Quasi-stationary state

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... 

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),... 

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. 

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... 

In treating parallel reaction, two concepts are often used (i) the concept of rate-determining path, in which the fastest path is the rate-determining path, and (ii) the concept of steady state, also called the concept of quasi-stationary states of trace-level intermediates. 

If a system is disturbed by periodical variation of an external parameter such as temperature (92), pressure, concentration of a reactant (41,48,65), or the absolute configuration of a probe molecule (54,59), then all the species in the system that are affected by this parameter will also change periodically at the same frequency as the stimulation, or harmonics thereof (91). Figure 24 shows schematically the relationship between stimulation and response. A phase lag <)) between stimulation and response occurs if the time constant of the process giving rise to some signal is of the order of the time constant Inim of the excitation. The shape of the response may be different from the one of the stimulation if the system response is non-linear. At the beginning of the modulation, the system relaxes to a new quasi-stationary state, about which it oscillates at frequency cu, as depicted in Fig. 24. In this quasi-stationary state, the absorbance variations A(v, t) are followed by measuring spectra... 

Fig. 24. Schematic behavior of the absorbance of a species with a concentration in the system affected by a periodic (in this case sinusoidal) change of an external parameter. Thick solid line stimulation amplitude dashed line absorbance A(v,t) of a species in the system at a particular wavelength thin solid line mean absorbance, 4o(T ). At time t = 0, one parameter in the system starts to be modulated with a particular stimulation amplitude around a mean value. The concentration of the species (which are affected by the external stimulation) and hence the absorbance associated with these species are changing periodically at the same frequency as the stimulation. After an initial period, a quasi-stationary state is reached, at which the mean absorbance is constant. Typically, modulation experiments are performed at this stationary state by recording n time-resolved spectra at to, within the modulation period T.

The auxiliary conditions are electroneutrality, no electric current flow and a quasi-stationary state. 

One is in the presence of a two-time-scale description. In the short part, the system relaxes nonexponentially to a quasi-stationary state with the characteristic time r° [Eq. (4.220)] depending on the initial condition, and the spatial distribution shows large variations. Then, on a longer-time... 

The increase in the rate of oxidation of methanol with supercritical water concentration at 500 °C has been attributed to the increase in the concentration of OH radicals in a quasi-stationary-state.285... 

The decay of Nal can be described in an alternative way [K.B. Mpller, N.E. Henriksen, and A.H. Zewail, J. Chem. Phys. 113, 10477 (2000)]. In the bound region of the excited-state potential energy surface, one can define a discrete set of quasi-stationary states that are (weakly) coupled to the continuum states in the dissociation channel Na + I. These quasi-stationary states are also called resonance states and they have a finite lifetime due to the coupling to the continuum. Each quasi-stationary state has a time-dependent amplitude with a time evolution that can be expressed in terms of an effective (complex, non-Hermitian) Hamiltonian. 

When the reaction probability of the activated molecule A E) is sufficiently small, see Example 7.1 for an illustration, one can introduce the concept of quasi-stationary states. These states, n), associated with the slowly decaying bound states of A (E) are also called resonance states. The dynamics of the states is, essentially, given by [3] U(t) n) = e l(E lTnl 1 )tlh n). Note that, if n) was a true stationary state, i.e., an eigenstate of the full Hamiltonian Hi, then T = 0. The probability of observing the molecule in a quasi-stationary state is given by U(t) n) 2 = c that is, a... 

From the reaction mechanism, we can derive a reaction rate equation (kinetics) for the overall reaction. Below, we will discuss two such methods rate limiting step and quasi-stationary state (pseudo-steady-state). 

Quasi-Stationary State. The assumption in this method is that the concentration of intermediates is constant, after a short initial period. This means that the rate of formation of intermediates is equal to the rate of their disappearance/consumption. 

Secondly, it is worth noting the simplification of enthalpy balances which occurs using the quasi-stationary state approximation. The writing of the energy balance causes a reaction enthalpy flow ti to appear, where... 

The quasi-stationary state approximation consists of setting Rj = 0 for very reactive and short-lived intermediates such as free radicals. The result is that molar enthalpies of these intermediates do not appear in the calculation of H. Therefore, it is necessary to know neither their standard heats of formation, nor their heat capacities, values of which are not as well known as those of stable species. 

To illustrate the concepts of determining, non-determining and negligible processes, the mechanism of the pyrolysis of neopentane will be discussed briefly here. Neopentane pyrolysis has been chosen because it has been studied by various techniques batch reactor [105— 108], continuous flow stirred tank reactor [74, 109], tubular reactor [110], very low pressure pyrolysis [111], wall-less reactor [112, 113], non-quasi-stationary state pyrolysis [114, 115], single pulse shock tube [93, 116] amongst others, and over a large range of temperature, from... 

The mathematical techniques most commonly used in chemical kinetics since their formulation by Bodenstein in the 1920s have been the quasi-stationary state approximation (QSSA) and related approximations, such as the long chain approximation. Formally, the QSSA consists of considering that the algebraic rate of formation of any very reactive intermediate, such as a free radical, is equal to zero. For example, the characteristic equations of an isothermal, constant volume, batch reactor are written (see Sect. 3.2) as... 

Computations on free radical mechanisms by means of these criteria show that most reactions have been carried out under quasi-stationary state conditions and are not in serious error due to the use of QSSA. 

The concept of the simulated moving bed is based on the periodical change of the inlet and outlet along the fixed beds as it is shown in Figure 10.4. For practical reasons, the fixed bed is separated in a series of interconnected smaller beds. The inlets and outlets are moved periodically from one column connection to another via a complex valve system. The relative solid s movement is related to the shift period of the inlet-outlet lines. The monitoring of the system consists of maintaining a quasi stationary state where the profiles for A and B are fixed all along the bed and where A and B are separated (pure A and S withdrawn in... 

While studying stationary (see following) states of chemically reactive systems, we shall presume, as is done in traditional courses of chemical kinetics, that the relaxation of the concentrations of intermediates of chemical transformations to some quasi stationary state is much faster than the change of the concentrations of initial reactants (see Section 2.1). Therefore, for example, the concentration of reactive intermediates may be considered as an internal parameter in contrast to the external para meters that are the concentrations of initial reactants and final products that change considerably more slowly. 

A stationary state is by definition one whose description does not change with time. According to this definition it becomes impossible for a closed system which is undergoing chemical reaction to ever achieve a stationary state, because the composition of such a system is constantly changing with time. In this sense it is only open systems (i.e., systems in communication with infinite external reservoirs of mass, heat, etc.) which can achieve truly stationary states. However, we have used the description stationary state in discussing sequences of consecutive reactions occurring in closed systems. By this we have really meant a quasi-stationary state, ... 

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