Electronic equilibria

These reactions can be viewed as a competition between two kinds of atoms (or molecules) for electrons. Equilibrium is attained when this competition reaches a balance between opposing reactions. In the case of reaction (3), copper metal reacting with silver nitrate solution, the Cu(s) releases electrons and Ag+ accepts them so readily that equilibrium greatly favors the products, Cu+2 and Ag(s). Since randomness tends to favor neither reactants nor products, the equilibrium must favor products because the energy is lowered as the electrons are transferred. If we regard reaction (5) as a competition between silver and copper for electrons, stability favors silver over copper. 

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... 

If an electronic equilibrium is set up on the surface, the parameters ij°, rr, and r/+ are strictly fixed. Their values are determined by the position of the Fermi level at the crystal surface, which will be characterized here by the quantity ea or +. These latter quantities are the distances from the Fermi level to the bottom of the conduction band or, accordingly, to the top of the valency band in the plane of the surface. Evidently,... 

Let us determine the quantities if, r), and ij+ for an illuminated specimen (3, 4) The same quantities for a specimen in the dark are denoted by jj0°, rjo- and j o+ (hereafter in the text the subscript 0 signifies the absence of illumination). From the condition of electronic equilibrium for the levels A and D, representing a particle of the species under discussion, we have, respectively,... 

A qualitative understanding of OMTS may be obtained with reference to Fig. 9. In this example we consider the simplest junction configuration, wherein a molecule is in electronic equilibrium with one metal electrode, and a relatively large insulating gap separates it from the other electrode. We will consider more complex cases later. 

The Pr4+ ion is easily reduced to Pr3+, which can be considered in terms of the electronic equilibrium ... 

Generally, when two phases are in electronic equilibrium, eoifa — fa) = Hi — ji 2- In our case, the wire I is in equilibrium with the metal M, the latter is in equilibrium with the redox couple, and the platinum electrode II is in equilibrium with the reference couple (index ref ). 

An interesting correlation exists between the work function of a metal and its pzc in a particular solvent. Consider a metal M at the pzc in contact with a solution of an inert, nonadsorbing electrolyte containing a standard platinum/hydrogen reference electrode. We connect a platinum wire (label I) to the metal, and label the platinum reference electrode with II. This setup is very similar to that considered in Section 2.4, but this time the metal-solution interface is not in electronic equilibrium. The derivation is simplified if we assume that the two platinum wires have the same work function, so that their surface potentials are equal. The electrode potential is then ... 

The first and the last term can again be expressed through the work function differences, but not the second term, since this interface is not in electronic equilibrium ... 

At high current densities the transport of electrons and holes may be too slow to establish electronic equilibrium at the semiconductor surface. 

The two reference electrodes and the interface between the two solution are in electronic equilibrium, so that we can express the differences in the inner potential through the differences in the chemical potentials. We denote the chemical potential of the two metal electrodes as hm, those of the two reference systems as / ef and and those of the two redox couples as /u4edox and /ij edox We obtain ... 

In general, differences in chemical bonding and electron configuration between carbon atoms and dopants mandate the deviation from the geometric and electronic equilibrium structure of the aromatic layers in CNTs. As a consequence, topological defects such as Stone-Wales defects are formed with increased probability [37]. 

Fig. 2-81. Surface degeneracy caused by Fermi level pinning at a surface state of high state density (a) in flat band state (Ep ep), G>) in electron equilibrium (cp = cp). cp = surface Fermi level = surface ccmduction band edge level.

Fig. 6-48. Differential capacity of a space charge layer of an n-type semiconductor electrode as a function of electrode potential solid cunre = electronic equilibrium established in the semiconductor electrode dashed curve = electronic equilibrium prevented to be established in the semiconductor electrode AL = accumulation layer DL = depletion layer IL = inversion layer, DDL - deep depletion layer.

When electronic equilibrium is established in the space charge layer, the concentration of interfacial electrons is given by n, = n exp (- e A /k T) and the concentration of interfacial holes is given by Pt = p exp(e A lk T) n and p are the concentrations of electrons and holes, respectively, in the semiconductor interior. In general, the ionization of surface atoms (Eqn. 9-24) is in quasiequilibrium so that the concentration of surface ions depends on the overvoltage... 

If the direct and reverse electron transitions (3) are in equilibrium (case when electron equilibrium at the surface is established), then a certain portion of the total number of acceptor levels A will be occupied by electrons, while a certain portion of the total number of donor levels D will be unoccupied that is, out of the total number N of the particles of a given kind chemisorbed on unit surface, a certain fraction of particles will be in a state of weak, strong acceptor, and strong donor bonding with the surface. Let us denote, respectively, by N°, N, N+ the number of particles per unit surface in each of these states and introduce the notation ... 

We see that once electron equilibrium is established, the relative amounts of the different forms of chemisorption on the surface, and hence the reactivity of the chemisorbed particles, are uniquely determined by the position of the Fermi level. This may be considered as the fourth important result of the electron theory. 

Suppose now that in addition to electron equilibrium on the surface we also have adsorption equilibrium between the surface and the gaseous phase. The condition of adsorption equilibrium (for the sake of simplicity we limit ourselves to the region of small coverage) has the form... 

Neglecting adsorption of the CO2 molecules and assuming that the different forms of chemisorption of CO2 are in equilibrium (case when electron equilibrium is established), we have... 

Once electronic equilibrium is established, the surface and the volume of the semiconductor have a common Fermi level, i.e., the same electrochemical potential (depicted by the horizontal line FF in Fig. 22). However, owing to the bending of the bands the position of the Fermi level in the energy spectrum of the crystal (its position relative to the energy bands) will, generally speaking, depend on the distance from the surface. We shall characterize the position of the Fermi level by its distance from the top of the valence band, denoted by e+. Evidently, + = We intro-... 

To determine n, we utilize the condition of electron equilibrium on the surface. In the absence of illumination, this is of the form (principle of detailed balance)... 

Upon illumination the condition of electron equilibrium takes the form... 

At the R/M interface there is electronic equilibrium, and their electrochemical potentials are the same. Therefore, = ]uR, and from the definition of n [seeEq. (6.34)],... 

Consider first the arrangement shown in Fig. A6.1. By adjusting the potentiometer to a reading V such that the galvanometer G indicates no current flowing, conditions of electronic equilibrium are created between a and P, i.e.,... 

The latter reaction will occur (at temperatures sufficiently high to overcome the activation energy in Fig. 1) because adsorption of hydrogen (Type A) will upset the electronic equilibrium between the number of electrons leaving the O sites and those returning from the zinc oxide. There will be fewer empty O sites after the Type A adsorption, hence more electrons will leave the O sites than return. As more 0 sites are formed, more hydrogen will be adsorbed. 

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