Nicholson Shain solution

The first reaction of oxidation of ferrocene occurs at an electrode followed by oxidation of GO(red) in solution with the pesudo-first-order rate constant f=/j[GO(red)]. The values k(la (a = nFv/RT v is a scan rate) estimated from the experimentally measured i ji ratios (Fig. 2) using the Nicholson and Shain approach (73) have been... 

The quasireversible LSV case was treated by Matsuda and Ayabe [389], who used a series sum as an approximation to the integral equation obtained from the Laplace-transform solution of the problem. The result depends on the heterogeneous rate constant, both the peak current and the peak potential varying with this parameter. Basha et al. [82] tried to improve on the results but it seems that those of Nicholson and Shain [417] were better. These also provided results for the totally irreversible case, first described by Delahay [199]. For this, the y(ai)-function has a constant maximum, given to four figures in [73,74], 0.4958, from the tables in [417]. Peak potential varies with rate constant, as with the quasireversible case. 

It is well known that experimental CVs for species in solution phase frequently diverge from theoretical ones for -electron reversible couples. The divergence can be caused by a variety of factors deviations from reversibility, occurrence of coupled chemical reactions and/or surface effects, and resistive and capacitive effects (Nicholson and Shain, 1964 Nicholson, 1965a). These last effects will be briefly treated here because of their potential significance when microheterogenous deposits or more or less homogeneous coatings of microporous materials cover the electrode surface. 

The catalytic mechanism in solution phase described by Equations (3.1) and (3.2) is usually described in terms of a reversible electron transfer for the Cat system (Equation (3.3)) followed by a reaction operating under conditions of pseudo first-order kinetics (Nicholson and Shain, 1964). Thus, the shape of cyclic voltammo-grams (CVs) depends on the parameter X = kc, t, where k is the rate constant for reaction (3.2) and c at is the concentration of catalyst. For low A, values, the catalytic reaction has no effect on the CV response and a profile equivalent to a singleelectron transfer process is approached. For high X values, s-shaped voltammetric curves are observed that can be described by (Bard and Faulkner, 2001) ... 

Electrochemical data recorded under no steady-state conditions can also be used for studying electrocatalytic processes involving porous materials. In cases where the catalytic system can be approached by homogeneous electrocatalysis in solution phase, variation of cyclic voltammetric profiles with potential scan rate (Nicholson and Shain, 1964) and/or, for instance, square-wave voltammetric responses with square-wave frequency (O Dea et al., 1981 O Dea and Osteryoung, 1993 Lovric, 2002) can be used. This situation can, in principle, be taken for highly porous materials where substrate transport, as well as charge-balancing ion transport, is allowed. On first examination, the catalytic process can be approached in the same manner... 

A more rigorous interpretation of Equation 3 for a quiescent solution can be done by considering the net diffusion of electroactive species due to electrochemical reactions at the electrode. A general solution to the diffusion equation with a planar source is given by Nicholson and Shain (2) and has been expanded on by Imbeaux and Saveant (10) ... 

At any given point, x( t) is a pure number, so that (6.2.17) gives the functional relationship between the current at any point on the LSV curve and the variables. Specifically, i is proportional to Cq and The solution of (6.2.15) has been carried out numerically [Nicholson and Shain (3)], by a series solution [Sevcik (2), Reinmuth (4)], analytically in terms of an integral that must be evaluated numerically [Matsuda and Ayabe (5), Gokhshtein (6)], and by related methods (7, 8). The general result of solving... 

In cyclic voltammetry, the anodic portion on the reverse scan is not affected as much as the forward response by the coupled reaction (Figure 12.3.2). The ratio of (with /pa measured from the extension of the cathodic curve as described in Section 6.5) increases with increasing scan rate as shown in the working curve in Figure 12.3.6 (25). The actual i-E curves can be drawn using series solutions or a table given by Nicholson and Shain (25) or by digital simulation. 

This expression is similar to the well-established integral equation describing semi-infinite diffusion of solution phase species originally proposed by Nicholson and Shain.65 On the other hand, when a) is small the current-potential expression reduces to... 

Eddowes and Hill found [48,49] that essentially reversible cyclic voltammetry of horse mitochondrial cytochrome c could be achieved with a Au electrode onto which was adsorbed, from the same solution, the reagent 4,4 -bipyridyl. The result is shown in Fig. 4. The criteria described by Nicholson and Shain [50] for a one-electron process controlled by linear diffusion of species to a planar electrode surface are met very closely indeed. The value of E°, given by (Ep -f- Epa)/2, was 255 mV, in good agreement with values determined by potentiometry. It could be argued that free, reduced 4,4 -bipyridyl played no part in the mechanism, since its reduction potential is much lower than that of cytochrome c. It was proposed [7] that the organic adsorbate allowed electron transfer to occur directly by providing, at the electrode surface, functionalities with which the protein could interact specifically and reversibly and thereupon donate or accept electrons rapidly. It was thus termed a promoter as opposed to a mediator, in which the latter is considered to convey electrons in bulk solution. 

Reinmuth has examined chronopotentiometric potential-time curves and proposed diagnostic criteria for their interpretation. His treatment applies to the very limited cases with conditions of semi-infinite linear diffusion to a plane electrode, where only one electrode process is possible and where both oxidized and reduced forms of the electroactive species are soluble in solution. This approach is further restricted in application, in many cases, to electrode processes whose rates are mass-transport controlled. Nicholson and Shain have examined in some detail the theory of stationary electrode polarography for single-scan and cyclic methods applied to reversible and irreversible systems. However, since in kinetic studies it is preferable to avoid diffusion control which obscures the reaction kinetics, such methods are not well suited for the general study of the mechanism of electrochemical organic oxidation. The relatively few studies which have attempted to analyze the mechanisms of electrochemical organic oxidation reactions will be discussed in detail in a following section. 

Cyclic voltammogram of mesoporous carbon coated glassy carbon electrode surface (0.2 cm ), recorded in N2-purged 0.5 M H2SO4 solution at 23°C and ambient pressure. Carbon loading = 100 ]ig.cm . Potential scan rate = 50 mV.S". Source Nicholson, R. S. and I. Shain. 1964. Analytical Chemistry, 36, 706-723. With permission.)... 

At small values of voltage sweep rate, typically below 1 mV/s, the capacity effects are small and in most cases can be ignored. At greater values of sweep rate, a correction needs to be applied to interpretations of ip, as described by Nicholson and Shain. With regard to the correction for ohmic drop in solution, typically this can he handled adequately by careful cell design and positive feedback compensation circuitry in the electronic instrumentation. 

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