Current Equations - Mass Transport
Given the reduction reaction
at a cathode, we know that at a given applied potential the rate and thus the current density of the reaction is proportional to the concentration of the species being reduced, O. If the reduced product R remains in solution and the reaction is reversible a simultaneous anodic current density will be a function of the concentration of R at the electrode surface.
If we assume an initial concentration of species O in the solution, cO, and an initial concentration of the product R as zero (cR = 0) and an initial current density of zero, we have the bulk concentration of O in the bulk of the solution, cO(bulk), at the electrode surface. There is no net current and no net reaction. If we step the potential to a more negative potential the reduction reaction will be initiated. After the initial (and brief) charging current required to charge the electrode, the rate will be a function of the potential and the concentration of O. Immediately the concentration of O at the electrode surface cO(x=0) will begin will begin to decrease as O is reduced to R. Thus the forward rate will decrease with time as cO(x=0) decreases. Also the anodic component of the current will increase with the appearance of R at the electrode as the reaction proceeds.
At this time a concentration gradient will have developed for both O and R and diffusion begins. For now we will ignore the contribution of migration where mass transfer of charged species is a function of the electric field or potential gradient. The concentration gradient of species O shortly after the potential step is shown in Figure 1. It is assumed that the solution is quiescent (not stirred).
Figure 1.
Provided that the potential is only so negative a condition will be reached where rate of the reduction reaction (electron transfer) matches the flux of O by diffusion to the electrode surface. This is not a steady state as the concentration gradient profile will change over time. As the gradient decreases the flux will decrease. Thus the current density will decrease with time.
As the electrode potential is shifted more negative a potential will be reached where any O that arrives at the the electrode surface x=0 will immediately be reduced. The reaction rate will then be limited entirely by the flux of O to the electrode. The current density is diffusion controlled at this point. Shifting the potential more negative will not increase the current unless it initiates another reaction. In this case as well the concentration gradient profile will change with time (Figure 2).
Figure 2.
Here too the current density will decrease with time as the concentration gradient decreases. Again, we are assuming no convection. If the solution is stirred or is flowing the depth of the diffusion region will be limited relatively constant.