Geometric rate equations

Rate equations expressing the a—time variations resulting from the inward advance of a reaction interface from the existing surfaces of other reactant shapes follow directly by the application of simple geometric considerations. The approach can also include quantitative allowance for any... 

When reaction is absent from certain crystallographic surfaces, the formulation of rate equations based on geometric considerations proceeds exactly as outlined above, but includes only the advance of interfaces into the bulk of the reactant particle from those crystallographic surfaces upon which the coherent reactant/product contact is initially established. When reaction occurs only at the edges of a disc or plate-like particle... 

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 ... 

Mark Z. Lazman and Gregory S. Yablonsky, Overall Reaction Rate Equation of Single-Route Complex Catalytic Reaction in Terms of Hyper geometric Series... 

Finally, the kinetics approximations cannot be forgotten. Usually, in order to put in evidence structure-activity relationships, a simple parameter, the TOF, is used. The TOF, which reflects the rate per accessible site, contains the combination of all the adsorption and surface reaction elementary steps. Each of these steps is dependent on adsorption and/or rate constants. For that reason, the significance of TOF dependence as a function of structural parameters, e.g., the particle size, is not obvious since the rate equation can be particle-size-dependent [17]. Moreover, the adsorption and surface reaction steps may exhibit very different sensitivities to electronic and geometrical features. 

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. 

The rate equations which have found most widespread application to solid state reactions are summarized in Table 3.3. Other functions can be found in the literature. The expressions are grouped according to the shape of the isothermal a-time curves as acceleratory, sigmoid or deceleratory. The deceleratory group is further subdivided according to the controlling factor assumed in the derivation, as geometrical, diffusion or reaction order. 

Some of the isothermal relative rate [= (dfli d/V(da/d/LJ - time plots corresponding to the rate equations in Table 3.3. (a) sigmoid models (b) geometrical and reaction order (RO) models. (NOTE (i) the relative rate - time plots for the third-order rate equation and the difhision models are too deceleratory for usehil comparison and (ii) calculation on an arbitrary basis, e.g. that a = 0.98 at / = 100 min., results in plots of relative rates against a, in place of time, having similar shapes). 

Reaction mechanism as currentiy used in solid state chemistry is an ambiguous term. Sometimes it is used to refer to the geometrical progress of the reactant-product interface, i.e. the fit of data to a particular rate equation (Table 3.3.), rather than the chemical steps taking place at the interface. Because the first interpretation is experimentally more accessible, many studies do not proceed further. Throughout this book the reaction mechanism refers to the sequence of chemical steps through which the reactant is transformed into products. 

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. 

The principles underlying the formulation of rate equations applicable to the decompositions of solids are presented in Section 5.4. In summary, these result in the replacement of the concentration terms generally applicable in homogeneous rate processes by geometric or diffusion parameters. It is possible, in principle, to formulate a further set of kinetic models that describe concurrent reactions proceeding... 

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. 

In Equation (6.5) the first term on the ri -hand side represaits the geometric change in the bed element due to solids dq>osition, tiie second the decrease in internal suifiice area (due to clogging) and the final term the flow through the bed at maxinnmi loading. Equations (6.4) or (6.5) can be evaluated enq>irically and used with the mass balance and rate equations, Equation (6.1) and (6.2), to obtain dq>osited sofids concentration as a function of position and time. 

This demonstrates that the saturation parameter can also be expressed as the square of the ratio y yi of the Rabi frequency Q 2 at resonance (co = cji>]2) and the geometric mean of the relaxation rates of 1) and 2). In other words, when the atoms are exposed to light with intensity I = 7s, their Rabi frequency is 2 2 = y K2- The saturation parameter defined by (7.6) for the open two-level system is more general than that defined in (3.69) for a closed two-level system. The difference lies in the definition of the mean relaxation probability, which is / = (7 i-h 7 2)/2 in the closed system but 7 = R R2j R +R2) in the open system. We can close our open system defined by the rate equations (7.3) by setting C = R2N2, C2 = R Ni, and N + N2 = N = const, (see Fig. 7.1b). The rate equations then become identical to (3.66) and 7 converts to R,... 

Thus, in a steady cylindrical radial flow without mudcake effects, Vt front displacement is obtained because of geometric divergence (Equation 6-15 assumes a constant volume flow-rate with constant porosity together with r = r yr at t = 0). We emphasize that this Vt behavior differs from the Vt obtained... 

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