Simplified rate expression, reaction kinetics

Reaction Kinetics Analyses Based on a Simplified Rate Expression... 

Since steps 2-4 are quasi-equilibrated under the conditions of the kinetic studies, the following simplified rate expression may be derived to described the hydrogenation/dehydrogenation reactions ... 

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. 

The example just discussed illustrates the ambiguity of simplified rate expressions, a very general problem of the kinetic analysis of sequences. More specifically, it shows that in the case of reactions taking place at the surface of solid catalysts, sequences of the type (4.5.3) that postulate adsorption equilibrium between the surface and reactants or products, lead to rate equations (4.5.6) which can also be obtained from sequences of the type... 

If 1/2/Cj [olefin] >> k [9-BBN] Eq. 4.6 reduces to Eq. 4.1, the reaction behaves like a unimolecular reaction and exhibits first-order kinetics. However, if 1/2kj [olefin] k [9-BBN], Eq. 4.6 reduces to Eq. 4.2. The reaction exhibits three-halves-order kinetics. Consequently, kinetics reveals that the hydrobora-tion with dimeric 9-BBN of representative alkenes proceeds through prior dissociation of the dimer to the monomer, leading to simplified kinetic expression, as expressed in Eqs. 4.1 and 4.2 for reactive alkenes and for less reactive alkenes, respectively. However, for olefins like 2-methyl-2-butene and ds-3-hexene 1/2 k [olefin] /c i [9-BBN]j and the kinetics fail to follow the simplified rate expression Eqs. 4.1 and 4.2. 

Those simplified models are often used together with simplified overall reaction rate expressions, in order to obtain analytical solutions for concentrations of reactants and products. However, it is possible to include more complex reaction kinetics if numerical solutions are allowed for. At the same time, it is possible to assume that the temperature is controlled by means of a properly designed device thus, not only adiabatic but isothermal or nonisothermal operations as well can be assumed and analyzed. 

The catalytically active phase was assumed to be exclusively a-Fe, and Fe304 was assumed not to be active for the Fischer-Tropsch reaction. Kinetic parameters for the simulations were obtained independently in separate oxidation/reduction studies. Balancing the oxidation and reduction rates for the CO/CO2 and the H2/H2O systems independently and describing the rate of synthesis in Fischer-Tropsch reactions by a standard expression, Caldwell could predict the oscillations with a simplified model for a tubular reactor fairly well. 

The reason many global reactions between stable reactants and products have complex mechanisms is that these unstable intermediates have to be produced in order for the reaction to proceed at reasonable rates. Often simplifying assumptions lead to closed-form kinetic rate expressions even for very complex global reactions, but care must be taken when using these since the simplifying assumptions are valid over limited ranges of compositions, temperature, and pressure. These assumptions can fail completely—in that case the full elementary reaction network has to be considered, and no closed-form kinetics can be derived to represent the complex system as a global reaction. 

The steady-state approach generally yields complex rate expressions. A simplification is obtained by the introduction of one or several rate-determining step(s) and ( wasi-equilibrium steps, and further by the initial reaction rate approach. For complex reaction schemes, identifying the most abundant reaction intermediates ("mari") and making use of the site balance can simplify the kinetic models and rate expressions. 

Kinetic modeling of diesel autothermal reforming is extremely complicated. Diesel fuel consists of a complex variable mixture of hundreds of hydrocarbon compounds containing paraffins, isoparaffins, naphthenes, aromatics, and olefins. To simplify the model, a steady-state power law rate expression for the diesel reforming over each type of catalyst used in this study was developed. A linearized least-squares method of data analysis was used to determine the power law parameters from a series of diesel ATR experiments. The power law rate model for diesel autothermal reaction may be written as ... 

The QSSA is a useful tool in reaction analysis. Material balances for hatch and plug-flow reactors are ordinary differential equations. By applying Equation 5.81 to the components that are QSSA species, their material balances become algebraic equations. These algebraic equations can be used to simplify the reaction expressions and reduce the number of equations that must be solved simultaneously. In addition, appropriate use of the QSSA can eliminate the need to know several difficult-to-measure rate constants. The required information can be reduced to ratios of certain rate constants, which can be more easily estimated from data. In the next section we show how the QSSA is used to develop a rate expression for the production of a component from a statement of the elementary reactions, and illustrate the kinetic model simplification that results from the QSSA model reduction.. 

Equation (8-105) is an implicit expression for rj, since rj is a function of (j), and (f> also a function of rj, so a trial procedure is necessary to determine a final value for rj. The overall rate for mth-order kinetics can be obtained from equation (8-102) once a value of r] has been determined. This approach, using the overall effectiveness, incorporates all the transport resistances and thus simplifies the calculation required to obtain an overall rate of reaction. 

In general, a mechanism for any complex reaction (catalytic or non-catalytic) is defined as a sequence of elementary steps involved in the overall transformation. To determine these steps and especially to find their kinetic parameters is very rare if at all possible. It requires sophisticated spectroscopic methods and/or computational tools. Therefore, a common way to construct a microkinetic model describing the overall transformation rate is to assume a simplified reaction mechanism that is based on experimental findings. Once the model is chosen, a rate expression can be obtained and fitted to the kinetics observed. 

Kinetics On the basis of simplified assumptions several investigators (7,11,28,29) proposed the following rate expression for the vapor phase reaction on Fe catalysts... 

The simplified isotherms and rate expressions developed in this section are extremely useful despite the implicit assumption that a single state exists for the adsorbed species. Real surfaces are heterogeneous on an atomic scale with a variety of distinguishable adsorption sites. Gas molecules adsorbed at each type of site may display a wide distribution of excited rotational, vibrational, and electronic states. Experimentally, we can measure meaningful rate and adsorption equilibrium constants provided that adsorption and desorption are fast compared with surface reactions so that an adsorption equilibrium exists. In this circumstance the kinetic parameters are an ensemble average over all surface sites and states of the system. 

The power law expression was widely adopted in the literature for CO oxidation [25-27]. This form is simplified from a Langmuir-Hinshelwood (L-H) expression and not suitable for small CO concentrations [30]. Therefore a full L-H expression for CO oxidation is necessary to account for a wide range of CO concentrations (Equation 27.4). The H2 oxidation was previously modeled using empirical power law rate expressions by others [29]. However, in PrOx in the presence of CO, the rate-limiting CO desorption strongly inhibits H2 and O2 adsorption and the subsequent H2 oxidation. Hence the incorporation of Pco in the H2 oxidation rate expression is necessary (Equation 27.5). The kinetics of the r-WGS reaction were well studied previously [31], in which an empirical reversible rate expression [32] is attractive due to its relative simplicity and its appropriateness in PrOx kinetic studies, as demonstrated previously [29]. 

To simplify our discussion of kinetics, let s first assume that the reactions are irreversible and then consider the mathematical expressions or rate laws governing these classes of reaction. To do so, we need only examine the rate-determining step for the particular type of substitution. In the case of an Sjvjl reaction, this step is the formation of a carbocation from the precursor R-L (Eq. 14.2). The rate of the overall reaction is then proportional only to the concentration of substrate, as expressed in Equation 14.24. We see that the rate is first order in the concentration of R-L, expressed as [R-L], and zeroeth order in that of Nu , that is, [Nu ] , which means that the rate is independent of its concentration. Adding the two exponents for the concentration gives the overall order of the Sj l reaction, which is seen to be first order. A simplified version of the rate law is seen in Equation 14.25, and this is the form in which it is normally written. 

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