Quasi-stationarity

So far the quasi-steady-state hypothesis introduced in 1913 has remained the most favourable approach to operating with chemical kinetic equations. In short (and not quite strictly), its most applicable version can be formulated as follows. During the reaction, the concentrations of some (usually intermediate) compounds are the concentration functions of the other (usually observed) substances and adapt to their values as if they were steady-state values. 

As usual, this hypothesis is associated with the names of Bodenstein and Semenov. The latter introduced a concept of partial quasi-stationarity realized for some intermediates. Christiansen described the history of the problem as follows [36] ... the first who applied this theory was S. Chapman and half the year later Bodenstein referred to it in his paper devoted to hydrogen reaction with chlorine. His efforts to confirm his viewpoint were so energetic that this theory is quite naturally associated with his name . 

In 1940 Frank-Kamenetskii made an attempt to formulate mathematical conditions for the applicability of this approach [37]. A strict formulation for the problem of a mathematical status for the principle of quasi-stationarity was suggested by Sayasov and Vasilieva [38] in terms of the theory of singularly perturbed differential equations. 

Substantiation for this hypothesis is constructed on the availability in the initial set of differential equations with a small parameter s standing before some derivatives. We will write this set as 

Sometimes to reduce the system examined to such a form it is necessary to pass to some new (usually dimensionless) variables or to a new time scale. For example, if the initial set is of the form 

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A] = b/a (equation (A3.4.145)) is stationary and not [A ] itself This suggests d[A ]/dt < d[A]/dt as a more appropriate fomuilation of quasi-stationarity. Furthemiore, the general stationary state solution (equation (A3.4.144)) for the Lindemaim mechanism contams cases that are not usually retained in the Bodenstein quasi-steady-state solution. 

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 ]. 

The assumption of quasi-stationarity can sometimes be justified if there is no significant chain branching, for example in FIBr fomiation at 200-3 00°C ... 

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]

Let us emphasize one typical inaccuracy met in the description of the quasi-stationarity hypothesis for chemical systems. It is suggested that the rate of changing the amount of intermediate particles (fast sub-system) tends to or even equals zero. But this is not true since it is not difficult to obtain an expression for y by differentiating the relationship g(x, y) = 0 and using an implicit function theorem... 

The introduction of a quasi-stationarity hypothesis was motivated previously by the fact that concentrations of the intermediates are low and so are the rates of their variations. First, however, rates are often not low... 

Specificity of a concrete system accounts for the source of the appearance of a small parameter and for its type. For homogeneous reactions, a small parameter is usually a ratio of rate constants for various reactions some reactions are much faster than the others. For just such a small parameter Vasiliev et al. [25] distinguished a class of chemical kinetic equations for which the application of the quasi-stationarity principle is correct (they considered a closed system). 

For catalytic reactions the fast and slow variables usually considered are the concentrations of surface intermediates on catalysts and gas-phase reactants, respectively. (In the case of high-vacuum conditions, "a vice versa quasi-stationarity is possible, see below.) But in the equations for heterogeneous catalytic reactions (119)... 

It is also possible to consider the case e - oo. It is symmetric to the above cases (a -(c) accurate to the substitution of cg, 6g and Nfnl by cs, bs and N ot, respectively. This case corresponds to catalytic reactions carried out under high-vacuum conditions. For this case one can observe a "reverse quasi-stationarity , i.e. a fast "adapting of the concentrations for gaseous substances cg to those of surface substances cs. 

If we assume that vin = 0 (the system is closed) we obtain the case considered previously. Note that then the asymptotes given are not applicable since a linear part in the equation for cA becomes singular and the major contribution is made by the terms of the order 0(e). For this case (vin = 0), the quasi-stationarity in the system becomes possible. Proceeding from the assumption that at bs -> 0 we will have v Jba = const., it is possible to go to a new time scale z = bst and obtain... 

Let us now consider a version of the reverse quasi-stationarity bs - go, S, V, bg, and vin being constant. Here a "fast subsystem is of the form... 

In conclusion, it must be noted that the equations to describe the transient behaviour of heterogeneous catalytic reactions, usually have a small parameter e = Altsot/Alt t. Here Atsot = bsS = the number of active sites (mole) in the system and Nfot = bg V = gas quantity (mole). Of most importance is the solution asymptotes for kinetic equations at A/,tsot/7Vtflt - 0, 6S, bg and vin/S being constant. Here we deal with the parameter SjV which is readily controlled in experiments. The case is different for the majority of the asymptotes examined. The parameters with respect to which we examine the asymptotes are difficult for control. For example, we cannot, even in principle, provide an infinite increase (or decrease) of such a parameter as the density of active sites, bs. Moreover, this parameter cannot be varied essentially without radical changes in the physico-chemical properties of the catalyst. Quasi-stationarity can be claimed when these parameters lie in a definite range which does not depend on the experimental conditions. 

To answer the question whether quasi-stationarity can be observed in our kinetic model at e - 0 it is first necessary to examine a subsystem of fast motions ("a fast subsystem ) so as to establish if it has a unique and stable steady-state solution. 

Except for Stage III, we cannot expect stationarity to hold Eq. (11) any more. However, from the temporal evolutions of M(t) (Fig. 2), we may expect quasi-stationarity in Stage I,... 

We envisage, within this time-expansion of the (56) or (57) [which possibility follows from the (quasi)stationarity concept), the infinite cycle in the observed machine TM arises, based on its Self-Description [ ]=[d(TM)]. But, following the Auto-Reference construction, it runs in the double-machine (TM,M =1M) = OO, 2... 

Long-term trends observed by recent measurements cover a few decades, at most. For much longer periods, the supposition of a quasi-stationarity of the atmosphere makes it likely that a longer trend will not continue indefinitely, but that eventually a new steady state is reached or the trend reverses. In recent years it has become possible to investigate the abundance of trace substances and their variations in past epochs by the exploration of deep ice cores from the great ice sheets of Greenland and Antarctica. These provide a record of atmospheric conditions dating back at least 70,000 years. 

Plonsey, R. and Heppner, D. 1967. Consideration of quasi-stationarity in electrophysiological systems. Bull. Math. Biophys. 29 657. 

The condition for the quasi-stationarity of a system on the basis of an intermediate substance can be written (see Chapter 1) as... 

This equation, as it was shown in Chapter 1, limits the number of linearly independent elements of the vector and in the quasi-stationarity system the number s of... 

However, at the same time, in accordance with equation (2.22) all elements of the matrix are equal to zero so on the left ftom the vector r in equation (2.26) is a null matrix. In. such a way, the condition of the quasi-stationarity of a system upon the intermediate substances in a form of equation (2.26) does not limit the elements of the vector r. 

So, neither the condition of quasi-stationarity in the form (2.26) nor the conservation law in the form (2.27) limit in any way the elements of the rates of route r vector all of them are linearly independent in the sense that the rate upon the any route can not be founded as a linear combination of the rates upon the rest routes. This conception can be considered as a principle of the route independency in a non-equilibrium quasi-stationary system according to which... 

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