Reversibility degree

In agreement with Eq. (1.189), the reversibility degree exhibited by the current-potential response will be determined not only by the value of the rate constants but also by the ratio Rt = k°/mi (with k° being the heterogeneous rate constant for the charge transfer reaction). Thus, for high values of/ Eq. (1.189) becomes... 

Equations (3.17) and (3.18) hold for electrochemical reactions of any reversibility degree. By comparing these equations with Eq. (3.19) corresponding to a reversible process, it can be inferred that the current for a non-reversible process is expressed as the reversible current modulated by F function (that contains the kinetic influence through the dimensionless parameter /), which increases with % from zero to the unity (see Fig. E.l of Appendix E). Hence, small values of % cause a strong kinetic influence and large values of x give rise to a reversible behavior. 

Note that the usual definition of the mass transfer coefficient is related to limiting diffusion conditions or nemstian conditions (mo, no = yDo/nt for a planar electrode see Sect. 1.8.4). The definition given in Eq. (3.43) is general for any reversibility degree of the electrode process at planar electrodes. 

The criterion discussed above is based on the dependence of the surface concentration of the oxidized species with the reversibility degree of the electrode process. So, for a totally irreversible process, the rate of depletion of the surface concentration Cq is much smaller than the mass transport rate process, and therefore, at the formal potential its value should be coincident with the bulk concentration (co(2,°)/coi — l)- In contrast- for reversible electrode reactions, cb(x°)/co = 0.5 (see Eq. (2.20) of Sect. 2.2 for = 0 and y = 1). In order to verify this behavior, the variation of the surface concentration of species O at the formal potential calculated as a function of has been plotted in Fig. 3.5b. From this figure, it can be deduced that at the irreversible limit (i.e., = 0.17),... 

As indicated in Sect. 3.2.1.4, it will be assumed that an irreversible process corresponds to/ phe < 0.05 and a reversible one to/ phe > 10. On the basis of these limits, it is clear that the lower the electrode radius, the higher the value of k° needed to consider the process reversible is. For example, for D = 10 5 cm2 s 1, reversible processes are observed in microelectrodes for k° > 10 4// s. This means that for rs = 10 cm a value of k° >0.1 cm s 1 is enough, whereas for rs = 10 5 cm it is necessary for k° > 10 cm s-1. This behavior is in agreement with the enhanced irreversibility observed for microelectrodes, as discussed above. The accuracy of this approach is based on the dependence of the surface concentration of the oxidized species with the reversibility degree. Under these conditions, at... 

The validity of analytical solution (4.120) has been studied by comparison with numerical calculations [45] and an excellent agreement between analytical and numerical results was obtained for any electrode size, for any length of the potential pulses, and whatever the reversibility degree of the electrode process. 

As can be deduced from Figs. 4.27 and 4.28, the value of the oxidative limiting current provides a simple criterion to distinguish between the EC and E mechanisms independently of the reversibility degree of the latter. Thus, when a follow-up... 

Fig. 5.12 CV response corresponding to a charge transfer process of different reversibility degrees taking place at a planar electrode, calculated numerically by following the procedure given in [21, 22] The values of the voltammetric dimensionless rate constant plane aPPCar tllC g C...

For nonplanar electrodes there are no analytical expressions for the CV or SCV curves corresponding to non-reversible (or even totally irreversible) electrode processes, and numerical simulation methods are used routinely to solve diffusion differential equations. The difficulties in the analysis of the resulting responses are related to the fact that the reversibility degree for a given value of the charge transfer coefficient a depends on the rate constant, the scan rate (as in the case of Nemstian processes) and also on the electrode size. For example, for spherical electrodes the expression of the dimensionless rate constant is... 

From Eqs. (5.92)-(5.94), it is clear that K°phe ss < x°phe < xplane, that is, the maximum value of the dimensionless rate constant is that corresponding to a planar electrode (macroelectrode). For smaller electrodes, /c(sphc decreases until it becomes identical to the value corresponding to a stationary response, xpphe ss. In practice, this means that the decrease of the electrode size will lead to the decrease of the reversibility degree of the observed signal. It can be seen in the CV curves of Fig. 5.14, calculated for k ) = 10 eras 1 and v = 0.1 Vs-1, that the decrease of rs causes an increase and distortion of the dimensionless current similar to that observed for Nemstian processes (see Fig. 5.5), but there is also a shift of the curve toward more negative potentials (which can be clearly seen in Fig. 5.14b). 

The total amount of adsorbed species remains constant during the experiment, so, whatever the reversibility degree of the charge transfer process ... 

From the curves in this figure, it can be deduced that the current decreases and the charge increases faster the higher ki l is (i.e., the higher the reversibility degree... 

From the above, it is clear that a simple visual inspection of the charge-time curves allows us to deduce the reversibility degree of the process. This characteristic behavior, in which the discrete nature of the potential is more evident for reversible processes, cannot be obtained when a continuous potential-time perturbation is applied, as in Cyclic Voltammetry, since this kind of perturbation gives rise to continuous charge-potential curves, whatever the reversibility degree of the response (see Sect. 6.4.2). 

A stationary behavior is attained when the condition kc 1 s 1 holds. Under these conditions, the terms 9m become null (see Eqs. (6.200) and (6.201)) and Eqs. (6.202) and (6.205) for the current and converted charge take the following simpler form, whatever the reversible degree of the electrode reaction ... 

Note that in agreement with Eqs. (6.218) and (6.220), the absolute value of the slope of these linear regions does not depend on the reversibility degree of the electron transfer step and its measurement will allow us to obtain the catalytic rate constant. 

Cyclic Square Wave Voltammetry (CSWV) is very useful in determining the reversibility degree and the charge transfer coefficient of a non-Nemstian electrochemical reaction. In order to prove this, the CSWV curves of a quasi-reversible process with Kplane = 0.03 and different values of a have been plotted in Fig. 7.17. In this figure, we have included the net current for the first and second scans (Fig. 7.17b, d, and f) and also the forward, reverse, and net current of a single scan (first or second, Fig. 7.17a, c, e) to help understand the observed response. 

So, a totally irreversible process could be mistaken for a quasi-reversible one with a 0.5 (Fig. 7.17f). In order to discriminate the reversibility degree of the electrochemical reaction, it is necessary to take into account that for a quasi-reversible process the peak corresponding to more cathodic potentials in the second scan (denoted as RC by [29]) is higher than that located at more anodic ones (denoted as RA by [29]) when a 3> 0.5, whereas the opposite is observed for a fully irreversible electron transfer for any value of a (see also Table insert, Fig. 7.20). 

From the / w/Gf (F Fc° ) curves in Fig. 7.57a, c, it can be seen that the current increases with kc, whatever the reversibility degree of the electrochemical step. Under reversible conditions (Fig. 7.57a), these curves present a peak-shaped feature centered at the formal potential of the immobilized electro-active species (dotted line). When the charge transfer step is quasi-reversible, the current curves show one or two peaks depending on the values of kc and sw. Typically, an increase of kc gives rise to a single peak located at more negative potentials than Fl° (see Fig. 7.57c). No simple expressions for the peak parameters can be obtained in these conditions. 

The most sensitive techiuques are based on the reversibility degree of the redox reactions of certain reactants present in solution. The reactions of TCNQ [60] and ferrocene derivatives [47,53] can be carried out quite reversibly on the freshly formed surfaces of encapsulated ceramic HTSC electrodes and high-quality films in dry acetonitrile. On addition of water, the reversibility is upset due to the formation of low-conductivity degradation products on the surface. The increase in the potential difference between anodic and cathodic peaks correlates with the degradation rate. According to [484] reversible responses to similar reactants can also be obtained on electrodes made of high-quality ceramics. 

Despite the dependence of ip on charge-transfer kinetics (eqns [5] and [6]), the sensitivity of LSV is almost independent of the reversibility degree (only a decrease of 25% is found on passing from a Nernstian to a totally irreversible process with a = 0.5). This fact makes LSV the most sensitive voltammetric technique for analytes involved in irreversible processes because pulsed voltammetric methods or alternating current voltammetry provide for these processes very low signals. 

The simulation techniques were also applied to the reduction of Re H, for which neither the abovementioned concentration dependence nor the medium basicity effect were observed. However, the simplest EC scheme did not provide an acceptable agreement with the experimental cyclic voltaimnograms, and a bimolecular chemical reaction (involving an hydrogen atom abstractor) should be involved, although the detailed analysis was precluded by difficulties encountered in getting a reproduceable reversibility degree for the cathodic wave. 

Figure 10.6 shows the plot of the cathodic and anodic components of the current, together with the relevant sum, for two different reversibility degrees. 

Fig. 10.6 Total currents with relevant cathodic and anodic components, i.e., forward and backward currents, respectively, as a function of the overvoltage for two different reversibility degrees, as simply computed by the Butler-Volmer expression in Eq. (10.27) in plot (b) the total current coincides with the anodic (red line) or cathodic (blue line) component, for positive and negative values of q, respectively...

---