Two-ion case steady state

When the electrolyte contains only one ion, necessarily an anion, in addition to the electroactive ion M +, the electroneutrality condition reduces to the simple 

The results attaching to the steady state may be deduced as special cases of the equations developed in Sect. 4.4, but it is instructive to develop these results ab initio for this so-called unsupported case. To begin, we shall not even assume the Nernst—Einstein relationship (Sect. 2.5) between the diffusion coefficient and mobility. 

Each ion experiences both a migratory and a diffusive flux and therefore 

This equation applies to ion j = 1 or / = 2, both in the steady state and in the transient state that precedes it. In the ultimate steady state, all dependence on time must have disappeared, so the general result reduces to 

One can argue from the conservation equation (26) that, in the steady state, because concentrations do not change with time, the fluxes do not change with distance. Hence Jj(x) in eqn. (59) is a constant, independent of x. The same conclusion can be arried at non-mathematically by thinking carefully about the implications of a steady state. Since Js x) is not a function of x, it must equal its value at x = 0 or L, as given in eqns. (55) and (56). 

---