Electron transfer Fermi energy

The term electrocatalysis refers to the acceleration of the rate of an electrochemical reaction taking place at a solid electrode. The main role of the catalyst material is to lower the activation energy for electron transfer between energy levels of electroactive species on the electrolyte side and electronic states at the Fermi level in the metal. Although the electrode itself does not undergo any chemical transformation, it participates in the reaction indirectly by acting as a reservoir of electrons. Moreover, the catalyst surface provides active sites for the adsorption of reaction intermediates. 

Unlike the values of values of electron work function always refer to the work of electron transfer from the metal to its own point of reference. Hence, in this case, the relation established between these two parameters by Eq. (29.1) is disturbed. The condition for electronic equilibrium between two phases is that of equal electrochemical potentials jl of the electrons in them [Eq. (2.5)]. In Eig. 29.1 the energies of the valence-band bottoms (or negative values of the Fermi energies) are plotted downward relative to this common level, in the direction of decreasing energies, while the values of the electron work functions are plotted upward. The difference in energy fevels of the valence-band bottoms (i.e., the difference in chemical potentials of the... 

Only if one takes into account the solvent dynamics, the situation changes. The electron transfer from the metal to the acceptor results in the transition from the initial free energy surface to the final surface and subsequent relaxation of the solvent polarization to the final equilibrium value Pqj,. This brings the energy level (now occupied) to its equilibrium position e red far below the Fermi level, where it remains occupied independent of the position of the acceptor with respect to the electrode surface. 

The physical picture of the transition is different here from that for nonadiabatic reaction. Equation (34.34) shows that the probability of electron transfer becomes equal to 1 when the acceptor energy level passes a small energy interval Ae 1/(2jiYlzP) near the Fermi level. However, unUke the nonadiabatic case,... 

In the electron transfer theories discussed so far, the metal has been treated as a structureless donor or acceptor of electrons—its electronic structure has not been considered. Mathematically, this view is expressed in the wide band approximation, in which A is considered as independent of the electronic energy e. For the. sp-metals, which near the Fermi level have just a wide, stmctureless band composed of. s- and p-states, this approximation is justified. However, these metals are generally bad catalysts for example, the hydrogen oxidation reaction proceeds very slowly on all. sp-metals, but rapidly on transition metals such as platinum and palladium [Trasatti, 1977]. Therefore, a theory of electrocatalysis must abandon the wide band approximation, and take account of the details of the electronic structure of the metal near the Fermi level [Santos and Schmickler, 2007a, b, c Santos and Schmickler, 2006]. 

Figure 1.7 A schematic representation of electron transfer between Fe(H20) + and a metal electrode. The figure represents (i) the total electronic energy of the Fe(H20)3 + ion together with the energy of the electron at the Fermi level of the metal (ii> the total electronic energy of the Fc(H20)jr + ion, plotted vs. the Fe-O bond distance in the hydrates.

To be specific, let us consider electron transfer from the reduced form of the reactant to the metal electrode. The electron may be transferred to any empty state on the metal denoting by e the difference in energy between the final state of the electron and the Fermi level, the energy of activation for the transfer is ... 

There is a fundamental difference between electron-transfer reactions on metals and on semiconductors. On metals the variation of the electrode potential causes a corresponding change in the molar Gibbs energy of the reaction. Due to the comparatively low conductivity of semiconductors, the positions of the band edges at the semiconductor surface do not change with respect to the solution as the potential is varied. However, the relative position of the Fermi level in the semiconductor is changed, and so are the densities of electrons and holes on the metal surface. 

When electron transfer is forced to take place at a large distance from the electrode by means of an appropriate spacer, the reaction quickly falls within the nonadiabatic limit. H is then a strongly decreasing function of distance. Several models predict an exponential decrease of H with distance with a coefficient on the order of 1 A-1.39 The version of the Marcus-Hush model presented so far is simplified in the sense that it assumed that only the electronic states of the electrode of energy close or equal to the Fermi level are involved in the reaction.31 What are the changes in the model predictions brought about by taking into account that all electrode electronic states are actually involved is the question that is examined now. The kinetics... 

Rose and Benjamin (see also Halley and Hautman ) utilized molecular dynamic simulations to compute the free energy function for an electron transfer reaction, Fe (aq) + e Fe (aq) at an electrodesolution interface. In this treatment, Fe (aq) in water is considered to be fixed next to a metal electrode. In this tight-binding approximation, the electron transfer is viewed as a transition between two states, Y yand Pf. In Pj, the electron is at the Fermi level of the metal and the water is in equilibrium with the Fe ion. In Pf, the electron is localized on the ion, and the water is in equilibrium with the Fe" ions. The initial state Hamiltonian H, is expressed as... 

The main shortcoming of the molecular dynamics approach discussed in the previous section is that it ignores the fact that an electron transfer at the solution/metal interface occurs between an ion in a well-defined electronic state and a continuum of electronic states in the metal. For example, depending on the ion s orbital energy, the reorganization free energy and the overpotential, the electron could be transferred from, or to, any level around the Fermi level of the metal. Therefore, a sum over all these possibilities must be performed. Analytical theories of electron transfer at the solution/metal interface recognized this issue very early on, and the reader is referred to many excellent expositions on this sub-... 

In the case of nonpolaiizable interfaces, the inner and the outer potential differences, 4>a/b and v a/b, are determined by the equilibrium of chai transfer that occurs across the interface. Figure 4—8 shows the electron energy levels in two sohd metals A and B before and after they are brought into contact with each other. As a result of contact, electrons in a metal B of the hi er electron level (the lower work function ) move into a metal A of the lower electron level (the higher work fiuiction), and the Fermi levels of the two metals finally become equal to each other in the state of electron transfer equilibrium. The electrochemical... 

For an electronic electrode at which the transfer of redox electrons is in equilibrimn (OX i + e(jj) = RED q), as shown in Fig. 4-17, the Fermi level EpdUEDoxs) of redox electrons e(REDox., s in hydrated redox particles equals the Fermi level cp(M) of electrons e,io in the electrode the energy for the electron transfer across the electrode interface is, then, zero (a M/s) = 0). Consequently, the electron level u M/aAo in the electrode equals the electron level a, s/v) in the aqueous solution, i.e. the redox electron level a KEoax s) of hydrated redox particles. 

Further, the electron level of adsorbed particles differs from that of isolated adsorbate i>articles in vacuum as shown in Fig. 5-5, this electron level of the adsorbate particle shifts in the course of adsorption by a magnitude equivalent to the adsorption energy of the particles [Gomer-Swanson, 1963]. In the illustration of Fig. 5-5, the electron level of adsorbate particles is reduced in accordance with the potential energy curve of adsorption towards its lowest level at the plane of adsorption where the level width is broadened. In the case in which the allowed electron energy level of adsorbed particles, such as elumo and ehcmio, approaches the Fermi level, ep, of the adsorbent metal, an electron transfer occurs between... 

Such an interfacial degeneracy of electron energy levels (quasi-metallization) at semiconductor electrodes also takes place when the Fermi level at the interface is polarized into either the conduction band or the valence band as shown in Fig. 5-42 (Refer to Sec. 2.7.3.) namely, quasi-metallization of the electrode interface results when semiconductor electrodes are polarized to a great extent in either the anodic or the cathodic direction. This quasi-metallization of electrode interfaces is important in dealing with semiconductor electrode kinetics, as is discussed in Chap. 8. It is worth noting that the interfacial quasi-metallization requires the electron transfer to be in the state of equilibrimn between the interface and the interior of semiconductors this may not be realized with wide band gap semiconductors. 

Fig. 8-1. Potential energy barrier for tunneling transfer of electrons across an interface of metal electrode (a) cathodic electron transfer from an occupied level of metal electrode to a vacant level of l drated oxidant particles, (b) anodic electron transfer fiom an occupied level of hjrdrated reductant particles to a vacant level of metal electrode. M. = electrode surface OHP = outer Helmholtz plane cfuh = Fermi level of electnms in metal electrode. [From Gerischer, I960.]...

Figures 8-5 and 8-6 are energy diagrams, as functions of electron energy e imder anodic and cathodic polarization, respectively, for the electron state density Dyf.t) in the metal electrode the electron state density AtEDox(c) in the redox particles and the differential reaction current ((e). From these figures it is revealed that most of the reaction current of redox electron transfer occurs in a narrow range of energy centered at the Fermi level of metal electrode even in the state of polarization. Further, polarization of the electrode potential causes the ratio to change between the occupied electron state density Dazc/itnu md the imoccupied...

Figure 8-11 shows as a function of electron energy e the electron state density Dgdit) in semiconductor electrodes, and the electron state density Z e) in metal electrodes. Both Dsd.t) and AKe) are in the state of electron transfer equilibrium with the state density Z>bei)ox(c) of hydrated redox particles the Fermi level is equilibrated between the redox particles and the electrode. For metal electrodes the electron state density Ai(e) is high at the Fermi level, and most of the electron transfer current occurs at the Fermi level enio. For semiconductor electrodes the Fermi level enao is located in the band gap where no electron level is available for the electron transfer (I>sc(ef(so) = 0) and, hence, no electron transfer current can occur at the Fermi level erso. Electron transfer is allowed to occur only within the conduction and valence bands where the state density of electrons is high (Dsc(e) > 0). 

Fig. 10-26. Energy diagram for a cell of photoelectrolytic decomposition of water consisting of a platinum cathode and an n-type semiconductor anode of strontium titanate of which the Fermi level at the flat band potential is higher than the Fermi level of hydrogen redox reaction (snao > epM+zHj) ) he = electron energy level referred to the normal hydrogen electrode ri = anodic overvoltage (positive) of hole transfer across an n-type anode interface t = cathodic overvoltage (negative) of electron transfer across a metallic cathode interface.

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