Semi-integration

The second strategy which may be used to learn about the kinetics of an electrode reaction is illustrated in Fig. 7. As before, a potential (constant or varying) is imposed on the cell and a current—time relationship is monitored. However, instead of assuming a particular kinetic law, one processes the experimental current by semi-integration (see Sects. 5.2 and 5.4), thus enabling the surface concentrations to be calculated directly. Hence, the kinetics can be elucidated by a study that involves only the... 

The d1/2/df1/2 and d-1/2/df1/2 operators are respectively the semidifferentiation and semi-integration operators [81]. These are analogues of the familiar differentiation and integration operators of the calculus. Since they are unfamiliar to many chemists, Table 6 has been included to illustrate some of their definitions and properties. The semi-... 

Two approaches have been used to semi-integrate electrochemical currents analog [43, 82—84] and digital [55, 60, 85]. Each has been used satisfactorily in experimental voltammetry. We discuss analog methods first. 

In order to appreciate the principles of analog semi-integration, let us first review the classical operational amplifier circuits shown in Fig. 30. The output of circuit (a) is a voltage proportional to the instantaneous value of the current input... 

In eqn. (160), we write the unusual operator notation d-1/df-1 for indefinite integration to emphasize the analogy with semi-integration. When the feedback loop of the operational amplifier contains a resistor,... 

Turning now to digital methods of semi-integration, we shall describe a technique by which the semi-integral m may be determined from a set of equally spaced current values, i 0, ii, i2,. . . , . . . , . If A is the time... 

The so-called Reimann—Liouville definition [81] of a semi-integral is... 

Fig. 36. Linear interpolation used in digital semi-integration.

This formula, termed the RL-algorithm, based as it is on a connect-the-dots approximation, is the semi-integration equivalent of the trapezoidal formula of integration. 

Thus, the kinetics can be elucidated by a study that involves only the electrical variables (potential and current) together with the semi-integral of the current. 

A similar approach is that adopted by Bond et al. [92]. These authors used cyclic voltammetry (Sect. 3.4) as a means of obtaining two values each of the current (7 and 7) and its semi-integral (in and in) at each potential E. They demonstrated that these four values could be combined into a quantity... 

Through, for reasons of mathematical simplicity, planar electrodes are assumed in most voltammetric derivations, they are seldom used in practice because of the difficulties of their fabrication. The favorite, and most convenient, electrode of experimental voltammetrists is the static mercury drop. This spherical electrode cannot always be adequately approximated by a planar model. In particular, eqn. (139) does not hold for a spherical electrode and hence semi-integration cannot be used to generate accurate values of surface concentrations. Indeed, even eqn. (137) is invalid, the concentrations of O and R not being linearly related at the surface of a spherical electrode. 

The integrals in eqns. (181), (182), and (187) are technically known as convolution integrals the procedure which uses them has been called spherical convolution [93]. Spherical convolution may be used to determine the concentrations of O and R at the surface of a spherical electrode in a fashion analogous to the use of semi-integration to determine concentrations at the surface of a planar electrode. 

Convolution potential sweep voltammetry (CPSV) refers to the mathematical transformation of LSV current—potential data resulting in curves with shapes like conventional polarograms which are suitable for logarithmic analysis. The method was first proposed for the study of electrode kinetics by Imbeaux and Saveant [74] but is equivalent in all respects to a semi-integral technique reported earlier by Oldham [75— 77]. A very readable description of the method has been presented by Bard and Faulkner [21]. 

Methods to compute jF (t) values reliably have been discussed extensively by the above-mentioned authors, especially by Oldham who stated that eqn. (99) can be considered as the semi-integral of jF t), generated by the operator d-1/2/df-1/2. Consequently, various rules derived for semiintegration and semi-differentiation can be applied, placing the principle in a more general context [80]. 

Although being of great fundamental importance, it should not be ignored that practical application of the semi-integral analysis requires separation of the faradaic current density jF, i.e. subtraction of the charging current density jc, from the overall current density, j, as well as perfect instrumental compensation or numerical subtraction of the ohmic potential drop jARn in order to obtain the interfacial potential E. 

Figure 20.6 Typical working curves for the first-order EC mechanism. The curve shown for f = i shows the behavior of the current ratio [i(2tf)/i(tf)] as a function of ktf. That given for f = Q shows the charge ratio, while that given for f = m shows the current semi-integral ratio over the same range. [From Ref. 10.]...

An answer to this lies in the transformation of the linear sweep response into a form which is readily analysable, i.e. the form of a steady-state voltammetric wave. Two independent methods of achieving this goal have been described the convolution technique by Saveant and co-workers11,12, and semi-integration by Oldham13. In this section we describe the convolution technique, and demonstrate the equivalence of the two approaches at the end. 

The mathematically most sophisticated technique involves computing convolution or semi-integrals of the voltammetric data obtained in digital form (Imbeaux and Saveant, 1973 Nadjo et al., 1974 Oldham and Spanier, 1970 Oldham, 1972, 1973). The latter results in the transformation of the LSV wave into a form resembling a polarogram (Fig. 19) which is amenable... 

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