Quasi-stationary

The system of coupled differential equations that result from a compound reaction mechanism consists of several different (reversible) elementary steps. The kinetics are described by a system of coupled differential equations rather than a single rate law. This system can sometimes be decoupled by assuming that the concentrations of the intennediate species are small and quasi-stationary. The Lindemann mechanism of thermal unimolecular reactions [18,19] affords an instructive example for the application of such approximations. This mechanism is based on the idea that a molecule A has to pick up sufficient energy... 

If the excitation energy required to fomi activated species A is much larger than k T its concentration will remain small. This is fulfilled if k Following Bodenstein, [A ] is then assumed to be quasi-stationary,... 

This leads to the quasi-stationary rate constant of equation (A3,4,133) if 4k + k +k f, which is... 

For this is identical to tire quasi-stationary result, equation (A3,4,133). although only tlie ratio [A ]/... 

In general, the assumption of quasi-stationarity is difficult to justify a priori. There may be several possible choices of intemiediates for which the assumption of quasi-stationary concentrations appears justified. It is possible to check for consistency, for example [A ]qj, [Pg.792]

Submitting the main topic, we deal with models of solids with cracks. These models of mechanics and geophysics describe the stationary and quasi-stationary deformation of elastic and inelastic solid bodies having cracks and cuts. The corresponding mathematical models are reduced to boundary value problems for domains with singular boundaries. We shall use, if it is possible, a variational formulation of the problems to apply methods of convex analysis. It is of importance to note the significance of restrictions stated a priori at the crack surfaces. We assume that nonpenetration conditions of inequality type at the crack surfaces are fulfilled, which improves the accuracy of these models for contact problems. We also include the modelling of problems with friction between the crack surfaces. 

J. G. Tolpin, G. S. John, and E. Field The Elucidation of Reaction Mechanisms by the Method of Intermediates in Quasi-Stationary Concentrations J. A. Christiansen... 

Photodiodes are the modem analogues to photocells. They increase their electrical resistance under light impact which, as part of an electric circuit, can be measured easily. Many current instruments display diode arrays instead of a single diode. Tens of photodiodes are arranged in a tight area. They are exposed to the sample bound spectrum where they respond to the color that corresponds to their positions in the diode array. A rapid, periodically performed electrical interrogation of all diodes (sequence periodicity in the order of milliseconds) reveals a quasi-stationary stable spectrogram. More sophisticated than photodiodes are phototransistors. They amplify internally the photoelectric effect, but the sensitivity of a photomultiplier cannot be achieved. 

Provided that chemiluminescence intensity Iql is proportional to the rate of peroxyl radicals termination, that is Icl [PO ]2, which is often assumed in the literature, chemiluminescence intensity should achieve some quasi-stationary level when hydroperoxide concentration becomes stationary and its decay should correspond to consumption of oxidizable groups, PH, in a polymer. At the same time, the chemiluminometric curves of type (a), which are typical with an autoaccelerating increase of the light emission (Figure 4) are relevant for... 

The molecular time scale may be taken to start at 10 14 s following energy absorption (see Sect. 2.2.3). At this time, H atoms begin to vibrate and most OH in water radiolysis is formed through the ion-molecule reaction H20+ + H20 H30+ + OH. Dissociation of excited and superexcited states, including delayed ionization, also should occur in this time scale. The subexcitation electron has not yet thermalized, but it should have established a quasi-stationary spectrum its mean energy is expected to be around a few tenths of an eV. 

Let us consider the case when the diffusion coefficient is small, or, more precisely, when the barrier height A is much larger than kT. As it turns out, one can obtain an analytic expression for the mean escape time in this limiting case, since then the probability current G over the barrier top near xmax is very small, so the probability density W(x,t) almost does not vary in time, representing quasi-stationary distribution. For this quasi-stationary state the small probability current G must be approximately independent of coordinate x and can be presented in the form... 

The use of the differential recurrence relations to calculate the mean first passage time is based on the observation that if in Eq. (5.48) one ignores the term sY(x, s) (which is tantamount to assuming that the process is quasi-stationary, i.e., all characteristic frequencies associated with it are very small), then one has... 

The theory of shape selection has been examined by many investigators concerned with solidification from the melt, and its status has recently been reviewed by Caroli and Muller-Krumbauer [63], The problem is to find stable, quasi-stationary solutions to the diffusion equation where a propagating branch maintains a constant shape and velocity. If the interface is assumed to have a uniform concentration, a family of such solutions exists, but there is no unique solution owing to the lack of a characteristic length. The solutions fix the peclet number. 

Since peroxyl radicals are also removed as a result of disproportionation, the reaction conditions are quasi-stationary for which the equality Vj = 2fc7[InH] [R02 ] I 2A 6[R02 ]2 is more appropriate. In this case, the rate of initiated chain oxidation of RH is equal to ... 

At high temperatures or in the presence of catalysts, hydroperoxide decomposes at a high rate, so that, after t t, inhibited oxidation becomes a quasi-stationary process with balanced rates of ROOH formation and decomposition. In this case, kdr 1, where kd is the overall specific rate of ROOH decomposition with allowance made for its decomposition... 

Lichtner, P. C., 1988, The quasi-stationary state approximation to coupled mass transport and fluid-rock interaction in a porous medium. Geochimica et Cosmochimica Acta 52, 143-165. 

Nevertheless, very-long-lived quasi-stationary-state solutions of Schrodinger s equation can be found for each of the chemical structures shown in (5.6a)-(5.6d). These are virtually stationary on the time scale of chemical experiments, and are therefore in better correspondence with laboratory samples than are the true stationary eigenstates of H.21 Each quasi-stationary solution corresponds (to an excellent approximation) to a distinct minimum on the Born-Oppenheimer potential-energy surface. In turn, each quasi-stationary solution can be used to construct an alternative model unperturbed Hamiltonian //(0) and perturbative interaction L("U),... 

We consider now the situation that component A and B are not volatile and that the volume of the stagnant film is small compared with the bulk of the liquid. This means that the bulk concentration of component A and B can be assumed to be constant. In fact we consider a quasi stationary process. Then the boundary conditions of equations (23), (24) and (25) are ... 

Anderson and Hartmann were the first to show that a series of experiments could be interpreted by supposing the system to be described by a quasi-stationary density matrix... 

Finally, we attack the problem of the transport coefficients, which, by definition, are calculated in the stationary or quasi-stationary state. The variation of the distribution functions during the time rc is consequently rigorously nil, which allows us to calculate these coefficients from more simple quantities than the generalized Boltzmann operators which we call asymptotic cross-sections or transport operators. 

In addition to the general problem of the kinetics of the approach towards equilibrium, the statistical mechanics of irreversible phenomena concern in particular the study of transport phenomena. The latter are calculated in a stationary or quasi-stationary form (the distribution functions do not vary or vary in hydrodynamic fashion). Therefore, let us consider (see, for... 

Generally, near the stoichiometric surface, both terms in (5.268) are large in magnitude and opposite in sign (Peters 2000). A quasi-stationary state is thus quickly established wherein the accumulation term on the left-hand side is negligible. The stationary laminar flamelet (SLF) model is found by simply neglecting the accumulation term, and ignoring the time dependency of x( , t) ... 

Assuming rapid attainment of a quasi stationary state and that k2[H202] k4[H02], the relation between kh and kt is... 

The influence of the appearance of such exotic states like quarks in stellar matter is topic of the study of quasi-stationary simulations of the evolution of isolated compact stars [15, 12, 7, 23] and accreting systems, where one companion is a superdense compact object [9,27], In this work we investigate the observability of the hadron-quark deconfinement phase transition in the dynamical evolution of a neutron star merger. 

Quasiperiodic behavior, Pt, 37 240-242 Quasi-stationary concentrations, reaction mechanisms by intermediates in, 5 311... 

A new approach to the explanation of fine structure has been offered by Hayasi (S). Like Kossel, he assumes bound states, but of short lifetime and of positive energy. These states, which he refers to as quasi-stationary, are stabihzed by Bragg reflections from important planes in the crystal. In the theory of Kronig the Bragg reflections of the photoelectron lead to minima in the absorption coefficient, in that of Hayasi to maxima. 

Europium also does not modify the mantle thermoluminescence perceptibly. There are no sharp emission lines in the red due to a conceivable luminescence of a quasi-stationary concentration of Uo built up by the flame, in contradistinction to cathodo-luminescence. It must be remembered that the Boltzmann population of this state is only 4.10 even at 2000 K. 

In flames, hydrogen seems to be more efficient than coal gas to produce candoluminescence, possibly because of appreciable quasi-stationary concentrations 62) of highly reactive H or OH. Hence, it might be true that selected mixed oxides would be particularly suitable for light emission in flames of pure hydrogen. 

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