Theoretical working curves

Cyclic voltammetry and derivative cyclic voltammetry (DCV) can be used to study the homogeneous kinetics of the reactions of B generated 

In practice, experimental data are usually plotted in conjunction with the theoretical working curve for the particular mechanism that the process in question is believed to follow. Whether or not the experimental data are consistent with the mechanism is then concluded on the basis of how well the data fit the working curve. Very often this aspect of the use of the theoretical working curve is highly subjective and it can be difficult for the reader of a paper to assess just how good the mechanism assignment really is. 

A more quantitative method for the analysis of the data using theoretical working curves has recently been proposed by Parker [67]. The method is illustrated by the three-dimensional plot of the theoretical working curve shown in Fig. 14. The observable O, Ri in this case, is represented as the Z axis, the theoretical variables Vtheory (kCA/a for CV) on the X axis and the corresponding experimental variables (CA/v for CV) 

An extensive set of theoretical data for CV and DCV is available for a number of different electrode mechanisms. Theoretical data were obtained at intervals of 0.050 from Ri equal 0.100 to 0.900 [68]. The data were obtained by digital simulation and a linear interpolation, 

In Ri = m In fesim + c, was used to obtain the even values of Ri. It was necessary to have data quite close on either side of the desired Rj and three sets of calculations were required to find values within 0.001 of those listed. 

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Fig. 14. An illustration of the normalized theoretical working curve with the linear projection on to the X—Y plane.

The first two columns of Table 17 represent the data that would normally be used to construct the theoretical working curve. The third column, labelled uN, represents the experimental v normalized to the... 

Quantity Reaction order approach Theoretical working curve... 

In this case, analysis has to proceed by fitting experimental data from a wide range of rotation speeds to theoretical working curves. 

Table 4.1 Heterogeneous rate constant (k°), electron transfer coefficient (a), and formal potential ) corresponding to the best fit of theoretical working curves (Eq. 4.120) to the RPV experimental results [48]... 

The theoretical working curve 163 Analysis at a constant value of the observable 166 Reaction orders and LSV response 170 Detailed analysis of LSV waves 172... 

It has also been shown that the electrode response of some processes can appear to fit theoretical working curves in which the reaction order in the intermediate differs from the true value (Parker, 1981b). For example, the deprotonation of hexamethylbenzene radical cation studied by derivative cyclic voltammetry gave data which fitted theoretical data for a simple first order decomposition of the intermediate. However, the observed first order rate constants were found to vary significantly with the substrate concentration indicating a higher order reaction. A method was proposed to treat... 

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. 

The question of dimerization mechanism has also surfaced in the study of reactions of cation radicals. The anodic oxidation of 4,4 -dimethoxystilbene is accompanied by the formation of dimeric products, the nature of which depends upon the nucleophile involved in the reaction as indicated in Scheme 5 (Parker and Eberson, 1969 Eberson and Parker, 1970 Steckhan, 1978 Burgbacher and Schafer, 1979). The reaction was studied by spectroelectro-chemistry (Steckhan, 1978) in acetonitrile containing methanol. Competing mechanisms were proposed involving the reactions of the radical cation (D7) with methanol (134) and by dimerization (135) giving rise to rate law (136). Theoretical working curves were used to find the best fit of the data... 

A special case of totally irreversible substrate kinetics (Kf s = 0) was treated numerically (3). Later, the entire family of theoretical working curves (IT vs. Kb S) calculated for various L was fit to Eq. (34) (12) ... 

The mechanism was deduced from rotating disc data. This involves producing a mathematical model of the expected responses for the various nuances of the mechanism and comparing the experimental result to theoretical working curves. The method has been fully described by Compton and coworkers [60]. 

FIGURE 11-1. Theoretical working curve for fluorescence intensity versus atom concentration. [From J. D. Winefordner and T. J. Vickers, Atomic Fluorescence Spectrometry as a Means of Chemical Analysis, Anal. Chem., 36,161 (1964). Used by permission of the American Chemical Society.]... 

Nevertheless, it must be pointed out that theoretical working curves for various mechanisms are often very similar. Two adjustable parameters included in the rate law (two rate constants or a rate- and equilibrium constants) make the working curves interchangeable at will. It is therefore reliable to combine several electrochemical methods and to examine effects of more variables, not only the potential-time-current relations, but also the influence of the reactant concentration, pH and temperature variations. The properties of the solvent used are worthy of interest, too. 

Derivative cyclic voltammetry (DCV) has been shown to offer great advantages in the detailed investigation of electrode reactions. The DCV reaction-order analysis conveniently gives access to overall kinetic rate laws. Arrhenius activation energies (E are readily obtained, even without a detailed knowledge about the reaction mechanism. Once the overall mechanism has been established, comparisons with theoretical working curves provide rate constants and the derived quantities AH and AS, ... 

Several methods have been utilized to determine the rate of the following chemical reaction from a series of CVs at different scan rates. The simplest involves a comparison of ip,e and i . The cathodic peak current is measured from the zero current baseline, while the anodic current baseline is established by the current at which the potential is switched. The experimental peak current ratios can then be compared to a previously calculated theoretical working curve to find the rate constant (for a first-order or pseudo-first-order reaction. Parker has emphasized the use of working curves based on derivative cyclic voltammetry, which discriminates to some degree against capacitive background current. ... 

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