where r i is the hydrated ionic radius, i.e. the distance of the closest approach of ions to the surface. The proton in the bulk has free energy µ H . After the protonation, there is a chemical free energy gain associated with the formation of a covalent bond between the proton and the site. This energy can be expressed in terms of the acid dissociation constant K a as k B T ln K a Λ 3 H . Furthermore when proton becomes bound to the site, it still interacts electrostatically with all the other deprotonated sites as well as with all the other ions inside the solution, resulting in an electrostatic energy qφ. Therefore,