Rate Expression Other Models

In many reactions it has been demonstrated that more than one site is involved in the catalytic process. This is particularly often the case for dissociation reactions. The same procedure as depicted above for a single site model can be used for the derivation of the rate expression for a dual site model, but the result is somewhat different. This is exemplified for the following dissociation reaction A 2B, which is thought to proceed according to the three step sequence  

By application of the steady-state hypothesis, a site balance and the assumption that the surface dissociation is rate determining while the other steps are in quasi-equilibrium, the following rate expression is derived  

Reactions of a molecule directly from the gas phase with a surface complex, as  

Other variants of kinetic models can be derived, of course. Froment and Bischoff [2] presented an extended treatment of this approach and it follows that rate expressions based on sequences of elementary steps, one of which is rate determining, can be represented by the following expression 

The kinetic factor always contains the rate constant of the rate determining step, together with the concentration of active sites and adsorption equilibrium constants. 

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A cure kinetics model relates chemical composition with time and temperature dining chemical reaction in the form of a reaction rate expression. Kinetic models may be phenomenological or mechanistic. A phenomenological model captures the main features of the reaction kinetics ignoring the details of how individual species react with each other. Mechanistic models, on the other hand, are obtained from balances of species involved in the reaction hence, they are better for prediction and interpretation of composition. Due to the complexity of thermosetting reactions, however, phenomenological models are the most common. 

Reference was made earlier to some specific numerical studies reported for the axial dispersion model employing rate expressions other than first order. Some results given by Fan and Balie for half-, second-, and third-order kinetics are illustrated in Figure 5.21. In these results the parameter R is defined as... 

Cropley made general recommendations to develop kinetic models for compUcated rate expressions. His approach includes first formulating a hyperbolic non-linear model in dimensionless form by linear statistical methods. This way, essential terms are identified and others are rejected, to reduce the number of unknown parameters. Only toward the end when model is reduced to the essential parts is non-linear estimation of parameters involved. His ten steps are summarized below. Their basis is a set of rate data measured in a recycle reactor using a sixteen experiment fractional factorial experimental design at two levels in five variables, with additional three repeated centerpoints. To these are added two outlier... 

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... 

Steps 1 through 9 constitute a model for heterogeneous catalysis in a fixed-bed reactor. There are many variations, particularly for Steps 4 through 6. For example, the Eley-Rideal mechanism described in Problem 10.4 envisions an adsorbed molecule reacting directly with a molecule in the gas phase. Other models contemplate a mixture of surface sites that can have different catalytic activity. For example, the platinum and the alumina used for hydrocarbon reforming may catalyze different reactions. Alternative models lead to rate expressions that differ in the details, but the functional forms for the rate expressions are usually similar. 

This result is experimentally indistinguishable from the general form, Equation (10.12), derived in Example 10.1 using the equality of rates method. Thus, assuming a particular step to be rate-controlling may not lead to any simplification of the intrinsic rate expression. Furthermore, when a simplified form such as Equation (10.15) is experimentally determined, it does not necessarily justify the assumptions used to derive the simplified form. Other models may lead to the same form. 

However, all rate data for this reaction are not explained simply by this rate expression. At pressures above 10 Torr the rate exhibits multiple steady states, long transients, and rate oscillations ]). Clearly other processes are Involved than those Implied by the simple one state, constant parameter LH model. 

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... 

By using these ratios and the relationships (3.18) and (3.19), we can alter the material balance expressions and the corresponding solutions of the reactor models if we use other rate expressions. It should be noted that in practice, in fixed-beds and slurries of porous particles, the external area of the particle and thus the parameters au and ac are used, respectively. 

Under the hypotheses of constant illumination intensity, fast reaction of tlie electron scavenger with photogenerated electrons, and steady-state conditions applied to ecB nd hvg, a functional form like Eq. (2) was obtained without invoking adsorption [34], The rate expression was given as in the EH model by reaction of surface-active species with the substrate, in which A LH= h, and where h is the surface concentration of any oxidative active species. No assumptions were made on the steady-state concentrations of conduction-band electrons, valence-band holes, and other transient species. The rate is given by... 

In the last 20 years, considerable efforts have been made to measure air-water exchange rates either in the laboratory or in the field. One central goal of these investigations was to check the validity of Eq. 20-19 or of alternative expressions. Thus let us see how corresponding forms of Eq. 20-19 look for other models. 

Although the simple rate expressions, Eqs. (2-6) and (2-9), may serve as first approximations they are inadequate for the complete description of the kinetics of many epoxy resin curing reactions. Complex parallel or sequential reactions requiring more than one rate constant may be involved. For example these reactions are often auto-catalytic in nature and the rate may become diffusion-controlled as the viscosity of the system increases. If processes of differing heat of reaction are involved, then the deconvolution of the DSC data is difficult and may require information from other analytical techniques. Some approaches to the interpretation of data using more complex kinetic models are discussed in Chapter 4. 

There remains one objection—of a less precise kind but felt by many chemists. It is that third-order kinetics as embodied in the representation of step (1) are intrinsically objectionable. If the equations had to be interpreted as representing elementary steps, this would be a weightier consideration, but it has also been asserted that the oscillatory properties of certain other model schemes collapse completely (King, 1983 Gray and Morley-Buchanan, 1985) if the third-order steps therein are replaced. Accordingly it is most desirable to establish whether oscillations and other exotic behaviour arising from a cubic rate-law of the form k ab2 can also arise from a series of successive second-order or bimolecular steps. Similar interests have been expressed previously by Tyson (1973) and Tyson and Light (1973). 

Homogeneous kinetics is used instead of diffusion kinetics to express the dependence of intraspur GH, on solute concentration. The rate-determining step for H2 formation is not the combination of reducing species, but first-order disappearance of "excited water." Two physical models of "excited water" are considered. In one model, the HsO + OH radical pair is assumed to undergo geminate recombination in a first-order process with H3O combination to form H2 as a concomitant process. In this model, solute decreases GH, by reaction with HsO. In the other model, "excited water" yields freely diffusing H3O + OH radicals in a first-order process and solute decreases GH, by reaction with "excited water." The dependence of intraspur GH, on solute concentration indicates th,o = 10 9 — 10 10 sec. 

The second model was discarded because the value of the rate constant k2 found by non-linear regression of the data was approximately zero which was not considered reasonable on a physical basis. Of the other two models, the second order rate expression was found to fit the data better. For denitrogenation only the first and second order models were examined, and in this case also, the latter model was found to represent the data better. 

Initially, an attempt was made to develop original global rate expressions for Reactions 14 to 16 from these rate data. It soon became clear, however, that the number of experimental points was too few to allow the attainment of this goal. Moreover, since a ternary system was being analyzed, the concentration profiles had an intricate form which made numerical differentiation to retrieve the rates somewhat inaccurate. It was therefore decided to use these rate data to check the overall rate expressions derived by other authors and used in the present model. 

After fitting data to the various rate expressions that have been derived it will often appear that several models do not differ much in their mean SSR value. Statistical tests, such as an F-test, cannot be directly performed on these mean SSR values since they are based on the same data set and are therefore not statistically independent. Therefore, other approaches have been devised to allow further selection between the possible candidates, and some of them are mentioned below [9],... 

P is the number of polymer molecules of degree of polymerization n, R is the number of radicals found in a volume V, R is the number of polymer radicals with degree of polymerization n found in a volume, V. For other definitions, please use the nomenclature associated with Table 15.2. Noting equation 15.14, the kinetics of polymer degradation are very complex. Only the most simple mechanisms have been thoroughly researched. These simplified reactions presented in Table 15.2 are sometimes zero order, more frequently first order, and infiiequently second order in polymer mass. These simplified rate expressions are typically used to model binder burnout. 

Poehlein and Degraff [336] extended the derivation of Gershberg and Long-field [330] to the calculation of both molecular weight and particle size distribution in the continuous emulsion polymerization of St in a CSTR. On the other hand, Nomura et al. [163] carried out the continuous emulsion polymerization of St in a cascade of two CSTRs and developed a novel model for the system by incorporating their batch model [ 14], which introduced the concept that the radical capture efficiency of a micelle relative to a polymer particle was much lower than that predicted by the diffusion entry model (pocd -°). The assumptions employed were almost the same as those of Smith and Ewart (Sect. 3.3), except that the model did not assume a constant value of p. The elementary reactions and their rate expressions employed in the first stage are as follows ... 

However, each set of factors entering in to the rate expression is also a potential source of scaleup error. For this, and other reasons, a fundamental requirement when scaling a process is that the model and prototype be similar to each other with respect to reactor type and design. For example, a cleaning process model of a continuous-stirred tank reactor (CSTR) cannot be scaled to a prototype with a tubular reactor design. Process conditions such as fluid flow and heat and mass transfer are totally different for the two types of reactors. However, results from rate-of-reaction experiments using a batch reactor can be used to design either a CSTR or a tubular reactor based solely on a function of conversion, -r ... 

An exception to this approach is to be found in the work of Cohen [44] who has used the transition state methods developed by Benson and his colleagues [38] as a framework for evaluating experimental data for bimolecular metathetical reactions such as those of H, O and OH with alkanes. Using a model of the transition state, a theoretical value of the pre-exponential factor in the rate expression is derived which may be combined with experimental data at one temperature to give the exponential term. The rate expression so derived may be used to calculate values at other temperatures. By treating families of reactions adjustments can be made to the transition states to make them compatible with the experimental data on the whole range of reactions considered. 

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