The statistics of the Debye–Hückel limiting law

The Debye–Hückel Limiting Law (DHLL) correctly predicts the thermodynamic behavior of dilute electrolyte solutions. Most articles and books explain this law using Peter Debye and Erich Hückel’s original formalism of linearizing the Poisson–Boltzmann equation for a simple electrolyte model. Brilliant in its own right, this approach does not fully explain which microstates contribute in the range of the Debye–Hückel theory. Notably, the original formalism does not establish the Energy Multiplicity Distribution (EMD), which is the energy distribution of a system’s microstates. This work establishes an analytical expression for the EMD that satisfies the DHLL. Specifically, an EMD that is proportional to exp(aUel3) satisfies the DHLL for a monovalent electrolyte solution. Here, Uel is the effective electrostatic energy due to ion–ion interactions. The proposed proportionality shows quantitative agreement with the simulated EMDs of a Coulomb lattice gas that corresponds to an aqueous sodium chloride solution at a concentration of 3.559 × 10−4 M. The lattice gas that is used does not incorporate solvent molecules, but the Coulomb interactions are scaled through a permittivity that emulates the solvent—similar to the Debye–Hückel theory. Moreover, this work explains the proportionality by partitioning Uel into a set of energy contributions using minimal spanning graphs. This discussion on the EMD is new in the field. It widens the scope of the Debye–Hückel theory and could lead to a new parameterization option for developing equations of state.