Ion–ion association is lost by linearizing the Poisson–Boltzmann equation when deriving the Debye–Hückel equation.

This article investigates the ion–ion association phenomenon in electrolyte solutions by analyzing the difference between the full nonlinear Poisson–Boltzmann (PB) equation and its linearized version (LPB). It demonstrates that the ion association constant, specifically the Ebeling association constant, can be derived from the higher-order electrostatic terms lost during linearization, linking ion pairing to these nonlinear effects. The study compares several models for mean ionic activity coefficients (MIACs), showing that incorporating ion association via the mass-action law (MAL) into the LPB framework (LPBα+MAL model) closely approximates the full PB results at very dilute conditions, while also explaining the negative deviations from the Debye–Hückel limiting law observed in electrolytes with small, highly charged ions. The findings suggest that neglecting either ion pairing or higher-order electrostatic interactions leads to incomplete thermodynamic descriptions of electrolyte solutions.