Quasi-equilibrium states, localized

Thus, the results stated above have shown that a change in the structure of epoxy polymers in the physical ageing process is restricted by reaching the quasi-equilibrium state, which is characterised by the balance between raising of the local order level (tendency to thermodynamically equilibrium structure) and entropic tautness of the chains. This approach allows to predict the change in the structure of polymers (and, hence, their properties) as a function of their initial structure, duration and temperature of physical ageing [101-103]. 

General first-order kinetics also play an important role for the so-called local eigenvalue analysis of more complicated reaction mechanisms, which are usually described by nonlinear systems of differential equations. Linearization leads to effective general first-order kinetics whose analysis reveals infomiation on the time scales of chemical reactions, species in steady states (quasi-stationarity), or partial equilibria (quasi-equilibrium) [M, and ]. 

Assume that the diffusion field is in a quasi-steady state and that local equilibrium is maintained at the surface and in the volume at a long distance from the surface, where yv = 0 and yA has the value characteristic of a flat surface. 

The use of the continuous quasi-equilibrium technique (see Chapter 3) made it possible to determine the corresponding adsorption isotherm in sufficient detail to reveal the sub-step shown in Figure 9.5 and located at the same 6 as the sharp calorimetric peak. The isotherm sub-step and associated calorimetric peak are evidently associated with an increase in packing density of the adsorbate. Rouquerol el al. (1977) and Grillet etal. (1979) concluded that these changes were due to a degenerated first-order transition from a hypercritical two-dimensional (2-D) fluid state to a 2-D localized state. 

From these time-scales, it may be assumed in most circumstances that the free electrons have a Maxwellian distribution and that the dominant populations of impurities in the plasma are those of the ground and metastable states of the various ions. The dominant populations evolve on time-scales of the order of plasma diffusion time-scales and so should be modeled dynamically, that is in the particle number continuity equations, along with the momentum and energy equations of plasma transport theory. The excited populations of impurities on the other hand may be assumed relaxed with respect to the instantaneous dominant populations, that is they are in a quasi-equilibrium. The quasi-equilibrium is determined by local conditions of electron temperature and electron density. So, the atomic modeling may be partially de-coupled from the impurity transport problem into local calculations which provide quasi-equilibrium excited ion populations and effective emission coefficients (PEC coefficients) and then effective source coefficients (GCR coefficients) for dominant populations which must be entered into the transport equations. The solution of the transport equations establishes the spatial and temporal behaviour of the dominant populations which may then be re-associated with the local emissivity calculations, for matching to and analysis of observations. 

Local Equilibrium Assumption. There exists a quasi-equilibrium between the reactant and a system crossing the TS from the reactant to the product. This precise representation was taken from an insightful article on the observability of the invariant of motion in the transition state by Marcus [16] The motions along the reaction coordinate at the transition state was... 

A simplified model(55(1972)) which explains the wide range of values, is that the electron in non-polar condensed media can exist in two states, a localized or trapped state whose mobility is comparable with that of small molecular ions and a quasi-free state the mobility of which is of the order of that in liquid xenon. In the quasi-free state the electron moves freely for much of its time in the uniform potential between molecules and only briefly interacts with the potential well surrounding molecules. Thus the localized and quasi-free states can be considered to exist in dynamic equilibrium and the mobility of the electron can be approximated by... 

The ocean is also by far the largest reservoir for most of the elements in the atmosphere-biosphere-ocean system. Perturbations caused by our increase in population and industrialization are passing through the ocean, and because the time-scale for ocean circulation is long (about 2000 years) relative to the time-scale of modern society, a new steady-state of quasi-equilibrium will slowly be established. Until that time, local concentrations of toxic chemicals, especially in estuaries and bays with restricted circulation, will be a major concern for mankind. 

Lichtner, P.C., 1991. The quasi-stationary state approximation to coupled mass transport and fluid-rock reaction local equilibrium revisited, in J. Ganguly, ed. Diffusion, Atomic Ordering and Mass Transport, Advances in Physical Geochemistry vol. 8, pp. 454-562. 

Equilibrium is assumed to prevail here at the boundaries of the membrane (e.g., the membrane is assumed to be thick or else it is a very thin film having a low ionic conductance so that the main resistance to the ionic fluxes is localized in the membrane proper and the equilibrium on the surfaces has enough time to be practically established) so that the boundary potential may be expressed by the Donnan ratios for the ions, and these ratios are determined from the fixed charge density in the membrane and the surrounding electrolyte concentrations. The diffusion potential, Eam, has a somewhat complicated expression. With the conditions of electroneutrality, zero electrical current, and a quasi-stationary state for each ion species, the expression for E m is... 

The quasi-equilibrium postulate states that systems are not far from equilibrium the gradients, or the thermodynamic forces are not too large. Within the system, local thermodynamic equilihrium holds. 

In Figure 3.5, there is a metastable solid solution M over the whole concentration interval, and an intermediate phase 1 (for instance, an ordered solution) is possible to appear, which is thermodynamically more favorable in a certain concentration interval. It is significant that at annealing of an A-B couple, the M phase lattice exists from the very beginning while phase 1 is absent. The system is not aware of a more favorable phase 1 up to fluctuation nucleation of a new lattice, so interdiflfusion takes place within the single metastable phase M hke any other quasiequilibrium process (each physically small volume has enough time to relax to an equilibrium state before its composition changes considerably). Since local relaxation reaches not truly equilibrium phases a, 1, P, but a metastable phase M, this process will be referred to as a meta-quasi-equilibrium. 

FIGURE 3.28 Schematics of the profile of the menisci in a flat capillary of the thickness 2H. (a) equUihrium state, no flow (h) a state of local equihhrium, when the flow is located in zone 2 ((1) the new quasi-equilihrium spherical menisci and quasi-equilibrium transition zone, (3) the equilibrium hquid film, which cannot be at the equilibrium with zone 1). 

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. 

Before the transfer starts, the energy distribution of electrons takes the form of a Fermi-Dirac distribution function. While the number of electrons is decreasing steadily with time, the distribution of electrons keep the form of a Fermi-Dirac distribution function. This constancy of the distribution is due to the fact that the capture rate of free electrons by the localized states is much faster than the loss of free electrons caused by the transfer when the occupation probability of localized states is not approximately one. Therefore, electrons are considered to be in their quasi-thermal equilibrium condition i.e., the energy distribution of electrons is described by quasi-Fermi energy EF. Then the total density t of electrons captured by the localized states per unit volume can be written as... 

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