Formal standard rate constant

Here, i is the faradaic current, n is the number of electrons transferred per molecule, F is the Faraday constant, A is the electrode surface area, k is the rate constant, and Cr is the bulk concentration of the reactant in units of mol cm-3. In general, the rate constant depends on the applied potential, and an important parameter is ke, the standard rate constant (more typically designated as k°), which is the forward rate constant when the applied potential equals the formal potential. Since there is zero driving force at the formal potential, the standard rate constant is analogous to the self-exchange rate constant of a homogeneous electron-transfer reaction. 

The Butler-Volmer rate law has been used to characterize the kinetics of a considerable number of electrode electron transfers in the framework of various electrochemical techniques. Three figures are usually reported the standard (formal) potential, the standard rate constant, and the transfer coefficient. As discussed earlier, neglecting the transfer coefficient variation with electrode potential at a given scan rate is not too serious a problem, provided that it is borne in mind that the value thus obtained might vary when going to a different scan rate in cyclic voltammetry or, more generally, when the time-window parameter of the method is varied. 

The strength of metal ion solvation affects not only the half-wave potentials but also the rates of electrode reactions of metal ions. For the reduction of a given metal ion, the reaction rate tends to decrease with increasing strength of solvation. The linear relation in Fig. 8.5 was obtained for the reduction of a sodium ion AG°v(Na+) is the solvation energy of Na+ and ks is the standard rate constant at the formal potential [23 a].2 For alkali metal ions in the same solvent, the rate... 

So far the attention has been on the nuclear reorganization barrier. Nevertheless, other important factors previously hidden in the pre-exponential factor (and ultimately in the standard rate constant) have to be considered, namely, the fundamental question of the magnitude of the electronic interaction between electroactive molecules and energy levels in the electrode (i.e., the degree of adiabaticity) and its variation with the tunneling medium (electrode-solution interface), the tunneling distance, and the electrode material. Thus, within the transition-state formalism, the rate constant for electron transfer can be expressed as the product of three factors [39—42] ... 

Table 7.2 Values of the standard rate constant (k°), the transfer coefficient (a), the reorganization energy (2), and the formal potential (Ef, vs. Ag) corresponding to the theoretical curves shown in Fig. 7.21. Taken from [30]...

By using the - formal potential (Effi) and the -> standard rate constant (ks)... 

The current-potential relationship of the totally - irreversible electrode reaction Ox + ne - Red in the techniques mentioned above is I = IiKexp(-af)/ (1+ Kexp(-asteady-state voltammetry, a. is a - transfer coefficient, ks is -> standard rate constant, t is a drop life-time, S is a -> diffusion layer thickness, and

logarithmic analysis of this wave is also a straight line E = Eff + 2.303 x (RT/anF) logzc + 2.303 x (RT/anF) log [(fi, - I) /I -The slope of this line is 0.059/a V. It can be used for the determination of transfer coefficients, if the number of electrons is known. The half-wave potential depends on the drop life-time, or the rotation rate, or the microelectrode radius, and this relationship can be used for the determination of the standard rate constant, if the formal potential is known. 

For solution redox couples uncomplicated by irreversible coupled chemical steps (e.g. protonation, ligand dissociation), a standard (or formal) potential, E°, can be evaluated at which the electrochemical tree-energy driving force for the overall electron-transfer reaction, AG c, is zero. At this potential, the electrochemical rate constants for the forward (cathodic) and backward (anodic) reactions kc and ka (cms-1), respectively, are equal to the so-called "standard rate constant, ks. The relationship between the cathodic rate constant and the electrode potential can be expressed as... 

In this approach the cation is considered as a reaction center, with the solvent molecules in the first coordination sphere as substituents. Combination of Eq. (24) with Eq. (56) suggests a linear dependence between the logarithm of the standard rate constant of a given system and its formal potential. 

Fig. 8. Dependence of the logarithm of the standard rate constant in mixed solvents related to the rate in aqueous solution (w) on the formal potential for the following systems (1) Pb(II)/Pb(Hg) (2) Zn(II)/Zn(Hg) and (3) Mn(II)/Mn(Hg).

Fig. 20. Variation of the product of the apparent standard rate constant Atq and the viscosity // of (O) the aqueous or ( ) the organic solvent phase with the formal potential difference the transfer of acetylcholine across the (O) (water-tsucrose)-1,2-dichloroethane or ( ) water-(nitrobenzene-Ketrachloromethane) interface. Concentration of sucrose (wt.%) (1) 4, (2) 10, (3) 20, (4) 30, and (5) (40) data taken from [124]. Concentration of tetrachloromethane (wt.%) (T) 10, (2 ) 23, (3 ) 47, (4 ) 57 and (5 ) 71 data taken from [139]. The broken line corresponds to a = 0.5.

TABLE 2. Formal potentials E°, transfer coefficients a and standard rate constants k° at 20 °C for electro-oxidation of 56 in DMF and dichloromethane solutions92... 

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). 

Assuming that standard rate constants (i.e., kf at AG° = 0) are similar, the differences in ET rates for these reactants may only be due to the different AE° values. The formal potentials of Ru(CN)s , Mo(CN)g , and Fe(CN)s- couples measured by cyclic voltammetry are 750, 590, and 235 mV versus Ag/AgCl, respectively. Accordingly, the feedback current obtained with Mo(CN)g in water was higher than with Ru(CN)g . With Fe(CN)s, the ET rate is much higher, and the overall process was diffusion-controlled at any [CKT]] ... 

Generally, the electrochemical rate constants are different for the surface-bound and solution-phase redox couples that, within the Butler-Volmer formalism, means they can have different values of the standard rate constant (fcg , fcg ), transfer coefficient (a °, and formal potential... 

In the case of irreversible reactions, the polarographic half-wave potential also depends on the standard potential (formal potential) however, the kinetics of the electrode reaction lead to strong deviation as an overpotential has to be applied to overcome the activation barrier of the slow electron transfer reaction. In the case of a totally irreversible electrode reaction, the half-wave potential depends on the standard rate constant ks of the electrode reaction, the transfer coefficient a, the number e- of transferred electrons, the diffusion coefficient T>ox, and the drop time t [7] as follows ... 

By using the -> formal potential Ef ) and the -5- standard rate constant (/Cs)... 

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