Equilibrium localized concept

Among the basic concepts to be introduced are ionic equilibrium, local equilibrium, local electro-neutrality, etc. 

For many systems it is known that there exist regions or environments in which the time-invariant condition closely approaches equilibrium. The concept of local equilibrium is important in examining complex systems. Local equilibrium conditions are expected to develop, for example, for kinetically rapid species and phases at sediment-water interfaces in fresh, estuarine, and marine environments. In contrast, other local environments, such as the photosyn-thetically active surface regions of nearly all lakes and ocean waters and the biologically active regions of soil-water systems, are clearly far removed from total system equilibrium. 

As shown above, the classical description has been undoubtedly useful nevertheless, it has some drawbacks both from the fundamental and practical points of view. It is based on the macro- and/or local- equilibrium hypothesis, which may be too restrictive for a wider class of phenomena where other variables, not found in equilibrium, may influence the thermodynamic equations in the situations taking place when we get out of equilibrium. The concept is consistent with the limiting case of linear and instantaneous relations between fluxes and forces, which, however, becomes unsatisfactory under extreme conditions of high frequencies and fast non-linear changes (explosions). Then the need arises to introduce the second derivatives ( ), such as 0 = 0 (T, T T , P,. ..) and the general thermodynamic potential, 0, is can be no longer assumed via a simple linear relation but takes up a new, non-stationary form 0 = - ST + VP + (d0/Fr)T r ,p+ (state variables become dependent on the temperature derivatives. 

The optimum value of c is determined by the variational principle. If c = 1, the UHF wave function is identical to RHF. This will normally be the case near the equilibrium distance. As the bond is stretched, the UHF wave function allows each of the electrons to localize on a nucleus c goes towards 0. The point where the RHF and UHF descriptions start to differ is often referred to as the RHF/UHF instability point. This is an example of symmetry breaking, as discussed in Section 3.8.3. The UHF wave function correctly dissociates into two hydrogen atoms, however, the symmetry breaking of the MOs has two other, closely connected, consequences introduction of electron correlation and spin contamination. To illustrate these concepts, we need to look at the 4 o UHF determinant, and the six RHF determinants in eqs. (4.15) and (4.16) in more detail. We will again ignore all normalization constants. 

For N2 molecules in the air at room temperature cr(d is of the order of the speed of sound, 370 ms-1, a is 0.43 nm2 and Z = 5 x 1028 cm-3 s 1. This is a very large number, which means that collisions between molecules occur very frequently and the energy can be averaged between them, ensuring the concept of local thermal equilibrium. Each molecule collides ZAK/NA times per second, which is about 5 x 109 s x once every 0.2 ns. However, in the diffuse ISM where the molecule density is of order 102 cm3 the collision frequency is 5 x 10-8 s-1 or a collision every 1.5 years. 

Two activity coefficient models have been developed for vapor-liquid equilibrium of electrolyte systems. The first model is an extension of the Pitzer equation and is applicable to aqueous electrolyte systems containing any number of molecular and ionic solutes. The validity of the model has been shown by data correlation studies on three aqueous electrolyte systems of industrial interest. The second model is based on the local composition concept and is designed to be applicable to all kinds of electrolyte systems. Preliminary data correlation results on many binary and ternary electrolyte systems suggest the validity of the local composition model. 

Figure 2,10 Schematic representation of concept of local equilibrium in disequilibrium processes. Reprinted from R F. Sciuto and G. Ottonello, Geochimica et Cosmochimica Acta, 59, 2207-2214, copyright 1995, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1 GB, UK.

Sciuto P. F. and Ottonello G. (1995a). Water-rock interaction on Zabargad Island (Red Sea)., a case study I), application of the concept of local equilibrium. Geochim. Cosmochim. Acta, 59 2187-2206. 

For liquid-vapor interfaces, the correlation length in the bulk is of t he order of atomic distance unless one is close to the critical point Hence the concept of local equilibrium is well justified in most practical circumstances For. solid surfaces above the roughening temperature, the concept also makes sense. Since the surface is rough adding (or removing) an atom to a particular part of the surface docs not disturb the local equilibrium state very much, and this sampling procedure can be used to determine the local chemical potential. This is the essence of the Gibbs-Thomson relation (1). 

The concept of transport resistances localized in the outermost regions of NS crystals was introduced in order to explain the differences between intracrystalline self-diffusion coefficients obtained by n.m.r methods and diffusion coefficients derived from non-equilibrium experiments based on the assumption that Intracrystalline transport is rate-limiting. This concept has been discussed during the past decade, cf. the pioneering work [79-81] and the reviews [2,7,8,23,32,82]. Nowadays, one can state that surface barriers do not occur necessarily in sorption uptake by NS crystals, but they may occur if the cross-sections of the sorbing molecular species and the micropore openings become comparable. For indication of their significance, careful analysis of... 

The Mott-like transition, a central concept for the description of the actinide metal series, causes the sudden increase of the atomic volumes, encountered when between Pu and Am (Fig. 3). All other properties indicate the onset of a 5f localized behaviour at Am (see Part V) the 5 f pressure, which had contained to smaller values the equilibrium interactinide distance, suddenly gives in, with the withdrawal of the 5f s within the atomic core. The occurrence of such a transition within a series characterized by an unsaturated shell, is a unique phenomenon of the actinide series. In lanthanides, it does not occur except perhaps under pressure in cerium metal the approaching of cerium atoms induces suddenly the itineracy of 4f orbitals and a sudden volume collapse - see Chap. C. Neither it occurs in d-transition metal series, where the atomic volumes have an almost parabolic behaviour when plotted vs. Z (see Fig. 3 and Chap. C). The current... 

The vacancy flux and the corresponding lattice shift vanish if bA = bB. In agreement with the irreversible thermodynamics of binary systems i.e., if local equilibrium prevails), there is only one single independent kinetic coefficient, D, necessary for a unique description of the chemical interdiffusion process. Information about individual mobilities and diffusivities can be obtained only from additional knowledge about vL, which must include concepts of the crystal lattice and point defects. 

In other cases, however, and in particular when sublattices are occupied by rather immobile components, the point defect concentrations may not be in local equilibrium during transport and reaction. For example, in ternary oxide solutions, component transport (at high temperatures) occurs almost exclusively in the cation sublattices. It is mediated by the predominant point defects, which are cation vacancies. The nearly perfect oxygen sublattice, by contrast, serves as a rigid matrix. These oxides can thus be regarded as models for closed or partially closed systems. These characteristic features make an AO-BO (or rather A, O-B, a 0) interdiffusion experiment a critical test for possible deviations from local point defect equilibrium. We therefore develop the concept and quantitative analysis using this inhomogeneous model solid solution. 

The basic mechanisms by which various types of interfaces are able to move non-conservatively are now considered, followed by discussion of whether an interface that is moving nonconservatively is able to operate rapidly enough as a source to maintain all species essentially in local equilibrium at the interface. When local equilibrium is achieved, the kinetics of the interface motion is determined by the rate at which the atoms diffuse to or from the interface and not by the rate at which the flux is accommodated at the interface. The kinetics is then diffusion-limited. When the rate is limited by the rate of interface accommodation, it is source-limited. Note that the same concepts were applied in Section 11.4.1 to the ability of dislocations to act as sources during climb. 

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