Formal potentials, conditional constants

The previous derivation was made under the implicit assumption that the activity coefficients of A and B are both equal to unity. This assumption matches the definition of E° as a standard potential. There are two cases of practical interest, where these conditions are not fulfilled. One is when the activity coefficients differ from unity but do not depend on the relative amounts of A and B in the film. This type of situation may arise when the interactions between the reactants are weak but the presence of the supporting electrolyte decreases the activity coefficients of A and/or B, yA and yB, to below 1 while they remain constant over the entire voltammo-gram. The only change required is thus to replace the standard potential by the formal potential ... 

The SECM can be used to measure the ET kinetics either at the tip or at the substrate electrode. In the former case, the tip is positioned in a close proximity of a conductive substrate (d < a). The substrate potential is kept at a constant and sufficiently positive (or negative) value to ensure the diffusion-controlled regeneration of the mediator at its surface. The tip potential is swept linearly to obtain a steady-state voltammogram. The kinetic parameters (k°, a) and the formal potential value can be obtained by fitting such a voltammogram to the theory [Eq. (22)]. A high value of the mass transfer coefficient (m) is achieved under steady-state conditions when d rate constants (k° > 1 cm-1 s) were measured with micrometersized SECM tips [92-94]. 

In the world of numerical analysis, one distinguishes formally between three kinds of boundary conditions [283,528] the Dirichlet, Neumann (derivative) and Robin (mixed) conditions they are also sometimes called [283,350] the first, second and third kind, respectively. In electrochemistry, we normally have to do with derivative boundary conditions, except in the case of the Cottrell experiment, that is, a jump to a potential where the concentration is forced to zero at the electrode (or, formally, to a constant value different from the initial bulk value). This is pure Dirichlet only for a single species simulation because if other species are involved, the flux condition must be applied, and it involves derivatives. Therefore, in what follows below, we briefly treat the single species case, which includes the Cottrell (Dirichlet) condition as well as derivative conditions, and then the two-species case, which always, at least in part, has derivative conditions. In a later section in this chapter, a mathematical formalism is described that includes all possible boundary conditions for a single species and can be useful in some more fundamental investigations. 

Conditional (apparent) equilibrium constants - Equilibrium constants that are determined for experimental conditions that deviate from the standard conditions used by convention in - thermodynamics. Frequently, the conditional equilibrium conditions refer to - concentrations, and not to - activities, and in many cases they also refer to overall concentrations of certain species. Thus, the formal potential, i.e., the conditional equilibrium constant of an electrochemical equilibrium, of iron(II)/iron(III) may refer to the ratio of the overall concentrations of the two redox forms. In the case of complex equilibria, the conditional - stability constant of a metal ion Mm+ with a ligand L" refers to the overall concentration of all complex species of Mm+ other than Conditional equilibrium... 

Formal potential — Symbol Efr (SI Unit V), has been introduced in order to replace the standard potential of -> cell reaction when the values of - activity coefficients of the species involved in the cell reaction are unknown, and therefore concentrations used in the equation expressing the composition dependence of ceii instead of activities. It also involves the activity effect regarding the -+ standard hydrogen electrode, consequently in this way the formal electrode potential is also defined. Formal potentials are similar to conditional (apparent) equilibrium constants (-> equilibrium constant), in that, beside the effect of the activity coefficients, side reaction equilibria are also considered if those are not known or too complex to be taken into account. It follows that when the logarithmic term which contains the ratio of concentrations in the -> Nernst... 

Formal potentials can be defined on different levels of conditions Thus the formal potential of the -> quinhydrone electrode may be defined (I) as including (a) the standard potential of the hydroquinone di-anion/quinone system, (b) the two acidity constants of the hydroquinone, and (c) the activity coefficients of the hydroquinone dianion and quinone, or, (II), it may also include (c) the pH value. In the latter case, for each pH value there is one formal potential, whereas in the first case one has one formal potential for all pH values, and an equation describing the dependence of the electrode potential as a function of that formal potential and the individual pH values. Formal potentials are strictly thermodynamic quantities, and no kinetic effects (e.g., by electrochemical -> irreversibility) are considered. 

A frequent complication is that several simultaneous equilibria must be considered (Section 3-1). Our objective is to simplify mathematical operations by suitable approximations, without loss of chemical precision. An experienced chemist with sound chemical instinct usually can handle several solution equilibria correctly. Frequently, the greatest uncertainty in equilibrium calculations is imposed not so much by the necessity to approximate as by the existence of equilibria that are unsuspected or for which quantitative data for equilibrium constants are not available. Many calculations can be based on concentrations rather than activities, a procedure justifiable on the practical grounds that values of equilibrium constants are obtained by determining equilibrium concentrations at finite ionic strengths and that extrapolated values at zero ionic strength are unavailable. Often the thermodynamic values based on activities may be less useful than the practical values determined under conditions comparable to those under which the values are used. Similarly, thermodynamically significant standard electrode potentials may be of less immediate value than formal potentials measured under actual conditions. 

Formal Potentials As with conditional constants, that is, constants valid under specifically selected conditions, for example, a given pH and a given ionic medium, conditional or formal potentials are of great utility. 

Figure 18. (A) Cyclic voltammetry of purified cytochrome c at doped indium oxide optically transparent electrodes. Solution contained 73 /uiM cytochrome c, 0.21 M Tris, 0.24 M cacodylic acid, pH 7.0, 0.20 M ionic strength. Electrode area = 0.71 cm. Potential scan rates in mV/s are (a) 100 (b) 50 (c) 20 (d) 10 (e) 5.0 (f) 2.0. (B) Derivative cyclic voltabsorptometry of purified cytochrome c at a tin-doped indium oxide optically transparent electrode. Same conditions as described above. Circles are calculated derivative cyclic voltabsorptometric responses for 73 /iM cytochrome c, formal potential = 0.260 V, n = 1.0, diffusion coefficient of oxidized and reduced cytochrome c = 1.2 x 10 cm /s, difference molar absorptivity at 416 nm = 57,000 cm" formal heterogeneous electron transfer rate constant = 1.0 x 10 cm/s, and electrochemical transfer coefficient = 0.5. Adapted from Reference (126) with permission.

When we considered metal complex formation equilibria, it was found very convenient to introduce the parameter, the conditional formation constant P. The use of p simplified the proper handling of side reactions, including proton transfer reactions of the ligand, complexation of the metal with other ligands present in the solution, and even those cases where the primary metal complex was itself involved. With redox equilibria there is also such a parameter, called the formal potential, E", enabling us to write the Nemst equation as... 

Although the standard potentials are the fundamental values for all thermodynamic calculations, in practice, one has more frequently to deal with the so-called formal potentials. The formal potentials are conditional constants, very similar to the conditional stability constants of complexes and conditional solubility products of sparingly soluble salts (see [2c]). The term conditional indicates that these constants relate to specific conditions, which deviate from the usual standard conditions. Formal potentials deviate from standard potentials for two reasons, i.e., because of nonunity activity coefficients and because of chemical side reactions . The latter should better be termed side equilibria however, this term is not in common use. Let us consider the redox system iron(II/ni) in water ... 

A comparison of experimental and simulated cyclic voltammograms obtained from oxidation of cis-Mn microparticles mechanically adhered to a GC electrode in [C2mim][N(T 2] and simulated data is shown in Fig. 14.12. Once again, aside from the larger magnitudes of the measured currents in the ionic liquid medium, the voltammetric characteristics obtained under dissolved or solid-state conditions are almost identical. The formal potentials ( j j and 2)7 and the ratio of equilibrium constants KJK ) Ki = kflk, K2 = kfi/k yi, kf and kfi are the rate constants of the forward reactions described in Fig. 14.13. k and bi are the rate constants of the backward reactions) derived from the voltammetric data were identical under dissolved and adhered solid-state conditions, as shown in Table 14.2. 

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