Pseudo-stationary states

We are left with the behaviour of the intermediates A and B. A common approach to kinetic models involving relatively reactive species is to apply the pseudo-stationary-state (PSS) hypothesis. 

The stationary-state concentrations of a and b are given by setting their rates of change simultaneously to zero. Thus 

Adding these two equations eliminates all terms containing ass and leads to the simple result 

Thus the intermediate B has a concentration that is directly proportional to that of the reactant P. The dependence of ass on p is slightly more complicated but can still be evaluated explicitly. As an example, for the values of the reaction rate constants and the initial value of p from Table 2.1, we find bss = 10 4moldm 3 and ass = 4 x 10 6moldm-3. 

In a real experiment the concentration of the reactant falls in time. We should, therefore, establish the way in which bss and ass vary with p. This, in fact, will turn out to be the basis of a particularly convenient approach in which we regard the pseudo-stationary-state concentrations as relatively simple functions of p, rather than the more complex functions of time which we derive later. The stationary-state loci are shown in Fig. 2.1. 

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Fig. 1.8. Consecutive first-order reactions with p0 = 0.1 mol dm-3, fcu = 5 x 10-3 s and k2 = 10"2s (a) the exponential decay of precursor reactant concentration, p (b) growth and decay of intermediate concentrations a(t) and b(t). Also shown in (b), as broken curves, are the pseudo-stationary-state loci, a (t) and b (t), given by eqns (1.31) and...

Fia. 2.2. Predicted pseudo-stationary-state evolution of the intermediate species concentrations a(t) and b(t), as given by eqns (2.15) and (2.16). Specific numerical values correspond to the rate data in Table 2.1. The time at which the two concentrations become equal and that at which a(t) attains its maximum are indicated. 

Fig. 2.4. Computed concentration.histories for autocatalytic model with rate constants given exactly as in Table 2.1 (a) exponential decay of precursor (b) intermediate concentrations a(t) and 6(r), showing initial pseudo-stationary-state behaviour but subsequent development of an oscillatory period of finite duration, 1752 s < t < 3940 s.

There is a period of time or, if we prefer, a range of reactant concentration over which the system spontaneously moves away from the pseudo-stationary state. The idea that stationary states may be unstable is not widely appreciated in chemical kinetics but it is fundamental to the analysis of oscillatory systems. 

At this stage, however, we may proceed to the important features of the model s behaviour qualitatively from Fig. 2.4. For some range of time or of reactant concentration the pseudo-stationary state described by eqns (2.15) and (2.16) become locally unstable. Let this range be denoted... 

As well as showing how a and b vary in time, the numerical traces can also be plotted in a different way. If we plot the two concentrations against each other (Fig. 2.5(c)), we find that they draw out a closed curve or limit cycle around which the system circulates. This limit cycle surrounds the unstable pseudo-stationary state appropriate to p. The amplitude of the oscillations is a measure of the size of this limit cycle. 

In addition to the general aims set out at the beginning of this chapter we have discovered a wealth of specific detail about the behaviour of the simple kinetic model introduced here. Most results have been obtained analytically, despite the non-linear equations involved, with numerical computation reserved for confirmation, rather than extension, of our predictions. Much of this information has been obtained using the idea of a pseudo-stationary state, and regarding this as not just a function of time but also as a function of the reactant concentration. Stationary states can be stable or unstable. 

We now turn again to the evaluation of the pseudo-stationary-state responses. 

These have only one true stationary-state solution, ass = pss = 0 (as t - go), corresponding to complete conversion of the initial reactant to the final product C. This stationary or chemical equilibrium state is a stable node as required by thermodynamics, but of course that tells us nothing about how the system evolves in time. If e is sufficiently small, we may hope that the concentrations of the intermediates will follow pseudo-stationary-state histories which we can identify with the results of the previous sections. In particular we may obtain a guide to the kinetics simply by replacing p by p0e et wherever it occurs in the stationary-state and Hopf formulae. Thus at any time t the dimensionless concentrations a and p would be related to the initial precursor concentration by... 

The traces in Fig. 3.9 were computed for a system with an uncatalysed reaction rate constant such that jcu > g, and hence there are no oscillatory responses in the corresponding pool chemical equations. For ku < we may also ask about the (time-dependent) local stability of the pseudo-stationary state. The concentration histories may become unstable to small perturbations for a limited time period. For sufficiently small e this should occur whilst the group p0e cr lies within the range... 

Fig. 3.9. Comparison of exact concentration histories for species A, oc(t), with pseudo-stationary-state form (broken curves) showing the influence of the precursor decay rate constant c when the uncatalysed reaction rate is relatively large, k = i (a) s= 10 3 (b) = 10 2 (c) e= 10 1 ...

The second significant difference between the predictions and the actual results is that oscillations survive beyond the time. This arises because the pseudo-stationary state has focal character just after the second Hopf bifurcation (i.e. the slowly varying eigenvalues i1>2 are complex conjugates with now negative real parts) so there is a damped oscillatory return to the locus. In Fig. 3.10(a) this can be seen after t 3966, whilst t = 3891. 

We will regard a and 6 as functions of the reactant concentration as expressed in fi and assume that they change on a fast timescale compared with reactant consumption (small e), i.e. we apply the pseudo-stationary-state hypothesis. The pseudo-stationary-state condition da/dt = dO/dz = 0 yields the following simultaneous equations ... 

The dependence of the, pseudo-stationary-state concentration of the intermediate A on p/k shows a maximum value of e-1 when p/k = 1. The time dependence of ass is given by... 

First, can we expect any oscillatory behaviour Instability is possible only if k < e 2. This requirement is satisfied here. From the data in Table 4.4, the Hopf bifurcation points for this system occur for n = 0.207 and n = 0 058. For our example, the initial value /r0 = 0.5 exceeds the upper bifurcation point, so the system at first has a stable pseudo-stationary state to approach, with dss x 10 and ass x 4.54 x 10 4. From Fig. 4.3 we may also estimate that the approach to this state will be monotonic since the initial conditions lie outside the region of damped oscillations. 

Following this the pseudo-stationary state becomes unstable and the concentration and temperature histories are expected to move away into oscillatory... 

As seen in the previous chapter, the growth of observable excursions is not immediate and may take a considerable time. If oscillations do develop, we expect them to last until the pseudo-stationary state regains stability at time t ... 

We may also briefly consider the behaviour of the simple autocatalytic model of chapters 2 and 3 under reaction-diffusion conditions. In a thermodynamically closed system this model has no multiplicity of (pseudo-) stationary states. We now consider a reaction zone surrounded by a reservoir of pure precursor P. Inside the zone, the following reactions occur ... 

From the results of chapters 4 and 5, we can predict the behaviour of the system if it is well stirred. For some experimental conditions, represented by particular values for the dimensionless reactant concentration /t and the rate constant k, the system will have a uniform, stable stationary state (really only a pseudo-stationary state as /t is decreasing slowly because of the inevitable consumption of the reactant discussed previously). For other conditions, the stationary state loses its stability and stable uniform oscillations can be... 

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. 

After coke depositor on the metal, some bare metal islands remain uncovered in a pseudo stationary state. From this moment and up to the end of the operation cycle, coke is only deposited on the acid function. 

Almost all works on optimization of a multi-product microbial cell factory focussed on a single objective (e.g., Schmid et al., 2004 Visser et al, 2004 Vital-Lopez et al, 2006). A common feature in these works is the pseudo-stationary assumption. Enzymatic reaction kinetics in a microbial cell factory are reversible and interdependent. In reality, the fluxes due to enzymatic reactions are never stationary. Given the limitations of a model, it is necessary to assume a pseudo-stationary state where some variables fluctuate about an averaged steady state within certain bounds. 

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). 

On the basis of the considerations made above and the approximation of pseudo-stationary state conditions, the following kinetic equations can be written to describe the catalytic behaviour under the two conditions considered ... 

The reverse reaction (—3) is the main branching step in the Hj + O2 thermal chain reaction.) Assumption of a pseudo-stationary-state value of O, i.e. established rapidly by reactions (2) and (3), gave the result, —d[OH]/dt = —3fc2[OH] . At long reaction times, t, such that t 1/At2[OH]o, the result [O] = was obtained. Values for the second-order rate constant k were obtained near 300 K using e.p.r. and absorption spectrophotometry. Kaufman and Del Greco s revised value for k was 8 5 x 10 cm moH s which may be compared... 

Assuming a pseudo-stationary state, we can determine the concentration of the initiator ion as a function of the measurable concentrations h and M. Note that only a fraction of the initiator participates in the reaction, and thus a correction factor y is used. Therefore,... 

Then, in the pseudo-stationary state, we have tr = 0, and thus... 

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