Levich constant equations

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... 

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. 

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). 

RDEs provide well-controlled diffusion conditions (Figure 1.27). The flow of fhe electiolyfe gives access of fresh solution to the surface, whereas the electrochemical reaction at the electrode surface changes the electrolyte composition. Both effects compensate each other, leading to a constant thickness of fhe Nemst diffusion layer all over fhe disc with a laminar flow of the solution in vicinity of ifs surface. The analytical treatment of this convection and diffusion problem leads to Levich s equation ... 

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 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. 

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. 

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. 

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]... 

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. 

Estimation of diffusion layer thickness ( S ) S for a moving particle is related to the velocity of motion (u) of the particle through the water. For a sphere of radius a moving through the water at a constant velocity it can be shown using the equations given by Levich (34, p. 84-85) that the average dif-... 

Aj may be evaluated from x-ray and infrared (IR) data or from theoretical calculations. However, for organic outer sphere electron transfers, this contribution is usually much smaller than Ao. In our opinion one of the greatest merits of the Marcus [43] and Levich-Dogonadze [44] theories is that they allow rather correct predictions of Aq through simple equations. Thus for most outer sphere electron transfers, reasonably accurate values of the rate constants can be predicted. 

Levich did not, however, specify the means for evaluating dpidx. This may have resulted from his assumption that h was constant. However, as Yih noted, from integrating the z momentum equation, the pressure is given by the local hydrostatic condition... 

The approximate experimental determination of xl), is based on measurement of the velocity of a charged particle in a solvent subjected to an applied voltage. Such a particle experiences an electrical force that initiates motion. Since a hydrodynamic frictional force acts on the particle as it moves, a steady state is reached, with the particle moving with a constant velocity U. To calculate this electrophoretic velocity U theoretically, it is, in general, necessary to solve Poisson s equation (Equation 3.19) and the governing equations for ion transport subject to the condition that the electric field is constant far away from the particle. The appropriate viscous drag on the particle can be calculated from the velocity field and the electrical force on the particle from the electrical potential distribution. The fact that the sum of the two is zero provides the electrophoretic velocity U. Actual solutions are complex, and the electrical properties of the particle (e.g., polarizability, conductivity, surface conductivity, etc.) come into play. Details are given by Levich (1962) (see also Problem 7.8). 

A thin liquid film lies on the solid surface which forms the floor of a narrow horizontal slit. Through the slit, air is blown at a steady rate. The air is seen to exert a constant shear stress on the liquid surface, thus the film thickness varies linearly with the distance from the leading edge, which is also the contact line (Derjaguin et al., 1944 Levich, 1962). Very close to the contact line the profile changes to retain the equilibrium contact angle at the contact line. The equations of motion and continuity under the lubrication theory approximation reduce to (Neogi, 1982) (see Problem 7.13)... 

Equation (39) reverts to the Levich equation for an irreversible reaction when the heterogeneous rate constant k is replaced hy kobsF- Inversion of Eq. (39) thus gives the corresponding Koutecky-Levich equation. 

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