Linear diffusion to a plane electrode

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. 

Fig. 1.5 — Model for linear diffusion to a plane electrode for the electrode reaction O + nt R. (a) Pick s 1st law, (b) Pick s 2nd law.

So far, we have only considered situations where the mass transport conditions may be described by linear diffusion to a plane electrode. It is now appropriate to consider diffusion to a spherical(i.e. drop) and cylindrical (i.e. wire) electrodes. In order to understand these geometries, however, it is necessary to be clear how Pick s second law is derived for the case of linear diffusion to a plane electrode -the first law is just a commonsense statement that the flux of a species in any direction is proportional to the concentration difference across any plane in space. 

The equivalency found between the behavior of hemisphere and that of disk electrodes also exists between cylinder and band electrodes [29]. Diffusion to a cylinder electrode is linear and described by Equation 12.7, while diffusion to a band is nonlinear. A plane of symmetry passes through the center of the band and normal to its surface, so the nonlinear diffusion process can be broken down into two planar components, one in the direction parallel to the electrode surface, x, and the other in the direction perpendicular to the electrode surface, y. So Fick s second law for a band electrode is... 

This equation reflects the rate of change with time of the concentration between parallel planes at points x and (x + dx) (which is equal to the difference in flux at the two planes). Fick s second law is vahd for the conditions assmned, namely planes parallel to one another and perpendicular to the direction of diffusion, i.e., conditions of linear diffusion. In contrast, for the case of diffusion toward a spherical electrode (where the lines of flux are not parallel but are perpendicular to segments of the sphere), Fick s second law has the form... 

Consider the process of plating copper on a plane electrode. Near the electrode, copper ions are being discharged on the surface and their concentration decreases near the surface. At some point away from the electrode, the copper ion concentration reaches its bulk level, and we obtain a picture of the copper ion concentration distribution, shown in Fig. 6. The actual concentration profile resembles the curved line, but to simplify computations, we assume that the concentration profile is linear, as indicated by the dashed line. The distance from the electrode where the extrapolated initial slope meets the bulk concentration line is called the Nernst diffusion-layer thickness S. For order of magnitude estimates, S is approximately 0.05 cm in unstirred aqueous solution and 0.01 cm in lightly stirred solution. 

Substrate transport through the film may be formally assimilated to membrane diffusion with a diffusion coefficient defined as12 Ds = Dch( 1 — 9)/pjort. In this equation, the effect of film structure on the transport process in taken into account in two ways. The factor 1—0 stands for the fact that in a plane parallel to the electrode surface and to the coating-solution interface, a fraction 9 of the surface area in made unavailable for linear diffusion (diffusion coefficient Dcj,) by the presence of the film. The tortuosity factor,, defined as the ratio between the average length of the channel and the film thickness, accounts for the fact that the substrate... 

Compared to conventional (macroscopic) electrodes discussed hitherto, microelectrodes are known to possess several unique properties, including reduced IR drop, high mass transport rates and the ability to achieve steady-state conditions. Diamond microelectrodes were first described recently diamond was deposited on a tip of electrochemically etched tungsten wire. The wire is further sealed into glass capillary. The microelectrode has a radius of few pm [150]. Because of a nearly spherical diffusion mode, voltammograms for the microelectrodes in Ru(NHy)63 and Fe(CN)64- solutions are S-shaped, with a limiting current plateau (Fig. 33a), unlike those for macroscopic plane-plate electrodes that exhibit linear diffusion (see e.g. Fig. 18). The electrode function is linear over the micro- and submicromolar concentration ranges (Fig. 33b) [151]. 

Using a simple redox couple. Fig. 3.5 depicts the voltammetry obtained when using either a Basal Plane Pyrolytic Graphite (BPPG) (i) or (ii) an EPPG electrode of HOPG, and the responses are compared with numerical simulations (iii) assuming linear diffusion only, in that, all parts of the electrode surface are uniformly (incorrectly) electrochemically active. Two features of Fig. 3.5 are to be... 

The -vk contribution to / in Eq. (26) from the desorbing water molecules can be roughly estimated by assuming that the potential varies linearly with the distance x in the compact layer enclosed between the electrode surface plane x = 0 and the outer Helmholtz plane x = d. In the presence of a strong excess of a nonspecifically adsorbed supporting electrolyte or upon correction for the potential difference across the diffuse layer, the electric potential in the bulk... 

Because r — tq is the distance from the electrode surface, this profile strongly resembles that for the linear case (equation 5.2.12). The difference is the factor r /r and, if the diffusion layer is thin compared to the electrode s radius, the linear and spherical cases are indistinguishable. The situation is directly analogous to our experience in living on a spherical planet. The zone of our activities above the earth s surface is small compared to its radius of curvature hence we usually cannot distinguish the surface from a rough plane. 

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