrolyte activity coefficients: experimental trends and an electrostatic framework

The concentration dependence of mean ionic activity coefficients is conventionally described using the Debye-Hückel (DH) square-root scaling, although historical treatments by Bjerrum and Malmström proposed a cube-root dependence. Here, experimental datasets for multiple electrolytes are examined in both square-root and cube-root concentration coordinates. Over experimentally accessible finite-concentration ranges, cube-root coordinates produce approximately linear trends across several salts and independent datasets. Successful finite-concentration descriptions retain the same organization. Pitzer-Mayorga curves for several 1 : 1 electrolytes were nearly linear in cube-root coordinates over approximately 1-200 mM, with R² ≥ 0.995. Meanwhile, a nonlinear Poisson-Boltzmann benchmark for a 1 : 1 electrolyte had a cube-root alignment of R² ≥ 0.997 up to 1 M. The corresponding values were R² ≥ 0.957 for 2 : 2 and 3 : 3 electrolytes in the 1-60 mM range. To investigate a possible physical origin of this behavior, a Malmström-Bjerrum cube-root (MBC) framework is derived from Boltzmann-weighted electrostatic interactions at separations determined by the characteristic interionic spacing, ℓ ∝ c^-1/3. The model generates leading cube-root and constrained higher-order concentration terms and reproduces activity-coefficient data for multiple electrolytes up to concentrations approaching 1 M using one fitted effective ion-size parameter. The results support interionic spacing as a natural reduced coordinate for finite-concentration electrolyte nonideality.