A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 37
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Kinetics of Electrode Reactionsdistribution for O; hence rapid reduction would be seen. Likewise, an overpotential of+ 300 mV (case b) brings the Fermi level down to match the peak in the state distribution for R and enables rapid oxidation. An overpotential of -1000 mV (case c) createsa situation in which WO(A, E) overlaps entirely with filled states on the electrode, andfor 7] = +1000 mV (case d), WR(\, E) overlaps only empty states on the electrode.These latter two cases correspond to very strongly enabled reduction and oxidation,respectively.The lower frame of Figure 3.6.5 shows the very different situation for the fairly largereorganization energy of 1.5 eV.
In this case, an overpotential of —300 mV is not enoughto elevate the Fermi level into a condition where filled states on the electrode overlapWo(\, E), nor is an overpotential of +300 mV enough to lower the Fermi level into acondition where empty states on the electrode overlap WR(A, E). It takes 77 ~ -1000 mVto enable reduction very effectively, and 77 ~ +1000 mV to do the same for oxidation.For this reorganization energy, the anodic and cathodic branches of the i-E curve wouldbe widely separated, much as shown in Figure 3.4.2c.Since this formulation of heterogeneous kinetics in terms of overlapping state distributions is linked directly to the basic Marcus theory, it is not surprising that many of itspredictions are compatible with those of the previous two sections.
The principal difference is that this formulation allows explicitly for contributions from states far from theFermi level, which can be important in reactions at semiconductor electrodes (Section18.2) or involving bound monolayers on metals (Section 14.5.2).3.6.4Tunneling and Extended Charge TransferIn the treatments discussed above, the reactant was assumed to be held at a fixed, shortdistance, JC0, from the electrode.
It is also of interest to consider whether a solution speciescan undergo electron transfer at different distances from the electrode and how the electron-transfer rate might depend on distance and on the nature of the intervening medium.The act of electron transfer is usually considered as tunneling of the electron betweenstates in the electrode and those on the reactant. Electron tunneling typically follows anexpression of the form:Probability of tunneling °c ехр(-Дх)(3.6.37)where x is the distance over which tunneling occurs, and /3 is a factor that depends uponthe height of the energy barrier and the nature of the medium between the states.
For example, for tunneling between two pieces of metal through vacuum (73)1/21/3 « 4тг(2тФ) //г « 1.02 A " eV~1/2X Ф1/2(3.6.38)28where m is the mass of the electron, 9.1 X 10~ g, and Ф is the work function of the1metal, typically given in eV. Thus for Pt, where Ф = 5.7 eV, /3 is about 2.4 A " . Withinthe electron-transfer theory, tunneling effects are usually incorporated by taking the transmission coefficient, /<el, in (3.6.2) asKeiW = ке1°ехр(-/ЗД(3.6.39)where ке\(х) -> 1 when x is at the distance where the interaction of reactant with the electrode is sufficiently strong for the reaction to be adiabatic (48, 49).In electron-transfer theory, the extent of interaction or electronic coupling betweentwo reactants (or between a reactant and the electrode) is often described in terms of adiabaticity. If the interaction is strong, there is a splitting larger than 6T in the energy curvesat the point of intersection (e.g., Figure 3.6.6a).
It leads to a lower curve (or surface) pro-3.6 Microscopic Theories of Charge Transfer -I 131G°(q)(a)(b)Figure 3.6.6 Splitting of energy curves (energy surfaces) in the intersection region, (a) A stronginteraction between О and the electrode leads to a well-defined, continuous curve (surface)connecting О + e with R. If the reacting system reaches the transition state, the probability ishigh that it will proceed into the valley corresponding to R, as indicated by the curved arrow.(b) A weak interaction leads to a splitting less than 4T. When the reacting system approaches thetransition state from the left, it has a tendency to remain on the О + е curve, as indicated by thestraight arrow.
The probability of crossover to the R curve can be small. These curves are drawn foran electrode reaction, but the principle is the same for a homogeneous reaction, where the reactantsand products might be О + R' and R + O', respectively.ceeding continuously from О to R and an upper curve (or surface) representing an excitedstate. In this situation of strong coupling, a system will nearly always stay on the lowersurface passing from О to R, and the reaction is said to be adiabatic. The probability ofreaction per passage approaches unity for an adiabatic reaction.If the interaction is small (e.g,. when the reactants are far apart), the splitting of thepotential energy curves at the point of intersection is less than &T (Figure 3.6.6/?).
In thiscase, there is a smaller likelihood that the system will proceed from О to R. The reactionis said to be nonadiabatic, because the system tends to stay on the original "reactant" surface (or, actually, to cross from the ground-state surface to the excited-state surface). Theprobability of reaction per passage through the intersection region is taken into account byKQi < 1 (47, 48). For example, к е 1 could be 10~5, meaning that the reactants would, on theaverage, pass through the intersection region (i.e., reach the transition state) 100,000times for every successful reaction.In considering dissolved reactants participating in a heterogeneous reaction, one cantreat the reaction as occurring over a range of distances, where the rate constant falls offexponentially with distance.
The result of such a treatment (48, 74) is that electron transfer occurs over a region near the electrode, rather than only at a single position, such asthe outer Helmholtz plane. However, the effect for dissolved reactants should be observable experimentally only under rather restricted circumstances (e.g., D < 10~ 10 cm2/s),and is thus usually not important.On the other hand, it is possible to study electron transfer to an electroactive speciesheld at a fixed distance (10-30 A) from the electrode surface by a suitable spacer, such asan adsorbed monolayer (Section 14.5.2) (75, 76). One approach is based on the use of ablocking monolayer, such as a self-assembled monolayer of an alkane thiol or an insulating oxide film, to define the distance of closest approach of a dissolved reactant to theelectrode.
This strategy requires knowledge of the precise thickness of the blocking layerand assurance that the layer is free of pinholes and defects, through which solutionspecies might penetrate (Section 14.5). Alternatively, the adsorbed monolayer may itself132Chapter 3. Kinetics of Electrode ReactionsElectroactive CentersAlkylthiolchainsGold ElectrodeFigure 3.6.7 Schematic diagram of anadsorbed monolayer of alkane thiolcontaining similar molecules withattached electroactive groups held by thefilm at a fixed distance from the electrodesurface.contain electroactive groups.
A typical layer of this kind (75) involves an alkane thiol(RSH) with a terminal ferrocene group (-Fc), that is, HS(CH2)nOOCFc (often written asHSCnOOCFc; typically n = 8 to 18) (Figure 3.6.7). These molecules are often diluted inthe monolayer film with similar nonelectroactive molecules (e.g., HSC^CH3). The rateconstant is measured as a function of the length of the alkyl chain, and the slope of theplot of ln(£) vs. nor x allows determination of /3.For saturated chains, /3 is typically in the range 1 to 1.2 A" 1 .
The difference between this through-bond value and that for vacuum (through-space), ~ 2 A~ , reflectsthe contribution of the molecular bonds to tunneling. Even smaller /3 values (0.4 to 0.6A" 1 ) have been seen with тг-conjugated molecules [e.g., those with phenyleneethynyl(-Ph-C=C-) units] as spacers (77, 78). Confidence in the /3 values found in these electrochemical studies is reinforced by the fact that they generally agree with those foundfor long-range intramolecular electron transfer, such as in proteins.3.7 REFERENCES2.
H. S. Johnston, "Gas Phase Reaction Rate Theory," Ronald, New York, 1966.9. В. Е. Conway, "Theory and Principles of Electrode Processes," Ronald, New York, 1965,Chap. 6.10. K. J. Vetter, "Electrochemical Kinetics," Academic, New York, 1967, Chap.
2.3. S. Glasstone, K. J. Laidler, and H. Eyring, "Theory of Rate Processes," McGraw-Hill, NewYork, 1941.11. J. O'M. Bockris and A. K. N. Reddy, "ModernElectrochemistry," Vol. 2, Plenum, New York,1970, Chap. 8.4. H. Eyring, S. H. Lin, and S. M. Lin, "BasicChemical Kinetics," Wiley, New York, 1980,Chap. 4.12. T. Erdey-Gruz, "Kinetics of ElectrodeProcesses," Wiley-Interscience, New York,1972, Chap. 1.13. H. R. Thirsk, "A Guide to the Study of Electrode Kinetics," Academic, New York, 1972,Chap. 1.14. W. J.
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