The Debye Onsager Model for Conductivity

In chapter 3, it was shown that the Debye-Hiickel theory for ion-ion interactions is able to account for solution non-ideality in very dilute systems. The same model forms the basis for understanding the concentration dependence of the conductance observed for strong electrolytes. Thus, Onsager [9] showed in 1927 that the limiting conductance law for 1-1 electrolytes has the form 

The influence of the interionic forces is due to two phenomena, namely, the electrophoretic effect and the time-of-relaxation effect. The net ionic atmosphere around a given ion carries the opposite charge and therefore moves in a direction opposite to the central ion. The final result is an increase in the local viscosity, and retardation of the central ion. This is called the electrophoretic effect. The time-of-relaxation effect is also related to the fact that the ionic atmosphere around a given ion is moving and therefore disrupted from its equilibrium configuration. It follows that the ionic atmosphere must constantly be re-formed from new counter ions as the ion under observation moves through the solution. The net effect is that the electrical force on each ion is reduced so that the net forward velocity is smaller. 

The derivation of an expression for the electrophoretic effect is based on the Debye-Hiickel theory described earlier in section 3.8. On the basis of the Boltzmann distribution law, the local concentration of ion i at a distance r from a central reference ion j is 

In keeping with the Debye-Hiickel theory, it is assumed that the electrostatic energy of an individual ion is small with respect to so that only the first term in the expansion need be considered as a result 

For a binary electrolyte with one type of cation and anion, the incremental force is 

---