Redox particle

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... 

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. 

The electron state densities DredCe) and Z ox(e) in the donor and acceptor bands of hydrated redox particles are given by the product of the probability densities Wrbd(c) and Wcacie) and the concentrations Nkbd and Nox, respectvely, in Eqns. 2-48 and 2-49 ... 

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 2-40 shows the state density distribution curves of hydrated redox particles in two cases in which (a) Nked = JVox and (b) AThed < iVox-... 

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 ... 

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. 

Fig. 2-41. Electron energy levels of hydrated redox particles and intrinsic semiconductors.

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. 

Electrodes may be classified into the following two categories as shown in Fig. 4-3 one is the electronic electrode at which the transfer of electrons takes place, and the other is the ionic electrode at which the transfer of ions takes place. The electronic electrode corresponds, for instance, to the case in which the transfer of redox electrons in reduction-oxidation reactions, such as Fe = Fe + e,occurs and the ionic electrode corresponds to the case in which the transfer of ions, such as Fe , , = Fe, occiirs across the electrode interface. Usually, the former is found with insoluble electrodes such as platinum electrodes in aqueous solution containing redox particles and the latter is found with soluble metal electrodes such as iron and nickel. In practice, both electron transfer and ion transfer can take place simultaneously across the electrode interface. 

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 ... 

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. 

The electrode potential, E, represented by the real potential redox electrons in the hydrated redox particles in aqueous solution as shown in Fig. 4-18 and defined in Eqn. 4-19 ... 

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 electrode potential in the equilibrium of redox electron transfer may also be defined by the free enthalpy change in the reaction of the hydrated redox particles with the standard gaseous electron eisro) as shown in Eqn. 4—20 ... 

The frontier electron level of adsorbed particles splits itself into an occupied level (donor level) in a reduced state (reductant, RED) and a vacant level (acceptor level) in an oxidized state (oxidant, OX), because the reduced and oxidized particles differ from each other both in their respective adsorption energies on the interface of metal electrodes and in their respective interaction energies with molecules of adsorbed water. The most probable electron levels, gred and eqx, of the adsorbed reductant and oxidant particles are separated from each other by a magnitude equivalent to the reorganization energy 2 >. ki in the same way as occurs with hydrated redox particles described in Sec. 2.10. 

We consider, as an example, adsorbed redox particles comprising protons and hydrogen atoms as shown in Eqn. 5-53 ,... 

Fig. S-S8. Electron levels of dehydrated redox particles, H ld + bh /h = H,d, adsorbed on an interface of metal electrodes D = state density (electron level density) 6 = adsorption coverage shVi - most probable vacant electron level of adsorbed protons (oxidants) eH(d = most probable occupied electron level of adsorbed hydrogen atoms (reductants) RO.d = adsorbed redox particles.

In adsorption equUibrimn, the Fermi level c m) of electrons in the metal electrode equals the Fermi level ep(HyH ) oi redox electrons in the adsorbed redox particles the state density of the occupied electron level equals the state density of the vacant electron level at the Fermi level ( >b = Da). Assuming the Langmuir adsorption isotherm at low adsorption coverages and the Gaussian distribution for the state density, we obtain Eqn. 5-55 for the Fermi level ... 

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. 

Fig. 6-S8. Probability density for the energy level of interfadal redox electrons in adsorbed redox particles of proton-hydrogen and hydroxyl-hydroxide on the electrode interface of semiconductor ADS = adsorption > ost probable...

Figure 5-64 shows the band edge potential for compound semiconductor electrodes in aqueous solutions, in which the standard redox potentials (the Fermi levels) of some hydrated redox particles are also shown on the right hand side. In studying reaction kinetics of redox electron transfer at semiconductor electrodes, it is important to find the relationship between the band edge level (the band edge potential) and the Fermi level of redox electrons (the redox potential) as is described in Chap. 8. 

TABLE 6-L Die standard equilibrium potentials for redox electrode reactions of h rdrated redox particles at 25 C nhe = relative electrode potential referred to the normal hydrogen electrode. [Handbooks of electrochemistry.]... 

Table 6-1 shows the standard equilibrium potentials of several redox reactions of hydrated redox particles. 

The reaction of electron transfer at electrodes in aqueous electrolytes proceeds either with hydrated redox particles at the plane of closest approach of hydrated ions to the electrode interface (OHP, the outer Helmholtz plane) or with dehydrated and adsorbed redox particles at the plane of contact adsorption on the electrode interface (IHP, the inner Helmholtz plane) as shown in Fig. 7-2. 

Fig. 7-2. Electron transfer of hydrated redox particles and of dehydrated adsorbed redox particles across an electrode interface (a) electron transfer of hydrated redox particles, (b) electron transfer of dehydrated and adsorbed redox particles on electrodes. (RED., OX,q) = hydrated redox particles (RED.d, OX.d) = dehydrated and adsorbed redox particles on electrode OHP = outer Helmholtz plane, IHP = inner Helmholtz plane.

A typical reaction of electron transfer of hydrated redox particles is the ferric-ferrous redox reaction in Eqn. 7-5 ... 

As the adsorption affinity of redox particles on the electrode interface increases, the hydrated redox particles is adsorbed in the dehydrated state (chemical adsorption, contact adsorption) rather than in the hydrated state (ph3 ical adsorption) as shown in Fig. 7-2 (b). Typical reactions of redox electron transfer of dehydrated and adsorbed redox particles on electrodes are the hydrogen and the oxygen electrode reactions in Eqns. 7-6 and 7-7 ... 

The electron transfer of hydrated redox particles at the outer Helmholtz plane is occasionally called the outer-sphere electron transfer, while the electron transfer of dehydrated and adsorbed redox particles on electrodes is called the inner-sphere electron transfer. 

For the electron transfer of hydrated redox particles (the outer-sphere electron transfer), the electrode acts merely as a source or sink of electrons transferring across the compact double layer so that the nature of the electrode hardly affects the reaction kinetics this lack of influence by the electrode has been observed for the ferric-ferrous redox reaction. On the other hand, the electron transfer of adsorbed redox particles (the inner-sphere electron transfer) is affected by the state of adsorption so that the nature of the electrode exerts a definite influence on the reaction kinetics, as has been observed with the hydrogen electrode reaction where the reaction rate depends on the property of electrode. 

We consider a simple electron transfer reaction (an outer-sphere electron transfer) between h3 rated redox particles OX /RED and a metal electrode M as shown in Eqn. 8-1 ... 

Figure 8-1 shows the potential energy barrier for the transfer reaction of redox electrons across the interface of metal electrode. On the side of metal electrode, an allowed electron energy band is occupied by electrons up to the Fermi level and vacant for electrons above the Fermi level. On the side of hydrated redox particles, the reductant particle RED is occupied by electrons in its highest occupied molecular orbital (HOMO) and the oxidant particle OX, is vacant for electrons in its lowest imoccupied molecular orbital (LUMO). As is described in Sec. 2.10, the highest occupied electron level (HOMO) of reductants and the lowest unoccupied electron level (LUMO) of oxidants are formed by the Franck-Condon level sphtting of the same frontier oihital of the redox particles... 

The plane of closest approach of hydrated ions, the outer Helmholtz plane (OHP), is located 0.3 to 0.5 run away from the electrode interface hence, the thickness of the interfacial compact layer across which electrons transfer is in the range of 0.3 to 0.5 nm. Electron transfer across the interfacial energy barrier occurs through a quantum tunneling mechanism at the identical electron energy level between the metal electrode and the hydrated redox particles as shown in Fig. 8-1. 

Figure 8-2 illustrates the distribution of the state density of electrons in the metal electrode and in the redox particles on both sides of the electrode interface in equilibrium with redox electron transfer. 

Equation 2-51 gives the Fermi level of redox electrons as a function of the concentration of redox particles as shown in Eqn. 8-12 ... 

At the Fermi level of redox electrons the whole state density DraDosKeroasDox)) of redox particles is half occupied by electrons that provide the occupied state density I>KED(er( U Dox)) of reductant particles, and the remaining half is vacant for electrons in the imoccupied state density -DoxCennEDox)) of oxidant particles. Consequently, to a first approximation, the state densities Dred(e) and i ox(e) at enei level e near the Fermi level of redox electrons can be derived, respectively, to produce Eqns. 8-15 and 8-16 [Gerischer, I960] ... 

Furthermore, the total state density of redox electrons Z)nEDox(e) in hydrated redox particles near the Fermi level is e ressed in Eqn. 8-17 by the sum of... 

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