KINETIC MODELS FOR SOLID-STATE REACTIONS

KINETIC MODELS FOR SOLID STATE REACTIONS 3.1. INTRODUCTION 

Factors that have been identified as controlling the rates of most solid state reactions, alone or in combination, include the following. 

Other types of rate control have been proposed, usually with reference to particular reactions. Some of these are mentioned in the later chapters concerned with the thermal behaviour of specific compounds. 

Several kinetic models, based on quantitative consideration of the three factors above, have been developed and are discussed in this Chapter. Valuable reviews of the development of such models representing the way in which the extent of reaction, a, may be expected to vary with time, at constant temperature, have been given by Jacobs and Tompkins [1], Young [2], Delmon [3], Barret [4], Brown et al. [5], Sestak [6], Schmalzried [7] and, most recently, Jacobs [8]. 

The geometric models of solid state reactions are based upon the processes of nucleation and growth of product nuclei by interface advance. These processes are discussed individually in the next section, followed by a description of the ways in which these contributions are combined to give rate equations for the overall progress of reaction. 

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

Carter RE (1961) Kinetic model for solid-state reactions. J Chem Phys 34 2010-2015... 

R. E. Carter, Addendum Kinetic Model for Solid State Reactions, J. Chem. Phys. 35(3) 1137-38 (1961). 

Since the traditional kinetic models of solid-state reactions are often based on a formal description of geometrically well defined bodies treated under strictly isothermal conditions, they are evidently not appropriate to describe the real process, which requires accoimt to be taken of irregularity of shape, polydispersity, shielding and overlapping, unequal mixing anisotropy and so on, for sample particles under reaction. One of the measures which has been taken to solve the problem is to introduce an accommodation function a a) [32]. The discrepancy between the idealized /(a) and the actual kinetic model function h a) can be expressed as... 

It is clear that the experimental curves, measured for solid-state reactions under thermoanalytical study, cannot be perfectly tied with the conventionally derived kinetic model functions (cf. previous table lO.I.), thus making impossible the full specification of any real process due to the complexity involved. The resultant description based on the so-called apparent kinetic parameters, deviates from the true portrayal and the associated true kinetic values, which is also a trivial mathematical consequence of the straight application of basic kinetic equation. Therefore, it was found useful to introduce a kind of pervasive des-cription by means of a simple empirical function, h(a), containing the smallest possible number of constant. It provides some flexibility, sufficient to match mathematically the real course of a process as closely as possible. In such case, the kinetic model of a heterogeneous reaction is assumed as a distorted case of a simpler (ideal) instance of homogeneous kinetic prototype f(a) (1-a)" [3,523,524]. It is mathematically treated by the introduction of a multiplying function a(a), i.e., h(a) =f(a) a(a), for which we coined the term [523] accommodation function and which is accountable for a certain defect state (imperfection, nonideality, error in the same sense as was treated the role of interface, e.g., during the new phase formation). 

Chapter 10 begins a more detailed treatment of heterogeneous reactors. This chapter continues the use of pseudohomogeneous models for steady-state, packed-bed reactors, but derives expressions for the reaction rate that reflect the underlying kinetics of surface-catalyzed reactions. The kinetic models are site-competition models that apply to a variety of catalytic systems, including the enzymatic reactions treated in Chapter 12. Here in Chapter 10, the example system is a solid-catalyzed gas reaction that is typical of the traditional chemical industry. A few important examples are listed here ... 

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... 

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. 

A different model [11] that can be used to obtain the kinetics equation for a pyrolytic reaction is adapted from the theory developed for the kinetics of heterogeneous catalytic reactions. This theory is described in literature for various cases regarding the determining step of the reaction rate. The case that can be adapted for a pyrolytic process in solid state is that of a heterogeneous catalytic reaction with the ratedetermining step consisting of a first-order unimolecular surface reaction. For the catalytic reaction of a gas, this case can be written as follows ... 

This simplified model of the decomposition process is provided here to introduce the more detailed accounts given in the later chapters. The relatively simple processes now accepted as being involved in decompositions have provided a model for developing kinetic principles apphcable to a wider range of solid state reactions. 

The factors governing the slow thermal decompositions of inorganic azides have been discussed by Fox and Hutchinson [18]. They draw attention to the interest shown in early studies for fitting of kinetic results to rate equations based on nucleation and growth models. Support of kinetic interpretations by microscopic observations (e g., [21]), contributed significantly towards establishment of the role of the active, advancing interface in solid state reactions. The kinetic characteristics of some of the metal azides are summarized in Table 11.1. 

In a later study, Grenier-Loustalot et al. [40] examined the reaction of monoacids and monoamines as models for the condensation reaction. Peaks due the acid and amide were resolved in the solid-state NMR spectra, however, the linewidths in the spectra of these model compounds were appreciably smaller than those expected for the conformationally-complex poly(amic acid). In this paper, the authors used HPLC and FTIR to determine the effect of size and basicity of the amine, as well as the removal of water on the kinetics of reaction. Full imidization was achieved only on the addition of catalysts to the reaction mixture. 

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). 

Introduction Dehydrations of metal hydroxides are attractive model reactions for basic studies of the kinetics of solid-state reactions and these reactions are widely used for the commercial production of metal oxides [45]. However, as shown in the recent paper by L vov and Ugolkov [55], available data on the reaction mechanisms and kinetics are inconsistent. For example, the parameter E for the dehydration of Mg(OH)2, reported in different papers, varies from 53 to 372 kJ mol One of the factors responsible for the large scatter of the E values estimated from Arrhenius plots is the low precision and accuracy of this method, especially as applied to decomposition to gaseous and solid products. The results obtained in [55[ by the third-law method, as indicated below, are much more reliable. 

Introduction Alkaline-earth metal carbonates and especially calcite are the most popular reactants in studying the decomposition kinetics of solids. This is caused, not only by the use of carbonates as mineral raw materials for some industrial processes (in particular, production of lime), but also by the fact that the decomposition of calcite is a convenient model reaction for studying the kinetics and mechanism of solid-state reactions as a whole. An enormous number of papers, summarized in part in several monographs [43, 45, 99] and reviews [100-102], deal with the decomposition of carbonates. A series... 

The kinetic evolution is usually represented by a sigmoid-type curve. Such a typical curve is given in Figure 9 for the case of displacement reaction, Ni + CuO Cu -F NiO realized in planetary ball mill [58]. Author of this overview analyzed these experimental results by one of the most frequently used kinetic model applied to various solid-state reactions, namely Johnson-Mehl-Avrami equation ... 

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