Expression of the reaction rate

We have seen that the reaction rate could almost be expressed as the propagation rate of the reaction (see relation [12.3]). 

We can also express this rate in a pseudo-steady state mode, by writing that whenever the active center is formed by initiation, v numbers of active centers disappear. 

We consider the case of a single reactant A and product B, and only one active center (as in Table 12.1). Each active center disappears during a reaction and has 

In the case of two reactants, A and B, two products, F and G, and two alternating active centers (as in Table 12.2), an active center formed by initiation involves the 

These expressions imply that the breaking reactions do not involve reactants and the initiation reaction does not involve a reactant or, if aity of these conditions is not met, that the chains are long enough. 

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As a result, it was obtained an analytical expression of the reaction rate in terms of hypergeometric series with no classical simplifications about the "limiting step" or the "vicinity of the equilibrium". The obtained explicit equation, "four-term equation", can be presented as follows in the Equation (77) ... 

In this expression the acetylene is taken as the key component, A, and the reaction rate is calculated knowing that the limiting step in the mass transfer is the gas/liquid interface. The expression of the reaction rate may be expressed as ... 

The second part (sections H and I) is devoted to a detailed discussion of the dynamics of unimolecular reactions in the presence and the absence of a potential barrier. Section H presents a critical examination of the Kramers approach. It is stressed that the expressions of the reaction rates in the low-, intermediate-, and high-friction limits are subjected to restrictive conditions, namely, the high barrier case and the quasi-stationary regime. The dynamics related to one-dimensional diffusion in a bistable potential is analyzed, and the exactness of the time dependence of the reaction rate is emphasized. The essential results of the non-Markovian theory extending the Kramers conclusions are also discussed. The final section investigates in detail the time evolution of an unimolecular reaction in the absence of a potential barrier. The formal treatment makes evident a two-time-scale description of the dynamics. 

In practice, we need an expression of the reaction rate with the pressure of the two reactants. A general expression cannot be obtain but in two limiting cases it has been shown that simple expressions are available [124] ... 

We can avoid these conversion factors that relate the different rate expressions, by defining the rate in terms of an equivalent concentration instead of the molar concentration. If X is the number of equivalents per liter that reacted in a time t, then dX/dt is a convenient expression of the reaction rate. However, the definition of equivalent must be made explicit. 

The final kinetic equation. Inserting a monochromatic form of equation 6.118 into equation 6.109 we get an expression of the reaction rate per particle and monochromatic radiation as follows ... 

Under these conditions, an electric current is an expression of the reaction rate. Now the electrode is not necessarily in thermodynamic equilibrium with the analyte. It is not necessary to wait until the equihbrium has been estab-Ushed. Consequently, commonly the response time of amperometric sensors is magnitudes shorter than that of potentiometric sensors (Morf and de Rooij 1995). The strict concentration proportionality of the measured quantity (the Umiting current) is another appreciated property of this sensor group. 

Polymerization of fluorocarbon monomers proceeds by a free radical mechanism. Two types of polymerization are possible homopolymerization and heteropolymerization or copolymerization. Homopolymerization only involves one monomer while copolymerization requires more than one monomer. An example of the free radical reaction and the corresponding reaction rate is illustrated by homopolymerization in Sec. 5.2.1. Copolymerization involves two or more monomers with a significantly more complex expression of the reaction rate. Section 5.2.2 illustrates the derivation of the general copolymer equation. 

The Michaelis-Menten equation uses both and to simplify the expression of the reaction rate of this mechanism... 

This is the Rice-Ramsperger-Kassel-Marcus (RRKM) expression of the reaction rate constant at energy E [127]. For total energies E below the potential energy of the transitions state, i.e., for E < F(/ts), k E) vanishes. The canonical and the microcanonical rate constants are related by a Laplace transform, ... 

When a chemical reaction involves stoichiometric coefficients that are different from one (1), it is necessary to take the stoichiometry into account in the expression of the reaction rate. 

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... 

Before discussing such theories, it is appropriate to refer to features of the reaction rate coefficient, k. As pointed out in Sect. 3, this may be a compound term containing contributions from both nucleation and growth processes. Furthermore, alternative definitions may be possible, illustrated, for example, by reference to the power law a1/n = kt or a = k tn so that k = A exp(-E/RT) or k = n nAn exp(—nE/RT). Measured magnitudes of A and E will depend, therefore, on the form of rate expression used to find k. However, provided k values are expressed in the same units, the magnitude of the measured value of E is relatively insensitive to the particular rate expression used to determine those rate coefficients. In the integral forms of equations listed in Table 5, units are all (time) 1. Alternative definitions of the type... 

For A l, the Faradaic efficiency A has, as already noted, an interesting physical meaning50 For oxidation reactions it expresses the ratio of the reaction rates of normally chemisorbed atomic oxygen on the promoted... 

As in collision theory, the rate of the reaction depends on the rate at which reactants can climb to the top of the barrier and form the activated complex. The resulting expression for the rate constant is very similar to the one given in Eq. 15, and so this more general theory also accounts for the form of the Arrhenius equation and the observed dependence of the reaction rate on temperature. 

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 ... 

The simplest form of a physicochemical reaction takes place when one species simply changes to another. This can be written in a general way as A B. The rate of such a reaction is defined as the amount of reactant (the reacting species, A, in this case) or equivalently the product (B) that changes per unit time. The key feature here is the form of the rate law, i.e., the expression for the dependence of the reaction rate on the concentrations of the reactants. For a first-order reaction... 

The notation is such that (Qs,j) , (Qf,j) denotes the ith component ofthejth basis vector in the subspace of slow and fast reactions, respectively. The corresponding expansion coefficients are (y )j and (yf )j, respectively, and are expressed by the reaction rates via... 

The interpretation is straightforward. At reaction conditions the concentration in the film is lowered by reaction, and, as a consequence, the driving force for mass transfer increases. In a homogeneous system this results in high values of Ha. In a slurry reactor this enhancement can occur if the catalyst particles are so small that they accumulate in the film layer. Table 5.4-4 summarizes expressions for the reaction rate or enhancement factor for various regimes. 

Early studies of ET dynamics at externally biased interfaces were based on conventional cyclic voltammetry employing four-electrode potentiostats [62,67 70,79]. The formal pseudo-first-order electron-transfer rate constants [ket(cms )] were measured on the basis of the Nicholson method [99] and convolution potential sweep voltammetry [79,100] in the presence of an excess of one of the reactant species. The constant composition approximation allows expression of the ET rate constant with the same units as in heterogeneous reaction on solid electrodes. However, any comparison with the expression described in Section II.B requires the transformation to bimolecular units, i.e., M cms . Values of of the order of 1-2 x lO cms (0.05 to O.IM cms ) were reported for Fe(CN)g in the aqueous phase and the redox species Lu(PC)2, Sn(PC)2, TCNQ, and RuTPP(Py)2 in DCE [62,70]. Despite the fact that large potential perturbations across the interface introduce interferences in kinetic analysis [101], these early estimations allowed some preliminary comparisons to established ET models in heterogeneous media. 

An expression for the equilibrium occupancy of pARt can again be obtained using the methods outlined in Chapter 1. A potential complication is that this mechanism contains a cycle, so the product of the reaction rates in both clockwise and counterclockwise directions should be equal in order to ensure the principle of microscopic reversibility is maintained. In this case, microscopic reversibility is maintained. Thus,... 

The functional form of this rate expression is consistent with the behavior of the iridium system observed throughout the kinetic investigations. The coordination of nitrile to iridium is anticipated to produce more than a simple inhibitory effect. Being the dominant equilibrium in the mechanism, nitrile coordination may produce the observed first order dependence of the reaction rate with respect to hydrogen. Given Kcn[RCN] is the predominant term in the denominator, the rate expression may be reduced to the form of (8) which is first order with respect to both olefin and [H2]. 

The lability of the transferable electrons or atoms influences all the factors in the expression for the reaction rate constant,... 

Basic Concepts in Chemical Kinetics—Determination of the Reaction Rate Expression... 

To develop expressions for the reaction rate in variable volume systems, one need only return to the fundamental definition of the reaction rate (3.0.1) and combine this relation with equations 3.1.40 and 3.1.48. 

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