Electrode theoretical treatment

Theoretical treatment of polarographic curves for the calculation of values of jo has been described [65Hey, 66Hey], for an overview see [94Gal], a further evaluation procedure has been described [6801d]. Experimental details, in particular of solid electrodes in combination with a rotating disk electrode have been reported elsewhere [84Guy]. (Data obtained with this method are labelled PP.)... 

In the theoretical treatment of ion exchange polymers the roles of charge propagation and of migration of ions were further studied by digital simulation. Another example of proven 3-dimensional redox catalysis of the oxidation of Ks[Fe(CN)5] at a ruthenium modified polyvinylpyridine coated electrode was reported... 

Yet at 0.1 V/s the anodic peak due to the oxidation of the radical cation does not exhibit the shape characteristic of stripping of a solid phase. At faster scan rates the anodic peak broadens considerably and splits into two peaks the same behavior is noticeable in Figure 1. We do not have an explanation for this phenomenon. A recent theoretical treatment of redox molecules attached to electrode surfaces predicts that under certain conditions an anodic surface wave can broaden and split with increasing scan rate in a manner shown in Figure 3 (16). However the same theory predicts that the corresponding cathodic peak normalized to constant scan rate will increase with increasing scan rate. The latter prediction is not observed in our system. 

More complicated reactions that combine competition between first- and second-order reactions with ECE-DISP processes are treated in detail in Section 6.2.8. The results of these theoretical treatments are used to analyze the mechanism of carbon dioxide reduction (Section 2.5.4) and the question of Fl-atom transfer vs. electron + proton transfer (Section 2.5.5). A treatment very similar to the latter case has also been used to treat the preparative-scale results in electrochemically triggered SrnI substitution reactions (Section 2.5.6). From this large range of treated reaction schemes and experimental illustrations, one may address with little adaptation any type of reaction scheme that associates electrode electron transfers and homogeneous reactions. 

The transport of electro active species from the bulk of the solution to the electrode may be governed not only by diffusion but also by adsorption of the species on the electrode surface. When both the mechanisms are operative, the overall electrochemical process may give considerably complicated results. The theoretical treatment is complex and of limited interest to inorganic chemists, therefore, a qualitative approach will be adopted to identify the presence of adsorption phenomena. 

A theoretical treatment of combination titrations with an ideal indicator electrode was given by Meites et al. [89-91 ]. They have shown that the dilution effect causes a deviation of the titration curve inflection point from the equivalence point. However, this deviation is small compared with the error... 

The symmetry factor P is obviously a central entity in electrodics and a fundamental quantity in the theoretical treatment of charge transfer at surfaces, particularly in relating electrode kinetics to solid-state physics. 

As noted above, the charge transfer step can involve either electron transfer, as in the case, say, of a Pt(s)IFe3+(aq),Fe2+(aq) electrode, or ion transfer, as in the case, say, of a Zn(s)lZn2+(aq) system. While the theoretical treatment of the two forms of charge transfer are essentially different in nature, the final equations relating current and voltage turn out to be very similar. 

Theoretical treatments of charge transfer at electrodes were developed by Gurney, Horiuti, and Eyring and the more recent work of Gerischer, Marcus, Hush, and Levich, among others, permitted the study of simple redox electrode reactions under the same theoretical framework developed for homogeneous redox reactions in solution. 

Theoretical treatments for the analysis of complex electrode reactions in terms of elementary steps can be made, as in general chemical kinetics, by the steady state [3, 5, 7] or the quasi-equilibrium methods [4, 5, 47]. 

The theoretical treatment of mass transfer in LSV and CV assumes that only diffusion is operative. Supporting electrolyte concentrations of the order of 0.1 M are generally used at substrate concentrations of the order of 10-3 M, which should preclude the necessity of considering mass transfer by migration. Here, it is assumed that planar stationary electrodes are used under circumstances where diffusion can be considered to be semi-infinite linear diffusion. Other types of electrode may give rise to spherical, cyclindrical or rectangular diffusion and these cases have been treated. 

A natural extension of the VDME is the streaming mercury electrode [74] where a fine jet of mercury issues from the capillary and is limited by a glass plate. Although some theoretical treatments are available, the poor definition of the length, radius and surface velocity of the mercury... 

It is for these reasons that most of the theoretical treatments have, until recently, considered uniformly accessible electrodes, i.e. the DME and RDE, where 8K and 5N are independent of the electrode coordinate, and also fast homogeneous reactions (5K 5N ). 

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. 

The theoretical treatment becomes more complicated if the electrode regions are taken into account — for example, when the solid is deposited on a thin wire acting as the central electrode of a corona discharge. Such an arrangement was used... 

The electrostatic aspects of electrochemical systems will be introduced first and the electrochemical potential as a key concept is presented (Sects. 1.2-1.4). The electrochemical equilibrium is discussed and Nemst s equation and standard and formal electrode potentials are introduced (Sect. 1.5). The study of electrochemical interfaces under equilibrium ends with the phenomenological and theoretical treatment of the electrical double layer (Sect. 1.6). 

They are applicable to electrodes of any shape and size and are extensively employed in electroanalysis due to their high sensitivity, good definition of signals, and minimization of double layer and background currents. In these techniques, both the theoretical treatments and the interpretation of the experimental results are easier than those corresponding to the multipulse techniques treated in the following chapters. Four double potential pulse techniques are analyzed in this chapter Double Pulse Chronoamperometry (DPC), Reverse Pulse Voltammetry (RPV), Differential Double Pulse Voltammetry (DDPV), and a variant of this called Additive Differential Double Pulse Voltammetry (ADDPV). A brief introduction to two triple pulse techniques (Reverse Differential Pulse Voltammetry, RDPV, and Double Differential Triple Pulse Voltammetry, DDTPV) is also given in Sect. 4.6. 

Section 2.2.1 above introduced the concept of homogenous electron transfer in a generic donor acceptor supermolecule A-L-B. Here, the most commonly applied theoretical treatments of heterogeneous electron transfer are discussed. In this scenario, A is now an electrode characterized by continuous or semi-continuous... 

The exact solution of the convection-diffusion equations is very complicated, since the theoretical treatments involve solving a hydrodynamic problem, i.e., the determination of the solution flow velocity profile by using the continuity equation or -> Navier-Stokes equation. For the calculation of a velocity profile the solution viscosity, densities, rotation rate or stirring rate, as well as the shape of the electrode should be considered. 

Faradaic rectification — When the electrode potential of the working - electrode is modulated with a sinusoidal -> alternating current the mean potential is shifted from the DC value by a small increment in many cases when the AC modulation is sufficiently large. This effect has been named faradaic rectification, it is caused by the nonlinearity of the electrode response, in particular the variation of current with electrode potential [i]. A theoretical treatment for an electrode in contact with a solution containing a redox system has been provided [ii]. It was extended to reactions where one reactant is present in its element form dissolved in the liquid metallic phase (e.g., Cd2+ + 2e -> Cd(Hg)) [iii]. An improved evaluation technique has been proposed [iv], and some inherent problems have been reviewed [v]. A variant of this method applied to -> polarography has been described [vi]. Second and higher harmonics in - AC voltammetry (polarography) [vii] also arise from this nonlinearity, and hence these techniques also have some characteristics that resemble those found in - faradaic rectification voltammetry. 

The foregoing theoretical treatment implicitly assumes that the interaction between the reacting species and the electrode is sufficiently weak and non-specific so that the energetics of the elementary electron-transfer step are determined by the properties of the isolated reactant and the surrounding solvent ("weak-overlap pathway, Sect. 2.2). However, as noted in Sect. 2.2, the occurrence of inner-sphere pathways may not only alter the overall reaction energentics via stabilization of the precursor and successor states, but also via alterations in the shape of the electron-transfer barrier itself ("strong-overlap pathway). 

---