Convolution analysis

A transformation of the peak voltammogram to the sigmoidal shape shown in the preceding section, Fig. 5.13, is achieved by the convolution analysis method proposed by K. Oldham. The experimental function j(t) = j[T(E — Ej)/v] is transformed by convolution integration... 

The difference in temporal dispersion in fibers, associated with the Stokes shift and broad spectral content of fluorescence as compared with the excitation, still has to be fully investigated with respect to the effect on convolution analysis. 

Radical-anion complexes Scope of this review 91 Thermodynamic and kinetic methodologies Voltammetric methods 92 Homogeneous redox catalysis 94 Convolution analysis 98 Laser flash photolysis 102 Photoacoustic calorimetry 103 Thermochemical estimates 105 Fleduction of C—O and O O bonds 106 Reduction of ethers 107 Reduction of peroxides and endoperoxides Reduction of S—S and C—S bonds 136 Reduction of disulfides 137 Reduction of sulfides 150 Concluding remarks 157 Fleferences 160... 

One of the most powerful electrochemical methods for the study of the subtle details of heterogeneous ET processes and, particularly, dissociative ET processes is the method of convolution analysis. This method, which was... 

Equation (25) is general in that it does not depend on the electrochemical method employed to obtain the i-E data. Moreover, unlike conventional electrochemical methods such as cyclic or linear scan voltammetry, all of the experimental i-E data are used in kinetic analysis (as opposed to using limited information such as the peak potentials and half-widths when using cyclic voltammetry). Finally, and of particular importance, the convolution analysis has the great advantage that the heterogeneous ET kinetics can be analyzed without the need of defining a priori the ET rate law. By contrast, in conventional voltammetric analyses, a specific ET rate law (as a rule, the Butler-Volmer rate law) must be used to extract the relevant kinetic information. 

The convolution analysis is based on the use of convolution data and further manipulation to obtain information on the ET mechanism, standard potentials, intrinsic barriers, and also to detect mechanism transitions. It is worth noting that the general outlines of the methodology were first introduced in the study of the kinetics of reduction of terf-nitrobutane in dipolar aprotic solvents, under conditions of chemical stability of the generated anion radical. For the study of concerted dissociative ET processes, linear scan voltammetry is the most useful electrochemical technique. 

The final step of the convolution analysis is the determination of the transfer coefficient a. This coefficient, sometimes called the symmetry factor, describes how variations in the reaction free energy affect the activation free energy (equation 26). The value of a does not depend on whether the reaction is a heterogeneous or a homogeneous ET (or even a different type of reaction such as a proton transfer, where a is better known as the Bronsted coefficient). Since the ET rate constant may be described by equation (4), the experimental determination of a is carried out by derivatization of the ln/Chet-AG° and thus of the experimental Inkhei- plots (AG° = F E — E°)) (equation 27). 

Fig. 6 Representative examples of the steps involved in the convolution analysis approach to obtaining the potential dependence of the heterogeneous rate constant. From top to bottom (a) background-subtracted cyclic voltammograms as a function of scan rate (left to right 0.5, 1, 2, 5, lOVs " ) (b) corresponding convolution curves (c) corresponding potential dependence of logkhet obtained using equation (25). Figures shown are for the reduction of (MeS)2 in DMF/0.1 M TBAP at a glassy carbon electrode.

The reduction of benzyl aryl ethers has been thoroughly investigated by voltammetric reduction, homogeneous redox catalysis,and currently, by convolution analysis. A family of ethers activated by proper substitution on the phenoxy side were chosen to provide a wide variation in the ET and bond cleavage properties of the molecule. ... 

The possibility of studying the heterogeneous ET to dialkyl peroxides at the mercury electrode has provided the opportunity to test the dissociative ET theory using experimental activation/driving force relationships. It was thus possible to observe parabolic patterns in agreement with the theory and to use the potential dependence of a to determine the double-layer corrected values. Thus, using the convolution analysis approach, values for E roor/rovro were determined for a number of peroxides in both acetonitrile and DMF solutions. Representative results are summarized in Table 4. 

Using the values of E°roor/rovro- determined from thermochemical cycles or convolution analysis, the log fchom can be plotted as a function of the reaction free energy shown in Figs 8 and 9. The data convincingly illustrate the parabolic nature of the expected quadratic-activation driving force relationship and thus equation (7). Because of the low BDE(O-O), and thus low intrinsic barrier, the parabolic nature is much more apparent in these systems than in others that undergo dissociative ET. 

This aspect was investigated later by comparing the heterogeneous ET kinetics of di-rert-butyl peroxide with that of the less hindered di-n-butyl peroxide by convolution analysis. A plot comparing the results of a logarithmic... 

The sensitivity of a to the ET mechanism is particularly useful in revealing deviation from the parabolic behavior expected on the basis of equation (7). This is exemplified by the application of the a convolution analysis to the reduction of a series of perbenzoates in DMF at the glassy carbon electrode. ... 

These nonresonant interactions significantly complicate the analysis at short times. As a result, independent measurements the of the ti = 0 and r3 = 0 points on the time scale are important ingredients in the analysis. If these parameters are removed from the fitting, the convolution analysis of the Raman echo signals is stable and unique, even at short times. 

The first half of this chapter covers basic concepts concerning the potential dependence of rates of electrochemical charge-transfer reactions, which will be familiar to specialists in the field. However, this material is included for the more general reader in order to provide a basis for following the more convoluted analysis required for dealing with complex, multistep electron-transfer reactions that are treated in the later part of this article in some detail. 

Bond AM, Mahon PJ, Oldham KB, Zoski CG (1994) Investigation of the influence of residual uncompensated resistance and incomplete charging current correction on the calculation of electrode kinetics when global and convolution analysis methods are used. J Electroanal Chem 366 15-27... 

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