Single kinetic equation equations

The occurrence of self-acceleration during curing of epoxy resins and epoxy-based compounds was proven by rheokinetic and calorimetric methods.53 This phenomenon can be treated formally in terms of an induction period (when the reaction is very slow in the initial stage of a process), followed by a constant rate. However, it seems preferable to use a single kinetic equation incorporating the self-acceleration effect to describe reaction as a whole. Such a kinetic equation contains only a limited number of constants (K and co in Eq. (2.33)) and allows easy and unambiguous interpretation of their dependencies on process factors. 

Most of the experimental studies of irreversible bimolecular ionization (3.52) break down into two parts accordingly to the methods and goals of the investigation. One part concentrates on the stationary or time-resolved fluorescence of reactants that can be analyzed with a single kinetic equation for the excitation concentration like Eq. (3.2). The other part is confined to the... 

How do the thermodynamic and kinetic parameters of elementary steps affect the most important parameters of a catalyst that is the rate of the catalytic reaction with the given transformation scheme Is it possible to apply a single kinetic equation in the case of multiphcity of the allowed reaction pathways in the presence of the catalyst ... 

Although the literature contains a very large number of research articles concerned with the kinetics and mechanisms of reactions involving solids, there are comparatively few authoritative, critical and comprehensive reviews of the formidable quantity of information which is available. Probably the most important general account of the field is the book Chemistry of the Solid State, edited by Gamer [63]. Chapters 7—9 are particularly relevant in the present context as they provide a systematic exposition of the kinetic equations applicable to the decomposition of single solids (Jacobs and Tompkins [28]) and their application to endothermic [64] and exothermic [65] reactions. 

Analyses of rate measurements for the decomposition of a large number of basic halides of Cd, Cu and Zn did not always identify obedience to a single kinetic expression [623—625], though in many instances a satisfactory fit to the first-order equation was found. Observations for the pyrolysis of lead salts were interpreted as indications of diffusion control. More recent work [625] has been concerned with the double salts jcM(OH)2 yMeCl2 where M is Cd or Cu and Me is Ca, Cd, Co, Cu, Mg, Mn, Ni or Zn. In the M = Cd series, with the single exception of the zinc salt, reaction was dehydroxylation with concomitant metathesis and the first-order equation was obeyed. Copper (=M) salts underwent a similar change but kinetic characteristics were more diverse and examples of obedience to the first order, the phase boundary and the Avrami—Erofe ev equations [eqns. (7) and (6)] were found for salts containing the various cations (=Me). 

The retarding influence of the product barrier in many solid—solid interactions is a rate-controlling factor that is not usually apparent in the decompositions of single solids. However, even where diffusion control operates, this is often in addition to, and in conjunction with, geometric factors (i.e. changes in reaction interfacial area with a) and kinetic equations based on contributions from both sources are discussed in Chap. 3, Sect. 3.3. As in the decompositions of single solids, reaction rate coefficients (and the shapes of a—time curves) for solid + solid reactions are sensitive to sizes, shapes and, here, also on the relative dispositions of the components of the reactant mixture. Inevitably as the number of different crystalline components present initially is increased, the number of variables requiring specification to define the reactant completely rises the parameters concerned are mentioned in Table 17. 

When this holds, the kinetic equations reduce to single exponentials. Chipperfield6 demonstrates that approximate adherence to Eq. (4-25) suffices to fit 20 absorbancetime pairs spaced at equal times over the first 75 percent of the reaction with correlation coefficients better than 0.999. 

The kinetic information is obtained by monitoring over time a property, such as absorbance or conductivity, that can be related to the incremental change in concentration. The experiment is designed so that the shift from one equilibrium position to another is not very large. On the one hand, the small size of the concentration adjustment requires sensitive detection. On the other, it produces a significant simplification in the mathematics, in that the re-equilibration of a single-step reaction will follow first-order kinetics regardless of the form of the kinetic equation. We shall shortly examine the data workup for this and for more complex kinetic schemes. 

An accurate calculation of the heat conductivity requires solving a kinetic equation for the phonons coupled with the multilevel systems, which would account for thermal saturation effects and so on. We encountered one example of such saturation in the expression (21) for the scattering strength by a two-level system, where the factor of tanh((3co/2) reflected the difference between thermal populations of the two states. Neglecting these effects should lead to an error on the order of unity for the thermal frequencies. Within this single relaxation time approximation for each phonon frequency, the Fermi golden rule yields, for the scattering rate of a phonon with Ha kgT,... 

Frost and Schwemer have developed a time-ratio technique based on equations 5.4.21 and 5.4.16 in order to facilitate the calculation of second-order rate constants for the class of reactions under consideration. Data for A/A0 versus t at various values of k are presented in Table 5.2, and time ratios are given in Table 5.3. The latter values may be used to determine k by using various time ratios from a single kinetic run if one recognizes that (tf/rf) = t1/t2). Once k has been determined, Table 5.2 may be used to determine the t values at a given A/A0 and k. Equation 5.4.18 may then be used to determine... 

In the special case of an ideal single catalyst pore, we have to take into account that diffusion is quicker than in a porous particle, where the tortuous nature of the pores has to be considered. Hence, the tortuosity r has to be regarded. Furthermore, the mass-related surface area AmBEX is used to calculate the surface-related rate constant based on the experimentally determined mass-related rate constant. Finally, the gas phase concentrations of the kinetic approach (Equation 12.14) were replaced by the liquid phase concentrations via the Henry coefficient. This yields the following differential equation ... 

These assumptions are partially different from those introduced in our previous model.10 In that work, in fact, in order to simplify the kinetic description, we assumed that all the steps involved in the formation of both the chain growth monomer CH2 and water (i.e., Equations 16.3 and 16.4a to 16.4e) were a series of irreversible and consecutive steps. Under this assumption, it was possible to describe the rate of the overall CO conversion process by means of a single rate equation. Nevertheless, from a physical point of view, this hypothesis implies that the surface concentration of the molecular adsorbed CO is nil, with the rate of formation of this species equal to the rate of consumption. However, recent in situ Fourier transform infrared (FT-IR) studies carried out on the same catalyst adopted in this work, at the typical reaction temperature and in an atmosphere composed by H2 and CO, revealed the presence of a significant amount of molecular CO adsorbed on the catalysts surface.17 For these reasons, in the present work, the hypothesis of the irreversible molecular CO adsorption has been removed. 

In previous chapters, we deal with simple systems in which the stoichiometry and kinetics can each be represented by a single equation. In this chapter we deal with complex systems, which require more than one equation, and this introduces the additional features of product distribution and reaction network. Product distribution is not uniquely determined by a single stoichiometric equation, but depends on the reactor type, as well as on the relative rates of two or more simultaneous processes, which form a reaction network. From the point of view of kinetics, we must follow the course of reaction with respect to more than one species in order to determine values of more than one rate constant. We continue to consider only systems in which reaction occurs in a single phase. This includes some catalytic reactions, which, for our purpose in this chapter, may be treated as pseudohomogeneous. Some development is done with those famous fictitious species A, B, C, etc. to illustrate some features as simply as possible, but real systems are introduced to explore details of product distribution and reaction networks involving more than one reaction step. 

The E-Z Solve software may also be used to solve Example 12-7 (see file exl2-7.msp). In this case, user-defined functions account for the addition of fiesh glucose, so that a single differential equation may be solved to desenbe the concentration-time profiles over the entire 30-dry period. This example file, with die user-defined functions, may be used as the basis for solution of a problem involving the nonlinear kinetics in equation (A), in place of the linear kinetics in (B) (see problem 12-17). 

Many kinetic equations can be suitably linearized to the form of Eq. (20). For example, Eq. (1) can be transformed logarithmically, or Eq. (2) can be transformed reciprocally. Two equations proposed for describing pentane-isomerization data (Cl, Jl) are the single site... 

Sparks (1989) discusses the application of various kinetic equations to earth materials based on the analysis of a large number of reported studies. Even though different equations describe rate data satisfactorily. Sparks (1989) uses hnear regression analysis to show that no single equation best describes every study. 

The lumped kinetic model can be obtained with further simplifications from the lumped pore model. We now ignore the presence of the intraparticle pores in which the mobile phase is stagnant. Thus, p = 0 and the external porosity becomes identical to the total bed porosity e. The mobile phase velocity in this model is the linear mobile phase velocity rather than the interstitial velocity u = L/Iq. There is now a single mass balance equation that is written in the same form as Equation 10.8. 

For the analysis of nonlinear cycles the new concept of kinetic polynomial was developed (Lazman and Yablonskii, 1991 Yablonskii et al., 1982). It was proven that the stationary state of the single-route reaction mechanism of catalytic reaction can be described by a single polynomial equation for the reaction rate. The roots of the kinetic polynomial are the values of the reaction rate in the steady state. For a system with limiting step the kinetic polynomial can be approximately solved and the reaction rate found in the form of a series in powers of the limiting-step constant (Lazman and Yablonskii, 1988). 

We assume the system under consideration to be a single domain. Then the orientational state of the system can be specified by the order parameter tensor S defined by Eq. (63), The time evolution of S is governed by the kinetic equation, Eq. (64), along with Eqs. (62) and (65). This kinetic equation tells us that the orientational state in the rodlike polymer system in an external flow field is determined by the term F related to the mean-field potential Vscf and by the term G arising from the external flow field. These two terms control the orientation state in a complex manner as explained below. 

Combining substrate-induced diastereoselection and mutual diastereoselectivity, as illustrated for the crotylzincation of the alkenyllithium derived from 209, led to excellent results as the gewt-dimetallic species 217 was obtained in a highly stereoselective fashion. The stereochemical outcome was explained by the addition of the kinetically reactive cisoid metallotropic form of the crotylzinc reagent anti to the propyl group in the chelated allyl alkenylzinc intermediate. After hydrolysis, compound 218 was obtained as a single diastereomer (equation 106)148,149. 

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