A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 53
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In addition, the wave is broadened bykinetic effects, as one can see clearly in Figure 5.5.3. The displacement is an overpotentialand is proportional to the required kinetic activation. For small k°, it can be hundreds ofmillivolts or even volts. Even so, kf is activated exponentially with potential and can become large enough at sufficiently negative potentials to handle the diffusion-limited flux ofelectroactive species; thus the wave eventually shows a plateau at /d, unless the backgroundlimit of the system is reached first.(e) Totally Irreversible ReactionsThe very displacement in potential that activates kf also suppresses k^\ hence the backward component of the electrode reaction becomes progressively less important at potentials further to the negative side of E° .
If k° is very small, a sizable activation of kf isrequired for all points where appreciable current flows, and къ is suppressed consistentlyto a negligible level. The irreversible regime is defined by the condition that kb/kf ~ 0(i.e., в « 0) over the whole of the voltammetric wave. Then (5.5.11) becomesi = FAkfiSnp [^J erfc(5.5.27)196 • Chapter 5. Basic Potential Step Methodsand (5.5.24) has the limiting form4- = Fx (A) = 7г1/2 A exp (A2) erfc (A)(5.5.28)where A has become kftl/2/Dol/2.The half-wave potential for an irreversible wave occurs where Fi(X) = 0.5, which iswhere A = 0.433.
If kf follows the usual exponential form and t = r, thenexp[-a/(£ 1/2 - £0')] = 0.433^(5.5.29)By taking logarithms and rearranging, one obtainsF-Fo'+ RTь/2.3U°r 1 / 2\I2Dwhere the second term is the displacement required to activate the kinetics. Obviously(5.5.30) provides a simple way to evaluate k° if a is otherwise known.(f) Kinetic RegimesConditions defining the three kinetic regimes can be distinguished in more precise termsby focusing on the particular value of A at E°', which we will call A0.
Since kf = къ = к0and в = 1 at E°\ A0 = (1 + g)k°Tl/2/Dol/2, which can be taken for our purpose as2k r 1 ' /DQ1'2. It is useful to understand A0 as a comparator of the intrinsic abilities of kinetics and diffusion to support a current. The greatest possible forward reaction rate atany potential is kf CQ, corresponding to the absence of depletion at the electrode surface.At E = E°\ this is £°CQ and the resulting current is FA$CQ. The greatest current supportable by diffusion at sampling time т is, of course, the Cottrell current. The ratio of thetwo currents is тт1/2к°т1/2/О^2, or (тг1/2/2) А0.If a system is to appear reversible, A0 must be sufficiently large that FX(A) is essentially unity at potentials neighboring E0'.
For A0 > 2 (or k0rl/2/Dol/2> 1), F^A 0 ) exceeds0.90, a value high enough to assure reversible behavior within practical experimental limits. Smaller values of A0 will produce measurable kinetic effects in the voltammetry. Thuswe can set A0 = 2 as the boundary between the reversible and quasireversible regimes, although we also recognize that the delineation is not sharp and that it depends operationally on the precision of experimental measurements.Total irreversibility requires that kb/kf ~ 0 (в ~ 0) at all potentials where the currentis measurably above the baseline. Because в is also exp[/(£ — £ ° ) ] , this condition simply implies that the rising portion of the wave be significantly displaced from E° in thenegative direction.
If £ 1 / 2 - E0' is at least as negative as -4.6RT/F, then kb/k{ will be nomore than 0.01 at Ец2, and the condition for total irreversibility will be satisfied. The implication is that the second term on the right side of (5.5.30) is more negative than-4.6RT/nF, and by rearrangement one finds that log A0 < -2a + log(2/2.31). The finalterm can be neglected for our purpose here, so the condition for total irreversibility becomes log A0 < -2a. For a = 0.5, A0 must be less than 0.1.In the middle ground, where 10~ 2a < A0 < 2, the system is quasireversible, and one cannot simplify (5.5.24) as a descriptor of either current decay or voltammetric wave shape.It is important to recognize that the kinetic regime, determined by A0, depends notonly on the intrinsic kinetic characteristics of the electrode reaction, but also on the experimental conditions.
The time scale, expressed as the sampling time т in voltammetry, is aparticularly important experimental variable and can be used to change the kinetic regimefor a given system. For example, suppose one has an electrode reaction with the following(not unusual) properties: k° = 10~2 cm/s, a = 0.5, and/) o = DR= Ю~5 cm2/s. For sam-5.5 Sampled-Current Voltammetry for Quasireversible and Irreversible Electrode Reactions ^ 197pling times longer than 1 s, A0 > 2 and the voltammetry would be reversible. Samplingtimes between 1 s and 250 ^is correspond to 2 ^ A0 ^ 0.1 and would produce quasireversible behavior. Values of r smaller than 250 [is would produce total irreversibility.5.5.2General Current-Time Behavior at a Spherical ElectrodeAs a prelude to a treatment of steady-state voltammetry in quasireversible and totally irreversible systems, it is useful to develop a very general description of current flow in a stepexperiment at a spherical electrode.
In Section 5.4.2(a) the basic diffusion problem wasoutlined, and the following relationships arose without invoking a particular kineticcondition.A(s) _ {slD )m0l/2r(5.5.31)rOe-(s/DR)l/2(r-ro)(5.5.32)where f = (D0/DR)l/2 andr0(s/Do)l/2ro(s/DR)(5.5.33)1/2Now we are interested in determining the function A(s) for a step experiment to an arbitrary potential, but where the electron-transfer kinetics are described explicitly in terms ofkf and къ.
By so doing, we will be able to use the results to define current-time responsesfor any sort of kinetic regime, whether reversible, quasireversible, or irreversible. Theproblem is developed just as in the sequence from (5.5.3) to (5.5.9), but in this instance,the results arekfr0фrsCR(r,s) =Do'оsDo1kfr0Vo£2T' ^o 'Dome-(s/DR)rs(r-r0)1/2DAs always, the current is proportional to the difference between the rates of the forwardand backward reactions. In transform space,(5.5.36)By substitution from (5.5.34) and (5.5.35) and algebraic rearrangement, one obtains thefollowing general expression for the current transform:-FADOC%8 +1\K)(5.5.37)198 • Chapter 5. Basic Potential Step Methodswhere 8 and к are two important dimensionless groups:(5.5.38)(5.5.39)Although the development of (5.5.37) from (5.5.34)-(5.5.36) is not obvious, it is straightforward.
The steps are left to Problem 5.8(a). As one proceeds, it is useful to recognizethat в is not only exp[/(£—E°)], but also къ/kf. Also, one can see by inspection of(5.5.33) thatУ = •8 +11(5.5.40)Equation 5.5.37 is powerful because it compactly describes the current response forall types of electrode kinetics and for all step potentials at any electrode where either linear diffusion or spherical diffusion hold. The principal restriction is that it describes onlysituations where R is initially absent, but the following extension, covering the case whereboth О and R exist in the bulk, is readily derived by the same method [Problem 5.8(b)]:Ks) =nFADo{C% rosИ(5.5.41)It is easy to see that (5.5.37) is the special case of (5.5.41) for the situation whereIn other words, all of the current-time relationships that we have so far consideredin this chapter are special-case inverse transformations of (5.5.41).
Because (5.5.37)and (5.5.41) contain the transform variable s not only explicitly, but also implicitly in 8and y, a general analytical inversion is beyond our reach; however one can readily derive the special cases using either (5.5.37) or (5.5.41) as the starting point. The trick isto recognize 8, к, and в as manifesting comparisons that divide important experimentalregimes.In Section 5.4.2(a) we developed the idea that 8 expresses the ratio of the electrode'sradius of curvature to the diffusion-layer thickness. When 8 »1, the diffusion layer issmall compared to TQ, and the system is in the early transient regime where semi-infinitelinear diffusion applies. When 8 «1, the diffusion layer is much larger than r 0 , and thesystem is in the steady-state regime.We now recognize к as the ratio of kf to the steady-state mass-transfer coefficientm0 = Do/r0.
When к «1, the interfacial rate constant for reduction is very small compared to the effective mass-transfer rate constant, so that diffusion imposes no limitationon the current. At the opposite limit, where к »1, the rate constant for interfacial electron transfer greatly exceeds the effective rate constant for mass transfer, but the interpretation of this fact depends on whether kb is also large.99Note that к is also the ratio of the largest current supportable by the kinetics divided by the largest currentsupportable by diffusion; thus it is analogous to the parameter Л used to characterize kinetic effects on systemsbased on semi-infinite linear diffusion.5.5 Sampled-Current Voltammetry for Quasireversible and Irreversible Electrode Reactions199As a first example of the way to develop special cases from (5.5.37), let us considerthe limit of diffusion control. For this case, kf is very large, so к —> °° and в = k^lkf —> 0.Thus,id(s) =FAD0C%(5.5.42)For the early transient regime, 8 »1 [see Section 5.4.2(a)], and (5.5.42) becomes(5.2.10), which is inverted to the Cottrell equation, (5.2.11).
For the steady-state regime,8 «1 and (5.5.42) collapses to a form that is easily inverted to the relationship for thesteady-state limiting current (5.3.2). Actually, (5.5.42) can be inverted directly to the fulldiffusion-controlled current-time relationship at a sphere, (5.2.18). All of these relationships also hold for a hemisphere of radius TQ, which has half of both the area and the current for the corresponding sphere.It is similarly easy to derive other prior results from (5.5.41), including the early transient and steady-state responses for a reversible system [(5.4.17) and (5.4.54)] and theearly transient responses for systems with quasireversible or irreversible kinetics[(5.5.11), (5.5.12), and (5.5.28)].
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