Keywords: electrochemistry, diffusion, semiintegration, semidifferentiation, fractional-operators, electrodes, concentration, flux.

Fick's laws govern the diffusion of chemical species in aqueous solution and other media. In one dimension, Fick's second law for the diffusion of a species S takes the form of the partial differential equation

Here c_S, a function of both distance x and time t, is the concentration of species S. D_S is a characteristic constant, the diffusivity of S. Consider conditions in which, prior to the time t=0, species S is uniformly distributed throughout the medium, with concentration c_S^b, then Fick's second law can be written as

In this format, the operator acts on the concentration difference [c_S-c_S^b], generating zero. In the spirit of the fractional calculus, we can split the operator into two subsidiary operators, acting sequentially:

It is now pertinent to enquire whether there is any circumstance in which the first of these subsidiary operators, acting alone, can convert the concentration difference [c_S-c_S^b] into zero:

It turns out that all that is needed to validate this simplification is for the diffusion field to be semiinfinite. In any such case, Fick's second law, which is second-order in distance and first-order in time, can be replaced by relationship (4), which is first-order in distance and half-order in time. Moreover, one can combine Fick's first law with equation (4) into

where j_S is the flux density of species S.

In a typical electrochemical experiment, a reaction such as

occurs at its an electrode. Species S, the substrate, reaches the electrode by diffusion and species P, the product, leaves by the same diffusive mechanism. The electrons that are liberated constitute the flow of electricity that is manifested as an electric current I. Equation (6) shows that the flux densities of all three participants in the reaction must be mutually proportional at the electrode surface. In fact:

In expression (7), A is the electrode area and F is Faraday's constant (96485 C mol^-1). Representing the concentration of species S at the electrode surface by c_S^s, combination of this result with equation (5) leads to

with a similar result for the product species P. In words, equation (8) reveals that the current flowing through the electrochemical cell is related in a simple fashion to the semiderivative of the concentration of species S at the electrode. An even more useful result comes from semiintegrating equation (8) and its P-analogue. One finds that

showing how measuring the electrochemical current I and semiintegrating it permits one to access to the concentrations of the species at the electrode surface. This ability has proved to be a great asset to electrochemists.

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