The reaction rate

While the forward reaction rate for the net reaction may depend on the concentration of both reactants and products, the forward rate of each elementary step can depend only on the concentration of reactants for this step. 

This leads to the expression of the rate of as the number of times the reaction proceeds per second, the turnover frequency. The rate is thus a rate for the reaction, not the rate 

If ni is the number of moles produced of product number i and Vi is the stochiometric coefficient for product number i, the turnover frequency is 

This turnover frequency is obviously the same for all products. The reactants have negative stochiometric coefficients and are produced with negative rate, so the turnover frequency is actually the same for all reactants and products. 

The rate at which reactants are converted to products is a key quantity for characterizing a chemical reaction. The rate of a stoichiometric reaction may be dehned by dividing the observed rate of consumption of moles of each individual reactant and the rate of formation of moles of each individual product by their respective stoichiometric coefficients, where the rates are time derivatives. Division by the stoichiometric coefficients reduces the individual species rates to a common value, R, having the dimensions of moles per unit time. For the stoichiometric reaction of equation (1) the rate is defined by 

The species concentration, [A,], is dehned by [A,] — n,/F. At constant volume equation (6) reduces to 

This equation suffices to describe reaction rates in systems where the [A,] are spatially uniform. If there are concentration non-uniformities an average rate may be obtained by integrating the rate over the reactor 

In open chemical reaction systems where matter is exchanged with the surroundings, concentration changes may also occur by forced flow, convection, and diffusion, and these must be taken into account in an equation of change. In most research on reaction rates experiments are designed to exclude transport effects on the rates of concentration change, since these may compUcate, or even obscure, the principal objectives of the work. The interaction of transport and kinetics may be very important in practical chemical reaction systems. In cases where this is so reaction models become much more complex than otherwise. 

Having defined reaction increments df, we can now define the rate of reaction as 

This expression (11.9) is written in terms of the absolute number of moles of A, B, and so on, ( a, b, ), but by considering a fixed volume we could change these to concentration terms. Thus, 

evidently, the rate of reaction can be determined by measuring the concentration of any of the reactants or products as a function of time. With one important stipulation the reaction we have written must be what is actually happening. 

If we measure the rate of change of concentration of products and reactants in many ordinary chemical reactions, we find that the relationship in (11.10) is often not obeyed. This is because the reaction does not actually proceed as written, at the molecular level. For example, Reaction (11.5), taken literally, indicates that a molecule of A reacts with 2 molecules of B, and at that instant, 3 molecules of C and 4 molecules of D are formed. But this might not be what happens at all, and in view of the improbability of three molecules (A + 2 B) meeting at a single point, it probably is not in this case. The reaction as written may well represent the overall result of a series of elementary reactions. Thus A and B may in fact react to form a number of intermediate species such as X and Y, which then react with each other or with A or B to form C and D. In thermodynamics, the existence of such intermediate species is not important to the study of the overall reaction, as long as equilibrium is attained, but, in kinetics, these intermediate species contribute to the overall rate of reaction and may actually be ratecontrolling, even though their concentrations may be small. 

The reaction mechanism is the description of an overall reaction in terms of the separate elementary reactions that are involved. 

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For homogeneous reactions we obtain the conventional definition of the reaction rate u. as rate of conversion per volume... 

The slopes of the fimctions shown provide the reaction rates according to the various definitions under the reaction conditions specified in the figure caption. These slopes are similar, but not identical (nor exactly proportional), in this simple case. In more complex cases, such as oscillatory reactions (chapter A3.14 and chapter C3.6). the simple definition of an overall rate law tluough equation (A3.4.6) loses its usefiilness, whereas equation (A3.4.1) could still be used for an isolated system. 

If certain species are present in large excess, their concentration stays approximately constant during the course of a reaction. In this case the dependence of the reaction rate on the concentration of these species can be included in an effective rate constant The dependence on the concentrations of the remaining species then defines the apparent order of the reaction. Take for example equation (A3,4.10) with e. The... 

In the thennodynamic fomiiilation of TST the pressure dependence of the reaction rate coefficient defines a volume of activation [24, 25 and 26]... 

Berezhkovskii A M and Zitserman V Yu 1993 Multi-dimensional Kramers theory of the reaction rate with highly... 

This equation represents the reaction rate at total energy E with a fixed energy in the reaction coordinate and may be written as... 

The simplest manifestation of nonlinear kinetics is the clock reaction—a reaction exliibiting an identifiable mduction period , during which the overall reaction rate (the rate of removal of reactants or production of final products) may be practically indistinguishable from zero, followed by a comparatively sharp reaction event during which reactants are converted more or less directly to the final products. A schematic evolution of the reactant, product and intenuediate species concentrations and of the reaction rate is represented in figure A3.14.2. Two typical mechanisms may operate to produce clock behaviour. 

Flere, A and B are regarded as pool chemicals , with concentrations regarded as imposed constants. The concentrations of the intemiediate species X and Y are the variables, with D and E being product species whose concentrations do not influence the reaction rates. The reaction rate equations for [X] and [Y] can be written in the following dimensionless fomi ... 

Under diffusion-controlled dissolution conditions (in the anodic direction) the crystal orientation has no influence on the reaction rate as only the mass transport conditions in the solution detennine the process. In other words, the material is removed unifonnly and electropolishing of the surface takes place. 

Fast transient studies are largely focused on elementary kinetic processes in atoms and molecules, i.e., on unimolecular and bimolecular reactions with first and second order kinetics, respectively (although confonnational heterogeneity in macromolecules may lead to the observation of more complicated unimolecular kinetics). Examples of fast thennally activated unimolecular processes include dissociation reactions in molecules as simple as diatomics, and isomerization and tautomerization reactions in polyatomic molecules. A very rough estimate of the minimum time scale required for an elementary unimolecular reaction may be obtained from the Arrhenius expression for the reaction rate constant, k = A. The quantity /cg T//i from transition state theory provides... 

A point in case is provided by the bromination of various monosubstituted benzene derivatives it was realized that substituents with atoms carrying free electron pairs bonded directly to the benzene ring (OH, NH2, etc) gave 0- and p-substituted benzene derivatives. Furthermore, in all cases except of the halogen atoms the reaction rates were higher than with unsubstituted benzene. On the other hand, substituents with double bonds in conjugation with the benzene ring (NO2, CHO, etc.) decreased reaction rates and provided m-substituted benzene derivatives. 

The reaction rate equations give differential equations that can be solved with methods such as the Runge-Kutta [14] integration or the Gear algorithm [15]. 

When a chemical reaction takes place at the solid surface, we expect a smooth variation in gas composition in the macropores on a scale comparable with the whole pellet, provided the reaction rate is not too high. 

Without assuming something specific about the reaction rate, it is not possible to go further and actually determine the micropore fluxes. We shall therefore consider the particular case of a first order reversible isomeri-... 

In particular if the reaction rate depends only on Cj, which is the case, for example, if the reaction is irreversible with mass,-action kinetics, then these reduce further to a pair of equations, namely... 

Equation (12.29) then represents the material balance on species A, while equation (12.30) represents the overall material balance. The reaction rate... 

Apart from this simple result, comparison of stability predictions for the two limiting situations can be made only by direct numerical computation, and for this purpose a specific algebraic form must be assumed for the reaction rate function, and a specific shape for che catalyst pellet. In particular, Lee and Luss considered a spherical pellet and a first order... 

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