Cubic autocatalysis

Figure C3.6.14 Space-time (y,t) plot of the minima (black) in the cubic autocatalysis front ( )(y,t) in equation C3.6.16 showing the nature of the spatio-temporal chaos.

Autocatalysis will play a central role in driving the oscillations and other non-linear phenomena of interest in this book. Usually, an autocatalytic process will be combined in a larger mechanism with other steps. Before considering such systems, however, we investigate the properties and behaviour of autocatalysis on its own—in particular how the concentrations and rate vary with time and with respect to each other. We start with quadratic autocatalysis, and then look at the cubic form. 

Cubic autocatalysis and clock reactions in closed vessels... 

The induction period, followed by a sharp increase in rate is, however, the most characteristic feature of autocatalysis in closed vessels. One manifestation of this behaviour is the clock reaction . An experimental system which is a typical chemical clock and which also exhibits cubic autocataiysis is the iodate-arsenite reaction. In the presence of excess iodate, the system which is initially colourless eventually undergoes a sudden colour change to brown (or blue in the presence of starch). The potential of an iodide-sensitive electrode shows a barely perceptible change during most of the induction period, but then rises rapidly, reaching a peak at the point of colour change. 

Returning to the simple cubic autocatalysis model above, we shall be more interested later in the relationship between the rate and the extent of conversion. This is shown for various values of b0 in Fig. 1.7. If b0 = 0 (Fig. 1.7(a)), the rate curve is both a minimum and zero at no conversion (i.e. there is a double root at the origin) and has a further zero at complete conversion (a = 0, a0 — a = a0). The rate has a maximum value of (4/27)/ccao occurring two-thirds of the way across the diagram (a = a0). There is also a point of inflection at 50 per cent conversion, a0 — a — a0. 

Again if reaction (1.21) is held in a pre-equilibrium state, the overall rate of conversion of A to B can show a cubic form. This realization of cubic autocatalysis seems to be of importance for the iodate-arsenite and iod-ate-hydrogen sulphite reactions. There the corresponding elementary steps include... 

The species HOI is then rapidly converted to iodide, by reaction first with I" to produce I2 which is then reduced by the arsenite of hydrogen sulphite. Thus, identifying A with the reactant iodate and B with iodide, the system shows a cubic autocatalysis with rate proportional to [IOJ][I-]2 at constant pH. 

Fig. 1.9. Consecutive first-order reactions and cubic autocatalysis, showing pseudosteady-state predictions for the intermediate concentrations. Initial concentrations and rate constants are given in Table 1.1.

Fig. 1,10. Actual time-dependent concentration of intermediate A for consecutive first-order reactions with cubic autocatalysis showing pseudo-steady-state behaviour, pre-oscillatory evolution, an oscillatory period, and then the return to pseudo-steady-state behaviour.

Fig. 1.13. Cubic autocatalysis with relatively high inflow concentration of the catalyst, b0 =

Fig. 1.15. Flow diagram for cubic autocatalysis showing different stabilities of multiple stationary-state intersections.

The application of cubic autocatalysis to model chemical and biochemical schemes in thermodynamically closed systems is dealt with in the following references. 

The previous chapter has provided some indication of the behaviour which can be exhibited by the simple cubic autocatalysis model. In order to make a full analysis, it is convenient both for algebraic manipulation and as an aid to clarity to recast the rate equations in dimensionless terms. This is meant to be a painless procedure (and beloved of chemical engineers even though traditionally mistrusted by chemists). We aim wherever possible to make use of symbols which can be quickly identified with their most important constituents thus for the dimensionless concentration of A we have a, with / for the dimensionless concentration of B. Once this transformation has been achieved, we can embark on a quite detailed and comprehensive analysis of the behaviour of this prototype chemical oscillator. 

The reaction rate equations for the cubic autocatalysis model of the previous chapter are... 

The evolution in time of the concentration of the species A and of the temperature rise AT, for the example data in Table 4.1, is shown in Fig. 4.1. The behaviour is in many ways similar to that of the isothermal cubic autocatalysis model of the previous chapters. The concentration of the precursor P decreases exponentially throughout the reaction. The temperature excess jumps rapidly to approximately 80 K, from which value it begins to decay approximately exponentially. At the same time, the concentration of the intermediate A rises relatively slowly to values of the order of 10"i mol dm-3. After approximately 15 s, the concentration of A and the... 

Figure 6.6(b) is better approximated by a cubic form, rate ocy2(l — y). Cubic autocatalysis has already provided us with behaviour of interest in chapter 2. In the remainder of this chapter we consider the stationary-state responses of schemes with this feedback mechanism in flow reactors. We will consider three models, with increasingly varied possible behaviour first an autocatalytic step on its own next we allow the autocatalytic species to undergo a subsequent reaction finally we add an uncatalysed reaction in competition with the autocatalysis. The local stability of such systems is... 

Fig. 6.6. Dependences of reaction rate R on extent of reaction y typical of self-accelerating (autocatalytic) systems (a) prototype quadratic autocatalysis (b) prototype cubic autocatalysis.

Fig. 6.7. The onset of stationary-state multiplicity represented on a flow diagram for cubic autocatalysis with P0 = (a) the reaction rate curve R (b) three typical flow lines L, with... 

Fig. 6.15. Flow diagram representation of the origin of isola solutions for cubic autocatalysis with decay. The gradient of the flow line L at first decreases with increasing residence time non-zero intersections appear as R and L become tangential and these move apart as L becomes less steep. After L has attained its minimum slope the non-zero intersections move closer together again, and merge (and disappear) as R and L attain tangency for the second time. The intersection at zero extent of reaction exists for all residence times.

Fig. 6.16. (a) Flow diagram representation of the different stationary-state loci for cubic autocatalysis with inflow and decay of B (b) unique response, minimum slope L, (c) birth of isola as isolated point, minimum slope L , (d) isola, minimum slope L2 (e) transition from isola to mushroom, minimum slope L > (f) mushroom, minimum slope L3. 

Fig. 6.22. The 14 different qualitative forms for the stationary-state locus for cubic autocatalysis with reversible reactions and inflow of all species, with c0 < a0 the broken line represents the equilibrium composition which is approached at long residence times. (Reprinted with permission from Balakotaiah, V. (1987). Proc. R. Soc., A41J, 193.)...

This reduction to a single variable is similar to that possible in the case of cubic autocatalysis when the species B is infinitely stable (i.e. k2 - 0) in chapter 6. In fact there are many qualitative parallels between the adiabatic non-isothermal reaction and autocatalysis without decay, as we shall see later. 

In between these tangencies, the curves R and L have three intersections, so the system has multiple stationary states (Fig. 7.3(b)). We see the characteristic S-shaped curve, with a hysteresis loop, similar to that observed with cubic autocatalysis in the absence of catalyst decay ( 4.2). 

Comparisons between properties of the reaction rate curve R for cubic autocatalysis without decay and the first-order non-isothermal system in an... 

If the temperature difference 0C between the heat bath and the inflow is greater than zero, we can have the opposite effect to Newtonian cooling, with a net flow of heat into the reactor through the walls. With his possibility, two more stationary-state patterns can be observed, giving a total of seven different forms—the same seven seen before in cubic autocatalysis with the additional uncatalysed step (the two new patterns then required negative values for the rate constant) or with reverse reactions included and c0 > ja0 ( 6.6). 

As the simplest example, let us consider the isothermal cubic autocatalysis of 6.2, where there is catalyst inflow but no decay. In terms of the extent of conversion x = 1 — a55, the appropriate stationary-state condition is... 

As an example we can again take our cubic autocatalysis with inflow of B, so Fx and Fz are as given above, leading to... 

Fig. 7.6. The unfolding of a hysteresis loop as condition (7.46) is satisfied. For the FONI model this occurs as 0ad decreases through 4 for cubic autocatalysis the unfolding corresponds to fl0...

Singularity theory for cubic autocatalysis with uncatalysed reaction... 

The previous two chapters have considered the stationary-state behaviour of reactions in continuous-flow well-stirred reactions. It was seen in chapters 2-5 that stationary states are not always stable. We now address the question of the local stability in a CSTR. For this we return to the isothermal model with cubic autocatalysis. Again we can take the model in two stages (i) systems with no catalyst decay, k2 = 0 and (ii) systems in which the catalyst is not indefinitely stable, so the concentrations of A and B are decoupled. In the former case, it was found from a qualitative analysis of the flow diagram in 6.2.5 that unique states are stable and that when there are multiple solutions they alternate between stable and unstable. In this chapter we become more quantitative and reveal conditions where the simplest exponential decay of perturbations is replaced by more complex time dependences. 

Equation (8.13) gives the appropriate form for l/treiax( = — X) for the cubic autocatalysis model with no inflow of autocatalyst. The condition for the turning point in the stationary-state locus (there is only one) is = 4. 

Fig. 8.3. The approach to, or departure from, stationary-state solutions following small perturbations for simple cubic autocatalysis again showing the instability of the middle branch. The turning points (ignition and extinction) have one-sided stability as perturbations in one direction decay back to the saddle-node point, but those of the opposite sign depart for the other...

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