Redox electron level

Fig. 2-39. Gaussian normal distri bution of the probabili density of redox electron levels due to thermal fluctuation of hydrate structures epd)Bx>X) = standard Fermi level of redox electrons.

As the localized electron level of hydrated redox particles distributes itself in a rather wide range, we may assume the presence of energy bands in which the redox electron level fluctuates the reductant particles form a donor band, and the oxidant particles form an acceptor band. The donor and acceptor bands overlap in the tailing of their probability densities as shown in Fig. 2-39. 

In the fluctuation band of electron energy of hydrated redox particles, the donor band of the reductant is an occupied band, and the acceptor band of the oxidant is a vacant band. The level erotsDcno at which the donor state density equals the acceptor state density (Aai/e) = Dox(e)) is called the Fermi level of the redox electron by analogy with the Fermi level e, of metal electrons [Gerischer, 1961]. From Eqns. 2—48 and 2—49 with f BED(e) =-DoxCe), we obtain the Fermi level Tiixxox.) (the redox electron level) as shown in Eqn. 2-51 ... 

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. 

It, thus, follows that the electrode potential in electron transfer equilibriiun represents the redox electron level of the redox particles in aqueous electrolyte solution. Further, it follows from Eqn. 4-19 that the electrode potential in the transfer equilibrium of redox electrons is characteristic of individual redox reactions but independent of the nature of the electrode materials. 

The same approach may also apply to the adsorption of redox particles other than the adsorption of proton-hydrogen atom on metal electrodes. To understand electrosorption phenomena, various concepts have been proposed such as the charge transfer coefficient and the adsorption valence [Vetter-Schultze, 1972]. The concept of the redox electron level in adsorbed particles introduced in this textbook is usefiil in dealing with the adsorption of partially ionized particles at electrodes. 

The interface of semiconductor electrode adsorbs both ionized basic hydroi groups -OH2 (b) and associated acidic hydroxyl groups -OH(a) in acidic solutions these hydroxyl groups introduce two-dimensional redox electron levels as identified in Eqns. 5-88 and 5-89 ... 

Next, we consider the anodic reaction current of redox electron transfer via the conduction band, of which the exchange reaction current has been shown in Fig. 8-16. Application of a slight anodic polarization to the electrode lowers the Fermi level of electrode fix>m the equilibrium level (Ep(sc)( n = 0) = eiiOTSDca)) to a polarized level (ep(8C)( n) = ep(REDox)- n)withoutchanging at the electrode interface the electron level relative to the redox electron level (the band edge level pinning) as shown in Fig. 8-20. As a result of anodic polarization, the concentration of interfacial electrons, n, in the conduction band decreases, and the concentration of interfadal holes, Pm, in the valence band increases. Thus, the cathodic transfer current of redox electrons, in, via the conduction band decreases (with the anodic electron im ection current, ii, being constant), and the anodic transfer current of redox holes, (p, via the valence band increases (with the cathodic hole injection... 

Under the condition of band edge level pinning, where the interfacial electron level of electrode relative to the redox electron level of redox particles is imchanged, the level differences of ej - ered and ej. - eqx remain constant irrespective of any change of the electrode potential. Consequently, the anodic transfer current of redox electrons, in(T ), in Eqn. 8—60 is independent of the overvoltage and remains equal to the exchange current ia.o as expressed in Eqn. 8-62 ... 

We now consider a cathodic transfer of electrons from the conduction band of electrode to the vacant redox electron level in a hydrated oxidant particle to form a hydrated reductant particle in solution OX , + ecB- RED . Equation 8-72 expresses this reaction current, due to the direct transfer of electrons from the conduction band to the oxidant particle based on Eqn. 8-61 as follows ... 

Fig. 8-33. Energy diagram showing a shift of redox electron level due to complexation of reductant and oxidant particles (1) afSnity for complexation is greater with oxidants than with reductants, (2) affinity for complexation is greater with reductants than with oxidants. COMPLEX z ligand-coordinated complex redox particles HYDRATE = simply hydrated redox particles W = probability density of electron states e., ) - standard Fermi level of hydrated redox particles - standard Fermi level of ligand-coordinated...

Contact adsorption shifts the redox electron level. 

Figure 8-37 shows a shift in the redox electron level of a reductant from the hydrated state, which can not anodically iiyect electrons into the conduction band because its electron levels are located within the band gap, to the adsorbed state, which then can iiyect electrons into the conduction band of semiconductor electrodes... 

The direct transfer of electrons from the frontier orbital of hydrated hydrogen molecules to the frontier orbital of hydrated o Q n molecules does not take place because its activation energy is high but the indirect transfer of electrons via both the electron level of metallic electrodes and the redox electron level of adsorbed reaction intermediates proceeds at an appreciable rate on metal electrodes. 

The reorganization free energy /.R represents the electronic-vibrational coupling, ( and y are fractions of the overpotential r] and of the bias voltage bias at the site of the redox center, e is the elementary charge, kB the Boltzmann constant, and coeff a characteristic nuclear vibration frequency, k and p represent, respectively, the microscopic transmission coefficient and the density of electronic levels in the metal leads, which are assumed to be identical for both the reduction and the oxidation of the intermediate redox group. Tmax and r max are the current and the overvoltage at the maximum. 

Relatively little is understood in the presence of non planar-non ideal interfaces, where electronic levels located in the band gap region act as recombination centers. Colloidal materials, low cost polycrystalline materials and films, interpenetrating networks of absorber and charge collecting phases (e.g., as in the DSSC cells), and the presence of redox active adsorbing species, all give rise to... 

The value of xh o is important for estimating theoretically the energy levels of hydrated ions and redox electrons in aqueous solutions. 

Fig. 2-35. Localized electron levels of gaseous redox particles, Fe /Fe I = ionization energy of Fe A = electron affinity of Fe (STO) = standard gaseous electrons tsro = standard gaseous electron level (reference zero level).

The electron level in hydrated redox particles consists of the energy AGmt (< 0) required for the standard gaseous electron to combine with or to be released from the gaseous redox partides and the energy AG ,(>0) required for the redox particles to form their hydrate structures. Since the donor and acceptor levels of gaseous redox particles Pefi j/Fe, equal each other, the difference between the... 

Fig. 2-36. Electron energy levels in hydrated oxidant Fe and reduc-tantFe AG = energy to organize hydrate structures dGj t = energy required for dehydrated redox ions to donate or accept gaseous electrons ep.2> o = most probable electron donor level of Fe Spe +.A = most probable electron acceptor level of Fe Hj05,2.,p,j = hydrated structures cgn) = standard gaseous electron level (s 0).

Fig. 2-40. Distribution of electron state density of hydrated redox particles (a) oxidant concentration JVox equal to reductant concentrationNRED. (b) oxidant concentration iVox greater than reductant concentration NgEo cnsEDox) = Fermi level of redox electrons.

Figure 2-41 compares the electron level diagram of intrinsic semiconductors with that of hydrated redox particles at the standard concentration. The two diagrams resemble each other in that the Fermi level is located midway between the occupied level and the vacant level. It is, however, obvious that the occupied and vacant bands for semiconductors are the bands of delocalized electron states, whereas they are the fluctuation bands of localized electron states for hydrated redox particles. 

The most probable donor level, ered, the most probable acceptor level, eox, and the standard Fermi level, e redox) of redox electrons are characteristic of individual redox particles but the Fermi level, e m dox), of redox electrons depends on the concentration ratio of the reductant to the oxidant, which fact is similar to the Fermi level of extrinsic semiconductors depending on the concentration ratio of the donor to the acceptor. 

The Fermi level ekredox) (= P.(redox)) of redox electrons may also be obtained thermodynamically from the reaction equUibriiun (RED = OX , + edanox)), i.e. p(REDox) = P.(iaD0X) = Pred Pox, as shown in Eqn. 2—53 ... 

The energy balance in the foregoing reaction cyde gives the real potential a<(NHE) of the equilibrium redox electron in the reaction of normal hydrogen electrode as shown in Fig. 2-44 represents the Fermi level croniB) of the... 

As some numerical energy values such as given in the foregoing involve a certain degree of inaccuracy, there have been several values reported for the Fermi level of the equilibrium redox electron of NHE. For instance, the value of a NHE) = ewNHE)= -4.44 eV has been reported in the International Union of Pure and Applied Chemistry (lUPAC) [Trasatti, 1986]. 

The nonpolarizable electrode may also be defined as the electrode at which an electron or ion transfer reaction is essentiaUy in equilibrium i. e. the electron or ion level in the electrode is pinned at the electron level of hydrated redox particles or at the hydrated ion level in aqueous electrolyte. In order for the electrode reaction to be in equilibrium at the interface of nonpolarizable electrode, an appreciable concentration of redox particles or potential determining ions must exist in the electrolyte. 

Next, we consider the interface M/S of a nonpolarizable electrode where electron or ion transfer is in equilibrium between a solid metal M and an aqueous solution S. Here, the interfadal potential is determined by the charge transfer equilibrium. As shown in Fig. 4-9, the electron transfer equilibrium equates the Fermi level, Enn) (= P (M)), of electrons in the metal with the Fermi level, erredox) (= P s)), of redox electrons in hydrated redox particles in the solution this gives rise to the inner and the outer potential differences, and respectively, as shown in Eqn. 4-10 ... 

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