From the Beginning to Bjerrums Model

The history of the ion-pair concept begins in the 1880s [50] with the theory of Arrhenius that described the electrolytic dissociation in solution as a function of electrolyte concentration and nature. This theory was completely eclipsed by the success of the D-H treatment of electrolyte activity in the 1920s [33] that also paved 

For distances larger than q, the usual D-H treatment applies to ions that are considered free. The cut-off distance is reasonable, because at that separation distance, the thermal energy is half the work necessary to separate the ion-pair. The ions in the couple cannot get closer than a distance of closest approach, a, which is at least the sum of the crystal radii, but it is not larger than the sum of the solvated ion radii. Since the neat charge of the duplex is zero, the ion-pair is not acted upon by coulombic fields, it does not migrate, and it does therefore not contribute to the electrical conductivity of the solution. 

However, as an ionic dipole, it may rotate in an external electrical field. On the average, a precise population of these duplexes exists, although the formation and dissociation of ion-pairs are incessant. The degree of association can be easily obtained if the medium dielectric constant and the ion sizes, and the electroly tel concentration 

If a is the fraction of free ions and (1 - a) is the fraction of paired ions we have 

The contribution of the coulombic interaction to the fonnatirm of stable ion-pairs is by far the most important and for a long time was regarded as the only one. Denison and Ramsey [56] put forth a thermodynamic approach to ion-pairing that considers only ions in contact to be ion-pairs this is tantamount to note the cut-off distance asR = a.A Bom cycle was then used to calculate the Gibbs energy of separating ions from contact to infinity and the equilibrium sanctions the following association equilibrium constant 

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