Levich equation and

It can be shown by rearranging the Levich equation and inserting via the Butler-Volmer equation that the Koutecky-Levich equation takes the following form ... 

Albery and Hadgraft [16] have taken the trouble to validate aspects of the procedures and mathematics of RDC work, particularly the applicability of the Levich equation and rotating disk hydrodynamics. The work reported here confirms and extends their conclusions, by explicit discussion of concentration dependence, and use of a wider range of substances than carboxylic acids and nicotinate esters. This contrasts with results from... 

Diffusion coefficient of the substrate (Dg) and diffusion coefficient of the electron-exchange (D ) were calculated from cyclic and disk current voltammograms by using the Koutecky-Levich equation and Fick s first law (14, 15) (Table II). Dg in the polymer domains was estimated as 10 - 10 cm /sec, much smaller than in solution (10 cm /sec). Dg is affected by charge density of the polymer domain, e.g., the diffusion of cations is suppressed in the positively charged domain composed of cationic polyelectrolyte, while anions moves faster. A larger Dg value was observed, of course, for the porous film and not for the film with high density. On the other hand, Dg in the polymer domain was also very small, i.e. 10" - 10" cm /sec. This may be explained as follows. An electron-transfer reaction always alters the... 

RDE is a commonly used technique for investigating the ORR in terms of both the electron transfer process on electrode surface and diffusion—convection kinetics near the electrode. To make appropriate usage in the ORR study, fundamental understanding of both the electron transfer process on electrode surface and diffusion—convection kinetics near the electrode is necessary. In this chapter, two kinds of RDE are presented, one is the smooth electrode surface, and the other is the catalyst layer-coated electrode. Based on the electrochemical reaction 0 + ne R), the RDE theory, particularly those of the diffusion—convection kinetics, and its coupling with the electron-transfer process are presented. The famous Koutecky—Levich equation and its... 

This expression is known as the Levich equation, and it provides an excellent test that the current is entirely mass transport controlled a plot of /vs should be linear and pass through the origin, and the slope of such a plot may be used to estimate the diffusion coefficient for the electroactive species, e.g. Fig. 4.10. Except when a chemical reaction limits the current density, the Levich equation will describe the rotation rate dependence of the anodic and cathodic limiting currents at high positive and high negative overpotentials respectively. 

Derive the Levich equation for the limiting current at the rotating disk electrode [based on combining equations (4-4) and (1-12)]. 

Although, in theory, the Koutecky-Levich equation can be applied to estimate n y and k at any part of the voltammogram (provided that the conditions stated above are satisfied), for practical reasons only limiting (plateau) currents can be acquired with adequate reproducibility to yield suitable Koutecky-Levich plots. 

At the RDE the velocity profile obtained by Karman and Cochran (see ref. 124) and depicted in Fig. 3.68a leads via solution of its differential convection-diffusion equation to the well known Levich equation ... 

In work by Okada et al. (03) on a rotating-disk flow, Eqs. (10a) and (10b) in Table VII, the electrolyte was completely enclosed between the rotating disk and the counterelectrode. Mass transfer was measured at the rotating as well as at the stationary disk, and the distance between disks was varied. At low rotation rates, the flux at the rotating disk was higher than predicted by the Levich equation, Eq. (la) in Table VII. The flux at the stationary disk followed a relation of the Levich type, but with a constant roughly two-thirds that in the rotating-disk equation. 

Equation (2.161) expresses the relative contributions of mass transport and kinetics to the observed current and is one expression of the Koutccky- Levich equation. 

In filtration, the particle-collector interaction is taken as the sum of the London-van der Waals and double layer interactions, i.e. the Deijagin-Landau-Verwey-Overbeek (DLVO) theory. In most cases, the London-van der Waals force is attractive. The double layer interaction, on the other hand, may be repulsive or attractive depending on whether the surface of the particle and the collector bear like or opposite charges. The range and distance dependence is also different. The DLVO theory was later extended with contributions from the Born repulsion, hydration (structural) forces, hydrophobic interactions and steric hindrance originating from adsorbed macromolecules or polymers. Because no analytical solutions exist for the full convective diffusion equation, a number of approximations were devised (e.g., Smoluchowski-Levich approximation, and the surface force boundary layer approximation) to solve the equations in an approximate way, using analytical methods. 

The use of the dip-coating technique allows to obtain different overlay thicknesses by acting on the solution viscosity and extraction speed as stated by the Landau-Levich equation (see (3.14)). In particular, thicker overlays can be obtained by increasing the extraction speed and/or by increasing the solution viscosity. 

It is interesting to note that there is no complete symmetry between the role of substrate diffusion and electron transport in their combination with the catalytic reaction, as can be seen in the structures compared in the equations and also in the fact that linear Koutecky-Levich plots are not obtained in all cases, as noted in Table 4.1. 

In a detailed rotating-disk electrode study of the characteristic currents were found to be under mixed control, showing kinetic as well as diffusional limitations [Ha3]. While for low HF concentrations (<1 M) kinetic limitations dominate, the regime of high HF concentrations (> 1 M) the currents become mainly diffusion controlled. However, none of the relevant currents (J1 to J4) obeys the Levich equation for any values of cF and pH studied [Etl, Ha3]. According to the Levich equation the electrochemical current at a rotating disk electrode is proportional to the square root of the rotation speed [Le6], Only for HF concentrations below 1 mol 1 1 and a fixed anodic potential of 2.2 V versus SCE the traditional Levich behavior has been reported [Cal 3]. 

Channel techniques employ rectangular ducts through which the electrolyte flows. The electrode is embedded into the wall [33]. Under suitable geometrical conditions [2] a parabolic velocity profile develops. Potential-controlled steady state (diffusion limiting conditions) and transient experiments are possible [34]. Similar to the Levich equation at the RDE, the diffusion limiting current is... 

To appreciate that there are two commonly employed ways of expressing a frequency of rotation, i.e. both angular and linear (oj and /, respectively), and that the Levich equation is formulated in terms of angular frequency. 

To appreciate that deviation from the Levich equation is likely to stem from non-limiting currents (the overpotential rj is not extreme enough), breakdown of mass transport ( j is too extreme) and turbulent flow. 

Note from equation (7.1) that the Levich equation was derived in terms of electrochemical units, so we recall that Canaiyte is expressed in mol cm , A in cm and D in cm s . If we prefer other units then we must alter the constant of 0.620. 

Assuming that A, co, D and v remain unchanged, we can rewrite the Levich equation (equation (7.1)) as follows ... 

Thirdly, we need to appreciate how the current term in the Levich equation represents a faradaic current, and hence the stipulation that we remove all dissolved oxygen from the solution before our analyses commence. Furthermore, the current is a limiting one, so we will commonly perform a few sample experiments before the analysis (usually at fixed frequency) by slowly increasing the potential until a limiting current is reached. 

In order to prevent such invalidation, we must produce voltanunograms such as those shown in Figure 7.3 (each at constant /) and then determine which potential ranges allow the reliable use of the Levich equation at our RDE for each rotation speed. 

Above a certain rotation speed, the solution flow suddenly becomes turbulent, and flow is irreproducible, because eddy currents and vortices form around the edges of the electrode. Such eddies cannot be modelled, implying that the Levich equation breaks down, and becomes unusable. 

The Levich equation (equation (7.1)) implies that faster rotation speeds allow for larger disc currents (see above), itself implying that it would be better to employ data obtained at higher rotation speeds o> when constructing Tafel plots such as those described in the previous sections, because the larger currents decrease the attendant errors associated with measurement of t o and a. 

Provided that the flow is laminar, and the counter electrode is larger than the working electrode, convective systems yield very reproducible currents. The limiting current at a rotated disc electrode (RDE) is directly proportional to the concentration of analyte, according to the Levich equation (equation (7.1)), where the latter also describes the proportionality between the limiting current and the square root of the angular frequency at which the RDE rotates. 

When [W(CN)s] " was coimmobilized with BOD and poly(L-lysine) on carbon felt sheet of 1-mm thickness on an RDE, a current density of 17 mA/ cm was observed at 0.4 V and 4000 rpm in oxygen-saturated phosphate buffer, pH 7. The authors partially attribute the high current density to convective penetration of the oxygen-saturated solution within the porous carbon paper electrode. This assertion is justified by calculation of an effective electrode area based on the Levich equation that exceeds the projected area of the experimental electrode by 70%. ° This conclusion likely applies to any... 

The basic assumption is that the rotating filter creates a laminar flow field that can be completely described mathematically. The thickness of the diffusion boundary layer (5) is calculated as a function of the rotational speed (to), viscosity, density, and diffusion coefficient (D). The thickness is expressed by the Levich equation, originally derived for electrochemical reactions occurring at a rotating disk electrode ... 

Frumkin and Nekrassow then applied Levich s equation to an analysis of intermediate production when the intermediate could leave the electrode surface, with the possibility of reacting again at the ring or leaving for the bulk. Damjanovic et aL developed the Ivanov and Levich equation to include a term, x, the ratio of the velocity of the two parallel reactions (A) and (B), thus increasing the helpful information obtained by using the equation. Damjanovic et al. s equation for the ratio of disk current to ring current is... 

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