A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 28
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Since the barrier for reduction is raised and that for oxidation is lowered,the net transformation is conversion of Na(Hg) to Na + + e. Setting the potential to avalue more negative than Eeq, raises the energy of the electron and shifts the curve forNa + + e to higher energies, as shown in Figure 3.3.1c. Since the reduction barrier dropsand the oxidation barrier rises, relative to the condition at E e q , a net cathodic currentflows.
These arguments show qualitatively the way in which the potential affects the netrates and directions of electrode reactions. By considering the model more closely, we canestablish a quantitative relationship.3.3.2One-Step, One-Electron ProcessLet us now consider the simplest possible electrode process, wherein species О and R engage in a one-electron transfer at the interface without being involved in any other chemical step,O + e*±R(3.3.2)КSuppose also that the standard free energy profiles along the reaction coordinate have theparabolic shapes shown in Figure 3.3.2.
The upper frame of that figure depicts the fullpath from reactants to products, while the lower frame is an enlargement of the regionnear the transition state. It is not important for this discussion that we know the shapes ofthese profiles in detail.In developing a theory of electrode kinetics, it is convenient to express potentialagainst a point of significance to the chemistry of the system, rather than against an arbitrary external reference, such as an SCE.
There are two natural reference points, viz.the equilibrium potential of the system and the standard (or formal) potential of thecouple under consideration. We actually used the equilibrium potential as a referencepoint in the discussion of the preceding section, and we will use it again in this section.However, it is possible to do so only when both members of the couple are present, sothat an equilibrium is defined.
The more general reference point is E° . Suppose theupper curve on the О + е side of Figure 3.3.2 applies when the electrode potential isequal to E°'. The cathodic and anodic activation energies are then AG$C and AGj$arespectively.If the potential is changed by AE to a new value, E, the relative energy of the electronresident on the electrode changes by -FkE = —F(E — E°); hence the О + е curvemoves up or down by that amount. The lower curve on the left side of Figure 3.3.2 showsthis effect for a positive A£.
It is readily apparent that the barrier for oxidation, AG*, hasbecome less than AG^a by a fraction of the total energy change. Let us call that fraction3.3 Butler-Volmer Model of Electrode Kinetics - 95Reaction coordinateReaction coordinateFigure 3.3.2 Effects of a potentialchange on the standard free energies ofactivation for oxidation and reduction.The lower frame is a magnified picture ofthe boxed area in the upper frame.I — a, where a, the transfer coefficient, can range from zero to unity, depending on theshape of the intersection region. Thus,AG\ =(3.3.3)- (1 - a)F(EA brief study of the figure also reveals that at potential E the cathodic barrier, AGf, ishigher than AGlc by aF(E - £ ° ) ; therefore,aF(E -(3.3.4)Now let us assume that the rate constants kf and kb have an Arrhenius form that canbe expressed ask{ = Af exp (-AGl/RT)|(3.3.5)(3.3.6)Inserting the activation energies, (3.3.3) and (3.3.4), giveskf = Afexp (-AGyRT)exp[-af(E- E0)]kb = Abexp (-AG^ a //?Dexp[(l - a)f(E -E0)](3.3.7)(3.3.8)where/= F/RT.
The first two factors in each of these expressions form a product that isindependent of potential and equal to the rate constant at E = E°'.4Now consider the special case in which the interface is at equilibrium with a solutionin which CQ = Cf. In this situation, E = E0' and kfC% = kbC^, so that k{ = kb. Thus, E0'is the potential where the forward and reverse rate constants have the same value. That4In other electrochemical literature, kf and k^ are designated as kc and k.d or as kox and kxt&. Sometimes kineticequations are written in terms of a complementary transfer coefficient, /3 = 1 —a.96 v Chapter 3.
Kinetics of Electrode Reactionsvalue is called the standard rate constant, k0.5 The rate constants at other potentials canthen be expressed simply in terms of k°:(3.3.9)(3.3.10)Insertion of these relations into (3.2.8) yields the complete current-potential characteristic:(3.3.11)This relation is very important. It, or a variation derived from it, is used in the treatment of almost every problem requiring an account of heterogeneous kinetics. Section 3.4will cover some of its ramifications. These results and the inferences derived from themare known broadly as the Butler-Volmer formulation of electrode kinetics, in honor of thepioneers in this area (17, 18).One can derive the Butler-Volmer kinetic expressions by an alternative methodbased on electrochemical potentials (8, 10, 12, 19-21).
Such an approach can be moreconvenient for more complicated cases, such as requiring the inclusion of double-layereffects or sequences of reactions in a mechanism. The first edition develops it in detail.63.3.3The Standard Rate ConstantThe physical interpretation of k° is straightforward. It simply is a measure of the kineticfacility of a redox couple. A system with a large k° will achieve equilibrium on a shorttime scale, but a system with small k° will be sluggish.
The largest measured standard rateconstants are in the range of 1 to 10 cm/s and are associated with particularly simple electron-transfer processes. For example, the standard rate constants for the reductions andoxidations of many aromatic hydrocarbons (such as substituted anthracenes, pyrene, andperylene) to the corresponding anion and cation radicals fall in this range (22-24).
Theseprocesses involve only electron transfer and resolvation. There are no significant alterations in the molecular forms. Similarly, some electrode processes involving the formation of amalgams [e.g., the couples Na+/Na(Hg), Cd 2+ /Cd(Hg), and Hg 2 2+ /Hg] are ratherfacile (25, 26).
More complicated reactions involving significant molecular rearrangementupon electron transfer, such as the reduction of molecular oxygen to hydrogen peroxide orwater, or the reduction of protons to molecular hydrogen, can be very sluggish (25-27).Many of these systems involve multistep mechanisms and are discussed more fully inSection 3.5.
Values of k° significantly lower than 10~9 cm/s have been reported (28-31);therefore, electrochemistry deals with a range of more than 10 orders of magnitude inkinetic reactivity.Note that kf and kb can be made quite large, even if k® is small, by using a sufficientlyextreme potential relative to E°'. In effect, one drives the reaction by supplying the activation energy electrically. This idea is explored more fully in Section 3.4.5The standard rate constant is also designated by £ s h or ks in the electrochemical literature.
Sometimes it is alsocalled the intrinsic rate constant.6First edition, Section 3.4.3.3 Butler-Volmer Model of Electrode Kinetics97£=00+E=EFigure 3.3.3 Relationship ofthe transfer coefficient to theangles of intersection of thefree energy curves.Reaction coordinate3.3.4 The Transfer CoefficientThe transfer coefficient, a, is a measure of the symmetry of the energy barrier. Thisidea can be amplified by considering a in terms of the geometry of the intersection region, as shown in Figure 3.3.3. If the curves are locally linear, then the angles в and фare defined byt<m6 = aFE/xХшф = (1 -a)FE/x(3.3.12)(3.3.13)hencea =tan вгшф+гтв(ЗЗЛ4)If the intersection is symmetrical, ф = в, and a = x/2. Otherwise 0 ^ a < У2 o rV2 < a < 1, as shown in Figure 3.3.4.
In most systems a turns out to lie between0.3 and 0.7, and it can usually be approximated by 0.5 in the absence of actualmeasurements.The free energy profiles are not likely to be linear over large ranges of the reactioncoordinate; thus the angles в and ф can be expected to change as the intersection betweenreactant and product curves shifts with potential.
Consequently, a should generally be ao+o+Reaction coordinateFigure 3.3.4 The transfer coefficient as an indicator of the symmetry of the barrier to reaction.The dashed lines show the shift in the curve for О + е as the potential is made more positive.98Chapter 3. Kinetics of Electrode Reactionspotential-dependent factor (see Section 6.7.3). However, in the great majority of experiments, a appears to be constant, if only because the potential range over which kineticdata can be collected is fairly narrow. In a typical chemical system, the free energies ofactivation are in the range of a few electron volts, but the full range of measurable kineticsusually corresponds to a change in activation energy of only 50-200 meV, or a few percent of the total.
Thus, the intersection point varies only over a small domain, such as, theboxed region in Figure 3.3.2, where the curvature in the profiles can hardly be seen. Thekinetically operable potential range is narrow in most systems, because the rate constantfor electron transfer rises exponentially with potential.
Not far beyond the potential wherea process first produces a detectable current, mass transfer becomes rate-limiting and theelectron-transfer kinetics no longer control the experiment. These points are discussed inmuch detail throughout the remainder of this book. In a few systems, mass transfer is notan issue and kinetics can be measured over very wide ranges of potential. Figure 14.5.8provides an example showing large variations of a with potential in a case involving asurface-bound electroactive species.3.4 IMPLICATIONS OF THE BUTLER-VOLMER MODELFOR THE ONE-STEP, ONE-ELECTRON PROCESSIn this section, we will develop a number of operational relationships that will prove valuable in the interpretation of electrochemical experiments.
Each is derived under the assumption that the electrode reaction is the one-step, one-electron process for which theprimary relations were derived above. The validity of the conclusions for a multistepprocess will be considered separately in Section 3.5.3.4.1Equilibrium Conditions. The Exchange Current (8-14)At equilibrium, the net current is zero, and the electrode is known to adopt a potentialbased on the bulk concentrations of О and R as dictated by the Nernst equation.
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