Irreversible Waves, Slope

E vs. log(id-i)/f which should be linear with a slope of 59.1/n mV at 25 °C if the wave is reversible. This method relies however upon a prior knowledge of n, and if this is not known then the method is not completely reliable as theory predicts that when the electron transfer process itself is slow, so that equilibrium at the electrode between the oxidized and reduced forms of the couple is established slowly and the Nemst equation cannot be applied, then an irreversible wave is obtained and a similar plot will also yield a straight line but of slope 54.2/ana mV at 25 °C (a = transfer coefficient, i.e. the fraction of the applied potential that influences the rate of the electrochemical reaction, usually cu. 0.5 na = the number of electrons transferred in the rate-determining step). Thus a slope of 59.1 mV at 25 °C could be interpreted either as a reversible one-electron process or an irreversible two-electron process with a = 0.45. If the wave is irreversible in DC polarography then it is not possible to obtain the redox potential of the couple. 

If the electron transfer is irreversible, the slope of the log plot will be > 59/nmV i.e., the wave will appear more drawn out (Fig. 2). The equation analogous to (d) that describes the irreversible wave is ... 

Although a large wave slope is a clear indicator that a system is not showing clean reversible behavior, it does not necessarily imply that one has an electrode process controlled by the kinetics of electron transfer. Electrode reactions frequently include purely chemical processes away from the electrode surface. A system involving chemical complications of this kind can show a wave shape essentially identical with that expected for a simple electron transfer in the totally irreversible regime. For example, the reduction of nitrobenzene in aqueous solutions can lead, depending on the pH, to phenylhydroxy-lamine (32) ... 

Wave-slope plot. Totally irreversible steady-state voltammograms give linear plots of E vs. log [(/d — /)//] in accord with (5.5.48). The slope provides a and the intercept at E yields l if DqIvq is known. This approach involves the assumption that Butler-Volmer kinetics apply. For a totally irreversible wave based on early transients, the wave-slope plot is predicted to be slightly curved consequently it does not have quantitative utility. 

Since a is usually between 0.3 and 0.7, both the wave slope and the Tomes criterion for a totally irreversible system are normally significantly larger than for a reversible system. These figures of merit are not without ambiguity, however. Consider the predicted wave slope of 63.8 mV for a = 0.85. Within the precision of normal measurements, one could diagnose the system as either reversible or irreversible. It is always a good idea to examine reversibility by a method, such as cyclic voltammetry, that allows a view of the electrode reaction in both directions. 

The irreversible cathodic/anodic polarographic wave is shifted to more negative/positive potentials with respect to the wave produced by a reversible process of the same standard potential E . If the electrode process is quasi-reversible the shift of the wave is no longer as pronounced as it is in the case of the totally irreversible reaction. The slope of the logarithmic presentation A ln[(ii — T)/T]/zlE is always lower than the slope of the reversible process involving the same number of electrons. At the foot of the irreversible wave virtually no current is observed. At the plateau of the wave the current becomes diffusion controlled. The T — E curves are like the curves recorded under steady state conditions at microelectrodes (cf. Fig. 9, curves 2 and 3). 

The half-wave potentials of (FTF4)Co2-mediated O2 reduction at pH 0-3 shifts by — 60 mV/pH [Durand et ah, 1983], which indicates that the turnover-determining part of the catalytic cycle contains a reversible electron transfer (ET) and a protonation, or two reversible ETs and two protonation steps. In contrast, if an irreversible ET step were present, the pH gradient would be 60/( + a) mV/pH, where n is the number of electrons transferred in redox equilibria prior to the irreversible ET and a is the transfer coefficient of the irreversible ET. The —60 mV/pH slope is identical to that manifested by simple Ee porphyrins (see Section 18.4.1). The turnover rate of ORR catalysis by (ETE4)Co2 was reported to be proportional to the bulk O2 concentration [Collman et ah, 1994], suggesting that the catalyst is not saturated with O2. 

As ksh in this instance is very small, then according to the Butler-Volmer formulation (eqn. 3.5) the reaction rate of the forward reaction, K — 8,he "F(E 0)/flr, even at E = E°, is also very low. Hence Etppl. must be appreciably more negative to reach the half-wave situation than for a reversible electrode process. Therefore, in the case of irreversibility, the polarographic curve is not only shifted to a more negative potential, but also the value of its slope is considerably less than in the case of reversibility (see Fig. 3.21). In... 

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. 

Both wave heights and half-wave potentials are dependent on pH in reduction of the arylarsonic acids, and it was found that a plot of 1,2 vs pH in the pH range 1.0-2.0 was linear for all of the derivatives except the nitro-substituted one with slopes equal to the slopes of the log [(i i — OA] vs E plots. This behaviour corresponds to the involvement of one proton prior to the irreversible electron transfer step . The initial step, equations 16 and 19, in the reduction process therefore includes the microscopic steps described by equations 23 and 24. 

HOMO) electron [16,29]. Thus, the potential at which the electrochemical wave is observed may correlate with E , since it depends on the three figures. Such a case is presented in Fig. 17 for the series of akylbenzenes [29] already mentioned in this chapter, in which the peak potentials of the chemically irreversible voltammograms (at 0.1 Vs ) correlate with E° with a slope close to unity. 

The variation of the half wave potential with pH also yields information about the reversibility of the electrode process (2.2). The slope of the plot of half wave potential against pH for a reversible process will be 0.059 pin V/pH unit, n is usually 1). That is for a reversible process the slope will be a simple ratio or multiple of 0.059 V. For an irreversible process the slope is given by 0.059 pan V/pH unit, that is no longer a simple ratio or multiple of 0.059 V since the transfer coefficient a has a value significantly less than 1. 

V showed irreversible in the CV time scale, a controlled potential coulometry led reversibly to a stable product with an electronic spectrum very different from the one obtained at pH = 12.8 or in acetonitrile. The half-wave potential of the couple correlated with the pH with a slope of 60 mV per pH unit, suggesting a proton-coupled 1-electron reduction (Equation 28). 

The ratio of transition times for second and first waves suggests the formation of Ti(I) as the product of this second process. The negative slope ofh vs. ( ) plot could be caused by its irreversible disproportionation according to the Fisher-Dracka reaction mechanism [16]. To check this assumptirm, the data were plotted as the function /r = (/v 2 6.17). It is linear, which allows us... 

Hence, for a reversible system, the well-known linear relation is obtained between the potential E and log (/iim -///). Other equations have been derived for those reversible systems that involve semiquinone formation, dimerization, or the formation of complex compounds with mercury. Logarithmic analysis of the polarographic wave is often the only proof of reversibility which is considered but recently several authors, in particular Zuman and Delahay, " have pointed out that it is inadequate to assume that an electrode process is reversible on this evidence alone. For a reversible reaction, plots of E vs. In (/lim - ///) give the electron number z from the slope of the plot, RT/zF, A clearer indication of irreversibility is the evaluation of slopes of log i-E curves for higher concentrations (for i < /lim). Irreversible processes will give Tafel behavior. 

---