Reversible autocatalytic reaction

A mechanism of this reaction may be represented by the following equation  

A kinetic equation, provided that [A] is maintained constant, is of the form x = klax — k 1x2 (6.28) 

Note that equation (6.28) has a form of the transcritical bifurcation, (5.69). Equation (5.69) may describe the catastrophe occurring on a change of sign of the parameter a. However, an analogous coefficient in equation (6.28), k, is always non-negative. Therefore, in the chemical system described by (6.28) a catastrophe cannot take place. The stationary state Xj = 0 is unstable and the state x2 = kla/k 1 is stable in the entire range of variability of the control parameters klt k t, a. 

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Reversible autocatalytic reaction with a following reaction... 

Note that equation (6.123b) is also obtained by accounting for diffusion in the kinetic equation (6.28) for a reversible autocatalytic reaction and renormalizing the x-variable, x - (ki/k i)x. 

Hydrolysis is a significant threat to phosphate ester stabiHty as moisture tends to cause reversion first to a monoacid of the phosphate ester ia an autocatalytic reaction. In turn, the fluid acidity can lead to corrosion, fluid gelation, and clogged filters. Moisture control and filtration with Fuller s earth, activated alumina, and ion-exchange resias are commonly used to minimise hydrolysis. Toxicity questions have been minimised ia current fluids by avoiding triorthocresyl phosphate which was present ia earlier natural fluids (38). 

Consecutive reactions are those in which the product of one reaction is the reactant in the next reaction. These are also called series reactions. Reversible (opposing) reactions, autocatalytic reactions, and chain reactions can be viewed as special types of consecutive reactions. 

Increased conversion and product purity are not the only benefits of simultaneous separation during the reaction. The chromatographic reactor was also found to be a very suitable tool for studying kinetics and mechanisms of chemical and biochemical reactions. Some recent publications describe the results on investigation of autocatalytic reactions [135], first-order reversible reactions [136], and estimation of enantioselectivity [137,138]. It is beyond the scope of this chapter to discuss the details, but the interested reader is referred to an overview published by Jeng and Langer [139]. 

Gray, B. F., Scott, S. K. and Gray, P., 1984, Multiplicity for isothermal autocatalytic reactions in open systems the influence of reversibility and detailed balance. J. Chem. Soc. Faraday Trans. 1 80, 3409. 

Enantiomers of XL and XD are produced from the reactants S and T, as shown in reactions (1) and (3), respectively. They are also produced by the autocatalytic reactions (2) and (4). The reaction rate constants in reactions (1) and (3) and in reactions (2) and (4) are identical. In reaction (5), the two enantiomers react to produce component P. Obviously, at equilibrium, XL = XD, and the system will be in a symmetric state. If we control the incoming flows of T and S and outgoing flow of P, and assume that the reverse reaction in (5) can be ignored, then we have the following kinetic equations... 

Example 8 The simplest autocatalytic reaction with the reversibility of all steps... 

When the removal of the reaction products from the surface is the slowest step, the conditions described above must be modified (Levich 1962). The concentration far from the surface is small compared with that at the surface, and the direction of the diffusion flux is reversed from that given by Eq. (4.1.14). Moreover, under some conditions a steady solution for slow product removal may not be achievable as, for example, with an autocatalytic reaction where the product catalyzes the reaction, causing the process to accelerate. 

In order to understand the complexity in oscillatory reactions, it would be worthwhile to examine their relationship with different types of chemical reactions [10], which have been summarized in Fig. 9.8 in increasing order of complexity viz., irreversible reactions -> reversible reactions parallel reaction consecutive reaction -> autocatalytic reactions damped oscillations aperiodic oscillations spatio-temporal oscillations chaotic oscillations. Further, Fig. 9.8 shows the concentration... 

Figure 6 shows the effect of reversibility of autocatalytic reactions on the yield of B under periodic operation. Increasing backward reaction rate constant group (R3) lowers the higher yield obtained under periodic operation. 

Chapter 2 covers the basic principles of chemical kinetics and catalysis and gives a brief introduction on classification and types of chemical reactors. Differential and integral methods of analysis of rate equations for different types of reactions—irreversible and reversible reactions, autocatalytic reactions, elementary and non-elementary reactions, and series and parallel reactions are discussed in detail. Development of rate equations for solid catalysed reactions and enzyme catalysed biochemical reactions are presented. Methods for estimation of kinetic parameters from batch reactor data are explained with a number of illustrative examples and solved problems. 

Since saturation of a reproducing population density is what happens in real ecosystems rather than unlimited reproduction, (2.55) is often used as a simulation model, the first time more than 100 years ago by VERHULST (1845, 1847). The stabilization tendency of reverse rates in autocatalytic reactions and their indirect but meaningful ecological implications is quite a general phenomenon as we will see later in Chapter 6. Corresponding terms as generated by the reverse rates of reaction fluxes such as (2.53) are less familiar in population dynamics... 

In the language of network topology, feedback manifests itself as a closed loop of bonds, junctions and elements. The reverse conclusion, however, does not hold the networks of the pore and carrier models and that of active transport in Sections 5.1 to 5.5 actually involve closed loops, but nevertheless show a unique globally and asymptotically stable steady state. The simplest case of a nontrivial feedback loop, i.e., a feedback loop which really leads to multiple steady states or limit cycles, is an autocatalytic reaction... 

Three further sets of circumstances are particularly illuminating one where the autocatalytic reaction is reversible (A+2B 3B) one where the catalyst B itself decays according to a first-order or second-order law and finally, one where cubic and quadratic autocatalysis occur simultaneously. The last case is discussed in section 5. 

Example L2.2.A Determination of the Activation Energy L2.3 Typical Rate Equations for Simple Reactions L2.3.1 Reversible Eirst-Order Reactions L2.3.2 Second-Order Reversible Reactions L2.3.3 Autocatalytic Reactions L2.4 Kinetic Analysis... 

NAGY-UNGVARAI - With growing redox potential the overall autocatalytic reaction step in the FKN model scheme becomes slower, because the reverse step of the rate-determining reaction can be neglected in case of the slow... 

Very recently, autocatalysis has been foimd in organocatalytic Matmich reactions with chiral amines that are themselves made by catalysis with the amino acid proline (26). So far, no asynunetric amplification was observed but the stereoselectivity of the reaction catalysed by the enantiopure product was still very high. Like other autocatalytic reactions known where the product formed is of lower ee than the initially present product-catalyst, corrsecutive cycles of autoeatalysis will inevitably lead to an erosion of errantiomeric prrrity. Let us therefore look at other means of asynunetric amplification that can provide the reverse from low to high enantiomerio purity. 

The study of the behavior of reactions involving a single species has attracted theoretical interest. In fact, the models are quite simple and often exhibit IPT. In contrast to standard reversible transitions, IPTs are also observed in one-dimensional systems. The study of models in ID is very attractive because, in some cases, one can obtain exact analytical results [100-104]. There are many single-component nonequilibrium stochastic lattice reaction processes of interacting particle systems [100,101]. The common feature of these stochastic models is that particles are created autocatalytically and annihilated spontaneously (eventually particle diffusion is also considered). Furthermore, since there is no spontaneous creation of particles, the zero-particle... 

The flow diagram technique can also be used to illustrate clearly the rather limited effects on autocatalytic systems which arise from the reversibility of chemical reactions. For this we replace step (6.8) by its reversible counterpart... 

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,...

The first reaction is autocatalytic and its reverse is neglected. Solve the equation. [GARDINER, p. 274.]... 

In both these cases there is an autocatalytic element, i.e. one which is both the product of the reaction and which tends to increase its rate. This is the substance B in the first case and heat in the second. It is this element of feedback that is the source of the interesting behaviour. The first two terms in both of these equations represent the access to the site of reaction, in this case the stirred tank, in accordance with the criterion of actuality. The feasibility of these simple reaction schemes can be established by showing that they can be embedded in a fully reversible mechanism and the simple system recovered by limiting processes that do not violate the laws of thermodynamics or kinetics (for example, the Wegscheider condition). Yablonskii and his coworkers (Bykov et al. 1978,1979a, b, c Yablonskii Bykov 1979 Gol dshtein et al. 1986) have considered a number of simple models from which it is clear that the autocatalytic feature is essential. In the bimolecular surface reaction the autocatalytic role is played by the vacant sites which are indeed the product of the main reaction which releases those previously held by adsorbate and, at the same time, are a positive influence on the rate of reaction. 

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