Rate constants electroactive reaction,

In this equation, aua represents the product of the coefficient of electron transfer (a) by the number of electrons (na) involved in the rate-determining step, n the total number of electrons involved in the electrochemical reaction, k the heterogeneous electrochemical rate constant at the zero potential, D the coefficient of diffusion of the electroactive species, and c the concentration of the same in the bulk of the solution. The initial potential is E/ and G represents a numerical constant. This equation predicts a linear variation of the logarithm of the current. In/, on the applied potential, E, which can easily be compared with experimental current-potential curves in linear potential scan and cyclic voltammetries. This type of dependence between current and potential does not apply to electron transfer processes with coupled chemical reactions [186]. In several cases, however, linear In/ vs. E plots can be approached in the rising portion of voltammetric curves for the solid-state electron transfer processes involving species immobilized on the electrode surface [131, 187-191], reductive/oxidative dissolution of metallic deposits [79], and reductive/oxidative dissolution of insulating compounds [147,148]. Thus, linear potential scan voltammograms for surface-confined electroactive species verify [79]... 

Here we introduce a reversible reaction coupled to the reactant, A, and assume that Y is not electroactive at the potential at which A is reduced. The manner in which the diagnostics respond to this coupled reaction depends in a complex way on the position of the equilibrium (Keq = k,/kb = [A]/[Y]) and the values of the rate constants compared to the CV sweep rate [16]. For illustrative purposes, we pick a case in which K, kf, and kb are all moderate, and note the effect of v on the current function, X, of compound A. 

For the determination of rate constants of the reactions that take place in the bulk of the solution, the change in the limiting current with time is measured. For this purpose, at least one of the components of the reaction mixture (either a reactant, intermediate, or product) must be electroactive and give a measurable polarographic wave. It is advantageous, if this wave (as most polarographic waves are) is a linear function of concentration1. 

When dehydration occurs as a consecutive reaction, its effect on polarographic curves can be observed only, if the electrode process is reversible. In such cases, the consecutive reaction affects neither the wave-height nor the wave-shape, but causes a shift in the half-wave potentials. Such systems, apart from the oxidation of -aminophenol mentioned above, probably play a role in the oxidation of enediols, e.g. of ascorbic acid. It is assumed that the oxidation of ascorbic acid gives in a reversible step an unstable electroactive product, which is then transformed to electroinactive dehydroascorbic acid in a fast chemical reaction. Theoretical treatment predicted a dependence of the half-wave potential on drop-time, and this was confirmed, but the rate constant of the deactivation reaction cannot be determined from the shift of the half-wave potential, because the value of the true standard potential (at t — 0) is not accessible to measurement. 

When considering the case of electroactive species in solution, the electron transfer reaction takes place over a range of several angstroms and the heterogeneous rate constant can be written as [46] ... 

The potentials Ex and E2 should be chosen in such way that at Ex no electrode process occurs and at E2 the electrode reaction of an electroactive species takes place. If the rate of the electrode process is controlled only by diffusion, the Cottrell equation [Eq. (3.6)] can be applied. Therefore, the observed current should be a linear function of t m with the intercept at the origin (a test for diffusion control). The diffusion coefficient of the electroactive species is directly proportional to the slope of the curve. The heterogeneous rate constant of a kinetically limited electrode reaction (kc or k3) also can be evaluated. 

If the electron-transfer step in an electrode reaction is preceded by a chemical reaction that involves proton transfer, the polarographic current often will be a complex function of the concentration of the electroactive species, the hy-dronium ion concentration, and the rate constants for proton and electron transfer. Currents controlled by the rate of a chemical reaction are called kinetic currents and often are observed in the reduction of electroactive acids (e.g., pyruvic acid), in which the protonated form of the acid is more easily reduced than the anion. A polarogram of pyruvic acid in unbuffered solution exhibits two waves whose relative wave heights depend on the concentration of pyruvic acid and the solution pH.59... 

If the equilibrium constant of the chemical reaction (such as complex stability constant, hydration-dehydration equilibrium constant, or the piCa of the investigated acid-base reaction) is known, limiting currents can be used to calculate the rate constant of the chemical reaction, generating the electroactive species. Such rate constants are of the order from 104 to 1010 Lmols-1. The use of kinetic currents for the determination of rate constants of fast chemical reactions preceded even the use of relaxation methods. In numerous instances a good agreement was found for data obtained by these two independent techniques. 

Studies on the electrochemical behavior of ferrocene encapsulated in the hemi-carcerands 61 and 62, indicated that encapsulation induces substantial changes in the oxidation behavior of the ferrocene subunit [98]. In particular, encapsulated ferrocene exhibits a positive shift of the oxidation potential of c. 120 mV, probably because of the poor solvation of ferrocenium inside the apolar guest cavity. Lower apparent standard rate constants were found for the heterogeneous electron transfer reactions, compared to those found in the uncomplexed ferrocene under identical experimental conditions. This effect may be due to two main contributions (i) the increased effective molecular mass of the electroactive species and (ii) the increased distance of maximum approach of the redox active center to the electrode surface. 

The opportunity of obtaining direct electrochemistry of cytochrome c and other metalloproteins at various electrode materials such as modified gold and pyrolytic graphite has led to numerous reports of heterogeneous electron transfer rates and mechanisms between the protein and the electrode. In all the reports, Nicholson s method (37) was employed to calculate rate constants, which were typically within the range of 10" -10 cm sec with scan rates varying between 1 and 500 mV sec This method is based on a macroscopic model of the electrode surface that assumes that mass transport of redox-active species to and from the electrode occurs via linear diffusion to a planar disk electrode and that the entire surface is uniformly electroactive, i.e., the heterogeneous electron transfer reaction can take place at any area. 

In this equation, and represent the surface concentrations of the oxidized and reduced forms of the electroactive species, respectively k° is the standard rate constant for the heterogeneous electron transfer process at the standard potential (cm/sec) and oc is the symmetry factor, a parameter characterizing the symmetry of the energy barrier that has to be surpassed during charge transfer. In Equation (1.2), E represents the applied potential and E° is the formal electrode potential, usually close to the standard electrode potential. The difference E-E° represents the overvoltage, a measure of the extra energy imparted to the electrode beyond the equilibrium potential for the reaction. Note that the Butler-Volmer equation reduces to the Nernst equation when the current is equal to zero (i.e., under equilibrium conditions) and when the reaction is very fast (i.e., when k° tends to approach oo). The latter is the condition of reversibility (Oldham and Myland, 1994 Rolison, 1995). 

The case of charge transfer process preceded by a first-order bulk-surface reaction has been described by Guidelli (1971). Here, the parent electroinactive species is transformed into an electroactive species both through a homogeneous chemical reaction taking place within a thin solution layer adjacent to the electrode (with rate constant, Ay) and through a heterogeneous chemical reaction catalyzed by the electrode surface (with rate constant The chronoamperometric current becomes ... 

For interest, Figs. 18(b) and 19(a) show standard rate constant-potential curves obtained by analysing the same current-potential and impedance-potential data, but assuming that PdCl2 is the electroactive species, i.e. that the deposition-dissolution reactions occurs by the reaction scheme... 

The measurement of peak currents in CV is imprecise because the correction for charging current is typically uncertain. For the reversal peak, the imprecision is increased further because one cannot readily define the folded faradaic response for the forward process (e.g., curve 1, 2, or 3 in Figure 6.5.2) to use as a reference for the measurement. Consequently, CV is not an ideal method for quantitative evaluation of system properties that must be derived from peak heights, such as the concentration of an electroactive species or the rate constant of a coupled homogeneous reaction. The method s power lies in its diagnostic strength, which is derived from the ease of interpreting qualitative and semi-quantitative behavior. Once a system is understood mechanistically, other methods are often better suited for the precise evaluation of parameters. 

Studies of tethered electroactive species are less sensitive to pinholes than experiments with solution reactants and blocking layers, although heterogeneity and roughness of the substrate and film defects can still play a role. The rate constant, k, in this case has units of a first-order reaction (s ). Rate constants can be determined by a voltammetric method as described earlier for electroactive monolayers (Section 14.3.3). In addition potential-step chronoamperometry can be employed, in which case the current follows a simple exponential decay (88, 90, 91) ... 

Fig. 2.18 An equivalent circuit representing an electrode/solution interface. The electrode surface is covered by a monolayer of a redox-active species. e ac potential across the faradaic unit of equivalent circuit, Ca double-layer capacitance, Rs -uncompensated solution resistance, Zf impedance representing solely the electron transfer reaction process of the monolayer, )> ac current due to the faradaic process, Z, total impedance of the whole system, ks. heterogeneous electron transfer rate constant of the monolayer of electroactive species, R charge transfer resistance, Q capacitance associated with the redox reaction of the adsorbed species.

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