Non-equilibrium chemical potential

It is known that the surface energy depends not only on the composition of the surface layer, but also on that of the bulk phases [130]. To formulate the Gibbs law for the non-equilibrium chemical potential, additional so-called cross-chemical potentials (the partial derivatives of the surface free energy with respect to the component concentrations in the bulk phases) have been introduced. Rusanov and Prokhorov [131] derived the Gibbs equation and the expression for the free energy of the surface layer in terms of the ordinary chemical potentials by dividing the transition layer adjacent to the surface into n thin layers. For each layer an equilibrium state was assumed. The expression for surface energy was derived by the summation of the equilibrium equations over all these layers. Further, the expression for the additional contribution to the surface tension due to the non-equilibrium diffusion layer was derived in [48, 132]... 

A corresponding relation can then be obtained for other thermodynamic properties and in particular for the non-equilibrium chemical potential in terms of the corresponding equilibrium function that is,... 

Following the procedure described in the previous section, an expression for the non-equilibrium chemical potential, Xi, of the penetrant species i in the mixture may be obtained by extending the expression of the free energy in equation 19 to nonequilibrium states according to the relation offered in equation 14 and using equation 12. We then obtain the following expression for the chemical potential of the penetrant species i ... 

Unlike the energy of an atom, which can be defined in terms of its local atomic environment, its chemical potential is a truly non-local quantity. In thermal equilibrium the chemical potential of each species is a constant throughout the system, whether atoms are at the interface or in the bulk. 

To this point, we have emphasized that the cycle of mobilization, transport, and redeposition involves changes in the physical state and chemical form of the elements, and that the ultimate distribution of an element among different chemical species can be described by thermochemical equilibrium data. Equilibrium calculations describe the potential for change between two end states, and only in certain cases can they provide information about rates (Hoffman, 1981). In analyzing and modeling a geochemical system, a decision must be made as to whether an equilibrium or non-equilibrium model is appropriate. The choice depends on the time scales involved, and specifically on the ratio of the rate of the relevant chemical transition to the rate of the dominant physical process within the physical-chemical system. 

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... 

Find the number density of positrons resulting from pair production by y-rays in thermal equilibrium in oxygen at a temperature of 109 K and a density of 1000 gmcm-3, using the twin conditions that the gas is electrically neutral and that the chemical potentials of positrons and electrons are equal and opposite. (At this temperature, the electrons can be taken as non-relativistic.) The quantum concentration for positrons and electrons is 8.1 x 1028 T93/2 cm-3, the electron mass is 511 keV and kT = 86.2 T9 keV. 

Figure 1.1 (a) Chemical potential diagrams for systems forming a complete range of solid solutions. The tangent shows at its terminal points, X, Y the chemical potentials of the elements x and y which are in equilibrium with the solid solution of composition M (b) Potentials for the system A-B, which forms the two stable compounds the stoichiometric A2B and the non-stoichiometric AB2. The graph shows that there are many pairs of potentials A, B in equilibrium with A2B and only one pair for a particular composition of AB2 AB is metastable with respect to decomposition to A2B and AB2... 

Since pyrolysis converts waste into CO, CH4, and H2, the product gases can be processed in an atmospheric pressure non-equilibrium plasma reformer to improve the energy-recovery potential of the product gas. Energy-recovery options include heat and chemical energy recovery. 

In thermal equilibrium, within a quantum statistical approach a mass action law can be derived, see [12], The densities of the different components are determined by the chemical potentials ftp and fin and temperature T. The densities of the free protons and neutrons as well as of the bound states follow in the non-relativistic case as... 

Within non-equilibrium thermodynamics, the driving force for relaxation is provided by deviations in the local chemical potential from it s equilibrium value. The rate at which such deviations relax is determined by the dominant kinetics in the physical system of interest. In addition, the thermal noise in the system randomly generates fluctuations. We thus describe the dynamics of a step edge by the equation. 

Solid state reactions occur mainly by diffusional transport. This transport and other kinetic processes in crystals are always regulated by crystal imperfections. Reaction partners in the crystal are its structure elements (SE) as defined in the list of symbols (see also [W. Schottky (1958)]). Structure elements do not exist outside the crystal lattice and are therefore not independent components of the crystal in a thermodynamic sense. In the framework of linear irreversible thermodynamics, the chemical (electrochemical) potential gradients of the independent components of a non-equilibrium (reacting) system are the driving forces for fluxes and reactions. However, the flux of one independent chemical component always consists of the fluxes of more than one SE in the crystal. In addition, local reactions between SE s may occur. 

Let us consider ionic systems. In non-equilibrium state, the potential drop across the interface differs from the equilibrium value A tpb (eq). If the adjacent phases a and P chemically buffer the interface on their respective sides, as is normally true considering the large number of particles in the bulk relative to the small number of interface particles, the overall potential drop, Atjb, is only due to the electric potential change 8[Pg.84]

The time evolution of a system may also be characterized according to the degree of perturbation from its equilibrium state. Linear theories hold if local equilibrium prevails, that is, each volume element of the non-equilibrium system can still be unambiguously defined by the usual set of (local) thermodynamic state variables. Often, a crystal is in (partial) equilibrium with respect to externally predetermined P and 7j but not with external component chemical potentials pik. Although P, T, and nk are all intensive functions of state, AP relaxes with sound velocity, A7 by heat conduction, and A/ik by matter transport. In solids, matter transport is normally much slower than the other modes of relaxation. 

When all the SE s of a solid with non-hydrostatic (deviatoric) stresses are immobile, no chemical potential of the solid exists, although transport between differently stressed surfaces takes place provided external transport paths are available. Attention should be given to crystals with immobile SE s which contain an (equilibrium) network of mobile dislocations. In these crystals, no bulk diffusion takes place although there may be gradients of the chemical free energy density and, in multicomponent systems, composition gradients (e.g., Cottrell atmospheres [A.H. Cottrell (1953)]). 

The most comprehensive description of the tunneling problem is based either on a self-consistent solution of the Lippman-Schwinger equation [3] or on the non-equilibrium Green s function approach [4-8]. Inelastic effects within e.g. a molecule-surface interface can be included by considering multiple electron paths from the vacuum into the surface substrate [9], The current between two leads with the chemical potentials /ja and hb is given by the energy integral ... 

However, this is still a non-equilibrium formulation of the problem, since the chemical potentials p account for the non-equilibrium condition of nonzero bias voltage. The only additional assumption in this formulation of the tunneling problem, compared to the general formulation above, is that the leads remain in thermal equilibrium. The expression can be calculated using a standard eigenvector expansion of surface and tip Green s functions ... 

We consider below what adjustments in concentration need to be made in order that non-equilibrium concentrations approach, and then become, equilibrium concentrations, driven by the need to minimise A G or the change in chemical potentials (Frame 5,27, 28, 29, 35, 37, 38 and 39). 

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