The final term is the excess free energy contribution, which comes from Debye-Hückel theory. The quantity κ = (4πλ B α ρ α z 2 α ) 1/2 is the Debye screening length of an electrolyte with species α with number concentration ρ α and valency z α . 71 λ B = e 2 /(4π 0 r k B T ) is the Bjerrum length, where e is the elementary charge, 0 is the permittivity of free space, r is the relative dielectric constant, and T is the temperature. The classical derivation of Debye-Hückel theory is found in most statistical mechanics textbooks, but yet it is instructive to demonstrate how this expression emerges from the cluster expansion approach that was originally used by To show how to perform this derivation, we re-write the f -bond by splitting up the energy into a hard-sphere portion and a Coulomb portion, βu α,γ = βu s + βu c,α,γ : 119