Kinetic Models for Reactions in Solids

There are significant differences between the kinetic models for reactions carried out in the solid phase and those taking place in gases or solutions. Therefore, it is appropriate to describe briefly some of the kinetic models that have been found to be particularly applicable to reactions in inorganic solids. 

As was discussed earlier in this chapter, the concept of a reaction order does not apply to a crystal that is not composed of molecules. However, there are numerous cases in which the rate of reaction is proportional to the amount of material present. We can show how this rate law is obtained in a simple way. If the amount of material at any time, t, is represented as W and if we let W0 be the amount of material initially present, the amount of material that has reacted at any time will be equal to (W0 — W). In a first-order reaction the rate is proportional to the amount of material. Therefore, the rate of reaction can be expressed as 

When integration is performed so that the amount of reactant is W0 at t = 0 and is W at time t, the result is 

We now need to transform the rate equation to one involving a, the fraction of the sample reacted, by making use of the relationship 

If the fraction reacted is a, then (1 - a) is the amount of sample unreacted, and we see that a plot of —ln(l - a) versus time would be linear with a slope of k. It should come as no surprise that reactions 

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Although not all facets of the reactions in which complexes function as catalysts are fully understood, some of the processes are formulated in terms of a sequence of steps that represent well-known reactions. The actual process may not be identical with the collection of proposed steps, but the steps represent chemistry that is well understood. It is interesting to note that developing kinetic models for reactions of substances that are adsorbed on the surface of a solid catalyst leads to rate laws that have exactly the same form as those that describe reactions of substrates bound to enzymes. In a very general way, some of the catalytic processes involving coordination compounds require the reactant(s) to be bound to the metal by coordinate bonds, so there is some similarity in kinetic behavior of all of these processes. Before the catalytic processes are considered, we will describe some of the types of reactions that constitute the individual steps of the reaction sequences. 

Because the radiation in most cases will not penetrate the entire sample, the concentration of the reactant is unlikely to approach zero at infinite time. A plot of remaining concentration vs. time will therefore level off at a value greater than zero. This should be taken into account when selecting the kinetic model for studies of solid-state degradation (Sande, 1996). The solid-state degradation will in some cases appear to consist of a series of consecutive processes with different mechanisms and rates (Carstensen, 1974). Such a stepwise change in reaction rate is most likely caused by an alteration in sample surface and fading of subsequent layers. The concept of reaction order may not be useful for photodecomposition in the solid state (De VUliers et al 1992). 

While these calculations provide information about the ultimate equilibrium conditions, redox reactions are often slow on human time scales, and sometimes even on geological time scales. Furthermore, the reactions in natural systems are complex and may be catalyzed or inhibited by the solids or trace constituents present. There is a dearth of information on the kinetics of redox reactions in such systems, but it is clear that many chemical species commonly found in environmental samples would not be present if equilibrium were attained. Furthermore, the conditions at equilibrium depend on the concentration of other species in the system, many of which are difficult or impossible to determine analytically. Morgan and Stone (1985) reviewed the kinetics of many environmentally important reactions and pointed out that determination of whether an equilibrium model is appropriate in a given situation depends on the relative time constants of the chemical reactions of interest and the physical processes governing the movement of material through the system. This point is discussed in some detail in Section 15.3.8. In the absence of detailed information with which to evaluate these time constants, chemical analysis for metals in each of their oxidation states, rather than equilibrium calculations, must be conducted to evaluate the current state of a system and the biological or geochemical importance of the metals it contains. 

Experimental determination of Ay for a reaction requires the rate constant k to be determined at different pressures, k is obtained as a fit parameter by the reproduction of the experimental kinetic data with a suitable model. The data are the concentration of the reactants or of the products, or any other coordinate representing their concentration, as a function of time. The choice of a kinetic model for a solid-state chemical reaction is not trivial because many steps, having comparable rates, may be involved in making the kinetic law the superposition of the kinetics of all the different, and often unknown, processes. The evolution of the reaction should be analyzed considering all the fundamental aspects of condensed phase reactions and, in particular, beside the strictly chemical transformations, also the diffusion (transport of matter to and from the reaction center) and the nucleation processes. 

A kinetic model for single-phase polymerizations— that is, reactions where because of the similarity of structure the polymer grows as a solid-state solution in the monomer crystal without phase separation—has been proposed by Baughman [294] to explain the experimental behavior observed in the temperature- or light-induced polymerization of substimted diacetylenes R—C=C—C=C—R. The basic feature of the model is that the rate constant for nucleation is assumed to depend on the fraction of converted monomer x(f) and is not constant like it is assumed in the Avrami model discussed above. The rate of the solid-state polymerization is given by... 

The several attempts, published in the literature, to describe the kinetics of vapour phase olefin (mostly ethylene) hydration can be classified into two groups according to the basic model used. One model, for reactions catalysed by phosphoric acid supported on solids, treats the kinetics as if the process were homogeneous acid catalysis and takes into account the acid strength of the supported acid. Thus, a semiempirical equation for the initial reaction rate [288]... 

The SNCR reaction has been studied extensively, and the detailed chemistry is fairly well established. Use the provided chemical kinetic model for SNCR (SNCR.mec [276]) together with a plug-flow code to assess the potential of SNCR in reducing NO in the municipal solid waste (MSW) facility, based on the available technical information. Determine the preferred agent injection location and the optimum NH3/NO ratio, with the restriction that the NH3 slip must not exceed 15 ppm. 

An essential step forward was also the development of kinetic models for electron tunneling reactions in solids [20-25]. Kinetic equations corresponding to these models were found to describe experimental data rather accurately. The agreement of experimental data with theory together with the absence of the temperature dependence for the reaction rate (which rules out its control by thermal diffusion) and with the evidence of considerable... 

Kinetics and Phase Behavior - Table IV represents a simplified picture of the situation however, some polymerizations go through several phase changes in the course of the reaction. For example, in the bulk polymerization of PVC, the reaction medium begins as a low viscosity liquid, progresses to a slurry (the PVC polymer, which is insoluble in the monomer, precipitates), becomes a paste as the monomer disappears and finishes as a solid powder. As might be expected, modelling the kinetics of the reaction in such a situation is not a simple exercise. 

For example, when we consider the design of specialty chemical, polymer, biological, electronic materials, etc. processes, the separation units are usually described by transport-limited models, rather than the thermodynamically limited models encountered in petrochemical processes (flash drums, plate distillations, plate absorbers, extractions, etc.). Thus, from a design perspective, we need to estimate vapor-liquid-solid equilibria, as well as transport coefficients. Similarly, we need to estimate reaction kinetic models for all kinds of reactors, for example, chemical, polymer, biological, and electronic materials reactors, as well as crystallization kinetics, based on the molecular structures of the components present. Furthermore, it will be necessary to estimate constitutive equations for the complex materials we will encounter in new processes. 

Heterogeneously catalyzed reactions are usually studied under steady-state conditions. There are some disadvantages to this method. Kinetic equations found in steady-state experiments may be inappropriate for a quantitative description of the dynamic reactor behavior with a characteristic time of the order of or lower than the chemical response time (l/kA for a first-order reaction). For rapid transient processes the relationship between the concentrations in the fluid and solid phases is different from those in the steady-state, due to the finite rate of the adsorption-desorption processes. A second disadvantage is that these experiments do not provide information on adsorption-desorption processes and on the formation of intermediates on the surface, which is needed for the validation of kinetic models. For complex reaction systems, where a large number of rival reaction models and potential model candidates exist, this give rise to difficulties in model discrimination. 

Figure 3 Idealized pH dependence of a ribozyme reaction. Ideal pH-species plots and pH-/cobs profiles according to a kinetic model for general acid/base catalysis. The solid lines depict a mechanism in which the species with the lower pKa (pKa,1) acts as the general base (shown by blue lines), and the species with the higher pKa (pKa,2) acts as the general acid (shown by red lines). The black line indicates the observed pH dependence of the reaction rate. The dotted lines simulate a mechanism in which the catalytic roles of the species with pKa,1 and pKa,2 have been switched. Adapted from References 34 and 35.

The rate equation specifies the mathematical fimction (g(ur) = ktox AodAt = k f(ur)) that represents (with greatest statistical accuracy. Chapter 3) the isothermal yield a) - time data for the reaction. For reactions of solids these equations are derived from geometric kinetic models (Chapter 3) involving processes such as nucleation and growth, advance of an interface and/or diflEusion. f( ir) and g(ar) are known as conversion functions and some of these may resemble the concentration functions in homogeneous kinetics which give rise to the definition of order of reaction. 

A number of workers have reported on kinetic models for plasma polymerization. Williams and Hayes (36) first suggested that the reaction occurred exclusively on solid surfaces within the reaction zone. Initially, monomer is adsorbed onto the electrode surface, where a portion is converted to free radical species after bombardment by ions and electrons produced in the plasma. Surface radicals then polymerize with adsorbed monomer to yield the thin film product. Based on this scheme, Denaro, et. al. derived a simple rate expression which showed reasonably good agreement with deposition rate data at various pressures and power levels (16). It is, however, unrealistic to assume that the plasma polymerization reactions occur exclusively on the surface. A more likely mechanism is that both gas phase and surface reactions proceed simultaneously in plasma polymer formation. 

Barrow [772] derived a kinetic model for sorption of ions on soils. This model considers two steps adsorption on heterogeneous surface and diffusive penetration. Eight parameters were used to model sorption kinetics at constant temperature and another parameter (activation energy of diffusion) was necessary to model kinetics at variable T. Normal distribution of initial surface potential was used with its mean value and standard deviation as adjustable parameters. This surface potential was assumed to decrease linearly with the amount adsorbed and amount transferred to the interior (diffusion), and the proportionality factors were two other adjustable parameters. The other model parameters were sorption capacity, binding constant and one rate constant of reaction representing the adsorption, and diffusion coefficient of the adsorbate in tire solid. The results used to test the model cover a broad range of T (3-80°C) and reaction times (1-75 days with uptake steadily increasing). The pH was not recorded or controlled. 

Kinetic models for catalytic reactions vary widely in sophistication, generality and accuracy. The simplest, least general and easiest to obtain are the power law kinetic models which are usually strictly empirical and contain a limited number of parameters. Consequently they are valid only in a narrow region of parameters and cannot be safely extrapolated outside this region. Kinetic models which take into consideration the interaction between the gas phase and the solid surface (whether through a reduction-oxidation mechanism or an adsorption-desorption mechanism) are more difficult to formulate and usually contain a large number of parameters. However, they are certainly more reliable, general and accurate than power law kinetics. 

Based on this elementary knowledge of the intrinsic kinetics of gas-solid catalytic systems, CSD kinetic models for some industrially important catalytic reactions can be described. Some of the cases will be presented briefly and others with interesting features will be presented in details together with their history of research. The cases with detailed exposition are chosen on the basis of their practical importance as well as the features that will help to highlight some of the important points that need to be emphasized in this chapter. A wide range of cases, with the exception of the partial oxidation reactions, will be discussed. 

In this section some of the kinetic models for a number of industrially important gas-solid catalytic reactions, are discussed. The historical evolution towards more rigorous CSD models are traced and the degree of reliability of the present models discussed. From the view point of this book which is aiming at introducing mathematical models for industrial catalytic reactors, the ultimate... 

The Langmuir isotherm is the most widely used in the derivation of kinetic models for gas-solid catalytic reactions. The derivation of a Langmuir isotherm depends on the kinetic method where the equilibrium conditions are determined by the equality of the rates of adsorption and desorption. At equilibrium the following relation holds,... 

Use of Kinetic Models for Solid State Reactions in Combustion Simulations... 

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