Stationary state approximation

The propagation of polymer chains is easy to consider under stationary-state conditions. As the preceding example illustrates, the stationary state is reached very rapidly, so we lose only a brief period at the start of the reaction by restricting ourselves to the stationary state. Of course, the stationary-state approximation breaks down at the end of the reaction also, when the radical concentration drops toward zero. We shall restrict our attention to relatively low conversion to polymer, however, to avoid the complications of the Tromms-dorff effect. Therefore deviations from the stationary state at long times need not concern us. 

We saw in the last chapter that the stationary-state approximation is apphc-able to free-radical homopolymerizations, and the same is true of copolymerizations. Of course, it takes a brief time for the stationary-state radical concentration to be reached, but this period is insignificant compared to the total duration of a polymerization reaction. If the total concentration of radicals is constant, this means that the rate of crossover between the different types of terminal units is also equal, or that R... 

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. 

With systems of this type, it is often found that some of the participating species are present in low concentrations only. If these species are very reactive (i.e. transient intermediates), the stationary state approximation may be applied. In making this approximation, it is assumed that the concentrations of the transient species remain constant. Thus, the kinetic scheme is simplified significantly and it may be possible to avoid the problem of stiffness . The validity of such assumptions must be examined carefully. As a simple example, consider the reaction scheme (17) for the case where Cb = 0. In applying the stationary state approximation to B, we have... 

One common use of the stationary state approximation is with chain reactions. The simplest cases have three types of constituent chemical step, viz. chain initiation, chain propagation and termination. The... 

Here, B and E are the chain carriers, D and F are the most abundant stable products. Other termination and propagation reactions, in which C may participate, are also possible. If only steps (44)—(47) occur and molecu-larity corresponds to reaction order, application of the stationary state approximation to B and E leads to the prediction that... 

From the stationary state approximation, d[A ]/dt <= 0 and d[A ]/dt = 0, the expressions for the steady state concentration of [A ] and [A ] can be set up. We can further define, 6m, and as quantum efficiencies of emission from dilute solution, from concentrated solution with quenching, and of excimer emission, respectively ... 

Occasionally the amount of solvento complex present is too great for the stationary state approximation to be valid and the analysis of the rate constants will be more complicated.441 A rapidly established equilibrium between the substrate and the solvento complex is not uncommon in labile systems with low concentrations of the nucleophile and leads to a rate law of the type... 

The stationary-state approximation Kinetic analysis of Equations 2.37-2.38, first step rate-determining, takes the following form. Because B is consumed as fast as it forms, its concentration is always very close to zero and therefore approximately constant. We assume that... 

This assumption is known as the stationary-state approximation, and is valid for highly reactive intermediates. We then write from the second step Equation 2.40... 

The stationary-state approximation thus allows us to equate A fA] with A 2[B]. The rates of formation and of disappearance of the reactive intermediate B are equal. We can therefore write instead of 2.40 the final rate equation 2.43 ... 

If, in the two-step mechanism in Equations 2.37-2.38, it is not justifiable to assume that B is consumed as fast as formed, [B] will increase and then decrease the rate of disappearance of A will not equal the rate of appearance of C, and the stationary-state approximation is not valid. This situation requires a more general approach.35... 

A steady-state (also named stationary-state) approximation on the concentrations of CH3 radicals and CuCH2+, namely, 9[CH3]/9f = 3[CuCH2 - ]ldt = 0, gives a stationary concentration of radicals ... 

The stationary-state approximation is applicable to all radical species ... 

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. 

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... 

For free radicals which are not involved in termination processes, i.e. those radicals which are the most reactive and, accordingly, the least concentrated, the QSSA can be applied even during the true induction period of the reaction. This is so for chain carrier radicals not involved in termination processes the concentrations of these radicals are not at all constant or slowly varying during the induction period however, the QSSA may be applied to them. For this reason, this special kind of QSSA will be termed pseudo-stationary state approximation (PSSA). As a consequence of the PSSA, the observation of a non-quasi-stationary behaviour for a radical concentration does not necessarily mean that the QSSA cannot be applied. This fact has probably played a role in the criticism of the QSSA. 

The loss reactions for CH D) are very fast, thus allowing us to treat its number density in the stationary-state approximation ... 

The Fe + reaction is complicated by the fact that the solution can become supersaturated with respect to O2 and also by the fact that at the higher pH s, Fe(OII)3 and Fe(OH)2 may precipitate. The production of O2 via long chains is possible only under conditions such that H2O2 is large with respect to both Fe++ and Fe ". Under these conditions we may neglect step 7 and use a stationary-state approximation for Fe" " [that is, d(Fc++)/ri/ =... 

Acetone photolysis at 253.7 and 313 nm studied to test 515 the validity of the stationary state approximation 193 and 248 nm photodissociation of acrylic acid and 516 methacrylic acid shown to result in decarboxylation and formation of carbenes. Energy disposal in the COj fragments probed using i.r. fluorescence... 

The stationary-state approximation applied to this mechanism gives a rate equation agreeing with equation (19), viz. 

For long chains (the observed chain length is 100 at 70 °C and initial [SjOl ] = 10 M) the stationary state approximation gives... 

The stationary-state approximation leads to a rate equation in agreement with the observed expression. 

Provided that inequality (68) is assumed, the stationary-state approximation applied to the concentrations of the radicals leads to the rate equation... 

The stationary-state approximation leads to the correct rate equation (87), below, provided that inequality (88) is assumed. The inequality is Justified by the known equilibrium data for the dioxalatocuprate(II) ion. 

The intermediate X, present in low concentration, is probably an aryl hydro-xylamine-O-sulphonate. The further oxidation products consistmainlyof polymeric humic acid. The stationary-state approximation applied to [X] shows that the ratio of the rate of formation of o-amino aryl sulphate to the rate of formation of other products is k3/k2[S20g ]", in qualitative agreement with the dependence of the yield of o-amino aryl sulphate on peroxodisulphate concentration, and the deviation from first-order kinetics with respect to peroxodisulphate towards the end of a run. 

To solve these equations and eliminate the reactive intermediates, we note the stationary-state approximation and the definition of the reactivity ratios r and r2 above. Then... 

This is the copolymer equation, which may also be derived by statistical means without invoking the stationary-state approximation (Odian, 1991). 

The lower limit of the rate constant k2,iso for reaction (1) was estimated under a few quantitative approximations, and the upper limit required die pseudo-stationary-state approximation for S04 radicals and SOs ions (Egn 4). 

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