Electronic equilibrium metallic phase

To understand why a stable equilibrium state of two metals in contact includes a contact potential, we can consider the chemical potential of the free electrons. The concept of chemical potential (i.e., partial molar Gibbs energy) applies to the free electrons in a metal just as it does to other species. The dependence of the chemical potential of free electrons in metal phase a on the electric potential 0 of the phase is given by the relation of Eq. 10.1.6 on page 287, with the charge number zi set equal to -1 ... 

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... 

At the contact of two electronic conductors (metals or semiconductors— see Fig. 3.3), equilibrium is attained when the Fermi levels (and thus the electrochemical potentials of the electrons) are identical in both phases. The chemical potentials of electrons in metals and semiconductors are constant, as the number of electrons is practically constant (the charge of the phase is the result of a negligible excess of electrons or holes, which is incomparably smaller than the total number of electrons present in the phase). The values of chemical potentials of electrons in various substances are of course different and thus the Galvani potential differences between various metals and semiconductors in contact are non-zero, which follows from Eq. (3.1.6). According to Eq. (3.1.2) the electrochemical potential of an electron in... 

The interfacial tension always depends on the potential of the ideal polarized electrode. In order to derive this dependence, consider a cell consisting of an ideal polarized electrode of metal M and a reference non-polarizable electrode of the second kind of the same metal covered with a sparingly soluble salt MA. Anion A is a component of the electrolyte in the cell. The quantities related to the first electrode will be denoted as m, the quantities related to the reference electrode as m and to the solution as 1. For equilibrium between the electrons and ions M+ in the metal phase, Eq. (4.2.17) can be written in the form (s = n — 2)... 

The standard electrode potential [1] of an electrochemical reaction is commonly measured with respect to the standard hydrogen electrode (SHE) [2], and the corresponding values have been compiled in tables. The choice of this reference is completely arbitrary, and it is natural to look for an absolute standard such as the vacuum level, which is commonly used in other branches of physics and chemistry. To see how this can be done, let us first consider two metals, I and II, of different chemical composition and different work functions 4>i and 4>ii-When the two metals are brought into contact, their Fermi levels must become equal. Hence electrons flow from the metal with the lower work function to that with the higher one, so that a small dipole layer is established at the contact, which gives rise to a difference in the outer potentials of the two phases (see Fig. 2.2). No work is required to transfer an electron from metal I to metal II, since the two systems are in equilibrium. This enables us calculate the outer potential difference between the two metals in the following way. We first take an electron from the Fermi level Ep of metal I to a point in the vacuum just outside metal I. The work required for this is the work function i of metal I. 

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 ). 

Since the electrochemical potential of electrons in metals is a function of the inner potential of the metal (P ca) = p. - inner potential difference, Mmb, across the interface where electron transfer is in equilibrium is represented by the difference in the chemical potential of electrons between the two metal phases A and B is shown in Eqn. 4-8 ... 

We have described above the relevant physics and electrostatics for a depleted semicondnctor/metal contact. It is also possible, however, that the initial electrochemical potential of the metal will be more negative than the Fermi level of the semicondnctor. Dnring the approach to equilibrium, electrons will then tend to flow into the semicondnctor from the metallic phase. This flow of electrons into the semicondnctor leads to a qnalitatively different energetic situation at the semicondnctor/metal junction relative to the situation in depletion. This situation is technologically desirable to form ohmic, electrically resistive contacts to semicondnctors. We discuss the charge-transfer eqnihbration for these types of contacts in this section. 

In Section 4.2, it was shown that the electrochemical potential is the relevant state function for use in discussions of electronic equilibrium and deviations from equilibrium. Consider, for example, two metal phases (A and B) so close that electrons may tunnel from one phase to the other. The difference in the electrochemical potential of the two phases quantifies the deviation from equilibrium this difference is very important for the rate of electron transfer. However, it is clear that thermodynamic considerations do not lead to predictions of the rate of electron transfer. 

The forward process describes the transfer of an electron from a reduced species, which is in equilibrium with the solvent molecules, to a metal phase, with electrochemical potential //. this results, eventually, in an oxidized species in equilibrium with the solvent molecules. When electronic equilibrium is reached between the metallic phase and the solution, the electrochemical potential in both phases is equal and given by... 

The first equation represents the equilibrium between hydrated Ag+ ions and Ag atoms in a single-crystal configuration. Alternatively, we may say that there is a heterogeneous thermodynamic equilibrium between Ag+ ions in the solid phase (where they are stabilized by the gas of free electrons) and Ag+ ions in the liquid phase (stabilized by interaction with water molecules). The forward reaction step corresponds to the anodic dissolution of a silver crystal. On an atomic level, one may say that a Ag" " core ion is transferred from the metallic phase to the liquid water phase. In an electrochemical cell, an electron flows from the Ag electrode (the working electrode) to the counter electrode each time that one Ag+ ion is transferred from the solid to the liquid phase across the electrochemical double layer. Although the electron flow is measured in the external circuit between the working... 

In subsequent sections of this chapter we concentrate on electronic-type metal-support interactions that are attributed to the collective electronic structure of the support and the metallic phases. The approach is based on the early work by Schwab [28] and Solymosi [29], and the interpretation of the observed chemisorptive and catalytic phenomena is, to a large extent, based on this early work. In order to obtain a more comprehensive understanding of the proposed interpretations, a brief review of important concepts of the electronic structure of metals and semiconductors, as well as of their contacts at equilibrium, is presented. 

As the equilibrium betv een the sample and the metal electrodes is maintained by the electron exchange, the equality of the electrochemical potentials of electrons in these phases leads to the follo ving expressions for the contact voltage drop and the heterogeneous thermovoltage coefficient ... 

For an inert metal in contact with a solution, the condition for electrical (or electronic) equilibrium is that the Fermi levels of the two phases be equal, that is,... 

To maintain equilibrium, there must also be a current in the opposite direction that opposes this forward rate, that is, electrons must also be able to leave the metal phase and enter the semiconductor conduction band. Because the electrons enter the empty states of the semiconductor, the concentration of these empty states can be taken as a constant. This leads to the expression... 

A more complicated example is provided by the contact of two electronic conductors, metals or semiconductors. At equilibrium the electrochemical potentials of electrons (i.e. their Fermi energies) must be equal in both bulk phases. The Galvani potential difference between them is determined by the bulk properties of the media in contact, i.e. by the difference between the chemical potentials of electrons ... 

According to the jellium model the metal can be considered as metal ions embedded in the electron plasma. The thermodynamic condition for electronic equilibrium between both phases is, from a chemist s point of view, equal values of the electrochemical potentials jUg of the electrons. 

From a physicist s point of view, the condition for electronic equilibrium is equal values of the Fermi energy E. Electronic equilibrium concerns all charged particles and might also be formulated for the metal ions. The equilibrium contact between a metal phase and an electrolyte phase is shown in Figure 3.1. 

Figure 3.1 Electronic equilibrium between a metallic phase and an electrolyte phase. The electronic energy states in the metal are described by the energy band (Section 2.9). The occupied states are and The density of states of electrons in the electrolyte are the energy distribution functions of the reduced and oxidized components of a redox system, e.g., Fe and Fe ions (Section 2.9.10). The equilibrium condition is equal values of the electrochemical potentials /x of the electrons in both phases. An alternative condition is equal values of the Fermi energy Ep in both phases.

Note the peculiarities of the work functions in a nonconducting medium (vacuum, pure solvent) and a conducting medium (electrolyte solution) when two metals contact each other, an electron equilibrium is always established between them, i.e., the condition te(l) = is met. The work function W is defined as the work of electron transfer from a metal to a point in the nonmetallic phase which is in the proximity to the interface at such a distance that the potential variation with distance can be ignored, i.e., beyond the superficial electric double layer, including the region in which the image forces are active ... 

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