Mechanism rate equation

No Process Scheme of reaction mechanism Rate equations... 

It is convenient to analyse tliese rate equations from a dynamical systems point of view similar to tliat used in classical mechanics where one follows tire trajectories of particles in phase space. For tire chemical rate law (C3.6.2) tire phase space , conventionally denoted by F, is -dimensional and tire chemical concentrations, CpC2,- are taken as ortliogonal coordinates of F, ratlier tlian tire particle positions and velocities used as tire coordinates in mechanics. In analogy to classical mechanical systems, as tire concentrations evolve in time tliey will trace out a trajectory in F. Since tire velocity functions in tire system of ODEs (C3.6.2) do not depend explicitly on time, a given initial condition in F will always produce tire same trajectory. The vector R of velocity functions in (C3.6.2) defines a phase-space (or trajectory) flow and in it is often convenient to tliink of tliese ODEs as describing tire motion of a fluid in F with velocity field/ (c p). 

Nitric acid being the solvent, terms involving its concentration cannot enter the rate equation. This form of the rate equation is consistent with reaction via molecular nitric acid, or any species whose concentration throughout the reaction bears a constant ratio to the stoichiometric concentration of nitric acid. In the latter case the nitrating agent may account for any fraction of the total concentration of acid, provided that it is formed quickly relative to the speed of nitration. More detailed information about the mechanism was obtained from the effects of certain added species on the rate of reaction. 

This mechanism leads to the rate equation (eq. 3) for hydrolysis and to an analogous expression for the esterification (13) ... 

The kinetics of the ethylene hydration reaction have been investigated for a tungstic oxide—siHca gel catalyst, and the energy of activation for the reaction deterrnined to be 125 kJ/mol (- 30 kcal/mol) (106,120). The kinetics over a phosphoric acid-siHca gel catalyst have been examined (121). By making some simplifying assumptions to Taft s mechanism, a rate equation was derived ... 

Some reactions apparently represented by single stoichiometric equations are in reahty the result of several reactions, often involving short-hved intermediates. After a set of such elementary reactions is postulated by experience, intuition, and exercise of judgment, a rate equation is deduced and checked against experimental rate data. Several examples are given under Mechanisms of Some Complex Reactions, following. 

Free radicals are molecular fragments having one or more unpaired electrons, usually short-lived (milhseconds) and highly reaclive. They are detectable spectroscopically and some have been isolated. They occur as initiators and intermediates in such basic phenomena as oxidation, combustion, photolysis, and polvmerization. The rate equation of a process in which they are involved is developed on the postulate that each free radical is at equihbrium or its net rate of formation is zero. Several examples of free radical and catalytic mechanisms will be cited, aU possessing nonintegral power law or hyperbohc rate equations. 

For the radiative mechanism of heat transfer to solids, the rate equation for parallel-surface operations is... 

In general, solutions are obtained by couphng the basic conservation equation for the batch system, Eq. (16-49) with the appropriate rate equation. Rate equations are summarized in Table 16-11 and 16-12 for different controlhng mechanisms. 

Asymptotic Solution Rate equations for the various mass-transfer mechanisms are written in dimensionless form in Table 16-13 in terms of a number of transfer units, N = L/HTU, for particle-scale mass-transfer resistances, a number of reaction units for the reaction kinetics mechanism, and a number of dispersion units, Np, for axial dispersion. For pore and sohd diffusion, q = / // p is a dimensionless radial coordinate, where / p is the radius of the particle, if a particle is bidisperse, then / p can be replaced by the radius of a suoparticle. For prehminary calculations. Fig. 16-13 can be used to estimate N for use with the LDF approximation when more than one resistance is important. 

Mechanism N Dimensionless rate equation Constant pattern Refs. 

For other mechanisms, the particle-scale equation must be integrated. Equation (16-140) is used to advantage. For example, for external mass transfer acting alone, the dimensionless rate equation in Table 16-13 would be transformed into the ( — Ti, Ti) coordinate system and derivatives with respect to Ti discarded. Equation (16-138) is then used to replace cfwith /ifin the transformed equation. Furthermore, for this case there are assumed to be no gradients within the particles, so we have nf=nf. After making this substitution, the transformed equation can be rearranged to... 

Various Langmiiir-Hinshelwood mechanisms were assumed. GO and GO2 were assumed to adsorb on one kind of active site, si, and H2 and H2O on another kind, s2. The H2 adsorbed with dissociation and all participants were assumed to be in adsorptive equilibrium. Some 48 possible controlling mechanisms were examined, each with 7 empirical constants. Variance analysis of the experimental data reduced the number to three possibilities. The rate equations of the three reactions are stated for the mechanisms finally adopted, with the constants correlated by the Arrhenius equation. 

In some cases, the exponent is unity. In other cases, the simple power law is only an approximation for an actual sequence of reactions. For instance, the chlorination of toluene catalyzed by acids was found to have CL = 1.15 at 6°C (43°F) and 1.57 at 32°C (90°F), indicating some complex mechanism sensitive to temperature. A particular reaction may proceed in the absence of catalyst out at a reduced rate. Then the rate equation may be... 

The latter kind of formulation is described at length in Sec. 7. The assumed mechanism is comprised of adsorption and desorption rates of the several participants and of the reaction rates of adsorbed species. In order to minimize the complexity of the resulting rate equation, one of the several rates in series may be assumed controlling. With several controlling steps the rate equation usually is not exphcit but can be used with some extra effort. 

This method estimates the reaction order based on the reaction stoichiometry and assumptions concerning its mechanism. The assumed rate equation is then integrated to obtain a relation between the composition and time. The following procedures are used for determining the rate equations ... 

In Chapter 1 we distinguished between elementary (one-step) and complex (multistep reactions). The set of elementary reactions constituting a proposed mechanism is called a kinetic scheme. Chapter 2 treated differential rate equations of the form V = IccaCb -., which we called simple rate equations. Chapter 3 deals with many examples of complicated rate equations, namely, those that are not simple. Note that this distinction is being made on the basis of the form of the differential rate equation. 

These steps may not proceed in the sequence shown, because a difficult kinetic problem may require cycling of attention among the steps as more is learned about the system, with corrections being made and tests of ideas being applied at each stage. In particular, steps 2 and 3 may be strongly interdependent. Our present concern is with these steps later chapters deal with step 4. Edwards et al., Bunnett, and Pearson have formulated provisional rules for proceeding from the rate equation to the mechanism, which includes step 4. 

Click Coached Problems for a self-study module on reaction mechanisms and rate equations. 

Rate constant The proportionality constant in the rate equation for a reaction, 288 Rate-determining step The slowest step in a multistep mechanism, 308 Rate expression A mathematical relationship describing the dependence of reaction rate upon the concentra-tion(s) of reactant(s), 288,308-309 Rayleigh, Lord, 190... 

The reaction mechanisms may assist us in obtaining a suitable rate equation. Based on the enzyme reaction mechanism given by (5.7.1.18) for the intermediate enzyme-substrate complex, the following equations are derived for ES ... 

The plotting of Dixon plot and its slope re-plot (see 5.9.5.9) is a commonly used graphical method for verification of kinetics mechanisms in a particular enzymatic reaction.9 The proposed kinetic mechanism for the system is valid if the experimental data fit the rate equation given by (5.9.4.4). In this attempt, different sets of experimental data for kinetic resolution of racemic ibuprofen ester by immobilised lipase in EMR were fitted into the rate equation of (5.7.5.6). The Dixon plot is presented in Figure 5.22. 

The development of methods for the kinetic measurement of heterogeneous catalytic reactions has enabled workers to obtain rate data of a great number of reactions [for a review, see (1, )]. The use of a statistical treatment of kinetic data and of computers [cf. (3-7) ] renders it possible to estimate objectively the suitability of kinetic models as well as to determine relatively accurate values of the constants of rate equations. Nevertheless, even these improvements allow the interpretation of kinetic results from the point of view of reaction mechanisms only within certain limits ... 

In eq. 8, the rate of polymerization is shown as being half order in initiator (T). This is only true for initiators that decompose to two radicals both of which begin chains. The form of this term depends on the particular initiator and the initiation mechanism. The equation takes a slightly different form in the case of thermal initiation (S), redox initiation, diradical initiation, etc. Side reactions also cause a departure from ideal behavior. 

In the modified Ridd mechanism for region B the deprotonation of the A-nitroso-anilinium ion Ar —NH2NO in Scheme 3-23 is rapid, and therefore does not influence the overall rate. However, the second-order term in the rate equation for region C (Scheme 3-25) is consistent with a mechanism in which the deprotonation of the A-nitrosoanilinium ion (Scheme 3-24) and of the C-nitrosoanilinium dication (Scheme 3-22) belongs to the rate-determining part of the reaction. 

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