Multi-ion case steady-state concentration profiles

4 MULTI-ION CASE STEADY-STATE CONCENTRATION PROFILES 

Transport by combined migration—diffusion in a finite planar geometry can achieve a true steady state when only two ions are present, as we saw in Sect. 4.2. The same holds true when there are three or more ions present. Under simplifying conditions [see eqn. (89) below], it is possible to predict the steady-state behaviour with arbitrary concentrations of many ions. However, the corresponding transient problem is much more difficult and we shall not attempt to derive the general transient relationship, as we were able to do in deriving eqn. (82) in the two-ion case. 

Since all time dependencies disappear in the steady state, the general equations (53) and (54) reduce to 

Recall that ct(x) and n(x) are the local total ionic concentration and the local ionic strength, respectively. Moreover we can argue, as we did in Sect. 4.2, that each Jj(x) must be independent of x and that all these terms must equal zero, except (x) which equals the constant — i/ZyAF. Hence 

Because the right side of eqn. (86) is a positive constant, it follows that the total ionic concentration, ci is a linear function of distance, as shown in Fig. 23. By integration of this equation, one can readily show that 

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