Adsorption controlled by transport

Feldberg (1972) has discussed this, in conjunction with redox processes we isolate adsorption here. If the adsorption step is fast (most often the case) then there is always equilibrium between the surface coverage (r, or 0) and the solution adjacent to the electrode, that is, Cq. The equilibrium is the appropriate adsorption isotherm, see Sect. 2.9, Eqs. 2.42 to 2.48, for example. The simulation then goes as follows at a given time t we have a known concentration profile, Cq, c, etc., from which (Eq. 4.9) g can be computed. This gradient causes a flux of the substance into or out of the adsorbed layer, expressed by Eq. 2.50 or 2.52, which can be discretised (dimensionless) into 

If the adsorbed substance is initially in solution at some concentration Cj then there will be a maximum 0 value, corresponding to that concentration for the adsorption isotherm in any case, 0 must never exceed unity. Because of accumulated errors or in order to know when to stop simulating, one may wish to test 0(T) against this maximum value, and it will thus be necessary to evaluate it. For some isotherms (e.g. the Frumkin isotherm) this is not straightforward, since they are almost all explicit for Cq, not 0. In these cases, a numerical solution must be carried out - for example, a search for that 0 that satisfies the isotherm for Cj.  

Another curious point is the length of time, in units of dimensionless T, the simulation may need to be run, and the dynamics of the concentration profile. Writing Eq. 2.54 in the form 

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