Levich constant

Hence the picture of the cathodic and anodic waves obtainable for a completely reversible redox couple by means of the RDE corresponds fully with that in Fig. 3.9 the value of i, i.e., the height of the sigmoidal waves, is linearly proportional to to1/2 and to C (see eqn. 3.89 and the Levich constant). If for a well chosen combination of C and E a plot of i against co1/2 deviates from a straight line passing through the origin, then in the kinetics of the electrode reaction we have to deal only with a rapid electron transfer (cf., Fig. 3.10) or even with a slow electron transfer (cf., Fig. 3.11), in which latter instance the transfer coefficient a plays an appreciable role (cf., eqns. 3.17 and 3.18). 

Thus, the true charge-transfer current can be calculated from the ordinate at the origin in the plot between the reciprocal of the measured current density, j"1, as a function of w. The slope (B 1) is the reciprocal value of the Levich constant, 0.620nFCJoj, because it is the only portion that strictly depends on the co value [107], where D, is the coefficient of diffusion of they-particle. With the currents corrected from the mass transport effects, we can depict the Tafel lines, from which the values of j0 and a can be calculated. 

This equation applies to the totally mass-transfer-limited condition at the RDE and predicts that //c is proportional to Cq and One can define the Levich constant as which is the RDE analog of the diffusion current constant or current function in voltammetry or the transition time constant in chronopotentiometry. 

Equation (9.46) shows that a plot of i vs. will be curved and tends toward the limit i = ik as straight line, where its slope can be used to determine Levich constant of B, from which the number of electrons involved in the reaction can be calculated using known values of concentration and the diffusivity of particular reactant in the medium under investigation. The intercept of the plot on the ordinate axis at < 2 = 0 gives the values of which can be used for further determination of the kinetic parameter k, according to Eq. (9.45). 

This case can also be approached using Kolmogoroff s (K9, H15) theory of local isotropic turbulence to predict the velocity of suspended particles relative to a homogeneous and isotropic turbulent flow. By examining this situation for spherical particles moving with a constant relative velocity, varying randomly in direction, Levich, (L3) has demonstrated that... 

Assuming that the spin conversion is a nonadiabatic process, the macroscopic rate constant may be expressed, following Levich [125], in terms of the thermally averaged transition probability, the averaging being extended over the initial vibronic levels, as ... 

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. 

Here we also consider sorption kinetics as the mass-transfer barrier to surfactant migration to and from the interface, and we follow the Levich framework. However, our analysis does not confine all surface-tension gradients to the constant thickness film. Rather, we treat the bubble shape and the surfactant distribution along the interface in a consistent fashion. 

The rate constant, k, may then be derived from variation of the plateau current with the rotation rate by means of the popular Koutecky-Levich plots, where the inverse of the plateau current is plotted against the inverse of the square root of the rotation rate (Figure 4.12). The intercept allows the determination of the kinetic constant kr°, and of the rate constant k, if the amount of catalyst on the electrode surface is known. 

In zone R, all three phenomena that take place in the film are fast compared to the diffusion of the substrate from the bulk of the solution to the film-solution interface. The concentrations of both Q and A are constant through the film. The RDEV response is similar to that of a monolayer coating (Section 4.3.2), except that more catalytic material is present on the surface of the electrode (it is multiplied by the number of layers in the multilayered coating). A linear Koutecky-Levich plot is obtained from the intercept, from which the kinetics of the catalytic reaction can be characterized. 

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. 

We should bear in mind that the Levich equation (equation (7.1)) was derived in terms of angular frequency, with the constant of 0.620 in this equation presupposing its continued use. 

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. 

Note that the slopes of the Br0nsted and Tafel plots, need not necesarily be constant over a large free energy range and, in fact, the Marcus—Levich theoretical treatment predicts a quadratic dependence [32c]. 

For a first-order regeneration mechanism, the effect will be to shift the Levich-type plot at the limiting current upwards by a constant amount. At the RDE, the intercept is given by... 

As noted earlier, the equation for turbulent film flow obtained by Levich (L8) [Eq. (71)] contains an unspecified constant and therefore cannot be readily compared with the other relationships for turbulent films. 

The modeling of the electrochemical response corresponding to the application of a constant potential to an RDE is similar to that discussed in the case of a DME since in this electrode it is imperative to consider the convection caused by the rotation of the electrode. This problem was solved by Levich under stationary conditions [76]. To do this, the starting point is the diffusive-convective differential... 

Two limiting situations may be identified r (1) the rate constant K is very small compared to aD, hence the process occurring in the interaction forces boundary layer controls the deposition rate, and (2) the rate constant K is very large hence the convective diffusion is the controlling factor. The first limiting case was treated by Hull and Kitchener (except for the variation of the diffusion coefficient) while the second was treated by Levich. In the present paper an equation is established which is valid for all values of the rate constant thus also incorporating both limiting situations. 

The present equations lead to the result of Levich when the apparent rate constant is large enough, and to a refined version (because it includes the effect of the distance dependent diffusion coefficient) of that of Hull and Kitchener when the apparent rate constant is small enough. 

Predicted values of the rate for all four speculations in Table I are much more sensitive to ionic strength than observed values. At high ionic strengths, the energy barrier disappears and the rate becomes equal to the maximum value (6110 particles/cm2 sec) predicted from Levich s equation (Eq, 3J). One possible explanation for the discrepancy in ionic strength dependence is that the Hamaker constant has a different value in each of the five electrolyte solutions tested. The Hamaker constant can be.affected by adsorbed layers of surfactant (18). Since the concentration of surfactant used in solutions of different ionic strengths varied between 1 X 10 4 and 4X 10-i 37/liter, the Hamaker constant could be affected differently. However, to obtain agreement between predicted and observed rates under speculation 1, the Hamaker constant would have to vary from 0.98 X10 13 erg... 

Due to the formation of an intermediate complex, this type of reaction mechanism was described as being analogous to Michaelis-Menten kinetics [39]. A common error made when examining the behaviour of systems of this type is to use the Koutecky-Levich equation to analyse the rotation speed-dependence of the current. This is incorrect because the Koutecky-Levich analysis is only applicable to surface reactions obeying strictly first-order kinetics. Applying the Koutecky-Levich analysis to situations where the surface kinetics are non-linear, as in this case, leads to erroneous values for the rate constants. Below, we present the correct treatment for this problem based on an extension of a model originally developed by Albery et al. [42]... 

A detailed examination of the mass transport effects of the HMRDE has been made. At low rotation speeds and for small amplitude modulations (as defined in Section 10.3.6.2) the response of the current is found to agree exactly with that predicted by the steady-state Levich theory (equations (10.15)-(10.17)) [27, 36, 37]. Theoretical and experimental application of the HMRDE, under these conditions, to cases where the electrode reaction rate constant was comparable to the mass-transfer coefficient has also been made [36]. At higher rotation speeds and/or larger amplitude modulations, the observed current response deviated from the expected Levich behaviour. 

As an alternative to EHD measurements, the presence of a surface film of constant thickness can be detected using Koutecky-Levich analysis. From equation (10.32),... 

Vandeputte et al. [122] used both of these equations to derive more accurate values of the constants than could be obtained from the Koutecky-Levich analysis alone, and hence derived rate constants for the predissociation of the thiosulfate complex involved in the electrodeposition of silver from thiosulfate solutions. 

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