Dimensionless limiting current

The dimensionless limiting current density N represents the ratio of ohmic potential drop to the concentration overpotential at the electrode. A large value of N implies that the ohmic resistance tends to be the controlling factor for the current distribution. For small values of N, the concentration overpotential is large and the mass transfer tends to be the rate-limiting step of the overall process. The dimensionless exchange current density J represents the ratio of the ohmic potential drop to the activation overpotential. When both N and J approach infinity, one obtains the geometrically dependent primary current distribution. 

In Fig. 3.14a, the dimensionless limiting current 7j ne(t)/7j ne(tp) (where lp is the total duration of the potential step) at a planar electrode is plotted versus 1 / ft under the Butler-Volmer (solid line) and Marcus-Hush (dashed lines) treatments for a fully irreversible process with k° = 10 4 cm s 1, where the differences between both models are more apparent according to the above discussion. Regarding the BV model, a unique curve is predicted independently of the electrode kinetics with a slope unity and a null intercept. With respect to the MH model, for typical values of the reorganization energy (X = 0.5 — 1 eV, A 20 — 40 [4]), the variation of the limiting current with time compares well with that predicted by Butler-Volmer kinetics. On the other hand, for small X values (A < 20) and short times, differences between the BV and MH results are observed such that the current expected with the MH model is smaller. In addition, a nonlinear dependence of 7 1 e(fp) with 1 / /l i s predicted, and any attempt at linearization would result in poor correlation coefficient and a slope smaller than unity and non-null intercept. 

In this chapter generalized mathematical models of three dimensional electrodes are developed. The models describe the coupled potential and concentration distributions in porous or packed bed electrodes. Four dimensionless variables that characterize the systems have been derived from modeling a dimensionless conduction modulus ju, a dimensionless diffusion (or lateral dispersion) modulus 5, a dimensionless transfer coefficient a and a dimensionless limiting current density y. The first three are... 

Weisz modulus = m potential in conductive solid phase (V) potential in solution phase (V) dimensionless potential variable = dimensionless potential atX= 1 dimensionless limiting current density = ///o overpotential (V) applied overpotential (V)... 

Dimensionless limiting current density A lcd (for rnass transfer controlled reactions)... 

A generalized equation for the limiting-current response of different detectors, based on the dimensionless Reynolds (Re) and Schmidt (Sc) numbers has been derived by Hanekamp and co-workers (62) ... 

From an analysis of the electrochemical mass-transfer process in well-supported solutions (N8a), it becomes evident that the use of the molecular diffusivity, for example, of CuS04, is not appropriate in investigations of mass transfer by the limiting-current method if use is made of the copper deposition reaction in acidified solution. To correlate the results in terms of the dimensionless numbers, Sc, Gr, and Sh, the diffusivity of the reacting ion must be used. 

Experimental data relative to unsteady-state mass transfer as a result of a concentration step at the electrode surface are not available. However, for a linear increase of the current to parallel-plate electrodes under laminar flow, Hickman (H3) found that steady-state limiting-current readings were obtained only if the time to reach the limiting current at the trailing edge of the plate (see Section IV,E), expressed in the dimensionless form of Eq. (18), is... 

In the above equations, a is the conical aspect ratio, r/h 7 is the ratio of the cone or hemisphere radius to the interelectrode distance, r/d and I, the dimensionless faradaic current (either Icon or hsph > t e rati° between the one-dimensional current contribution, ifLC ant t le limiting current for an isolated hemispherical electrode, i gph (see Eq. 5)(64) ... 

The plateau currents are thus a function of two dimensionless parameters, Jis/ik and 4/4(1 — k/i )- On this basis, a kinetic zone diagram may be established (Figure 4.19) as well as the expressions of the plateau currents pertaining to each kinetic zone (Table 4.1).17 Derivation of these expressions is described in Section 6.4.4. There are in most cases two successive waves, and the expressions of both limiting currents are given in Table 4.1. The general case corresponds to a situation where none of the rate-limiting factors... 

Since the form of the dimensionless convective-diffusion equation for tube and channel electrodes is exactly the same as for rotating electrodes, we can immediately conclude that the steady-state collection efficiency, N0, under conditions of uniform surface concentration at the generator electrode (which corresponds to the limiting current at the generator or to any point on a reversible wave) is, once again... 

The effects of the catalytic reaction on the CV curve are related to the value of dimensionless parameter A in whose expressions appear variables related to the chemical reaction and also to the geometry of the diffusion field. For small values of A, the surface concentration of species C is scarcely affected by the catalysis for any value of the electrode radius, such that r)7,> —> c c and the current becomes identical to that corresponding to a pseudo-first-order catalytic mechanism (see Eq. (6.203)). In contrast, for high values of A and f —> 1 (cathodic limit), the rate-determining step of the process is the mass transport. In this case, the catalytic limiting current coincides with that obtained for a simple charge transfer process. 

When a system is operating at the limiting current, rather than at an appreciable fraction of the limiting current, the problem is very much simplified. Such problems can be classified as mass-transport limited. Usually, the limiting current density is correlated with dimensionless numbers. Most forced-convection correlations take the form... 

The concentration profile expected for a system at one-half of the mass-transfer-limited current and for a concentration perturbation of 20 percent at the interface (see equation (11.44)) is presented in Figure 11.5 with dimensionless time as a parameter. At the higher frequency, the propagation of the disturbance away from... 

Figure 6.29 shows some example linear sweep voltammograms assuming different scan rates (Osrrefers to the dimensionless scan rate Osr = F/RT)(yrl/D)). As the experimental time scale decreases, the diffusional behavior changes from near-steady-state to near-planar diffusion. With respect to the different shapes of microparticles, the mass transport-limiting current was found to be fairly consistent that is, a difference of less than 2% for sphere and hemispheres of equal surface area. 

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