Chemical potential reference level

In general, AH n(0 K) can be ealculated directly from DFT or higher level quantum methods, but experimental results can be used where reliable computed values are not available. We can now substitute eqn (2.35) for each of the Ap terms and replace the chemical potential references with A/frxn(0 K) to obtain a complete expression for the oxygen chemical potential ... 

The chemical potential now includes any such effects, and one refers to the gmvochemicalpotential, the electrochemical potential, etc. For example, if the system consists of a gas extending over a substantial difference in height, it is the gravochemical potential (which includes a tenn m.gh) that is the same at all levels, not the pressure. The electrochemical potential will be considered later. 

Figure 5.18. Schematic representation of the density of states N(E) in the conduction band and of the definitions of work function d>, chemical potential of electrons p, electrochemical potential of electrons or Fermi level p, surface potential x> Galvani (or inner) potential

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... 

To understand the role of the noble metal in modifying the photocatalysts we have to consider that the interaction between two different materials with different work functions can occur because of their different chemical potentials (see [200] and references therein). The electrons can transfer from a material with a high Fermi level to another with a lower Fermi level when they contact each other. The Fermi level of an n-type semiconductor is higher than that of the metal. Hence, the electrons can transfer from the semiconductor to the metal until thermodynamic equilibrium is established between the two when they contact each other, that is, the Fermi level of the semiconductor and metal at the interface is the same, which results in the formation of an electron-depletion region and surface upward-bent band in the semiconductor. On the contrary, the Fermi level of a p-type semiconductor is lower than that of the metal. Thus, the electrons can transfer from the metal to the semiconductor until thermodynamic equilibrium is established between the two when they contact each other, which results in the formation of a hole depletion region and surface downward-bent band in the semiconductor. Figure 12.6 shows the formation of semiconductor surface band bending when a semiconductor contacts a metal. 

In general, the chemical potential of electrons, t., is characteristic of individual electron ensembles, but the electrostatic energy of-e< > varies with the choice of zero electrostatic potential. In electrochemistry, as is described in Sec. 1.5, the reference level of electrostatic potential is set at the outer potential of the electron ensemble. 

In electrochemistry, we deal with the energy level of charged particles such as electrons and ions in condensed phases. The electrochemical potential, Pi,of a charged particle i in a condensed phase is defined by the differential work done for the charged particle to transfer from the standard reference level (e.g. the standard gaseous state) at infinity = 0) to the interior of the condensed phase. The electrochemical potential may be conventionally divided into two terms the chemical potential Pi and the electrostatic energy Zi e as shown in Eqn. 1-21 ... 

The ion level in condensed phases has been represented by the real potential, a, referred to the standard gaseous state of the ion at the outer potential of the condensed phases. The reference level, then, is not common to all ions but differs with different ions. In chemical thermodynamics, the conventional energy scale is based on the assumption that all atoms in the stable form in the standard state are at the zero energy level, which is the thermodynamic reference level of energy for chemical substances. In the following, we discuss the relationship between the scale of the ion level represented by the real potential of ions and the conventional energy scale of particles in chemical thermodynamics. 

Referring to Figure 3, evidence exists for placement of the Fermi levels (chemical potentials) of the redox reactions involving Hzr H2O and O2 roughly at the positions shown relative to the energies of the conduction band minimum and valence band maximum of the semiconductor, E and E, respectively. This picture takes the electron in a vacuum at infinity as the zero of energy. On this basis, the Fermi level for the reaction... 

The reference state of A-electron theory becomes a reference vacuum state 4>) in the field theory. A complete orthonormal set of spin-indexed orbital functions fip(x) is defined by eigenfunctions of a one-electron Hamiltonian Ti, with eigenvalues ep. The reference vacuum state corresponds to the ground state of a noninteracting A-electron system determined by this Hamiltonian. N occupied orbital functions (el < pi) are characterized by fermion creation operators a such that a] ) =0. Here pt is the chemical potential or Fermi level. A complementary orthogonal set of unoccupied orbital functions are characterized by destruction operators aa such that aa < >) = 0 for ea > p and a > N. A fermion quantum field is defined in this orbital basis by... 

In electrochemistry we make it a rule that the standard chemical potential ju. of hydrogen ions is set zero as the level of reference for the chemical potentials of all other hydrated ions. The standard chemical potentials of various hydrated ions tabulated in electrochemical handbooks are thus relative to the standard chemical potential of hydrogen ions at unit activity in aqueous solutions. Table 9.3 shows the numerical values of the standard chemical potential, the standard partial molar enthalpy h°, and the standard partial molar entropy. 5 ,° for a few of hydrated ions. 

Let us assume internal equilibrium in Zf, which corresponds to the mutually open subsystems, Zfe=(Z/> Zf 2), with equalized chemical potentials, nf = nf = P.r = dE/dN, at the global chemical potential level. The internal stability refers to intra-5 (hypothetical) charge displacements, dA/y(A) = (A, — A), that preserve N. The corresponding quadratic energy change due to this polarizational displacement from the initial internal equilibrium state ... 

If the pH of the electrolyte and the pressure of hydrogen are constant, the chemical potential of protons and hydrogen molecules are also constant. This means that the chemical potential (the Fermi level) of the electrons is constant, which defines a reference for the potential. When a bias is applied, the potentials are hereafter measured relative to this reversible hydrogen electrode reference, which evidentially is independent of the electrode material. 

In metals the chemical potential remains equal to the Fermi energy to a high degree of accuracy up to room temperature and the term Fermi level usually denotes chemical potential in solid-state physics. In semiconductors at temperatures T 4= 0 the chemical potential may significantly differ from the -> Fermi energy and in several cases there are no available states at this energy, but even in this case it is also a widespread practice to refer to the chemical potential of a semiconductor as the Fermi level. 

Figure 2 translates the charge carrier formation reaction into an energy level diagram for various systems. In fact these levels refer to standard chemical potentials or (in the case of the Fermi-levels ) to full chemical potentials (see e.g. Refs.3,35). As long as —in pure materials— the gap remains large compared to RT, the Boltzmann-form of the chemical potential of the respective charge carrier (defect) is valid,... 

The reference energy EF is called the Fermi energy and corresponds to the chemical potential of the electrons in the solid, and the factor 2 in eqn. (1) takes account of the fact that each level may be occupied by two electrons of opposite spins. 

Chemical potential, like electrical and gravitational potentials, must be expressed relative to some arbitrary energy level. An unknown additive constant, or reference level, /tj, is therefore included in Equation 2.4. Because it contains an unknown constant, the actual value of the chemical potential is not determinable. For our applications of chemical potential, however, we are interested in the difference in the chemical potential between two particular locations (Fig. 2-6), so only relative values are important. Specifically, because fi is added to each of the chemical potentials being compared, it cancels out when the chemical potential in one location is subtracted from that in another to obtain the chemical potential difference... 

The additive constant term fij in Equation 2.4 is the chemical potential of species j for a specific reference state. From the preceding definitions of the various quantities involved, this reference state is attained when the following conditions hold The activity of species j is 1 (RT In cij = 0) the hydrostatic pressure equals atmospheric pressure (VjP = 0) the species is uncharged or the electrical potential is zero (ZjFE = 0) we are at the zero level for the gravitational term (rrijgh = 0) and the temperature equals the temperature of the system under consideration. Under these conditions, fij equals fij (Eq. 2.4). 

It is similarly practical to choose a convenient level of reference for the values of the chemical potential because differences of can be determined much more precisely than absolute values. Moreover, because we only need to compare potential values or their sums, it does not matter what the unit is at first. The p values could be expressed in different scales similarly to how temperature can be expressed (Celsius, Fahrenheit, Kelvin, etc.). We will use Gibbs", abbreviated to G, following a proposition of Wiberg [9] to honour Josiah Willard Gibbs (1839-1903) who first introduced the concept of chemical potential. For avoiding complications in further considerations we choose the unit to be Sl-consistent, which means 1 G coiresponds to 1 J mol . 

Next we enter into the question what reference states are suitable for the measurement of the potential differences. It is useful to refer to the conventional basic materials in chemistry, the elements. Because it is not possible to transfonn one element into another by chemical means (i.e., nuclear reactions are excluded), the values of the various elements are not related to each other. Therefore, the reference level for any... 

Reference Level Up to now, what we have been missing in order to make concrete predictions are the p values of the substances we have been dealing with. The chemical potential can be assigned an absolute zero value, just as temperature can. In principle, the absolute values could be used, but they are enormous. It would mean that in order to work with the tiny differences in potentials common in chemical and biological reactions, at least 11 digits would be necessary (the ratio between the potential differences and the absolute values is around one to one billion ). This alone would lead to numbers that are much too unwieldy, not to mention that the absolute values are not known accurately enough for this to be feasible. 

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