Rate Expressions, Simplifying using

The first thing to note is that stoichiometric quantities of reactants were used in this investigation. Because the reaction rate expression simplifies when stoichiometric quantities of reactants are used, the equations developed earlier in this chapter cannot be applied directly in the solution of this problem. Thus we will have to derive appropriate relations in the course of our analysis. 

If the adsorption of A is nearly equilibrated, the Langmuir adsorption isotherm can be used to find 0, and the final rate expression simplifies to ... 

As the solution of this set of equations is quite formidable even for the first-order case, the following simplified set of first-order reaction rate expressions were used to describe the HDA reactions (Chowdhury et al., 2002 Bhaskar et al., 2004) ... 

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. 

Using the concept of the rate-determining step significantly simplifies the overall rate expression. Therefore, it is widely used in the analysis of kinetie data, especially in the field of heterogeneous catalysis. 

These equations can be evaluated by using the rate expression for fnet given in Section III,B,3. As shown in that section, the results are equivalent to evaluating the absolute rate terms but the computation procedure is greatly simplified. 

In this section we discuss the mathematical forms of the integrated rate expression for a few simple combinations of the component rate expressions. The discussion is limited to reactions that occur isothermally in constant density systems, because this simplifies the mathematics and permits one to focus on the basic principles involved. We will again place a V to the right of certain equation numbers to emphasize that such equations are not general but are restricted to constant volume batch reactors. The use of the extent per unit volume in a constant volume system ( ) will also serve to emphasize this restriction. For constant volume systems,... 

The rate expression can be simplified by making use of the relationship between the fluorescence lifetime and the spectrum of the donor molecule and lumping together all the constants in one characteristic range R0. The rate of transfer is then... 

Following Carbeny (1976), in this book the term rate coefficient is used for the proportionality coefficients kt in the typical rate expression of the form r] = kJ(C). To simplify the following analysis, a first-order elementary reaction is considered. Then the intrinsic reaction rate can be... 

It is common within the industry to characterize chemical processes in terms of one or a few global reaction steps, assigning an Arrhenius rate expression to describe the rate of each reaction. If knowledge of the detailed chemistry is inadequate or the chemical scheme is to be combined with computational fluid dynamics for a complex flow description, a simplified chemistry may be necessary. It is important, however, to realize that such a chemical description can only be used for the narrow range of conditions (temperature, composition, etc.) for which it is developed. Any extrapolation outside these conditions may be erroneous or even disastrous. 

The derivation of the mechanistic rate expression is considerably simplified if the steady state treatment can be used (Sections 3.19, 3.19.1 and 3.20). When intermediate concentrations are not sufficiently low and constant, the steady state approximation is no longer valid. Numerical integration by computer of the differential equations involved in the analysis, or computer simulation, may have to be used. 

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. 

It is normally necessary to use a simplified or empirical expression for the reaction rate r, in terms of constants and reactant and product concentrations which can be assumed from the stoichiometry of a proposed reaction mechanism or developed purely empirically on the basis of experimental data. One of the key components of the rate expression is the specific rate constant kt which must almost always be determined directly from laboratory or other rate data. 

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 rate expression is simplified by eliminating surface concentrations of species through the use of appropriate equilibrium relationships. According to step (2) ... 

These simplifying assumptions must be adapted to some extent to explain the nature of some reactions on catalyst surfaces. The case of ammonia synthesis on supported ruthenium described in Example 5.3.1 presents a situation that is similar to rule 1, except the rate-determining step does not involve the mari. Nevertheless, the solution of the problem was possible. Example 5.3.2 involves a similar scenario. If a mari cannot be assumed, then a rate expression can be derived through repeated use of the steady-state approximation to eliminate the concentrations of reactive intermediates. 

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. 

We conclude the discussion with a brief description of several techniques to simplify the determination of the parameters of the rate expression. We can mix the reactants in proportions that are convenient for the determination of the individual orders or the overall orders. Commonly, one of the following mixtures is used ... 

We are using standard thermodynamic quantities here, because the free energy and the entropy of a species are concentration-dependent. The rate constant is not concentration-dependent in dilute systems thus the argument that leads to (3.1.10) needs to be developed in the context of a standard state of concentration. The choice of standard state is not critical to the discussion. It simply affects the way in which constants are apportioned in rate expressions. To simplify notation, we omit the superscript 0 from A// A5, and AG but understand them throughout this book to be referred to the standard state of concentration. 

The simplex method given by Nelder and Mead (1965), sometimes called the downhill simplex method, is one of the few robust and efficient methods that does not use any derivative information. This greatly simplifies computational requirements and reduces the chances of errors that can crop up in the differentiation of complex 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 ... 

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