Basic Kinetic Concepts and Situations

We will introduce basic kinetic concepts that are frequently used and illustrate them with pertinent examples. One of those concepts is the idea of dynamic equilibrium, as opposed to static (mechanical) equilibrium. Dynamic equilibrium at a phase boundary, for example, means that equal fluxes of particles are continuously crossing the boundary in both directions so that the (macroscopic) net flux is always zero. This concept enables us to understand the non-equilibrium state of a system as a monotonic deviation from the equilibrium state. Driven by the deviations from equilibrium of certain functions of state, a change in time for such a system can then be understood as the return to equilibrium. We can select these functions of state according to the imposed constraints. If the deviations from equilibrium are sufficiently small, the result falls within a linear theory of process rates. As long as the kinetic coefficients can be explained in terms of the dynamic equilibrium properties, the reaction rates are directly proportional to the deviations. The thermodynamic equilibrium state is chosen as the reference state in which the driving forces X, vanish, but not the random thermal motions of structure elements i. Therefore, systems which we wish to study kinetically must first be understood at equilibrium, where the SE fluxes vanish individually both in the interior of all phases and across phase boundaries. This concept will be worked out in Section 4.2.1 after fluxes of matter, charge, etc. have been introduced through the formalism of irreversible thermodynamics. 

Following the introduction of basic kinetic concepts, some common kinetic situations will be discussed. These will be referred to repeatedly in later chapters and include 1) diffusion, particularly chemical diffusion in different solids (metals, semiconductors, mixed conductors, ionic crystals), 2) electrical conduction in solids (giving special attention to inhomogeneous systems), 3) matter transport across phase boundaries, in particular in electrochemical systems (solid electrode/solicl electrolyte), and 4) relaxation of structure elements. 

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In the last four sections, we have illustrated some basic kinetic concepts. We will repeatedly meet the underlying kinetic situations in the following chapters. In one way or the other, they will serve as starting points when we later analyze and discuss more complicated kinetic problems in greater depth. 

In this first chapter, we will outline the scope of this book on the kinetics of chemical processes in the solid state. They are often different from the kinetics of processes in fluids because of structural constraints. After a brief historical introduction, typical situations of non-equilibrium crystals will be described. These will illustrate some basic concepts and our approach to understanding solid state kinetics. 

By necessity, the treatment of solid state kinetics has to be selective in view of the myriad processes which can occur in the solid state. This multitude is mainly due to three facts 1) correlation lengths in crystals are often much larger than in fluids and may comprise the whole crystal, 2) a structure element is characterized by three parameters instead of only by two in a liquid (chemical species, electrical charge, type of crystallographic site), and 3) a crystal can be elastically stressed. The stress state is normally inhomogeneous. If the yield strength is exceeded, then plastic deformation and the formation of dislocations will change the structural state of a crystal. What we aim at in this book is a strict treatment of concepts and basic situations in a quantitative way, so far as it is possible. In contrast, the often extremely complex kinetic situations in solid state chemistry and materials science will be analyzed in a rather qualitative manner, but with clearcut thermodynamic and kinetic concepts. 

In Chapter 3 we described the structure of interfaces and in the previous section we described their thermodynamic properties. In the following, we will discuss the kinetics of interfaces. However, kinetic effects due to interface energies (eg., Ostwald ripening) are treated in Chapter 12 on phase transformations, whereas Chapter 14 is devoted to the influence of elasticity on the kinetics. As such, we will concentrate here on the basic kinetics of interface reactions. Stationary, immobile phase boundaries in solids (e.g., A/B, A/AX, AX/AY, etc.) may be compared to two-phase heterogeneous systems of which one phase is a liquid. Their kinetics have been extensively studied in electrochemistry and we shall make use of the concepts developed in that subject. For electrodes in dynamic equilibrium, we know that charged atomic particles are continuously crossing the boundary in both directions. This transfer is thermally activated. At the stationary equilibrium boundary, the opposite fluxes of both electrons and ions are necessarily equal. Figure 10-7 shows this situation schematically for two different crystals bounded by the (b) interface. This was already presented in Section 4.5 and we continue that preliminary discussion now in more detail. 

In this chapter some of the theoretical concepts used in these models will be outlined. In particular, emphasis will be given to the chemical thermodynamic principles that can be used to predict the stable forms of a given element. Such chemical principles provide the theoretical foundation of the commonly used chemical models. These models can be used to predict the final extent of reaction but not the rate. It is probably fair to say that these laws as basic principles are indisputable scientific fact however, problems arise when we try to apply them to ill-defined complex natural media such as soils and soil solutions where some reactions are kinetically slow and practically irreversible. However inadequate our chemical models are in relation to real-world situations they are the best we have and can be used to give valuable insight and meaning into the processes we observe. 

Diffusion is the mass transfer caused by molecular movement, while convection is the mass transfer caused by bulk movement of mass. Large diffusion rates often cause convection. Because mass transfer can become intricate, at least five different analysis techniques have been developed to analyze it. Since they all look at the same phenomena, their ultimate predictions of the mass-transfer rates and the concentration profiles should be similar. However, each of the five has its place they are useful in different situations and for different purposes. We start in Section 15.1 with a nonmathematical molecular picture of mass transfer (the first model) that is useful to understand the basic concepts, and a more detailed model based on the kinetic theory of gases is presented in Section 15.7.1. For robust correlation of mass-transfer rates with different materials, we need a parameter, the diffusivity that is a fundamental measure of the ability of solutes to transfer in different fluids or solids. To define and measure this parameter, we need a model for mass transfer. In Section 15.2. we discuss the second model, the Fickian model, which is the most common diffusion model. This is the diffusivity model usually discussed in chemical engineering courses. Typical values and correlations for the Fickian diffusivity are discussed in Section 15.3. Fickian diffusivity is convenient for binary mass transfer but has limitations for nonideal systems and for multicomponent mass transfer. 

The book is therefore situated at the interface between physical chemistry (classical thermodynamics and statistical mechanics, chemical kinetics, transport phenomena) and the theory of reactors, themselves at the heart of chemical reaction engineering. It therefore possesses a marked pluridisciplinary character. However, in order to keep this book readable by newcomers to the fields both of GPTRs and the kinetic modelling of reactions, basic concepts, theories and laws of the underlying scientific disciplines are given. The main equations are illustrated by simple numerical applications in order to show how the data tables are used. 

There arc other criticisms of the Avrami method here. Negahban points out that modeling of crystallization kinetics using Avrami-type equations is incompatible with the entropy production inequahty (i.e., Clausius Duhem inequahty) that is basic to die termination of crystallization in real situations. Much of his work has concentrated on the crystallization of rubber (hterature results) with all its ancillary effects/property-wise material functions that arc amenable to simulation, hr the same marmer, the kinetic theory of crystallization, based upon concepts of molecular chain-folding and it s related parameters,at the same time recognizing some of the shortcomings involved. An attempt has been made to keep... 

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