Transport kinetics, semi-infinite diffusion

The basic strategy in the application of electroanalytical methods in studies of the kinetics and mechanisms of reactions of radicals and radical ions is the comparison of experimental results with predictions based on a mechanistic hypothesis. Thus, equations such as 6.28 and 6.29 have to be combined with the expressions describing the transport. Again, we restrict ourselves to considering transport governed only by linear semi-infinite diffusion, in which case the combination of Equations 6.28 and 6.29 with Fick s second law, Equation 6.18, leads to Equations 6.31 and 6.32 (note that we have now replaced the notation for concentration introduced in Equation 6.18 earlier by the more usual square brackets). Also, it is assumed here that the diffusion coefficients of A and A - are the same, i.e. DA = DA.- = D. 

In the description of adsorption, not only static (equilibrium) features but also kinetic aspects are of importance and we shall consider some of these aspects in sec. 1.5a. Some Information is already available in secs. I.6.5d and e. where diffusion transport to and from surfaces has been discussed. One of the more Important equations that we derived is (1.6.5.36), describing the increase of r with time t due to (semi-infinite) diffusion onto a flat surface. Replacing the concentration c in that equation by the number of moles n per volume V, we have for an ideal gas c = p/RT, so that [1.6.5.36) can be written as... 

In the real world, of course, no medium actually extends to infinity. However, infinite or semi-infinite boundary conditions are fully appropriate for many finite situations in which the length scale of the diffusion is much smaller than the thickness of the material. In such cases, the material appears infinitely thick relative to the scale of the diffusion—or, in other words, the diffusion process never reaches the far boundaries of the material over the relevant time scale of interest. Since typical length scales for solid-state diffusion processes are often on the micrometer scale, even diffusion into relatively thin films can often be treated using semi-infinite or infinite boundary condition approaches. Semi-infinite and infinite tfansient diffusion has therefore been widely applied to understand many real-world kinetic processes—everything from transport of chemicals in biological systems to the doping of semiconductor films to make integrated circuits. 

Reinmuth has examined chronopotentiometric potential-time curves and proposed diagnostic criteria for their interpretation. His treatment applies to the very limited cases with conditions of semi-infinite linear diffusion to a plane electrode, where only one electrode process is possible and where both oxidized and reduced forms of the electroactive species are soluble in solution. This approach is further restricted in application, in many cases, to electrode processes whose rates are mass-transport controlled. Nicholson and Shain have examined in some detail the theory of stationary electrode polarography for single-scan and cyclic methods applied to reversible and irreversible systems. However, since in kinetic studies it is preferable to avoid diffusion control which obscures the reaction kinetics, such methods are not well suited for the general study of the mechanism of electrochemical organic oxidation. The relatively few studies which have attempted to analyze the mechanisms of electrochemical organic oxidation reactions will be discussed in detail in a following section. 

Semi-infinite linear diffusion conditions The rate of an electrochemical process depends not only on electrode kinetics but also on the transport of species to/from the bulk solution. Mass transport can occur by diffusion, convection or migration. Generally, in a spectroeiectrochemicai experiment, conditions are chosen in which migration and convection effects are negligible. The solution of diffusion equations, that is the discovery of an equation for the calculation of oxidized form [O] and reduced form [R] concentrations as functions of distance from electrode and time, requires boundary conditions to be assumed. Usually the electrochemical cell is so large relative to the length of the diffusion path that effects at walls of the cell are not felt at the electrode. For semiinfinite linear diffusion boundary conditions, one can assume that at large distances from the electrode the concentration reaches a constant value. 

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