A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 35
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This is the inverted region, where an increase in the thermodynamic driving force leads to a decreasein the rate of electron transfer. There are two physical reasons for this effect. First, a largenegative free energy of reaction implies that the products are required to accept the liberated energy very quickly in vibrational modes, and the probability for doing so declines as- AG° exceeds A (see Chapter 18).
Second, one can develop a situation in the inverted region where the energy surfaces no longer allow for adiabatic electron transfer (see Section3.6.4). The existence of the inverted region accounts for the phenomenon of electrogenerated chemiluminescence (Chapter 18) and has also been seen by other means for severalelectron-transfer reactions in solution.Even though (3.6.10b) also has a minimum, an inverted-region effect should notoccur for an electrode reaction at a metal electrode. The reason is that (3.6.10b) was derived with the implicit idea that electrons always react from a narrow range of states onthe electrode corresponding to the Fermi energy (see the caption to Figure 3.6.2).
Eventhough the reaction rate at this energy is predicted to show inversion at very negativeoverpotentials, there are always occupied states in the metal below the Fermi energy, andthey can transfer an electron to О without inversion. Any low-level vacancy created in themetal by heterogeneous reaction is filled ultimately with an electron from the Fermi energy, with dissipation of the difference in energy as heat; thus the overall energy change isas expected from thermodynamics. A similar argument holds for oxidations at metals,where unoccupied states are always available. The ideas behind this discussion are developed much more fully in the next section.3.5 -3 2.5 -2 —AG*1.5 —1 —1.5 eV,0.5-3.5^ l0-0.5 -e V0.5Figure 3.6.3 Effect of AG° fora homogeneous electron-transferreaction on LG\ at several differentvalues of A.124 • Chapter 3.
Kinetics of Electrode ReactionsAn inverted region should be seen for interfacial electron transfer at the interfacebetween immiscible electrolyte solutions, with an oxidant, O, in one phase, and a reductant, R', in the other (69). Experimental studies bearing on this issue have just been reported (70).3.6.3A Model Based on Distributions of Energy StatesAn alternative theoretical approach to heterogeneous kinetics is based on the overlap between electronic states of the electrode and those of the reactants in solution (41, 42, 46,47, 71, 72). The concept is presented graphically in Figure 3.6.4, which will be discussedextensively in this section.
This model is rooted in contributions from Gerischer (71, 72)and is particularly useful for treating electron transfer at semiconductor electrodes (Section 18.2.3), where the electronic structure of the electrode is important. The main idea isthat an electron transfer can take place from any occupied energy state that is matched inenergy, E, with an unoccupied receiving state.
If the process is a reduction, the occupiedstate is on the electrode and the receiving state is on an electroreactant, O. For an oxidation, the occupied state is on species R in solution, and the receiving state is on the electrode. In general, the eligible states extend over a range of energies, and the total rate is anintegral of the rates at each energy.-2UnoccupiedStatesI(2)UT)V2DO(\,E)шш —41Electrode StatesReactant StatesFigure 3.6.4 Relationships among electronic states at an interface between a metal electrode anda solution containing species О and R at equal concentrations. The vertical axis is electron energy,E, on the absolute scale.
Indicated on the electrode side is a zone A4T wide centered on the Fermilevel, Ep, where/(E) makes the transition from a value of nearly 1 below the zone to a value ofvirtually zero above. See the graph of/(E) in the area of solid shading on the left. On the solutionside, the state density distributions are shown for О and R. These are gaussians having the sameshapes as the probability density functions, W0(X, E) and WR(\, E). The electron energycorresponding to the standard potential, E°, is -3.8 eV, and Л = 0.3 eV. The Fermi energycorresponds here to an electrode potential of -250 mV vs.
E°. Filled states are denoted on bothsides of the interface by dark shading. Since filled electrode states overlap with (empty) О states,reduction can proceed. Since the (filled) R states overlap only with filled electrode states, oxidationis blocked.3.6 Microscopic Theories of Charge Transfer < 125On the electrode, the number of electronic states in the energy range between Eand E 4- dE is given by Ap(E)dE, where A is the area exposed to the solution, and p(E) isthe density of states [having units of (area-energy)" 1 , such as с т ~ 2 е У - 1 ] . The totalnumber of states in a broad energy range is, of course, the integral of Ap(E) over therange. If the electrode is a metal, the density of states is large and continuous, but if itis a semiconductor, there is a sizable energy range, called the band gap, where the density of states is very small.
(See Section 18.2 for a fuller discussion of the electronicproperties of materials.)Electrons fill states on the electrode from lower energies to higher ones until all electrons are accommodated. Any material has more states than are required for the electrons,so there are always empty states above the filled ones. If the material were at absolutezero in temperature, the highest filled state would correspond to the Fermi level (or theFermi energy), Ep, and all states above the Fermi level would be empty. At any highertemperature, thermal energy elevates some of the electrons into states above Ep and creates vacancies below. The filling of the states at thermal equilibrium is described by theFermi function, /(E),/(E) = {1 + exp[(E - E F )/47]}" l(3.6.19)which is the probability that a state of energy E is occupied by an electron.
It is easy tosee that for energies much lower than the Fermi level, the occupancy is virtually unity,and for energies much higher than the Fermi level, the occupancy is practically zero (seeFigure 3.6.4). States within a few 4T of Ep have intermediate occupancy, graded fromunity to zero as the energy rises through Ep, where the occupancy is 0.5. This intermediate zone is shown in Figure 3.6.4 as a band 44T wide (about 100 meV at 25°C).The number of electrons in the energy range between E and E + dE is the number ofoccupied states, A/V0CC(E)dE, where Afocc(E) is the density functionAW(E)=/(E)p(E)(3.6.20)Like p(E), 7V0CC(E) has units of (area-energy)"1, typically c m " 2 eV~\ while/(E) is dimensionless.
In a similar manner, we can define the density of unoccupied states asAU>cc(E) = [1 ~/(E)]p(E)(3.6.21)As the potential is changed, the Fermi level moves, with the change being towardhigher energies at more negative potentials and vice versa. On a metal electrode, thesechanges occur not by the filling or emptying of many additional states, but mostly bycharging the metal, so that all states are shifted by the effect of potential (Section 2.2).While charging does involve a change in the total electron population on the metal, thechange is a tiny fraction of the total (Section 2.2.2).
Consequently, the same set of statesexists near the Fermi level at all potentials. For this reason, it is more appropriate to thinkof p(E) as a consistent function of E — Ep, nearly independent of the value of Ep. Since/(E) behaves in the same way, so do N0CC(E) and Nunocc(E).
The picture is more complicated at a semiconductor, as discussed in Section 18.2.States in solution are described by similar concepts, except that filled and emptystates correspond to different chemical species, namely the two components of a redoxcouple, R and O, respectively. These states differ from those of the metal in being localized. The R and О species cannot communicate with the electrode without first approaching it closely. Since R and О can exist in the solution inhomogeneously and our concern iswith the mix of states near the electrode surface, it is better to express the density of states126 • Chapter 3. Kinetics of Electrode Reactionsin terms of concentration, rather than total number.
At any moment, the removable electrons on R species in solution in the vicinity of the electrode20 are distributed over an energy range according to a concentration density function, £>R(A, E), having units of(volume-energy)"1, such as cm" 3 eV" 1 . Thus, the number concentration of R speciesnear the electrode in the range between E and E + dE is £>R(A, E)dE.
Because this smallelement of the R population should be proportional to the overall surface concentration ofR, CR(0, t), we can factor Z)R(A, E) in the following way:DR(A, E) = WACR(0, t)WR(\, E)(3.6.22)where NA is Avogadro's number, and WR(A, E) is a probability density function with unitsof (energy)"1. Since the integral of DR(A, E) over all energies must yield the total numberconcentration of all states, which is iVACR(0, t), we see that VFR(A, E) is a normalizedfunctionWR(\,E)dE = 1(3.6.23)0Similarly, the distribution of vacant states represented by О species is given by£>o(A, E) = NACo(0, t)Wo(\, E)(3.6.24)where W0(X, E) is normalized, as indicated for its counterpart in (3.6.23). In Figure 3.6.4,the state distributions for О and R are depicted as gaussians for reasons that we will discover below.Now let us consider the rate at which О is reduced from occupied states on the electrode in the energy range between E and E + dE.
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