Overall reaction model

Three modeling approaches can be applied to the esterification reaction kinetics the molecular species model, the functional group model, and the overall reaction model. These are schematically illustrated in Scheme 4.5. 

If there are only limited available experimental data such as those referring to the evaporated water, simple models such as the overall reaction model are quite... 

As our first approach to the model, we considered the controlling step to be the mass transfer from gas to liquid, the mass transfer from liquid to catalyst, or the catalytic surface reaction step. The other steps were eliminated since convective transport with small catalyst particles and high local mixing should offer virtually no resistance to the overall reaction scheme. Mathematical models were constructed for each of these three steps. 

Fiery1 252-254) studied only the last stage of the reactions, i.e. when the concentration of reactive end groups has been greatly decreased and when the dielectric properties of the medium (ester or polyester) no longer change with conversion. Under these conditions, he showed that the overall reaction order relative to various model esterifications and polyesterifications is 3. As a general rule, it is accepted that the order with respect to acid is two which means that the add behaves both as reactant and as catalyst. However, the only way to determine experimentally reaction orders with respect to add and alcohol would be to carry out kinetic studies on non-stoichiometric systems. 

In this section, only those studies, all of relatively recent date, that particularly emphasize the determination of rate-determining process steps and the application of the relatively advanced theoretical models discussed in Section IV will be reviewed. For earlier studies of overall reaction kinetics, the reader is referred to the publications of Hall et al. (HI) and Kolbel (K6). 

Measurements of overall reaction rates (of product formation or of reactant consumption) do not necessarily provide sufficient information to describe completely and unambiguously the kinetics of the constituent steps of a composite rate process. A nucleation and growth reaction, for example, is composed of the interlinked but distinct and different changes which lead to the initial generation and to the subsequent advance of the reaction interface. Quantitative kinetic analysis of yield—time data does not always lead to a unique reaction model but, in favourable systems, the rate parameters, considered with reference to quantitative microscopic measurements, can be identified with specific nucleation and growth steps. Microscopic examinations provide positive evidence for interpretation of shapes of fractional decomposition (a)—time curves. In reactions of solids, it is often convenient to consider separately the geometry of interface development and the chemical changes which occur within that zone of locally enhanced reactivity. 

The series nature of the model permits calculation of the overall reaction resistance (Rq ) simply by summing the individual resistances ... 

All these steps can influence the overall reaction rate. The reactor models of Chapter 9 are used to predict the bulk, gas-phase concentrations of reactants and products at point (r, z) in the reactor. They directly model only Steps 1 and 9, and the effects of Steps 2 through 8 are lumped into the pseudohomoge-neous rate expression, a, b,. ..), where a,b,. .. are the bulk, gas-phase concentrations. The overall reaction mechanism is complex, and the rate expression is necessarily empirical. Heterogeneous catalysis remains an experimental science. The techniques of this chapter are useful to interpret experimental results. Their predictive value is limited. 

Example 10.5 Derive a model of the Hougen and Watson type for the overall reaction 2A -> P, assuming that the surface reaction is the ratedetermining step. 

The concentration of gas over the active catalyst surface at location / in a pore is ai [). The pore diffusion model of Section 10.4.1 linked concentrations within the pore to the concentration at the pore mouth, a. The film resistance between the external surface of the catalyst (i.e., at the mouths of the pore) and the concentration in the bulk gas phase is frequently small. Thus, a, and the effectiveness factor depends only on diffusion within the particle. However, situations exist where the film resistance also makes a contribution to rj so that Steps 2 and 8 must be considered. This contribution can be determined using the principle of equal rates i.e., the overall reaction rate equals the rate of mass transfer across the stagnant film at the external surface of the particle. Assume A is consumed by a first-order reaction. The results of the previous section give the overall reaction rate as a function of the concentration at the external surface, a. ... 

At low temperatures the orders in CO and O2 are about -1 and while at high temperatures they become +1 and + (2, respectively. Hence, the orders of overall reactions should certainly not be treated as universal constants but rather as a convenient parameterization that is valid for a specific set of reaction conditions. We shall later see how these numbers become meaningful when we construct a detailed model for the overall process in terms of a number of elementary steps. The model should, naturally, be capable of describing what has been measured. 

Once the kinetic parameters of elementary steps, as well as thermodynamic quantities such as heats of adsorption (Chapter 6), are available one can construct a micro-kinetic model to describe the overall reaction. Otherwise, one has to rely on fitting a rate expression that is based on an assumed reaction mechanism. Examples of both cases are discussed this chapter. 

The first step in constructing a micro-kinetic model is to identify all the elementary reaction steps that may be involved in the catalytic process we want to describe, in this case the synthesis of ammonia. The overall reaction is... 

Table 10.4 lists the rate parameters for the elementary steps of the CO + NO reaction in the limit of zero coverage. Parameters such as those listed in Tab. 10.4 form the highly desirable input for modeling overall reaction mechanisms. In addition, elementary rate parameters can be compared to calculations on the basis of the theories outlined in Chapters 3 and 6. In this way the kinetic parameters of elementary reaction steps provide, through spectroscopy and computational chemistry, a link between the intramolecular properties of adsorbed reactants and their reactivity Statistical thermodynamics furnishes the theoretical framework to describe how equilibrium constants and reaction rate constants depend on the partition functions of vibration and rotation. Thus, spectroscopy studies of adsorbed reactants and intermediates provide the input for computing equilibrium constants, while calculations on the transition states of reaction pathways, starting from structurally, electronically and vibrationally well-characterized ground states, enable the prediction of kinetic parameters. 

The voltammograms at the microhole-supported ITIES were analyzed using the Tomes criterion [34], which predicts ii3/4 — iii/4l = 56.4/n mV (where n is the number of electrons transferred and E- i and 1/4 refer to the three-quarter and one-quarter potentials, respectively) for a reversible ET reaction. An attempt was made to use the deviations from the reversible behavior to estimate kinetic parameters using the method previously developed for UMEs [21,27]. However, the shape of measured voltammograms was imperfect, and the slope of the semilogarithmic plot observed was much lower than expected from the theory. It was concluded that voltammetry at micro-ITIES is not suitable for ET kinetic measurements because of insufficient accuracy and repeatability [16]. Those experiments may have been affected by reactions involving the supporting electrolytes, ion transfers, and interfacial precipitation. It is also possible that the data was at variance with the Butler-Volmer model because the overall reaction rate was only weakly potential-dependent [35] and/or limited by the precursor complex formation at the interface [33b]. 

Using a "home made" aneroid calorimeter, we have measured rates of production of heat and thence rates of oxidation of Athabasca bitumen under nearly isothermal conditions in the temperature range 155-320°C. Results of these kinetic measurements, supported by chemical analyses, mass balances, and fuel-energy relationships, indicate that there are two principal classes of oxidation reactions in the specified temperature region. At temperatures much lc er than 285°C, the principal reactions of oxygen with Athabasca bitumen lead to deposition of "fuel" or coke. At temperatures much higher than 285°C, the principal oxidation reactions lead to formation of carbon oxides and water. We have fitted an overall mathematical model (related to the factorial design of the experiments) to the kinetic results, and have also developed a "two reaction chemical model". 

The ultimate goal in reaction modeling is to discover the overall reaction that occurs within a system. Strangely, whereas the results of nearly every published study involving reaction modeling are presented in a spaghetti diagram (see, e.g., Fig. 13.1), few papers report the overall reaction. For this reason, some of the most... 

The procedure for determining the overall reaction, fortunately, is straightforward. The modeler plots the mole numbers of the species and minerals in the system against the mass of a reactant added to the system over the reaction path. The plot must be in linear coordinates with both axes in consistent units, such as moles or mmoles. 

Even neglecting the question of the precise steps that make up the overall reaction, our calculations are a considerable simplification of reality. The implicit assumption that iron in the fluid maintains redox equilibrium with the dissolved oxygen, as described in Chapter 7, is especially vulnerable. In reality, the ferrous iron added to solution by the dissolving pyrite must react with dissolved oxygen to produce ferric species, a process that may proceed slowly. To construct a more realistic model, we could treat the dissolution in two steps by disenabling the Fe++/Fe+++ redox couple. In the first step we would let pyrite dissolve, and in the second, let the ferrous species oxidize. 

Geochemists have long recognized the need for computational models to trace the progress of reaction processes, both natural and artificial. Given a process involving many individual reactions (possibly thousands), some of which yield products that provide reactants for others, how can we know which reactions are important, how far each will progress, what overall reaction path will be followed, and what the path s endpoint will be ... 

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