States local equilibrium

We consider a two state system, state A and state B. A state is defined as a domain in phase space that is (at least) in local equilibrium since thermodynamic variables are assigned to it. We assume that A or B are described by a local canonical ensemble. There are no dark or hidden states and the probability of the system to be in either A or in B is one. A phenomenological rate equation that describes the transitions between A and B is... 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

The third approach is called the thermodynamic theory of passive systems. It is based on the following postulates (1) The introduction of the notion of entropy is avoided for nonequilibrium states and the principle of local state is not assumed, (2) The inequality is replaced by an inequality expressing the fundamental property of passivity. This inequality follows from the second law of thermodynamics and the condition of thermodynamic stability. Further the inequality is known to have sense only for states of equilibrium, (3) The temperature is assumed to exist for non-equilibrium states, (4) As a consequence of the fundamental inequality the class of processes under consideration is limited to processes in which deviations from the equilibrium conditions are small. This enables full linearization of the constitutive equations. An important feature of this approach is the clear physical interpretation of all the quantities introduced. 

Fig. 5 Absorption and fluorescence emission spectra of the 3-hydroxychromone dye F4N1 in the absence (black) and presence (red) of a local electric field, which promotes the excitation charge transfer leading from the ground state to the N state. In the presence of the local electric field, the energy of the N state is reduced, causing a red shift of the N emission peak and an increase in its intensity relative to the T emission peak. The change in relative intensities of the N and T peaks reflects a shift in the excited state tautomeric equilibrium toward the N state...

This can be inserted into the current expressions (30) and (32), whose divergences enter into the time development equations for the densities. (Note that the denominator of (76) simplifies greatly in all cases except where the silicon is nearly intrinsic. Here again, if the charge states are equilibrated, n+ and n can be eliminated in favor of n0, via (3), (4), and (75). Whether AH and /or nDH must be retained as distinct variables in the system of differential equations, or whether they, too, can be eliminated in favor of 0, will depend on whether or not they are able to come quickly into local equilibrium with the monatomic species. 

The standard wall function is of limited applicability, being restricted to cases of near-wall turbulence in local equilibrium. Especially the constant shear stress and the local equilibrium assumptions restrict the universality of the standard wall functions. The local equilibrium assumption states that the turbulence kinetic energy production and dissipation are equal in the wall-bounded control volumes. In cases where there is a strong pressure gradient near the wall (increased shear stress) or the flow does not satisfy the local equilibrium condition an alternate model, the nonequilibrium model, is recommended (Kim and Choudhury, 1995). In the nonequilibrium wall function the heat transfer procedure remains exactly the same, but the mean velocity is made more sensitive to pressure gradient effects. 

A final variant of local equilibrium models is the dump option (Wolery, 1979). Here, once the equilibrium state of the initial system is determined, the minerals... 

As noted in Section 6.9, the chemical-reaction step is performed last in order to allow fast chemical reactions to return to their local equilibrium states. 

However, we have to reflect on one of our model assumptions (Table 5.1). It is certainly not justified to assume a completely uniform oxide surface. The dissolution is favored at a few localized (active) sites where the reactions have lower activation energy. The overall reaction rate is the sum of the rates of the various types of sites. The reactions occurring at differently active sites are parallel reaction steps occurring at different rates (Table 5.1). In parallel reactions the fast reaction is rate determining. We can assume that the ratio (mol fraction, %a) of active sites to total (active plus less active) sites remains constant during the dissolution that is the active sites are continuously regenerated after AI(III) detachment and thus steady state conditions are maintained, i.e., a mean field rate law can generalize the dissolution rate. The reaction constant k in Eq. (5.9) includes %a, which is a function of the particular material used (see remark 4 in Table 5.1). In the activated complex theory the surface complex is the precursor of the activated complex (Fig. 5.4) and is in local equilibrium with it. The detachment corresponds to the desorption of the activated surface complex. 

Mass and energy transport occur throughout all of the various sandwich layers. These processes, along with electrochemical kinetics, are key in describing how fuel cells function. In this section, thermal transport is not considered, and all of the models discussed are isothermal and at steady state. Some other assumptions include local equilibrium, well-mixed gas channels, and ideal-gas behavior. The section is outlined as follows. First, the general fundamental equations are presented. This is followed by an examination of the various models for the fuel-cell sandwich in terms of the layers shown in Figure 5. Finally, the interplay between the various layers and the results of sandwich models are discussed. 

Kinetic system, wherein the pathways along the system are moving toward some state of local equilibrium, which in tnm determines the rate of change along the pathway. In the context of a kinetic approach, which is relevant to geochemical processes, dissolntion-precipitation, exchange-adsorption, oxidation-reduction, vaporization, and formation of new phases, are discussed here. 

The earth s subsurface is not at complete thermodynamic equilibrium, but parts of the system and many species are observed to be at local equilibrium or, at least, at a dynamic steady state. For example, the release of a toxic contaminant into a groundwater reservoir can be viewed as a perturbation of the local equilibrium, and we can ask questions such as. What reactions will occur How long will they take and Over what spatial scale will they occur Addressing these questions leads to a need to identify actual chemical species and reaction processes and consider both the thermodynamics and kinetics of reactions. 

Systemic effect represents the critical effect and local effects are of secondary importance. Half-life of the substance under consideration is not too short it should be possible to assume a state of equilibrium in the body. 

To remove momentum fluctuations from the problem, BCAH assume that in the creeping flow limit, in which a system of small mass interacts strongly with a thermally equilibrated solvent, the distribution of values for the momenta for fixed coordinate values stays very near a state of local equilibrium, in which... 

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 above estimates suggest that, with the scaling (1.7) valid, it takes about time td for an ionic system to restore local electro-neutrality on the macroscopic length scale and to reach local equilibrium. By local equilibrium we merely imply a state with the normalized ionic fluxes ji, defined by... 

In a similar vein, Riemann s formalism finds useful application in expressing the global thermodynamic behavior of a system S. The metric geometry governed by M( ) represents thermodynamic responses (as before), while labels distinct states of equilibrium, each exhibiting its own local geometry of responses. The state-specifier manifold may actually be chosen rather freely, for example, as any/independent intensive variables (such as gi = T, 2 = P 3 = Mr, > = l c-p)- For our purposes, it is particularly convenient to... 

Let us begin the discussion of the last example of solid state kinetics in this introductory chapter with the assumption of local equilibrium at the A/AB and AB/B interfaces of the A/AB/B reaction couple (Fig. 1-5). Let us further assume that the reaction geometry is linear and the interfaces between the reactants and the product AB are planar. Later it will be shown that under these assumptions, the (moving) interfaces are morphologically stable during reaction. 

The increase A will occur at interface A/AB if LA/LR< 1, and it will occur at AB/B if La >Lr (Fig. 1-5). We conclude that parabolic rate laws in heterogeneous solid state reactions are the result of two conditions, the prevalence of a linear geometry and of local equilibrium which includes the phase boundaries. 

Let us now consider the equalization of the component concentrations in an inhomogeneous multicomponent system. We may start with Eqn. (4.33) which relates the component fluxes, jk, to the (n-1) independent forces, Vyq, of the n-compo-nent solid solution. In local equilibrium, the chemical potentials are functions of state. Hence, at any given P and T... 

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. 

Heterogeneous solid state reactions occur when two phases, A and B, contact and react to form a different product phase C. A and B may be either chemical elements or compounds. We have already introduced this type of solid state reaction in Section 1.3.4. The rate law is parabolic if the reacting system is in local equilibrium and the growth geometry is linear. The characteristic feature of this type of reaction is the fact that the product C separates the reactants A and B and that growth of the product proceeds by transport of A and/or B through the product layer. 

Chapters 6 and 7 dealt with solid state reactions in which the product separates the reactants spatially. For binary (or quasi-binary) systems, reactive growth is the only mode possible for an isothermal heterogeneous solid state reaction if local equilibrium prevails and phase transitions are disregarded. In ternary (and higher) systems, another reactive growth mode can occur. This is the internal reaction mode. The reaction product does not form at the contacting surfaces of the two reactants as discussed in Chapters 6 and 7, but instead forms within the interior of one of the reactants or within a solvent crystal. 

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