A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 36
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This is only a part of the total rate of reduction, so we call it a local rate for energy E. In a time interval Дг, electrons fromoccupied states on the electrode can make the transition to states on species О in the sameenergy range, and the rate of reduction is the number that succeed divided by Д*. This rateis the instantaneous rate, if At is short enough (a) that the reduction does not appreciablyalter the number of unoccupied states on the solution side and (b) that individual О molecules do not appreciably change the energy of their unoccupied levels by internal vibrational and rotational motion.
Thus At is at or below the time scale of vibration. The localrate of reduction can be written asLocal Rate(E) =Pred(E)AN0CC(E)dEredоссд^—(3.6.25)where AN0CCdE is the number of electrons available for the transition and Pred(E) is theprobability of transition to an unoccupied state on O.
It is intuitive that /\ed(E) is proportional to the density of states DO(A, E). Defining e r e d (E) as the proportionality function,we haveLocal Rate(E) =206rredred(E)£>o(A, E)ANocc(E)dE°'—^^-^—(3.6.26)In this discussion, the phrases "concentration in the vicinity of the electrode" and "concentration near theelectrode" are used interchangeably to denote concentrations that are given by C(0, i) in most mass-transfer andheterogeneous rate equations in this book.
However, C(0, t) is not the same as the concentration in the reactiveposition at an electrode (i.e., in the precursor state), but is the concentration just outside the diffuse layer. Weare now considering events on a much finer distance scale than in most contexts in this book, and thisdistinction is needed. The same point is made in Section 13.7.3.6 Microscopic Theories of Charge Transfer127where ered(E) has units of volume-energy (e.g., cm3 eV). The total rate of reductionis the sum of the local rates in all infinitesimal energy ranges; thus it is given by the integralRate = v \J —00ered(E)D o (A, E)AN0CC(E)dE(3.6.27)where, in accord with custom, we have expressed Ar in terms of a frequency, v = 1/Дг.The limits on the integral cover all energies, but the integrand has a significant value onlywhere there is overlap between occupied states on the electrode and states of О in the solution.
In Figure 3.6.4, the relevant range is roughly -4.0 to -3.5 eV.Substitution from (3.6.20) and (3.6.24) givesRate = vANKC'o(0, t) JJ —00s red (E)W 0 (A, E)/(E)p(E)dE(3.6.28)This rate is expressed in molecules or electrons per second. Division by ANA gives therate more conventionally in mol c m " 2 s" 1 , and further division by CQ(0, t) provides therate constant,Г 00(3.6.29)In an analogous way, one can easily derive the rate constant for the oxidation ofR. On the electrode side, the empty states are candidates to receive an electron; henceNunocc(E) is the distribution of interest.
The density of filled states on the solutionside is £>R(A, E), and the probability for electron transfer in the time interval Ar isP 0X (E) = 80X(E)Z)R(A, E). Proceeding exactly as in the derivation of (3.6.29), wearrive at(3.6.30)In Figure 3.6.4, the distribution of states for species R does not overlap the zone ofunoccupied states on the electrode, so the integrand in (3.6.30) is practically zero everywhere, and &ь is negligible compared to kf. The electrode is in a reducing condition withrespect to the O/R couple.
By changing the electrode potential to a more positive value,we shift the position of the Fermi level downward, and we can reach a position where theR states begin to overlap unoccupied electrode states, so that the integral in (3.6.30) becomes significant, and къ is enhanced.The literature contains many versions of equations 3.6.29 and 3.6.30 manifestingdifferent notation and involving wide variations in the interpretation applied to the integral prefactors and the proportionality functions e r e d (E) and e o x (E).
For example, itis common to see a tunneling probability, Kej, or a precursor equilibrium constant, A^oor /sTp R, extracted from the e-functions and placed in the integral prefactor. Often thefrequency v is identified with vn in (3.6.2). Sometimes the prefactor encompassesthings other than the frequency parameter, but is still expressed as a single symbol.These variations in representation reflect the fact that basic ideas are still evolving.The treatment offered here is general and can be accommodated to any of the extantviews about how the fundamental properties of the system determine v, e r e d (E), ande o x (E).With (3.6.29) and (3.6.30), it is apparently possible to account for kinetic effects ofthe electronic structure of the electrode by using an appropriate density of states, p(E), for128Chapter 3.
Kinetics of Electrode Reactionsthe electrode material. Efforts in that direction have been reported. However, one must beon guard for the possibility that ered(E) and eOx(E) also depend on p(E). 2 1The Marcus theory can be used to define the probability densities W0(A, E) andWR(A, E). The key is to recognize that the derivation leading to (3.6.10b) is based implicitly on the idea that electron transfer occurs entirely from the Fermi level. In the contextthat we are now considering, the rate constant corresponding to the activation energy in(3.6.10b) is therefore proportional to the local rate at the Fermi level, wherever it mightbe situated relative to the state distributions for О and R. We can rewrite (3.6.10b) interms of electron energy asE~ ^»(3.6.31)where E° is the energy corresponding to the standard potential of the O/R couple.
One caneasily show that AG| reaches a minimum at E = E° + A, where AGj = 0. Thus the maximum local rate of reduction at the Fermi level is found where E F = E° + Л. When theFermi level is at any other energy, E, the local rate of reduction at the Fermi level can be expressed, according to (3.6.2), (3.6.26), and (3.6.31), in terms of the following ratiosА Л^^ЫУLocal Rate (E F = E)VnKelLocal Rate (E F = E° + A)gredЕ- EJJ(3-63(E)Po(A,E)/(EF)p(EF)ered (E° + A) Do (A, E° + A)/(E F ) p (E F )Assuming that e r e d does not depend on the position of E F , we can simplify this toZ)0(A,E)Z) O (A,E°2_ exp Г- (Е-Е°-А) 1=(3.6.33)This is a gaussian distribution having a mean at E = E° + Л and a standard deviationof (2A^T) 1/2 , as shown in Figure 3.6.4 (see also Section A.3).
From (3.6.24),£>O(A, E)/D0(A, E° + Л) = Wo(\, E)/Wo(\, E° + A). Also, since Wo(\, E) is normalized, the exponential prefactor, WO(X, E° + A), is quickly identified (Section A.3) as(2тг)~ / times the reciprocal of the standard deviation; thereforeWo(\,E) = (4TT\£T)~1/2exp-(3.6.34)21Consider, for example, a simple model based on the idea that, in the time interval Ar, all of the electrons in theenergy range between E and E + dE redistribute themselves among all available states with equal probability. Arefinement allows for the possibility that the states on species О participate with different weight from those on theelectrode.
If the states on the electrode are given unit weight and those in solution are given weight /cred(E), then/cred (E)D0(A, E)5p(E) + /cred(E)£>0(A, E)dwhere 8 is the average distance across which electron transfer occurs, and /cred(E) is dimensionless and can beidentified with the tunneling probability, /cel, used in other representations of kf. If the electrode is a metal, p(E)is orders of magnitude greater than Kred(E)Z)0(A, E)8; hence the rate constant becomeswhich has no dependence on the electronic structure of the electrode.3.6 Microscopic Theories of Charge Transfer *i 129In a similar manner, one can show that(3.6.35)thus the distribution for R has the same shape as that for O, but is centered on E° - Л, asdepicted in Figure 3.6.4.Any model of electrode kinetics is constrained by the requirement that=е(3.6.36)which is easily derived from the need for convergence to the Nernst equation at equilibrium (Problem 3.16).
The development of the Gerischer model up through equations3.6.29 and 3.6.30 is general, and one can imagine that the various component functions inthose two equations might come together in different ways to fulfill this requirement. Bylater including results from the Marcus theory without work terms, we were able to definethe distribution functions, WO(X, E) and WR(A, E).
Another feature of this simpleGerischer-Marcus model is that sox(E) and ered(E) turn out to be identical functions andneed no longer be distinguished. However, this will not necessarily be true for relatedmodels including work terms and a precursor equilibrium.The reorganization energy, A, has a large effect on the predicted current-potentialresponse, as shown in Figure 3.6.5. The top frame illustrates the situation for A = 0.3 eV,a value near the lower limit found experimentally. For this reorganization energy, anoverpotential of —300 mV (case a) places the Fermi level opposite the peak of the state-2-3-4-5-._ERe° •~d—ажк=а-6Electrode StatesSolution StatesFigure 3.6.5 Effect of A on kinetics in theGerischer-Marcus representation. Top:A = 0.3 eV.
Bottom: A = 1.5 eV. Bothdiagrams are for species О and R at equalconcentrations, so that the Fermi levelcorresponding to the equilibrium potential,E F eq , is equal to the electron energy at thestandard potential, E° (dashed line). Forboth frames, E° = -4.5 eV. Also shown ineach frame is the way in which the Fermilevel shifts with electrode potential. Thedifferent Fermi levels are for (a) r] = -300mV, (b) г] = +300 mV, (c) rj = -1000mV, and (d) r) = +1000 mV. On thesolution side, Wo(\, E) and WR(X, E) areshown with lighter and darker shading,respectively.130 • Chapter 3.
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