The Levich equation

Levich [32] solved eqn. (1) for a channel electrode by invoking an approximation originally introduced in 1928 by Leveque in his theory of heat transfer in pipes [33]. In the present context, this simplification can be written as 

The establishment of Poiseuille fiow in a channel under a laminar regime. 

Assuming both diffusion in the direction of convective flow and diffusion-al side-edge effects to be negligible, Levich arrived at the equation (under the Leveque approximation) 

The missing links between eqns. (11)-(13) are produced in Appendix 1. The solution to the problem of the mass transport-limited current at the channel electrode requires eqn. (13) to be solved subject to the boundary conditions 

This may be evaluated by expanding the indefinite integral in eqn. (18) as a power series in t) and integrating term-by-term. This results in 

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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)]. 

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. 

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. 

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 ... 

The study of rotating disk electrode behavior provides a unique opportunity to develop a model that predicts the effect of diffusion and convection on the current. This is one of the few convective systems that have simple hydrodynamic equations that may be combined with the diffusion model developed herein to produce meaningful results. The effect of diffusion is modeled exactly as it has been done previously. The effect of convection is treated by integrating an approximate velocity equation to determine the extent of convective flow during a given At interval. Matter, then, is simply transferred from volume element to volume element in accord with this result to simulate convection. The whole process repeated results in a steady-state concentration profile and a steady-state representation of the current (the Levich equation). 

This expression is identical to the Levich equation within the reliability of the simulation. 

It should be mentioned that this situation is experimentally not accessible and is therefore performed by an extrapolation method. The value of 7l,2/7l,i at these slow rotation rates is equal to the value of n2/ni. The relationship between the logarithm of logco and log[log(7Lj2/7u)] was plotted for various concentrations of sodium dithionite and revealed that all the curves tend towards a value of -0.33 for log[log(7Lj2/7Ljl)] at slow rotation rates (e.g. co=10). This corresponds to a 7Lj2/7u ratio of approximately 3. In the Levich equation (Chapterl, Equationl.15), 7L is proportional to n thus it can be concluded that the combination of Equations (6.1) and (6.3) can explain the experimentally obtained results. 

In the preceding sections, it was mentioned several times that the limiting-current of the first wave of the sodium dithionite oxidation is suitable for electroanalytical purposes because of the transport-controlled nature of this limiting-current. Indeed, it was proven earlier that this limiting-current can be correlated with the Levich equation (1.15), showing a linear relationship between limiting-current and sodium dithionite concentration. 

To analyze the mediation process, rotating-disk measurements were carried out. As discussed above in Chapter 3, in such an experiment an electrode is rotated at speeds of up to several thousands of rpm in the presence of a redox species in solution. By a variation of the potential, a situation is reached where the current becomes independent of the potential applied. For a bare electrode, this limiting current, / lev, is given by the Levich equation, as follows ... 

Figure 9 Comparison of previously reported values of kSA for reduction by Fe° with external mass transport coefficients estimated for batch, column, and rotating disk electrode reactors. References for the overall rate coefficients are given in Fig. 1 of Ref. 101. Mass transport coefficients were estimated for the batch and column reactors based on empirical correlations discussed in Refs. 125 and 101. Mass transport coefficients for the RDE were calculated using the Levich equation [178].

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