Semi-integration current

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.]...

Thus, the current semi-empirical methods (MNDO, AMI and PM3) differ in the way in which core-repulsions are treated. Within the MNDO formalism the corerepulsion ( asmndo) is expressed in terms of two-centre, two-electron integrals (Eq. 5-4), where Za and Zb correspond to the core charges, Rab is the internuclear separation, and a a and aB are adjustable parameters in the exponential term [19]. 

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... 

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... 

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... 

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... 

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]. 

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. 

In its semi-integration mode, this macro takes the faradaic current of a linear sweep voltammogram, and transforms it into the underlying current- voltage curve corrected for the time-dependence of planar diffusion. The resulting curve is then amenable to further mathematical analysis along the lines pioneered by Koutecky, and summarized by, e.g., Heyrovsky ... 

The transformed current data can be used directly, by (6.7.2), to obtain Cq(0, t). Under conditions where Cq(0, 0 = 0 (i e under purely diffusion-controlled conditions), 7(0 reaches its limiting or maximum value, 7/ [or, in semi-integral notation, m(0max] where... 

Fig. 11.10 Linear sweep semi-integral and semi-differential curves of 10 SM Cd(II) in 1M NaCI at a DME. Average of 25 scans, (a) Linear sweep curve, (b) Linear extrapolation of background current, (c) Background-corrected linear sweep curve, (d) Semi-integral curve, (e) Least-squares fitted semi-integral curve, (f) Semi-differential curve. (Reproduced from [39] with permission of the American Chemical Society).

Another advantage of LOC devices is the possibility of integration of different steps of the entire analytical procedure on to a single device, resulting in a micro-Total Analysis Systems (p-TAS). Only a limited number of LOC device developments have reached the status of a complete laboratory, where all the stages of conventional laboratory systems are integrated on a chip. Most of the currently available systems are semi-integrated. 

Other procedures applied to the LSV technique Semidifferentiating (deconvolutive) procedure may be considered as a counterpart to semi-integrating (convolutive) transformation. The replacement of Pick s laws by formulations involving semidifferentiation was proposed 25 years ago [122]. Five years later [123] the deconvolutive transformation of recordered (sampled) currents represented by the equation... 

Figure 3 shows the corresponding semi-integrals of the current, the hysteresis of which proves that the reduction step is not fast. But the cathodic limiting plateau proves that mass transfer is limited by diffusion. From the limiting value of the semi-integral, the value of the diffiision coefficient of HfCU . could be estimated as 2.4 10 cm s The... 

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