Equation 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.

An instance of autocatalysis is the hydrolysis of methyl acetate, which is catalyzed by product acetic acid, A C. The rate equation may be... 

GouldConsistent biniolecular rate constants at various concentrations and times were found except in cases of autocatalysis. For the rate equations used and details on the limits of error, the reader is referred to the original publications. 

The increase in energy content of an atom, ion, or molecular entity or the process that makes an atom, ion, or molecular entity more active or reactive. In enzymology, activation often refers to processes that result in increased enzyme activity. For example, increasing temperature often can have a positive effect on enzyme activity (See Arrhenius Equation). Other examples of enzyme activation include (1) proteolysis of zymogens (2) alterations in ionic strength (3) alterations due to pH changes (4) activation in cooperative systems (5) lipid or membrane interface activation (6) metal ion effects (7) autocatalysis and (8) covalent modification. 

ENCOUNTER-CONTROLLED RATE SECOND-ORDER REACTiON CHEMICAL KINETICS ORDER OF REACTION NOYES EQUATION MOLECULARITY AUTOCATALYSIS FIRST-ORDER REACTION... 

This equation includes the particular cases of an nth order reaction model with g(a) = 1, and the autocatalytic model with g(a) = 1 + la, where / is the autocatalysis intensity. In general, the form of g(a) must be determined from experimental data. 

Asymmetric autocatalysis is defined as an enantioselective synthesis in which the chiral product acts as an asymmetric catalyst for its own production (equation 39). Asymmetric autocatalysis is an efficient method for the catalytic enantioselective automultiplication of a chiral molecule without the need for any other chiral auxiliary158. 

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... 

Because of the autocatalysis, this equation involves the concentrations of both A and B. A similar mass-balance equation could be,written for the autocatalyst in the form... 

Here F represents the functional form of the left-hand sides of the various stationary-state equations, x is the stationary-state solution such as the extent of reaction, the temperature excess, etc., and rres is the parameter we have singled out as the one which can be varied during a given experiment (the distinguished or bifurcation parameter). All the remaining parameters are represented by p, q, r, s,. For example, in eqn (7.21) the role of x could be played by the extent of reaction 1 — ass, with p = 0ad and q = tN for isothermal autocatalysis, x can again be the extent of reaction, with p = P0, q = k2, and r = jcu. 

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. 

Section we show that presence of two such intermediate stages is more than enough for the self-organization manifestation. Lotka [22] was the first to demonstrate theoretically that the concentration oscillations could be in principle described in terms of a simplest kinetic scheme based on the law of mass action [4], Its scheme given by (2.1.21) is similar to that of the Lotka-Volterra model, equation (2.1.27). The only difference is the mechanism of creation of particles A unlike the reproduction by division, E + A - 2A, due to the autocatalysis, a simpler reproduction law E —> A with a constant birth rate of A s holds here. Note that analogous mechanism was studied by us above for the A + B — B and A + B — 0 reactions (Chapter 7). 

Reactions (377) and (378) are usually not ranked among catalytic processes, although they are, as a matter of fact, autocatalytic since they occur on the surface of carbon, one of the participants of the reactions. True, it is more usual to speak of autocatalysis for reactions catalyzed by products rather than by reactants, but it is not essential in our case. Equations (377) and (378) may be read from right to left as the reactions are reversible. 

The reactions usually obey the equation for a unimolecular reaction -dM/dt= KM, where M is the mass of solid in grams and t is the time in secs. In some cases, however, the rate is proportional to the surface of the solid instead of its mass, and, in addition, the equation sometimes needs modifications because of such phenomena as autocatalysis... 

A similar analysis of reactions with autocatalysis by the final products [equations (15) and (16), see Figs. 2 and 3] is significantly simpler and we leave it for the reader. A simpler, but also weaker algebraic (rather than exponential, in accordance with Arrhenius law) dependence of the reaction rate on the concentration in an isothermic autocatalytic reaction causes much less complete combustion at the limit of extinction. 

The oxidation of L-norleucine, L-leucine, L-isoleucine, and r.-r-leucine with permanganate in strong acid medium showed an autocatalysis by Mn(II), except in the case of L-leucine. For the autocatalytic activity to initiate, a certain concentration of Mn(II) is required. Moreover, the autocatalytic phenomenon vanishes in concentrations of sulfuric acid that are greater than 4.3 mol dm-3. The oxidation showed a good correlation in a biparametric equation, with p = -4.57 and Sc = 2.23 at 318 K.43... 

The results of these experiments are recorded below, and show that reactant catalyst interaction must be important under the conditions of asymmetric autocatalysis. It raises a possibility that the true catalytic entity could be the association complex on the right-hand side of the equation. 

Commenting their discovery of the first case of chiral autocatalysis in organic chemistry, Soai et al. stated that it seems conceivable that the [Soai] reaction. .. may be an example of the scheme proposed by Frank [9]. Indeed, the so-called Frank model that has been developed in 1953 predicts the spontaneous amplification of an initial enantiomeric excess by means of a simple set of coupled differential equations [1] ... 

As for theoretical research on the chiral symmetry breaking, Frank was the first to show that a linear autocatalysis with an antagonistic nonlinear chemical reaction can lead to homochirality [16]. His formulation with rate equations corresponds to the mean-field analysis of the phase transition in a nonequilibrium situation [17], and other variants have been proposed [6, 18-23]. All these analyses have been carried out only for open systems where... 

If the erroneous linear catalysis (Eq. 3) is at work in addition to the linear autocatalysis (Eq. 2), rate equations are modified ... 

Another model for the chiral asymmetry amplification is a variant of the Frank model in a closed system [34]. In its simplest form, the model consists of reaction schemes of spontaneous production (Eq. 1), linear autocatalysis (Eq. 2), and heterodimer formation (Eq. 31). Then, the rate equations for monomers are the same as those of the original Frank model [ 16], but they are supplemented with the heterodimer formation ... 

The existence of bistability in the //under conditions under which chemical variable, on which the current depends, exhibits bistability as a function of DL. Thus, in S-NDR systems we have to require that the dynamic equations contain a chemical autocatalysis. As set forth below, m takes the role of the negative feedback variable. The positive feedback might be due to chemical autocatalytic reaction steps as is the case in Zn deposition [157, 158] or CO bulk oxidation on Pt [159], S-shaped current-potential characteristics may also arise in systems with potential-dependent surface phase transitions between a disordered (dilute) and an ordered (condensed) adsorption state due to attractive interactions among the adsorbed molecules. 

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