Autocatalytic term

In the more general case for liquid phase reactions or other cases where S = 0, the autocatalytic term in the reaction rate expression can be written as... 

Malkin s autocatalytic model is an extension of the first-order reaction to account for the rapid rise in reaction rate with conversion. Equation 1.3 does not obey any mechanistic model because it was derived by an empirical approach of fitting the calorimetric data to the rate equation such that the deviations between the experimental data and the predicted data are minimized. The model, however, both gives a good fit to the experimental data and yields a single pre-exponential factor (also called the front factor [64]), k, activation energy, U, and autocatalytic term, b. The value of the front factor k allows a comparison of the efficiency of various initiators in the initial polymerization of caprolactam [62]. On the other hand, the value of the autocatalytic term, b, describes the intensity of the self-acceleration effect during chain growth [62]. 

The role of the isothermal and pseudo-first-order reaction assumptions on the observed value of activation energy was assessed to allow comparison of our data to previous work by modifying Malkin s autocatalytic equation so that the autocatalytic term b is equal to zero. The values of the activation energy and front factor were calculated using short-time, low-conversion data. By making the autocatalytic term equal to zero, the modified Malkin autocatalytic model becomes a first-order rate reaction. Table 1.2 shows that by assuming a... 

Thus, the problem of exactness of kinetic systems which are not standard, for example (4.22) or (4.26), seems to appear. As will be shown, the systems containing autocatalytic terms or termolecular interactions may be modelled, with a desired accuracy, be means of the standard system (4.27). 

In numerous cases the standard system of kinetic equations (4.27) may be effectively replaced with a considerably simpler system of equations. This results in a number of advantages. Firstly, the reduced system of equations is easier to examined. Secondly, as we will show later, such a reduced system may contain autocatalytic terms and non-linear terms of order higher than... 

Figure 3 The left-hand-side of Equation (7) vs time (min) for the three diacids, indicating good agreement between the model incorporating an autocatalytic term and the data.

Uninhibited chloroprene suitable for polymerisation must be stored at low temperature (<10° C) under nitrogen if quaUty is to be maintained. Otherwise, dimers or oxidation products are formed and polymerisation activity is unpredictable. Insoluble, autocatalytic "popcorn" polymer can also be formed at ambient or higher temperature without adequate inhibition. For longer term storage, inhibition is required. Phenothiasine [92-84-2] / fZ-butylcatechol [2743-78-17, picric acid [88-89-17, and the ammonium salt of /V-nitroso-/V-pheny1hydroxy1 amine [135-20-6] have been recommended. 

Figure 2.4 illustrates the course of the reaction for various values of bolao. Inflection points and S-shaped curves are characteristic of autocatalytic behavior. The reaction rate is initially low because the concentration of the catalyst, B, is low. Indeed, no reaction ever occurs if bo = 0. As B is formed, the rate accelerates and continues to increase so long as the term ab in Equation (2.28) is growing. Eventually, however, this term must decrease as component A is depleted, even though the concentration of B continues to increase. The inflection point is caused by depletion of component A. 

Trace amounts of Cu(II) were reported to catalyze the oxidation of I-to I2 (156) and the phosphinate ion (H2P02) to peroxodiphosphate ion (PDP), which could be present as P20g, HP20 or H2P20f (757). Individual kinetic traces showed some unusual patterns in these reactions, such as the variation between first- and zeroth-order kinetics with respect to the formation of I2 under very similar conditions, or an autocatalytic feature in the concentration profiles of PDP, but these events were not studied in detail. The catalytic effect was interpreted in terms of a Cu(II) / Cu(I) redox cycle and the superoxide ion radical,... 

Also, long-term isothermal storage (or low heating rate) tests are used to investigate autocatalytic effects. For example, a sample is held above 100°C for 10 hours to simulate drying operations [137]. In layer tests, the substance layer is heated by hot air passing around it with a fixed velocity. 

At the point where amphiphiles were recruited to provide the precursors to cell membranes, stable lipid vesicles could have evolved [141] to enclose autocatalytic chiral hypercycles. Credible models for the subsequent evolution of vesicles containing self-replicating chiral molecules have appeared in the literature. [193,194] These vesicles could then emerge from the feldspar spaces [134,192] as micron-sized self-reproducing, energy-metabolizing vesicular systems protobacteria ready to face the hydrothermal world on their own terms. 

The dynamics of controlled systems is an open problem that has recently attracted the attention of scientific community [13]. In fact, oscillatory behavior in chemical systems is an interesting topic (which has been typically studied in autocatalytic reactions, e.g., the Lotka system see [44] and references therein). Dynamics of controlled systems can be explained in terms of interconnections. Indeed, by analogy with control systems, autocatalytic chemical systems can be described as examples of chemical feedback [44]. 

In this autocatalytic mechanism the olefin hydrogenation step is considered faster than the nucleation and growth steps. When the olefin hydrogenation is a rapid process, the equations can be deduced in terms of the two constants, ki and lc2, the values of which can be obtained from kinetic Equation 15.1 ... 

To clarify this point, it may be useful to keep in mind two practical cases, which have been mentioned previously a synthetic vesicle that absorbs a particular molecule from the medium, and by so doing, reproduces itself via an autocatalytic process and - second case - a bacterium that recognizes and absorbs sugar from the environment. At hrst sight, these two processes are similar however, we would commonly ascribe the dehnition of living to the bacterium, and generally not to the other case. Intuitively, the difference can be seen in terms of cognition. How can we clarify this point ... 

Micellar catalysis is a broad field (Fendler and Fendler, 1975 Rathman, 1996 Rispens and Engberts, 2001), and caution is needed when using this term. In fact, whereas the broad term catalysis is justihed when referring to an increase of the velocity of reachon, this does not always mean that the velocity constant is increased (namely that there is a decrease of the specific activation energy). Rather, the velocity effect can be due to a concentration effect operated by the surface of the micelles. This is also the case for the autocatalytic self-reproduction of micelles discussed in the previous chapter, where the lipophilic precursor of the surfactant is concentrated on the hydrophobic surface of the fatty acid micelles (Bachmann et al., 1992), a feature that has given rise to some controversy (Mavelli and Luisi, 1996 Buhse etal, 1991 1998 Mavelli, 2004). 

The terms in Equation 1.3 (Malkin s autocatalytic model) are described in Nomenclature. In Malkin s autocatalytic model, the concentration of the activator, [A], is defined as the concentration of the initiator times the functionality of the initiator. For a difimctional initiator [e.g., isophthaloyl-bis-caprolactam, the concentration of the activator (acyllactam) is twice the concentration of the initiator]. The term [C] is defined as the concentration of the metal ion that catalyzes the anionic polymerization of caprolactam. In a magnesium-bromide catalyzed system, the concentration of the metal ion is the same as the concentration of the caprolactam-magnesium-bromide (catalyst) because the latter is monofunctional. 

We have now seen how local stability analysis can give us useful information about any given state in terms of the experimental conditions (i.e. in terms of the parameters p and ku for the present isothermal autocatalytic model). The methods are powerful and for low-dimensional systems their application is not difficult. In particular we can recognize the range of conditions over which damped oscillatory behaviour or even sustained oscillations might be observed. The Hopf bifurcation condition, in terms of the eigenvalues k2 and k2, enabled us to locate the onset or death of oscillatory behaviour. Some comments have been made about the stability and growth of the oscillations, but the details of this part of the analysis will have to wait until the next chapter. 

To determine the response of the cubic autocatalytic system to perturbations in the vicinity of a turning point in the locus, we must return to eqn (8.6). The first two terms (not involving A a) again cancel exactly, because of the stationary-state condition. If we are also at an ignition or extinction point, the tangency condition in any of its forms discussed above ensures that the coefficient of the A a term is also zero. Thus the first non-zero term is that involving (Aa)2 ... 

Thus, with the simple cubic autocatalytic rate law, we have been able to find an analytical expression for the time and space dependence of a steady reaction-diffusion wave and make various quantitative and qualitative comments about the behaviour of the wave in terms of the kinetic and diffusion parameters. We now turn to the apparently simpler kinetics of a quadratic autocatalysis, hoping for similar rewards. 

We may comment further on the importance of the condition on the dimensionless decay rate k2. This term is the ratio of decay and autocatalytic rate constants, and the condition k2 < 1 becomes, in terms of the physical rate constants,... 

In recent years Emanuel, Neiman, and their respective schools have greatly contributed to the theory of antioxidant action by studying the phenomenon of the critical antioxidant concentration in terms of a degenerate branched chain reaction. The critical antioxidant concentration, a well-established feature of phenolic antioxidants, is one below which autoxidation is autocatalytic and above which it proceeds at a slow and steady rate. Since the theory allowed not only a satisfactory explanation of the critical antioxidant concentration itself but elucidation of many refinements, such as the greater than expected activity of multifunctional phenolic antioxidants (21), we wondered whether catalyst-inhibitor conversion could be fitted into its framework. If degenerate chain branching is assumed to be the result of... 

The delay period exhibited with sulfenamide cures is explained in terms of the formation of intermediates by reaction with activated sulfur (Scheme 5) (80MI11508). The 2-mercap-tobenzothiazole (31) produced reacts rapidly with the sulfenamide providing a more facile pathway for vulcanization (equation 11). The amine (38) produced also acts as a catalyst, so that the cure, once started, becomes autocatalytic (64MI11503). 

---