Decomposition model, first-order

Discussion of Results and Conclusions. The results of regression analysis show that a chemical reaction model, first order with respect to fractional carbon conversion, with a production and a decomposition step for each of CH4, C2H5, BTX and Oils, satisfactorily describes the dilute phase flash hydrogenation of both lignite and subbltumlnous coal. 

The approach taken in the development of an analytical model for the combustion of double-base propellants has been based on the decomposition behavior of the two principal propellant ingredients, nitrocellulose and nitroglycerin. The results of several studies reviewed by Huggett (HI2) and Adams (Al) show that nitrocellulose undergoes exothermic decomposition between 90° and 175°C. In this temperature range, the rate of decomposition follows the simple first-order expression... 

Sensitivity to step size was thought to be likely due to an unnecessary simplification in the original development of the model. The simplification was to consider initiator concentration constant over a small time increment. When instead the initiator was allowed to vary according to the usual first order decomposition path an analytical solution for the variation of polymer concentration could still readily be obtained and was as follows ... 

The Cu-, Co- and Fe-ZSM-5 catalysts are active systems for the decomposition of N2O, but their behaviour differs with respect to conditions and gas atmospheres. They all seem to obey a (nearly) first order dependency towards pmo> which can be rationalised by the two step kinetic model given by eqs. (2) and (3). A step like eq. (3) is quite well feasible, since the TM ions in ZSM-5 can be coordinated by several ligands simultaneously [18,22], The resulting rate expression is given by eq. (7). 

Figure B-1. Approximate 95% confidence region for first-order decomposition model. Reproduced from Kittrell (1970) with permission from Academic Press.

At first glance the process described in Equation (6) is bimolecular in [La3 + 33], but the kinetics strictly adhere to a first-order process for the loss of starting material. However, the subsequent observation that the products of the reaction actually catalyze the decomposition of starting material allow us to treat the kinetics at each pH according to a simple one-site binding model kobs = /S x [La3 + 33] init (Kd + [La3 + 33]... 

The reaction with CoII(ACAC)2 was studied in more detail and the rate law was established. The reaction was found to be first-order with respect to the substrate and the catalyst concentrations, and the partial pressure of 02. The corresponding kinetic model postulates reversible formation of a H2DTBC-Con(ACAC)2 02 adduct which undergoes redox decomposition in the rate-determining step. Hydrogen peroxide is also a primary... 

For fitting such a set of existing data, a much more reasonable approach has been used (P2). For the naphthalene oxidation system, major reactants and products are symbolized in Table III. In this table, letters in bold type represent species for which data were used in estimating the frequency factors and activation energies contained in the body of the table. Note that the rate equations have been reparameterized (Section III,B) to allow a better estimation of the two parameters. For the first entry of the table, then, a model involving only the first-order decomposition of naphthalene to phthalic anhydride and naphthoquinone was assumed. The parameter estimates obtained by a nonlinear-least-squares fit of these data, are seen to be relatively precise when compared to the standard errors of these estimates, s0. The residual mean square, using these best parameter estimates, is contained in the last column of the table. This quantity should estimate the variance of the experimental error if the model adequately fits the data (Section IV). The remainder of Table III, then, presents similar results for increasingly complex models, each of which entails several first-order decompositions. 

Fig. 10. Approximate 95 % confidence region for reparameterized first-order decomposition model.

Interaction of permeation with decomposition. The effects of detoxifying reactions can be considered using the multilaminate model and assuming that first order reaction takes place only in the aqueous phase with a rate constant K which... 

With this information in mind, we can construct a model for the deposition rate. In the simplest case, the rate of flux of reactants to the surface (step 2) is equal to the rate at which the reactants are consumed at steady state (step 5). All other processes (decomposition, adsorption, surface diffusion, desorption, and transport away from the substrate) are assumed to be rapid. It is generally assumed that most CVD reactions are heterogeneous and first order with respect to the major reactant species, such that a general rate expression of the form of Eq. (3.2) would reduce to... 

Proton transfer may proceed directly or via a six-membered cyclic transition state involving a molecule of water. A calculation of the intermediate zwitter-ionic concentration for the hydrolysis of methyl phosphate monoanion, based on the pKa values for methanol and methyl phosphate dianion, predicts the first-order rate coefficient for zwitterion decomposition to be ca. 10 sec-1 at 100°C. This value is in good agreement with the observed rate of hydrolysis and, considering the assumptions involved, with the rate of P-O bond fission of the presumed zwitterionic intermediate (2) formed in the Hg(II) catalyzed solvolysis of phosphoenolpyruvic acid, a model reaction for pyruvate kinase10. 

Equation (8) has been used to describe the progress of the homogeneous polymerization up to conversions of approximately 60%. Experimental and calculated conversion-time curves were in good agreement, even for the case of changing experimental conditions during the polymerization [51]. For the heterophase polymerization experimental and modeled conversion-time curves coincide well if a kinetic model based on first order initiator decomposition was applied and consideration of gel effect for conversions greater than 35% was included [13]. 

When several temperature-dependent rate constants have been determined or at least estimated, the adherence of the decay in the system to Arrhenius behavior can be easily determined. If a plot of these rate constants vs. reciprocal temperature (1/7) produces a linear correlation, the system is adhering to the well-studied Arrhenius kinetic model and some prediction of the rate of decay at any temperature can be made. As detailed in Figure 17, Carstensen s adaptation of data, originally described by Tardif (99), demonstrates the pseudo-first-order decay behavior of the decomposition of ascorbic acid in solid dosage forms at temperatures of 50° C, 60°C, and 70°C (100). Further analysis of the data confirmed that the system adhered closely to Arrhenius behavior as the plot of the rate constants with respect to reciprocal temperature (1/7) showed linearity (Fig. 18). Carsten-sen suggests that it is not always necessary to determine the mechanism of decay if some relevant property of the degradation can be explained as a function of time, and therefore logically quantified and rationally predicted. 

Assuming constant density and volume, the following modeling equations can be written, where it is assumed that each reaction is a first-order decomposition ... 

The atomic mechanism responsible for monomolecular reactions, including thermal decompositions, was first discussed by Polanyi and Wigner.26 Their model assumes that decomposition occurs when, due to energy fluctuations in the bonds of the molecule, the bond strength is exceeded or more precisely, the bond energy resides in harmonic vibrations and that decomposition occurs when their amplitude is exceeded. The resulting expression for the first-order Polanyi-Wigner rate constant is... 

Berner (1980) first introduced the following concept of a one-G model for determining first-order decay constants (k) of organic matter decomposition ... 

Figure 5.20 shows concentration-time profiles for the decomposition of hydrocortisone butyrate at 60°C in a buffered aqueous propylene glycol (50 w/w%, pH 7.6). Consecutive, irreversible, first-order kinetic models [i.e., Equation (5.119a), Equation (5.119b), and Equation (5.119c)] fit reasonably well with the experimental... 

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