On the rate constant for reactions between ions in solution: a simple unification of the Born and Debye-Hückel models

The rate of reaction between ions in solution depends on solvent properties like permittivity and ionic strength. The influence of a solution's ionic strength is described by the Brønsted-Bjerrum equation. We show how this equation can be derived directly from transition-state theory without introducing the concept of activity coefficients. The electrostatic contribution to the activation energy is extracted from the electric potential of an ion, according to the Debye-Hückel theory. The derivation also gives a contribution to the activation energy, which corresponds to the Born model for ions represented as spherical charged particles. That is, for a given solvent permittivity, the dependence on ion radii is also displayed. This approach provides enhanced physical insights into the description of reactions where ions are participating.