Steady-state hypothesis

This procedure constitutes an application of the steady-state approximation [also called the quasi-steady-state approximation, the Bodenstein approximation, or the stationary-state hypothesis]. It is a powerful method for the simplification of complicated rate equations, but because it is an approximation, it is not always valid. Sometimes the inapplicability of the steady-state approximation is easily detected for example, Eq. (3-143) predicts simple first-order behavior, and significant deviation from this behavior is evidence that the approximation cannot be applied. In more complex systems the validity of the steady-state approximation may be difficult to assess. Because it is an approximation in wide use, much critical attention has been directed to the steady-state hypothesis. 

The free-radical concentrations will be small—and the quasi-steady state hypothesis will be justified— whenever the initiation reaction is slow compared with the termination reaction, kj /f[CH3CHO]. 

Invoking the steady-state hypothesis for radicals, one can derive the following moment equations. 

The rates of the elementary steps can be formulated in a conventional manner, and the quasi-steady state hypothesis is applied to the adsorbed substrate (A ). The... 

If it is assumed that the rate-determining step is reaction (116), that reactions (118) and (119) are much more rapid than (115) and (116) and that the steady-state hypothesis can be applied, the resulting rate law corresponds exactly to the experimental one, provided that a = = 2 and k — k i. At the... 

Kinetic vs. material chain. Kinetically, a chain reaction exists throughout the "life" of the radical, that is, from the initiation of a radical up to its termination by recombination or by disproportionation. The lifetime of a radical determines the so-called kinetic chain length Lp defined as the number of monomers consumed per initiating radical. Lp, by definition, can be calculated from the ratio between the propagation rate Rp to the initiation rate R, or, using steady-state hypothesis (Equation (1)), from the ratio between propagation rate to the termination rate Rt (Equation (3)). 

The steady state hypothesis is applied to the two free radicals. 

Apply the steady state hypothesis to the free radicals CH3 and... 

If, at large time intervals, flux data do not change within the experimental error, the steady-state hypothesis is the simplest option. A possible check on that hypothesis would be to analyse the data in comparison with the transient behaviour as derived from the general expressions given above for given values of k and Ku. 

The kinetic arguments which these authors adduce to support their theory are not decisive - they are based on the steady state hypothesis which is by no means always valid for cationic polymerisations. There is no independent evidence for the various types of initiation which they give, and they are too general to be useful. Without going into a detailed discussion we would merely point out that under identical conditions (CH2C12 - TiCl4 -... 

The general rate equation resulting from the steady-state hypothesis is... 

For ease of solution, it is assumed that the spherical shape of the pellet is maintained throughout reaction and that the densities of the solid product and solid reactant are equal. Adopting the pseudo-steady state hypothesis implies that the intrinsic chemical reaction rate is very much greater than diffusional processes in the product layer and consequently the reaction is confined to a gradually receding interface between reactant core and product ash. Under these circumstances, the problem can be formulated in terms of pseudo-steady state diffusion through the product layer. The conservation equation for this zone will simply reflect that (in the pseudo-steady state) there will be no net change in diffusive flux so... 

For complex reactions, some of whose elementary steps are of comparable rates, but others are slower, the so-called steady state hypothesis (Refs 4, 6, 10 11) can occasionally lead to a simple theoretical description (mechanism) of the complex reaction. A famous example of this is the thermal decompn of N2Os, where the observed kinetics for this reaction are accurately first-order, even though the reaction is complex (Ref 10)... 

A simple example is the so-called Michaelis-Menten kinetics for enzymatic reactions A + E +C->B + E, which, when the pseudo-steady-state hypothesis is invoked, gives for the concentration of A, for instance, a,... 

However if c3 and c4 are constant then c2 = (k2 + k3)c4/k1c3 must be constant, and no reaction takes place. There is therefore a basic inconsistency in the attempt to make the mechanism SR account strictly for the reaction Si. In spite of this, such kinetic equations as (28) have been found to be extremely useful and quite accurate in kinetic studies. The chemical kineticist therefore claims that over an important part of the course of reaction c3 and c4 are approximately constant, or often that they are both small and slowly varying. This is called a pseudo-steady-state hypothesis and however pseudo it must appear to the mathematician it is sufficiently important to merit formalization. We shall therefore propound a formal definition and illustrate further how it may be used. 

Mobility, permeability, and the pseudo-steady-state hypothesis. Math. Biosci 13, 1-8 (1972). 

Intermediate B in Scheme 11.15 can be observed in principle but, if it undergoes rapid decay, it is usually expressed in concentrations very small compared with those of reactants and products, and it may be too dilute to be observed by instrumental techniques it is then usually called a transient intermediate. Under these conditions, the Bodenstein steady-state hypothesis applies and the rate equation for Scheme 11.15 can be solved to give Equation 11.10 (see Chapter 4) ... 

Example 4. Let us return to the catalytic isomerization reaction described in example 1 and give it a complete consideration without using the suggestion about the low amount of the catalyst and the quasi-steady state hypothesis (in contrast to example 3). Substances for this reaction are isomers Ai and A2 surface compounds A3 = Z (active size) A4 = A,Z A5 = A2Z. There exist two laws of conservation under conservation are the overall number of isomers (both in the gas and on the surface) and the overall number of active sites... 

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. 

In the intriguingly entitled publication "The steady-state approximation, fact or fiction by Farrow and Edelson [41] presents calculated data on the unsteady-state behaviour of a complex chemical reaction including 81 steps. The reaction mixture consists of 50 substances. Numerical calculation shows a great variety of unsteady-state characteristics of a complex reaction. This variety cannot be interpreted in the narrow framework of the quasi-steady-state hypothesis. Nevertheless, the authors discriminate between the ranges of parameters and time intervals within which this hypothesis is confirmed by numerical experiments. 

Considering that the term R /0 can be neglected and steady state hypothesis is applicable to R, Eq.(3) leads to ... 

An important assumption in our discussion is that the rate constant describes a simple reaction, reactants - products. Where this assumption is not justified, the role of the solvent in the kinetics is rarely amenable to detailed analysis, because the rate constant then represents a complex term in rate constants and equilibrium constants which describe different stages in the reaction. For this reason, solvent effects on acid catalysed hydrolysis of esters are difficult to analyse. Attention is also drawn to recent criticisms of the steady state hypothesis in complex reaction schemes (Farrow and Edelson, 1974). The development of numerical integration techniques, especially the method due to Gear (1971), could herald a new way of examining such schemes. 

Rate expression (3.9) has been derived for a relatively simple kinetic model by application of the site balance and the steady state hypothesis. More complex models will result in more complex expressions, which are hard to handle. Fortunately, some simplifications can be applied. 

By application of the steady-state hypothesis, a site balance and the assumption that the surface dissociation is rate determining while the other steps are in quasi-equilibrium, the following rate expression is derived ... 

---