Limiting Current
If our dissolved reactant is being consumed or transformed by electron transfer at the electrode the concentration near the electrode is diminished. A concentration gradient dc/dx will form. Given a reasonably high exchange current density the reaction rate and thus the current may become limited by rate at which the reactant arrives at the electrode by diffusion. The diffusion limited current density is given by
where F is the Faraday constant in coul mol-1, D is the diffusion coefficient in cm2 sec-1, and dc/dx is in mol cm-4. The current density i is in coul · cm-2 · sec-1.
(coul · mol-1)(cm2 · sec-1)((mol · cm-3)(cm-1)) = coul · cm-2 · sec-1.
In the presence of convection, stirring for example, the bulk concentration will be maintained up to the hydrodynamic stagnant layer at the surface of the electrode. It is in this stagnant layer that the concentration gradient exists. Although there is no sharp distinction between the stagnant and moving regions an approximation is used to give a definite linear quantity to this layer called the Nernst diffusion layer and symbolized by d. This is illustrated in Figure 1.
Figure 1.
If we assume this layer has a constant value as determined by the mechanics of the apparatus then we can see that the concentration gradient will be at a maximum when the concentration at the electrode (x=0) is zero. At this point the rate of diffusion has reached a maximum. This is the maximum diffusion limited current density.
The Butler -Volmer equation seems to indicate that with increasing field strength the current will increase without limit. Of course this is not right as the reaction soon becomes limited by the rate at which the reactant arrives at the electrode. The limiting current density can be written as
which is obviously at a maximum when the concentration at x=0 is zero. In this case the concentration goes from the bulk concentration to zero. The relationship between the Butler-Volmer equation curve and the effect of diffusion limited current is shown in Figure 2.
Figure 2.
In practice it is not easy to maintain a diffusion layer of constant thickness over time. The rotating disk electrode provides control over this.