Thermodynamic decomposition of Debye-Hückel activity coefficients: resolving attractive, repulsive, and entropic...

Ionic activity coefficients are central to the thermodynamics of electrolyte solutions, yet their thermodynamic structure at the level of individual ions is rarely made explicit. In this work, Debye-Hückel theory is formulated within classical partial molar thermodynamics, starting from the linearised Poisson-Boltzmann free energy functional. The resulting framework provides an explicit decomposition of the excess free energy into attractive and induced-repulsive energetic contributions, as well as an entropic contribution. These contributions are thermodynamically consistently split and defined at the level of excess partial molar quantities. This formulation reveals physical information that remains implicit in the conventional global free energy construction, while preserving the classical Debye-Hückel result. The derived expressions reproduce both the limiting law and extended Debye-Hückel equations, and establish a transparent thermodynamic connection between the mean-field description of ionic atmospheres and activity coefficients. More generally, the framework provides a systematic basis for analysing energetic and entropic contributions to electrolyte thermodynamics within mean-field theory.