Autocatalytic steps

For chaotic or oscillating behavior the mechanism must contain an autocatalytic step ... 

To visualize the situation one may take Y as a population of foxes, E as the foxes that have died, X as rabbits and A as carrots. Rabbits live very well on carrots and the population would grow exponentially if there are sufScient carrots. The fox population starts to grow when the rabbit population is high, untU there are more foxes than can be sustained by the rabbits. Famine sets in and the fox population diminishes, after which the rabbit population starts to grow again. In other words, X and Y are oscillating out of phase. It is essential that the mechanism contains autocatalytic steps, and that there is a continuous supply of reactant A, which keeps the system far away from equilibrium. 

How relevant are these phenomena First, many oscillating reactions exist and play an important role in living matter. Biochemical oscillations and also the inorganic oscillatory Belousov-Zhabotinsky system are very complex reaction networks. Oscillating surface reactions though are much simpler and so offer convenient model systems to investigate the realm of non-equilibrium reactions on a fundamental level. Secondly, as mentioned above, the conditions under which nonlinear effects such as those caused by autocatalytic steps lead to uncontrollable situations, which should be avoided in practice. Hence, some knowledge about the subject is desired. Finally, the application of forced oscillations in some reactions may lead to better performance in favorable situations for example, when a catalytic system alternates between conditions where the catalyst deactivates due to carbon deposition and conditions where this deposit is reacted away. 

In Fig. 6, separate regions of bi-stability, oscillations and single stable steady-states can be noticed. This cross-shaped phase diagram is common for many non-linear chemical systems containing autocatalytic steps, and this was used as an argument to suggest that the Cu(II) ion catalyzed autoxidation of the ascorbic acid is also autocatalytic. The... 

Once the door was opened to these new perspectives, the works multiplied rapidly. In 1968 an important paper by Prigogine and Rene Lefever was published On symmetry-breaking instabilities in dissipative systems (TNC.19). Clearly, not any nolinear mechanism can produce the phenomena described above. In the case of chemical reactions, it can be shown that an autocatalytic step must be present in the reaction scheme in order to produce the necessary instability. Prigogine and Lefever invented a very simple model of reactions which contains all the necessary ingerdients for a detailed study of the bifurcations. This model, later called the Brusselator, provided the basis of many subsequent studies. 

In this simplified version of the Brusselator model, the trimolecular autocatalytic step, which is a necessary condition for the existence of instabilities, is, of course, retained. However, the linear source-sink reaction steps A—>X—>E are suppressed. A continuous flow of X inside the system may still be ensured through the values maintained at the boundaries. The price of this simplification is that (36) can never lead to a homogeneous time-periodic solution. The homogeneous steady states are... 

The stationary-state response curves, or bifurcation diagrams shown in Figs 1.13(b) and 1.12(f), represent two of the simplest possible patterns monotonic variation and a single hysteresis loop respectively. These are the only qualitatively different responses possible for the cubic autocatalytic step on its own. They are also found for a first-order exothermic reaction in an adiabatic flow reactor (see chapter 6). With only slightly more complex chemical mechanisms a whole array of extra exotic patterns can be found, such as those displayed in Fig. 1.14. The origins of these shapes will be determined in chapter 4. 

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... 

The application of the flow diagram to a single cubic autocatalytic step... 

Fig. 6.11. The influence on the reaction rate curve R of increasing the degree of reversibility of the cubic autocatalytic step for a system with /i0 = 0 (a) Kc = 9, (b) Ke - 4, (c) = 1,...

Fig. 8.1. Indication of local stability or instability for the simple cubic autocatalytic step without decay solid curves indicate branches of stable stationary-state solutions, broken curves correspond to unstable states, (a) Stationary-state locus with no autocatalyst inflow, fl0 = 0, with one stable solution, 1 - = 0, corresponding to zero reaction (b) stationary-state locus...

We should first recall the stationary-state behaviour for this case. If the reaction rate constant for the catalyst decay step is large compared with that for the autocatalytic step, so that k2 > iV, the system can only ever have one stationary state. This state corresponds to no net conversion of A to B, so ass = 1. For slower decay rates, k2 < Vs non-zero stationary states exist over a range of residence times t 9 < ires < t+s in the form of an isola. The extents of conversion along the branches of the isola are given by... 

The autocatalytic reaction scheme A + 2B —> 3B, B —> C was introduced in 1983s and has proved itself to be fecund of useful applications in the study of reactor stability and chemical oscillations.6 We shall depart from their notation for we wish to be able to generalize to several species, Au and it is not desirable to use the concentration of A as a reference concentration when it is going to be varied. Similarly, the several species will have different rate constants for the several autocatalytic steps and therefore the first-order rate constant of B — C is most apt for the time scale. 

The Wicke and Eigenberger models are models for an ideal adsorption layer. They have been examined at the Institute of Catalysis, Siberian Branch of the U.S.S.R. Academy of Sciences [93-104,108,109] independently of Wicke and Eigenberger (the first publications refer to 1974). It was shown [93-96] that, for the detailed mechanisms of catalytic reactions either with the steps that are linear with respect to intermediates or with non-linear steps (but containing no interactions between various intermediates), the steady state of the reaction is unique and stable (autocatalytic steps are assumed to be absent). Thus the necessary condition for the multiplicity of steady states is the presence of steps for the interaction between various intermediates in the detailed reaction mechanism [93-100]. Special attention in these studies was paid to the adsorption mechanism of the general type permitting the multiplicity of steady states [102-104]... 

As already implied by the above scheme, it is essential that chiral amplification and symmetry breaking comprising the generic autocatalytic steps A + R -> 2R and A+S 2S, require some sort of cross-inhibition between the two enantiomers, for instance expressed by R + S —> P. Here A denotes an achiral substrate and P an unspecified product. In the absence of crossinhibition, the ee will at best stay at its initial value but amplification remains impossible [66]. 

The minimal model comprises an uncatalyzed and unspecific direct formation of R and S (ko), a simple and unspecified description of the autocatalytic steps assuming monomers as catalytic species (k ), and the monomer-dimer equilibria (k2, M and (M M in which different rates of dimer formation for homochiral and heterochiral species is allowed. This model translates into the following set of differential equations ... 

Further insight into the structural aspects of the basic reaction process was given by NMR studies, which indicate that additional equilibria between the alkoxide dimers and diisopropylzinc molecules should be taken into account yielding RR-Z, SS-Z, and RS-Z association complexes [33,83]. Further studies in such a direction combined with kinetic experiments will be needed to decide about the closer reaction network in the Soai reaction (especially about the autocatalytic steps) in order to shine light on the possible catalytic action of monomer or oligomer species. 

X]Ss = k1[A][B]/(k 1 + k2)-] This steady state is stable, although transitory, since perturbations of the steady-state concentration will decay back to the same concentration. Systems with autocatalytic steps do not always behave this way. Sufficiently large perturbations of their steady-states may lead to transitions to different steady-state concentrations. In other words, systems with autocatalytic steps may have multiple steady-states. 

Modeling of the Stirring Effect in the Autocatalytic Step of the Belousov—Zhabotinsky Reaction. 

The net reaction is A I B -> E I F. This reaction scheme has been developed by the Brussels School of Thermodynamics, and consists of a trimolecular collision and an autocatalytic step. This reaction may take place in a well-stirred medium leading to oscillations, or the diffusions of the components A and B may be considered. In the latter case, the system may produce Turing structures. 

Here, the initial and final concentrations of S, B, D, and P are maintained constant so that only the concentrations of X and Y are the independent variables. The autocatalytic step 2X + Y = 3X involves a trimolecular reaction. The overall reaction in Eq. (a) is... 

The phenomenon of self organization occurs at nonstabHities of the sta tionary state and leads to the formation of temporal and spatio temporal dissipative structures. Remember that oscillating instabilities of stationary states of dynamic systems can be observed for the intermediate nonlinear stepwise reactions only, when no fewer than two intermediates are involved (see Section 3.5) and at least one of the elementary steps is kinet icaUy irreversible. The minimal sufficient requirements for the scheme of a process with temporal instabilities are not yet strictly formulated. However, in aU known examples of such reactions, the rate of the kineti caUy irreversible elementary reaction at one of the intermediate steps is at least in a quadratic dependence on the intermediate concentrations. Among these reactions are autocatalytic steps. 

At approximately the same time, Lotka proposed his famous models of oscillating chemical reactions based on irreversible autocatalytic processes. The first model included one autocatalytic step and gave damped oscillations. The second model became a paradigm in oscillating chemistry. It consists of two consecutive autocatalytic steps, resulting in undamped oscillations. The Lotka models attracted great attention from theoretical biologists, because... 

The same authors also investigated zinc electrodeposition from acidic and alkaline electrolytes without and with inhibitors [6.82-6.86]. It was suggested that the deposition mechanism involves an autocatalytic step... 

The general consensus is to assume that the first active macromolecular species would be capable of self-reproduction, namely capable to induce the synthesis of copies of themselves via an autocatalytic step. Self-reproduction is in fact always to be connected with autocatalysis. Shnerior Lifson provides a nice numerical example to understand the importance of self-reproduction and autocatalysis suppose having a process which produces one molecule of product each microsecond. In a heterocataly tic reactions scheme, to produce one mole of product, 6 x 1 microseconds would be necessary, i.e. more than the age of the universe. Conversely, in an autocatalytic reaction scheme, where the number of reacting species would double every microsecond, the total time to attain a mole of product would be 79 microseconds. 

DePoy and Mason (1974) proposed a mechanism including an autocatalytic step which produces oscillations, and gave a mathematical model with product nonlinearity as in the case of the Lotka model, Lotka (1910-1, 2). 

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