Energy Levels in Electrolytes

The chemical potential of electrons for a redox couple is given by the Nemst equation 

The redox potential is generally referred to the standard hydrogen potential (SHE), which has an exactly defined energy, E y, relative to the energy of the free electron in vacuum or at infinity. Thus, electrode potentials of redox couples can be expressed on the absolute energy scale according to 

The negative sign of tPrcdox in the equation is due to the different signs in the conventional and the absolute electron energy scales. she has beenfound to be about-4.50eVreferred to the vacuum level so that the electron energy of any redox couple is  

FIGURE 1.4. Electron energy levels of a redox couple with respect to the standard hydrogen electrode and the vacuum level. 

The energy level, E , and energy level, E,, can be related to the redox energy level, Sredox. by a quantity called the reorientation energy X = l/2( ox - J red), which is determined by the relaxation process involved in the regrouping and reorientation of the solvation shell after electron transfer between the oxidized and reduced states. The value of X can be experimentally determined and is on the order of 0.5 to 1 The 

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Fig. 4-10. Electron energy levels in (a) an isolated solid metal and in (b) a metal electrode immersed in an electrolyte solution M = metal S = electrolyte solution e(STD) = gaseous electrons in the standard state e

Memming R (1978) The role of energy levels in semiconductor-electrolyte solar cells. J Electrochem Soc 125 117-123... 

Here, ka and kc are the electrochemical rate constants (cm s ) and CRed and cox are the concentrations of the redox ions just outside the electrochemical double layer. The anodic current is due to electron transfer from the reduced species to the empty states in the working electrode, the cathodic current is due to transfer from an occupied electron level in the metal to an unoccupied level corresponding to the oxidized species. We evaluate the electrochemical rate constants by taking into account all elastic tunneling events between the energy levels in the metal, g E), and those in the electrolyte, given by Wox E)cox and IFRed(F)cRed. The procedure is similar to that described in Section 4.6.2. Thus ... 

Figure 5. (a) Energy levels in a semiconductor (left-hand side) and a redox electrolyte (right-hand side) shown on a common vacuum reference scale. / and are the semiconductor electron affinity and work function, respectively, (b) The semiconductor-electrolyte interface before (left) and after (right) equilibration (i.e., contact of the two phases) shown for an n-type semiconductor, (c) As in (b) but for a p-type semiconductor. 

Similar to for the energy levels in semiconductors, the energy levels of electrons in electrolytes associated with ions are characterized by the redox potential, Eraiox-The redox potential describes the tendency of the species to give up or accept electrons and can be considered as the effective Fermi level of the solution. 

In an ideal case when surface states are absent and charge transfer proceeds directly between the energy levels in the bands and in the solution, according to Gerischer, an anodic current involving an electron transfer from a molecule in the electrolyte to the electrode and a cathodic current involving an electron transfer from the electrode to a molecule in the electrolyte are given by... 

It can be expected from the nature of silicon/electrolyte interfaces described in the previous sections that the surface states on silicon electrodes may have different physical and chemical characteristics such as type, quantity, distribution, transfer kinetics, and so on, depending on the surface condition. Table 2.12 shows examples of measurements of surface states reported in the literature. Thus, while the energy levels in bulk silicon and electrolyte can be described by a general theory, those of surface states can only be dealt with by specific theories applicable to the specific situations. 

Figure 13. Schematic outline of a dye-sensitized photovoltaic cell, showing the electron energy levels in the different phases. The system consists of a semiconducting nanocrystalline Ti02 film onto which a Ru-complex is adsorbed as a dye and a conductive counterelectrode, while the electrolyte contains an I /Ij redox couple. The cell voltage observed under illumination corresponds to the difference, AF, between the quasi-Fermi level of Ti02 and the electrochemical potential of the electrolyte. S, S, and S+ designate, respectively, the sensitizer, the electronically excited sensitizer, and the oxidized sensitizer. See text for details. Adapted from [69], A Flagfeldt and M. Gratzel, Chem Rev. 95, 49 (1995). 1995, American Chemical Society.

Suppose that A(Ab< ) A(A5) when the electrode potential varies. This inequality means that the potential drop across the Helmholtz layer remains practically unchanged (A(/ = Ab under electrode polarization. Therefore, the positions of all energy levels at the surface and, in particular, of band edges and remain the same with respect to the position of energy levels in the electrolyte solution and reference electrode (Figs. 4a and 4b). In this case, the band edges are said to be pinned at the surface. 

In electrochemistry, the electrode potential is defined by the electronic energy level in a solid electrode referred to the energy level of the standard gaseous electron just outside the surface of an electrolyte (aqueous solution) in which the electrode is immersed [6] ... 

Electron-transfer processes at the semiconductor/electrolyte interface are strongly affected by the density of available carriers (electrons and holes) in the semiconductor at the interface. The observed i-E behavior differs from that at metals and carbon (Chapter 3), where there is always a large density of carriers in the conductor. In the dark, electron-transfer processes involving species in solution with energy levels in the band gap of the semiconductor (Figure 18.2.5Z ) are usually dominated by the majority carrier. Thus, moderately doped w-type materials can carry out reductions, but not oxidations. That is, there are electrons available in the conduction band to transfer to an oxidized solution species, but few holes to accept an electron from a reduced species. The current for a reduction of species O at an w-type semiconductor is given by... 

Otherwise, the transfer of an electron from the metal into the electrolyte can occur, if the electron on an energy level E in the metal finds an unoccupied energy level at the same energy E in the electrolyte. The intensity of electron transfer is proportional to the density of unoccupied electron-energy levels in the metal, times the density of unoccupied energy levels in the electrolyte (the density of oxidized ions,... 

The charge transfer on a metal electrode depends on the overlap between occupied and unoccupied energy levels in metal and electrolyte. This can be found around the Fermi level (Chapter 6). The contact of an electrolyte with a redox system and a semiconductor... 

In Eq. (5), Cqx and c,.ed are the concentrations (roughly activities) of the oxidized and reduced species, respectively, in the redox couple. The parameter ( redox = / e,redox) defined by Eq. (5) can be identified with the Fermi level (fp,redox) in the electrolyte. This was the topic of debate some years back [23], although this premise now appears to be well founded. The task now is to relate the electron energy levels in the solid and liquid phases on a common basis. 

When designing the electrolyte solvent and salt compounds one can consult the molecular orbital methods with some programs such as the Gaussian [63]. The molecular orbital method calculates the most stable conformation as well as the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) levels of the compound concerned. Figure 2.11 indicates the relationship between molecular orbitals and their energy levels in a molecule. When the oxidation takes place an electron is ranoved from the HOMO. When the reduction occurs an electron is inserted into the LUMO. Therefore, the lower HOMO level... 

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