Rate equation, kinetic description

Here, we shall examine a series of processes from the viewpoint of their kinetics and develop model reactions for the appropriate rate equations. The equations are used to andve at an expression that relates measurable parameters of the reactions to constants and to concentration terms. The rate constant or other parameters can then be determined by graphical or numerical solutions from this relationship. If the kinetics of a process are found to fit closely with the model equation that is derived, then the model can be used as a basis for the description of the process. Kinetics is concerned about the quantities of the reactants and the products and their rates of change. Since reactants disappear in reactions, their rate expressions are given a... 

The quantitative description of enzyme kinetics has been developed in great detail by applying the steady-state approximation to all intermediate forms of the enzyme. Some of the kinetic schemes are extremely complex, and even with the aid of the steady-state treatment the algebraic manipulations are formidable. Kineticists have, therefore, developed ingenious schemes for writing down the steady-state rate equations directly from the kinetic scheme without carrying out the intermediate algebra." -" ... 

The kinetics of hydrogenation of phenol has already been studied in the liquid phase on Raney nickel (18). Cyclohexanone was proved to be the reaction intermediate, and the kinetics of single reactions were determined, however, by a somewhat simplified method. The description of the kinetics of the hydrogenation of phenol in gaseous phase on a supported palladium catalyst (62) was obtained by simultaneously solving a set of rate equations for the complicated reaction schemes containing six to seven constants. The same catalyst was used for a kinetic study also in the liquid phase (62a). 

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. 

A complex reaction requires more than one chemical equation and rate law for its stoichiometric and kinetics description, respectively. It can be thought of as yielding more than one set of products. The mechanisms for their production may involve some of the same intermediate species. In these cases, their rates of formation are coupled, as reflected in the predicted rate laws. 

Aiming at a computer-based description of cellular metabolism, we briefly summarize some characteristic rate equations associated with competitive and allosteric regulation. Starting with irreversible Michaelis Menten kinetics, the most common types of feedback inhibition are depicted in Fig. 9. Allowing all possible associations between the enzyme and the inhibitor shown in Fig. 9, the total enzyme concentration Er can be expressed as... 

The variation of efficiencies is due to interaction phenomena caused by the simultaneous diffusional transport of several components. From a fundamental point of view one should therefore take these interaction phenomena explicitly into account in the description of the elementary processes (i.e. mass and heat transfer with chemical reaction). In literature this approach has been used within the non-equilibrium stage model (Sivasubramanian and Boston, 1990). Sawistowski (1983) and Sawistowski and Pilavakis (1979) have developed a model describing reactive distillation in a packed column. Their model incorporates a simple representation of the prevailing mass and heat transfer processes supplemented with a rate equation for chemical reaction, allowing chemical enhancement of mass transfer. They assumed elementary reaction kinetics, equal binary diffusion coefficients and equal molar latent heat of evaporation for each component. 

The kinetics of template polymerization depends, in the first place, on the type of polyreaction involved in polymer formation. The polycondensation process description is based on the Flory s assumptions which lead to a simple (in most cases of the second order), classic equation. The kinetics of addition polymerization is based on a well known scheme, in which classical rate equations are applied to the elementary processes (initiation, propagation, and termination), according to the general concept of chain reactions. 

This section discusses the inhibition phenomenon with specific reference to its influence on the conversion rates of dialkyldibenzothiophenes. The kinetic description of inhibition effects of even the parent molecule, thiophene, is quite complicated, and the complications become even greater as the thiophene core is fused to other aromatic rings and/or substituted with alkyl groups. In commercial processes the fact that there are many different sulfur species that are simultaneously being converted makes describing inhibition with a single equation an almost impossible task. A particularly relevant comment to this effect was made by Stanislaus and Cooper and is quoted here (109). ... 

The problem discussed in Sect. 1.2.6, i.e. what composition the working surface has, also has its kinetic counterpart. If the number of active sites of a certain type depends on the partial pressures of some reaction components, then the question arises whether rate equations of type (2) are sufficient for the description of such changes. 

Different approaches to the kinetics of alcohol dehydration were attempted by two groups of authors [118,119]. In one case, it has been assumed that the active surface of alumina is formed either by free hydroxyl groups or by surface alkoxyl groups. The rate equation was then derived on the basis of the steady-state assumption a good fit to the experimental data was obtained [1118]. The second model was based on the fact that water influences the adsorption of an alcohol and diminishes the available surface. The surface concentrations of tert-butanol and water were taken from independent adsorption measurements and put into the first-order rate equation a good description of integral conversion data was achieved [119]. 

Several rate equations, some of them in the integrated form, have been used for a kinetic description of cracking. A critical comparison of them yielded the consistent kinetic model [206] discussed below. 

For a formal kinetic description of vapour phase esterifications on inorganic catalysts (Table 21), Langmuir—Hinshelwood-type rate equations were applied in the majority of cases [405—408,410—412,414,415]. In some work, purely empirical equations [413] or second-order power law-type equations [401,409] were used. In the latter cases, the authors found that transport phenomena were important either pore diffusion [401] or diffusion of reactants through the gaseous film, as well as through the condensed liquid on the surface [409], were rate-controlling. 

Unfortunately, many of the chemical processes which are important industrially are quite complex. A complete description of the kinetics of a process, including byproduct formation as well as the main chemical reaction, may involve several individual reactions, some occurring simultaneously, some proceeding in a consecutive manner. Often the results of laboratory experiments in such cases are ambiguous and, even if complete elucidation of such a complex reaction pattern is possible, it may take several man-years of experimental effort. Whereas ideally the design engineer would like to have a complete set of rate equations for all the reactions involved in a process, in practice the data available to him often fall far short of this. 

For the very restricted conditions where Eq. (5.2) provides a rigorous description of the reaction kinetics, the activation energy, E, is a constant independent of conversion. But in most cases it is found that E is indeed a function of conversion, E (x). This is usually attributed to the presence of two or more mechanisms to obtain the reaction products e.g., a catalytic and a noncatalytic mechanism. However, the problem is in general associated to the fact that the statement in which the isoconversional method is based, the validity of Eq. (5.1), is not true. Therefore, isoconversional methods must be only used to infer the validity of Eq. (5.2) to provide a rigorous description of the polymerization kinetics. If a unique value of the activation energy is found for all the conversion range, Eq. (5.2) may be considered valid. If this is not true, a different set of rate equations must be selected. 

The statement of kinetics in terms of Eqs (5.10)—(5.13) assumes that all initial, intermediate, and final species participating in the reaction scheme have been identified. This is not necessarily true if there are stable complexes formed by association of OH groups with e, al5 or a2 groups. In this case, constitutive equations giving the rate of generation of these new intermediates have to be added to the kinetic description. 

The kinetic description of chainwise polymerizations requires a set of rate equations accounting for inhibition, initiation, propagation, termination, and transfer steps. The polymerization rate is usually given by the propagation steps (one or more), because they consume functional groups much more frequently than any other step involving them e.g. initiation or transfer steps. However, it is always necessary to consider the overall set of kinetic equations to determine the evolution of the concentration of active species (free radicals, ions, etc.) that participate in the propagation step. 

Equations (10) are generally valid for both liquid and gas phases if reactions take place there. They represent nothing but a differential mass balance for the film region with the account of the source term due to the reaction. To link this balance to the process variables like component concentrations, some additional relationships - often called constitutive relations (see Ref. [16]) - are necessary. For the component fluxes Ni, these constitutive relations result from the multicomponent diffusion description (Eqs. (1), (2)) for the source terms, from the reaction kinetics description. The latter strongly depends on the specific reaction mechanism [27]. The reaction rate expressions lli usually represent nonlinear dependencies on the mixture composition and temperature of the corresponding phase. 

Catalytic reaction engineering is a scientific discipline which bridges the gap between the fundamentals of catalysis and its industrial application. Starting from insight into reaction mechanisms provided by catalytic chemists and surface scientists, the rate equations are developed which allow a quantitative description of the effects of the reaction conditions on reaction rates and on selectivities for desired products. The study of intrinsic reaction kinetics, i.e. those determined solely by chemical events, belongs to the core of catalytic reaction engineering. Very close to it lies the study of the interaction between physical transport and chemical reaction. Such interactions can have pronounced effects on the rates and selectivities obtained in industrial reactors. They have to be accounted for explicitly when scaling up from laboratory to industrial dimensions. 

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