Working curve approach

An example of the working curve approach, resulting from a study of the stereochemistry of the electrohydrodimerisation of cinnamic acid esters [38], is shown in Fig. 6.16. 

Fit the same mechanism for data collected at different scan rates. This is a very good indication of the correctness of the assumed mechanism. It is equivalent to a working curve approach. A divergence of a fitted parameter indicates an incomplete or incorrect mechanism. On the other hand, fitted parameters that are constant over scan rate indicate that the mechanism is consistent with experiment. Error estimates on the parameters can be obtained by standard deviation of parameters obtained for different data sets (both at the same scan rate and varying scan rate). 

In the case of 0-pipettes, the collection efficiency also decreases markedly with increasing separation. The situation becomes more complicated when the transferred ion participates in a homogeneous chemical reaction. For the pseudo-first-order reaction a semiquantita-tive description is given by the family of dimensionless working curves calculated for two disks (Fig. 6) [23]. Clearly, at any separation distance the collection efficiency approaches zero when the dimensionless rate constant (a = 2kr /D, where k is the first-order rate constant of the homogeneous ionic reaction) becomes 1. 

Work done by L. Mullins on the prestressing of filler-loaded vulcanisates showed that such prestressing gives a stress-strain curve approaching that of an unfilled rubber. This work has thrown much light on so called permanent set and the theory of filler reinforcement. See Stress Softening. 

Quantity Reaction order approach Theoretical working curve... 

The simulation of the ECE mechanism may also employ the double-potential-step technique, but a working curve can be constructed from single-potential-step data also. This is because some of the current that passes, as A is converted to B, is due to the electrolysis of C, the decomposition product of B. The greater the decomposition rate of B, the more current flows, approaching the rate of... 

As mentioned above, the characteristic feature of processes in this kinetic region is that the peak current ratio — z°x/Zped varies from about unity to zero. Thus, a procedure for studying the kinetics would be to record values of —/°x//ped at different sweep rates and compare these with a working curve for the proposed mechanism in a way analogous to that discussed for DPSCA above. However, a problem with this approach is the difficulty of defining a baseline for the reverse sweep (see below) and, for that reason, CV suffers from some limitations when used in quantitative work. This has led to the development of derivative cyclic voltammetry (DCV) [37]. 

Later [24], it was shown that the theory for the ErQ process under SECM conditions can be reduced to a single working curve. To understand this approach, it is useful first to consider a positive feedback situation with a simple redox mediator (i.e., without homogeneous chemistry involved) and with both tip and substrate processes under diffusion control. The normalized steady-state tip current can be presented as the sum of two terms... 

The inadequacy of theoretical working curves in electrode-mechanism analysis has prompted the development of an alternative approach which does not involve the use of theoretical data at all in the determination of the mechanism (Parker, 1981e). Theoretical data are used only after the mechanism has been established and then to evaluate rate constants. 

Alternatively, the determination of k may be based on measurements of Ep/4 — Ep mam where the quarter-peak potential Ep/4 is the value of E at i = ip,mam/4 [123] [see Fig. 17(a)]. The advantage of this approach is that measurements can be made even in cases where the prepeak appears only as a shoulder on the main peak, and where Ep.pre therefore cannot be determined. The working curve for the eCen mechanism is shown in Fig. 17(b) together with experimental data obtained for the protonation of the anthracene radical anion by benzoic acid. The rate constant resulting from these data is 2.7 X 10 M s [123]. 

Signal-time behavior of the ESR response following a current pulse has been calculated by Goldberg and Bard [367] for a number of mechanisms, including first-order decomposition, radical ion dimerization, and radical ion-substrate coupling, and working curves from which rate constants can be calculated were presented. Application of this approach, which is very similar to that taken, for example, in transmission spectroelec-trochemistry, was demonstrated for the reductive dimerization of a series of activated olefins (Fig. 58), a reaction that has been studied by a number of different electrochemical... 

One can see that the different working curves have substantially different curvatures, thus a unique curve can be found to obtain the best ht with the experimental data. One can also see that the current for a conical tip tends to reach some limiting values as L — 0 and as k — 0, the working curves for a conical tip approach that computed for a disk-shaped tip. 

Under these conditions (12.2.32) becomes identical to the equation for unperturbed reversal chronopotentiometry and T2 = /i/3 (see equation 8.4.9). When kt is large (kinetic zone), T2 approaches 0. The variation of T2lt with kt is shown in Figure 12.2.2 (20-22). Note that kinetic information can be obtained from reversal measurements only in the intermediate zone (0.1 kt 5). The actual value of k is obtained by determining T2//1 for different values of t and fitting the data to the working curve shown in Figure 12.2.2 (23). Kinetic information can also be obtained in the kinetic zone from the shift of potential with t however, fP must be known for the electron-transfer step before an actual value of k can be determined. 

Now suppose the tip-generated species is not stable and decomposes to an elec-troinactive species, such as in the case (Chapter 12). If R reacts appreciably before it diffuses across the tip/substrate gap, the collection efficiency will be smaller than unity, approaching zero for a very rapidly decomposing R. Thus a determination of /x//s as a function of d and concentration of O can be used to study the kinetics of decomposition of R. In a similar way, this decomposition decreases the amount of positive feedback of O to the tip, so that ij is smaller than in the absence of any kinetic complication. Accordingly, a plot of ij vs. d can also be used to determine the rate constant for R decomposition, k. For both the collection and feedback experiments, k is determined from working curves in the form of dimensionless current distance (e.g., did) for different values of the dimensionless kinetic parameter, K = kcP ID (first-order reaction) or = k a CQlD (second-order reaction). 

Research into this area is dominated by microelectrodes. At short times, the diffusion layer thickness is much smaller than the microelectrode radius and the dominant mass transport mechanism is planar diffusion. Under these conditions, the classical theories, e.g., that of Nicholson and Shain, can be used to extract kinetic parameters from the scan rate dependence of the separation between the anodic and cathodic peak potentials. Using this approach, the standard heterogeneous electron transfer rate constant, k°, may be determined from the published working curves relating AEp to a kinetic parameter The variation of AEp with o is determined and, from this, T is calculated. k° is then determined by the following equation ... 

---