The Singh and Dutt approximation

This approximation, as indicated previously (Sect. 1.2), involves replacing the concentration gradients in the convective term by average values as defined by eqns. (4) and (5). The effect of this approximation on the calculation of the diffusion-limited currents at a channel electrode can be illustrated as follows. The concentration gradient in eqn. (11) is replaced by 

The mass transport-limited current is then calculated as 

Notice that, except for the numerical factor of0.834 as compared with 0.925, this equation is identical to that derived by Levich [eqn. (27)]. Thus, in this application, the approximation entails an error of around 10%. 

Given this level of success in the description of the steady-state transport- 

Three-dimensional plots comparing the quantities (a) (gjMl - gjk)(Kll), proportional 5[Red]/3x, and (b) (gj k - 1)//, proportional to ([Red] - [Red] )//, over the cartesian (x, y) space of a channel electrode. The quantities have been multiplied by (-1) for ease of viewing. 

---

That is to say, no additional approximations are involved. Thus, the Singh and Dutt approximation is expected to work best for both time-dependent and steady-state behaviour with or without the presence of first-order kinetics, when the concentrations near the electrode are not too different from the concentration gradient averaged over the length of the electrode. Whilst this requirement might be considered to be contradictory to the essential non-uniformity of accessibility to the electrode, we shall see in Sect. 4.1 that this is not the case for steady-state problems. 

An alternative method is to solve eqn. (10) under Levich conditions, but invoking the Singh and Dutt approximation [61], The boundary conditions relevant to the problem for the reaction... 

The form of this equation is compared with that of Aoki et al. in fig. 18. Notice that, as the steady-state is approached, there is excellent agreement between the two methods but at short times (r < 0.4) the method utilising the Singh and Dutt approximation overestimates the quantity 7(t)/7(t - oo) by about 10%. Can we rationalise this Implicit in the analysis of the problem via the Singh and Dutt approximation is that the current ratio 7(t)//(t - oo) is defined with reference to "Singh and Dutt steady-state currents, which have been shown to be lower than the corresponding cur-... 

Fig. 18. Theoretical chronoamperomograms calculated using the Singh and Dutt approximation (—) [eqn. (73)] and compared with the rigorous theory of Aoki et al. [62] (-),...

As to why the Singh and Dutt approximation works to such a high degree of accuracy (when definition is with respect to "Levich steady-state cur-... 

Fig. 19. Theoretical chronoamperomograms. —, Calculated via the Singh and Dutt approximation, but referenced to "Levich steady-state values -, according to the theory of Aoki...

Levich approach requires Laplace transformation of the partial differential equations (77)-(79) with subsequent (difficult) inversions into real space, the merits of the Singh and Dutt approximation are apparent. 

Fig. 23. The variation of neB with K for a DISPl process, calculated (a) using the Singh and Dutt approximation (—) and (b) via the "Levich approach as described in ref. 64 (-).

Singh and Dutt, using the approximation described in Sect. 2.3, have theoretically predicted, and experimentally verified, the behaviour when very fast scans are applied in both the linear sweep and cyclic voltammetric modes, for reversible [22, 23], quasi-reversible [25], and irreversible [24] electrode kinetics. Very attractive agreement with experiment is typically found, of which Fig. 15 is representative. 

---