Kinetic rate equations, geometric with

It is usually assumed in the derivation of isothermal rate equations based on geometric reaction models, that interface advance proceeds at constant rate (Chap. 3 Sects. 2 and 3). Much of the early experimental support for this important and widely accepted premise derives from measurements for dehydration reactions in which easily recognizable, large and well-defined nuclei permitted accurate measurement. This simple representation of constant rate of interface advance is, however, not universally applicable and may require modifications for use in the formulation of rate equations for quantitative kinetic analyses. Such modifications include due allowance for the following factors, (i) The rate of initial growth of small nuclei is often less than that ultimately achieved, (ii) Rates of interface advance may vary with crystallographic direction and reactant surface, (iii) The impedance to water vapour escape offered by... 

The King and Altman Method. King and Altman developed a systematic approach for deriving steady-state rate equations, which has contributed to the advance of enzyme kinetics. The first step of this method is to draw an enclosed geometric figure with each enzyme form as one of the corners. Equation (5), for instance, can be rewritten as ... 

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. 

In this section several empirical rate expressions for Ziegler-Natta polymerizations will be presented and attempts to model the polymerization will be described. It is found that several models could be proposed to explain the same rate equations. The models are based on the assumption of a fixed geometric center that has a definable identity and activity invariant with time. These assumptions, however, are far too simplistic and only limited general agreement of the models with the observed kinetic behavior or good agreement only in specific cases could be expected. 

The retarding influence of the product barrier in many solid—solid interactions is a rate-controlling factor that is not usually apparent in the decompositions of single solids. However, even where diffusion control operates, this is often in addition to, and in conjunction with, geometric factors (i.e. changes in reaction interfacial area with a) and kinetic equations based on contributions from both sources are discussed in Chap. 3, Sect. 3.3. As in the decompositions of single solids, reaction rate coefficients (and the shapes of a—time curves) for solid + solid reactions are sensitive to sizes, shapes and, here, also on the relative dispositions of the components of the reactant mixture. Inevitably as the number of different crystalline components present initially is increased, the number of variables requiring specification to define the reactant completely rises the parameters concerned are mentioned in Table 17. 

In the broadest sense, I found the analogy with fluid mechanics to be very helpful. Just as kinematics provides the geometrical framework of fluid mechanics by exploring the motions that are possible, so also stoicheiometry defines the possible reactions and the restrictions on them without saying whether or at what rate they may take place. When dynamic laws are imposed on kinematic principles, we arrive at equations of motion so, also, when chemical kinetics is added to stoicheiometry, we can speak about reaction rates. In fluid mechanics different materials are distinguished by their constitutive relations and allow equations for the density and velocity to be formulated thence, various flow situations are examined by adding appropriate boundary conditions. Similarly, the chemical kinetics of the reaction system allow the rates of reaction to be expressed in terms of concentrations, and the reactor is brought into the picture as these rates are incorporated into appropriate equations and their boundary conditions. 

Values of measured for reversible reactions under conditions permitting absorption equilibration between gas and solid products are often comparable with the reaction enthalpy [1,47]. Under such conditions the identification of the rate expression, g(nr) = kt, on the single criterion that this equation gives the most acceptable correlation coefficient is not a sufficient foimdation to characterize a geometric reaction model. The use of additional information, for example microscopy, can provide confirmatory evidence concerning interface development. Similarly, the value of studies which conclude that kinetic results are satisfactorily described by equations based on diffusion models is increased considerably if the identity of the migrating species is established [53]. 

In analytical models, it is assumed that the particles in the initial powder compact are spherical with the same size and uniform packing, which is called the geometrical model [6]. With appropriate boundary conditions, the remainder of the powder system is considered as a continuum, having the same macroscopic properties, such as shrinkage and densification rate, as the isolated unit. The equations of the sintering kinetics can be derived from the established mass transport equations, which are solved under appropriate boundary conditions. 

Equations relating x and t have been derived for simple geometrical systems assuming (1) the reaction rate is phase-boundary controlled, (2) the reaction rate is proportional to the surface area of the fraction of unreacted material, and (3) the nucleation step occurs virtually instantaneously so that the surface of each particle is covered with a layer of product. The models developed from the foregoing boundary conditions are termed phase boundary or contracting volume kinetic models. For a sphere reacting from the surface inward ( ) the relationship between x and t is... 

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