Transfer coefficient potential variation

Mean bed particle size (thus, the in-bed heat-transfer coefficient) may vary for external reasons such as a change of feedstock supply or a deterioration in crusher performance. This potential source of variation should be considered before any decision to resurface is made. 

The transfer coefficient a has a dual role (1) It determines the dependence of the current on the electrode potential. (2) It gives the variation of the Gibbs energy of activation with potential, and hence affects the temperature dependence of the current. If an experimental value for a is obtained from current-potential curves, its value should be independent of temperature. A small temperature dependence may arise from quantum effects (not treated here), but a strong dependence is not compatible with an outer-sphere mechanism. 

In Section 1.4.4 we describe some typical examples of outer-sphere electron transfer kinetics, with particular emphasis on the variation of the transfer coefficient (symmetry factor) with the electrode potential (driving force). 

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. 

FIGURE 1.23. Variations of the transfer coefficient with the electrode potential derived from convolutive cyclic voltammetry of the following systems with double layer correction, t-nitrobutane in acetonitrile ( ), r-nitrobutane in DMF ( ), nitrodurene in acetonitrile + 2%H20 (a), nitrodurene in acetonitrile ( ), nitromesitylene in acetonitrile (y). Data from reference 64 and references therein. 

FIGURE 2.6. EC reaction scheme in cyclic voltammetry. Mixed kinetic control by an electron transfer obeying a MHL kinetic law (Xt — 0.7 eV, koo — 4 x 103 cms-1, implying that kg = 0.69 cms-1) and an irreversible follow-up reaction (from bottom to top, k+ = 103, 105, 107, 109s 1). Temperature, 25°C. a Potential-dependent rate constant derived from convolutive manipulation of the cyclic voltammetric data (see the text), b Variation with potential of the apparent transfer coefficient (see the text) obtained from differentiation of the curves in part a. 

FIGURE 3.15. Electrochemical reduction of 4-cyano-tert-butylperbenzoate in DMF showing the variation of the transfer coefficient, a, with the difference in standard potentials of the perbenzoate and the benzoate. Adapted from Figure 8 in reference 20, with permision from the American Chemical Society. 

The apparent transfer coefficient, as derived from the peak width and the variation of the peak potential with the scan rate, is small (between 0.2 and 0.3) in all cases. This rules out the occurrence of a stepwise mechanism (46, 47), in which the follow-up, bond-breaking step would have been... 

Evidence for the potential dependence of the transfer coefficient predicted by eqn. (155) is not very clear. However, variations beyond experimental error have recently been observed by Saveant and Tessier for a series of simple electron transfer reactions with organic molecules in different non-aqueous solvents [70a]. 

Fig. 1.16 (a) Variation of the reduction and oxidation rate constants with the applied potential according to the Marcus-Hush-Chidsey kinetic model, (b) Variation of the transfer coefficient with the applied potential in the Marcus-Hush-Chidsey model. Gray solid lines correspond to the values predicted by the BV model for a = 0.5... 

With respect to the effect of the reorganization energy on the rate constants, Fig. 1.16 shows that this is clearly different from that of the BV transfer coefficient. Thus, the /-value has the same influence on the variation of the cathodic and anodic rate constants with the applied potential. Thus, independently of the value of the... 

Figure 10. Variations of dimensionless potential O0 with dimensionless current density / for different values of transfer coefficient 0. 0 = 0.25, 0.5, 0.75 (from left to right) F = 0.1 s = 0.1 solid line the first order dashed line the second order numerical result for the first order O numerical result for the second order.

So, what is the variation of the rate constant with potential The charge transfer coefficient, or, anodic or cathodic, is given by... 

So, for very fast reactions, the theory predicts a variation of a with potential. There is some evidence that this occurs, but given the multistep nature of any electrode reaction no definitive conclusions can be taken, and mechanisms can be elaborated which have constant charge transfer coefficients. Indeed the fact that the enthalpic and entropic parts of the coefficients have different temperature dependences leads to the question as to what is the real significance of the charge transfer coefficient, a topic currently under discussion9. 

There is ample evidence that the plots of the logarithm of the apparent rate constant Tc against the potential difference (Tafel plots) for univalent ions have reciprocal slopes of about 118 mV per decade [42, 61, 124, 132, 142-144, 146]. This behavior is illustrated for several cations and anions in Fig. 13, which displays data obtained from equilibrium impedance measurements [132]. Tafel plots derived from dc or ac voltammetric measurements in a sufficiently broad potential range are usually curved [144, 145, 163]. Chronocoulometry has been claimed [124] to provide independently the rate constants for the forward and the backward ion transfer in Eq. (21) cf. also Girault s review [14]. However, this is impossible in principle, because these rate constants should always be related to each other by Eq. (22). The origin of the value of the apparent charge transfer coefficient and its variation with the potential has been always the key issue. 

Figure 16. Variations in the apparent heterogeneous rate constant k(E) in Eq. (110) with the electrode potential for the reduction of (CH3)3C — NO2 in DMF at the hanging mercury drop. A dashed line with slope 0.5 is positioned at E = E° to emphasize the curvature, (b) Resulting variations in the transfer coefficient a in Eq. (111). (Experimental data from Ref. 84b.)...

In the following we consider a simple electron transfer mechanism in order to discuss quantitatively the variations in the potential location of the steady-state voltammogram of the system according to the kinetics of the heterogeneous electron transfer. In the derivation of the kinetics we consider that the solution contains only the reactant at concentration C before the electrochemical experiment. Let E°, k, and a be the standard reduction potential, the standard heterogeneous rate constant, and the transfer coefficient of the electron transfer in Eq. (176). 

---