Levich equations

Solution From the Levich equation, (4-5), one can calculate first the disk current under the new conditions ... 

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 ... 

At the RRE the derivation123 of the Levich equation requires reconsideration of the convection-diffusion equation, which results in... 

Effective ionic diffusivities at a rotating-disk electrode are calculated from the Levich equation as derived for constant physical properties, used here in inverted form ... 

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. 

This equation is referred to as the Marcus-Levich equation in which the tunneling effect is included. 

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. 

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 find that the limiting current at a rotated disc electrode (RDE) is directly proportional to the concentration of analyte, according to the Levich equation. 

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. 

In order to maintain the constant of 0.620 in the Levich equation, the kinematic... 

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. 

Erom the traces in Figure 7.3, it can be seen that a limiting oxidative current can only be obtained if rde > 0.4 (V vs. SCE). The maximum current is a function of the applied potential, which means that we cannot employ the Levich equation if Erde < 0.4 (V vs. SCE). 

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 reason why the currents are smaller than those expected from the Levich equation is because turbulent flow results in the entrapment of air within the vortex around the electrode. In effect, the active area (the area in contact with solution) of the electrode decreases in a random way. 

Why does the Levich equation contain a diffusion coefficient D if the RDE is a system under convective control ... 

Yes - the limiting current lu does depend (albeit very slightly) on the potential. For this reason, it is the usual practice to determine I/,m at a fixed potential as well as at a fixed rotation speed (because um is a function of to from the Levich equation). 

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. 

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 ... 

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. 

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