Shoup-Szabo equation

Disk UMEs are easier to fabricate than hemispherical UMEs, and are therefore much more conunonly used. The Shoup and Szabo equation (1) for chronoamperometry at a disk UME... 

Note that D is now dependent on the square of the radius, so careful characterization of the UME s dimensions using a test compound is particularly important. An additional advantage of this normalization technique is that because the measurement of and i(l) can occur in the same experiment, calibration errors in the current amplifier are essentially eliminated. An alternative approach is to fit the normalized chronoamperometry data to the Shoup and Szabo equation (equation (19.7)), which only requires knowledge of r. ... 

The solution of equation Ti 4 is not easy and, therefore, approximate analytical solutions have been derived. The analytical expressions derived by Aoki and Osteryoung and, later on, by Shoup and Szabo (equations T2,4 e and respectively, in Table 15.2) are commonly employed for practical applications. They contain the parameter, t, that can be regarded as a dimensionless time ... 

In the context of nano-impact data such as that in Fig. 8.7a it is best expressed as a plot of the cumulative number of impacts as a function of time. This can then be compared with an integrated foim of the Shoup-Szabo expression. Fitting the latter to the former so as to determine C, the unknown concentration of nanoparticles, requires a knowledge of re (which can be found by independent electrochemical calibration) and D, the nanoparticle diffusion coefficient. Given the large size of the nanoparticles the latter can be reliably calculated from the Stokes-Einstein equation... 

Fletcher (1974) introduced unequal 8x intervals Whiting and Carr (1977) applied orthogonal collocation to electrochemistry Shoup and Szabo (1982) applied Gourlay s (1970) hopscotch method to electrochemistry and Heinze et al (1984) showed how to include the boundary value c in the implicit equations of the Crank-Nicolson method, thereby removing a major problem with that method. Britz (1988) applied simple explicit... 

This is the definition assumed in the work of Shoup and Szabo [30] and Aoki and coworkers [12, 41, 42] and also by Gavaghan in some recent works [44, 45]. The above three formulae (12.9), (12.11) and (12.12), are those for this definition of normalised time. One inconvenient side-effect of the definition is that, when one normalises the diffusion equation (12.2), using the new dimensionless variables definitions... 

The time-dependent current response obtained on the first step (see Fig. 2.29) was analysed via the use of the following equation, as proposed by Shoup and Szabo [12], which sufficiently describes the current response over the entire time domain, with a maximum error of less than 0.6 % ... 

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