Including electric potentials

In the presence of an electric potential V (e.g. from nuclei), the time-independent Dirac equation may be written as in eq. (8.8), where we have again explicitly indicated the electron mass. 

Here Tl and Ps are (large and small) two-component wave functions that include the a and spin functions. The latter equation can be solved for Ps. 

In the non-relativistic limit (c - o°) the K factor is 1, and the first term becomes (a p)(a p). Using the vector identity (a p)(a p) = p p + ia(p x p), this gives the non-relativistic kinetic energy pVlm, since the vector product of any vector with itself is zero (p X p = 0). The equation for the large component therefore reduces to the Schrddinger equation. 

The electron spin is still present in eq. (8.14), since Pl is a two-component wave function, but this can trivially be separated out since the operators do not contain any spin dependence. 

In the non-relativistic limit the small component of the wave function is given by eq. (8.15). 

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The first term of this Hamiltonian is a free-particle discrete-level model (10) with eafj including electrical potentials. And the second term describes all possible interactions between electrons and is equivalent to the real-space Hamiltonian... 

For all five adsorption models to be discussed, the surface offlie solid contiguous to the solution is viewed as a plane of constant thermodynamic chemical (including electrical) potential. Adsorption is regarded as those solute molecules which partition into a monomolecular surface layer. However, these five models have significant distinguishing characteristics in the way they approach adsorption, as shown heuristically in Fig. 31. 

This handbook deals only with systems involving metallic materials and electrolytes. Both partners to the reaction are conductors. In corrosion reactions a partial electrochemical step occurs that is influenced by electrical variables. These include the electric current I flowing through the metal/electrolyte phase boundary, and the potential difference A( = 0, - arising at the interface. and represent the electric potentials of the partners to the reaction immediately at the interface. The potential difference A0 is not directly measurable. Therefore, instead the voltage U of the cell Me /metal/electrolyte/reference electrode/Me is measured as the conventional electrode potential of the metal. The connection to the voltmeter is made of the same conductor metal Me. The potential difference - 0 is negligibly small then since A0g = 0b - 0ei ... 

The free energy difference for protons across the inner mitochondrial membrane includes a term for the concentration difference and a term for the electrical potential. This is expressed as... 

An expansion in powers of 1 /c is a standard approach for deriving relativistic correction terms. Taking into account electron (s) and nuclear spins (1), and indicating explicitly an external electric potential by means of the field (F = —V0, or —— dAjdt if time dependent), an expansion up to order 1/c of the Dirac Hamiltonian including the... 

Other variables come into play for specific kinds of problems. For example, surface area / s and surface tension 7 are used when surface effects are considered, and electrical potential E and quantity of electrical charge Q are included when electrical work is involved. 

Electrochemical cells can be constructed using an almost limitless combination of electrodes and solutions, and each combination generates a specific potential. Keeping track of the electrical potentials of all cells under all possible situations would be extremely tedious without a set of standard reference conditions. By definition, the standard electrical potential is the potential developed by a cell In which all chemical species are present under standard thermodynamic conditions. Recall that standard conditions for thermodynamic properties include concentrations of 1 M for solutes in solution and pressures of 1 bar for gases. Chemists use the same standard conditions for electrochemical properties. As in thermodynamics, standard conditions are designated with a superscript °. A standard electrical potential is designated E °. 

The electrical potentials assumed to exist at liquid-liquid interfaces, including inert gas or liquid dielectric environments are presented in Fig. 1. 

In electrochemical kinetics, the concept of the electrode potential is employed in a more general sense, and designates the electrical potential difference between two identical metal leads, the first of which is connected to the electrode under study (test, working or indicator electrode) and the second to the reference electrode which is in a currentless state. Electric current flows, of course, between the test electrode and the third, auxiliary, electrode. The electric potential difference between these two electrodes includes the ohmic potential difference as discussed in Section 5.5.2. 

Possible driving forces for solute flux can be enumerated as a linear combination of gradient contributions [Eq. (20)] to solute potential across the membrane barrier (see Part I of this volume). These transbarrier gradients include chemical potential (concentration gradient-driven diffusion), hydrostatic potential (pressure gradient-driven convection), electrical potential (ion gradient-driven cotransport), osmotic potential (osmotic pressure-driven convection), and chemical potential modified by chemical or biochemical reaction. 

The analysis of oxidation processes to which diffusion control and interfacial equilibrium applied has been analysed by Wagner (1933) who used the Einstein mobility equation as a starting point. To describe the oxidation for example of nickel to the monoxide NiO, consideration must be given to the respective fluxes of cations, anions and positive holes. These fluxes must be balanced to preserve local electroneutrality throughout the growing oxide. The flux equation for each species includes a term due to a chemical potential gradient plus a term due to the electric potential gradient... 

Inhalation of certain hydrocarbons, including some anesthetics, can make the mammalian heart abnormally sensitive to epinephrine, resulting in ventricular arrhythmias, which in some cases can lead to sudden death (Reinhardt et al. 1971). The mechanism of action of cardiac sensitization is not completely understood but appears to involve a disturbance in the normal conduction of the electrical impulse through the heart, probably by producing a local disturbance in the electrical potential across cell membranes. The hydrocarbons themselves do not produce arrhythmia the arrhythmia is the result of the potentiation of endogenous epinephrine (adrenalin) by the hydrocarbon. 

In the area of wind offshore energy, Garrad Hassan et al. (1995) place the electricity production potential at 10904 PJ/year including areas with a distance up to 30 km from the coast and a water depth of less than 40 m. The EWEA (2003) and Greenpeace (2001) apply further constraints leading to a significantly lower value for the offshore electricity potential (see Fig. 5.6). In this way they restrict the area available for offshore production to a water depth of 20 m and reduce the capacity density. 

The signals were recorded as electrical potential in millivolts. The well known Maxwell s formula and an adjustable empirical coefficients were used to obtain the equivalent volume fraction of liquid [43]. Since it is known that kinematic waves exist only in the frequency of some few Hertz, hardware low pass filter with 20 Hz cutting frequency was included for each channel in the electronic unit. The filter was tuned at 82.66 Db. Data were acquired by a computer at 100 Hz. The following comments regarding the above apparatus description are in order ... 

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