Analyses of Kinetic Data

The rate constant for the reaction A — B is kAli and that for the reaction A — C is kAC. Given that kAK is smaller than kAC, say kAli = 0.2 s 1 and kAC = 0.8 s one may be tempted to conclude that the observed rate constant for the formation of C is larger than that of B. This is not the case. The differential rate law for the disappearance of A is given by Equation 3.4  

This is a first-order rate law with a single observable rate constant /cobs = kAB + A ac. Integration gives 

The differential rate law for the appearance of compound B is dcB(0 = kABcA(t)dt. Insertion of cA(t) from Equation 3.5 and integration assuming cB(t — 0) = 0 gives 

Equation 3.7 Efficiency of a single step first order reaction 

When the decay of intermediate B is faster than its formation, kBC kAB, the absolute value of the pre-exponential term will become small and thus the transient concentration cB will be small at all times. For kBC 3 kAB, the appearance of product C will approach the first-order rate law, Equation 3.6 (replace the index B by C), because the transient concentration of intermediate B becomes negligible. This shows that observation of a first-order rate law for the reaction A — C does not guarantee that there is no intermediate involved, that is, that the observed reaction A — C is an elementary reaction. 

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Predicting the solvent or density dependence of rate constants by equation (A3.6.29) or equation (A3.6.31) requires the same ingredients as the calculation of TST rate constants plus an estimate of and a suitable model for the friction coefficient y and its density dependence. While in the framework of molecular dynamics simulations it may be worthwhile to numerically calculate friction coefficients from the average of the relevant time correlation fiinctions, for practical purposes in the analysis of kinetic data it is much more convenient and instructive to use experimentally detemiined macroscopic solvent parameters. 

Table 6-2. Least-Squares Analysis of Kinetic Data in Table 6-1...

In other instances, reaction kinetic data provide an insight into the rate-controlling steps but not the reaction mechanism see, for example, Hougen and Watson s analysis of the kinetics of the hydrogenation of mixed isooctenes (16). Analysis of kinetic data can, however, yield a convenient analytical insight into the relative catalyst activities, and the effects of such factors as catalyst age, temperature, and feed-gas impurities on the catalyst. 

The account of the formal derivation of kinetic expressions for the reactions of solids given in Sect. 3 first discusses those types of behaviour which usually generate three-dimensional nuclei. Such product particles may often be directly observed. Quantitative measurements of rates of nucleation and growth may even be possible, thus providing valuable supplementary evidence for the analysis of kinetic data. Thereafter, attention is directed to expressions based on the existence of diffuse nuclei or involving diffusion control such nuclei are not susceptible to quantitative... 

If chain transfer of the radical center to a previously formed polymer molecule is followed ultimately by termination through coupling with another similarly transferred center, the net result of these two processes is the combination of a pair of previously independent polymer molecules. T. G. Fox (private communication of results as yet unpublished) has suggested this mechanism as one which may give rise to network structures in the polymerization of monovinyl compounds. His preliminary analysis of kinetic data indicates that proliferous polymerization of methyl acrylate may be triggered by networks thus generated. 

The following example illustrates the use of the differential method for the analysis of kinetic data. It also exemplifies some of the problems... 

A General Integral Method for the Analysis of Kinetic Data—Graphical Procedure. 

Integral Methods for the Analysis of Kinetic Data—Numerical Procedures. While the graphical procedures discussed in the previous section are perhaps the most practical and useful of the simple methods for determining rate constants, a number of simple numerical procedures exist for accomplishing this task. 

Many computer libraries contain programs that perform the necessary statistical calculations and relieve the engineer of this burden. For discussions of the use of weighted least squares methods for the analysis of kinetic data, see Margerison s review (8) on the treatment of experimental data and the treatments of Kittrell et al. (9), and Peterson (10). 

These concentrations may be used in the various integral and differential methods for the analysis of kinetic data that have been described in previous sections. An example of the use of this approach is given in Illustration 3.5. 

Equations 5.1.5, 5.1.6, and 5.1.8 are alternative methods of characterizing the progress of the reaction in time. However, for use in the analysis of kinetic data, they require an a priori knowledge of the ratio of kx to k x. To determine the individual rate constants, one must either carry out initial rate studies on both the forward and reverse reactions or know the equilibrium constant for the reaction. In the latter connection it is useful to indicate some alternative forms in which the integrated rate expressions may be rewritten using the equilibrium constant, the equilibrium extent of reaction, or equilibrium species concentrations. 

In the analysis of kinetic data from reactions believed to be first-order in both directions, the equation that is most suitable for use depends on the pertinent equilibrium data available. Equations 5.1.17 and 5.1.11 are perhaps the most useful, but others may be more appropriate for use in some cases. The integrated forms permit one to determine the sum (kx + k x), while equilibrium data permit one to determine the equilibrium constant K = kl/k v Such information is sufficient to determine kx and /c x. 

Equation 8.3.4 may also be used in the analysis of kinetic data taken in laboratory scale stirred tank reactors. One may directly determine the reaction rate from a knowledge of the reactor volume, flow rate through the reactor, and stream compositions. The fact that one may determine the rate directly and without integration makes stirred tank reactors particularly attractive for use in studies of reactions with complex rate expressions (e.g., enzymatic or heterogeneous catalytic reactions) or of systems in which multiple reactions take place. 

The difference in mole fractions is most significant in the case of S02 where this difference is 15% of the bulk phase level. This result indicates that external mass transfer limitations are indeed significant, and that this difference should be taken into account in the analysis of kinetic data from this system. Note that there is a difference in nitrogen concentration between the bulk fluid and the external surface because there is a change in the number of moles on reaction, and there is a net molar flux toward... 

Swinboume, ES, Analysis of Kinetic Data, Nelson, Chicago, IL, 1971. 

The collection of kinetic modelling programs will be adapted in the subsequent chapter for the non-linear least-squares analysis of kinetic data and the determination of rate constants. 

Figure 5-56. ALS analysis of kinetic data Data ABC2. m analysed using constraints nonneg, m.

The hyperbolic model types have very commonly been used in the analysis of kinetic data, as discussed in Section I. Such applications are sometimes justified on the theoretical bases already alluded to, or simply because models of the form of Eq. (2) empirically describe the existing reaction-rate data. Considerably more complex models are quite possible under the Hougen-Watson formalism, however. For example, Rogers, Lih, and Hougen (Rl) have proposed the competitive-noncompetitive model... 

One of the widely used methods of analysis of kinetic data is based on extraction of the distribution of relaxation times or, equivalently, enthalpic barrier heights. In this section, we show that this may be done easily by using the distribution function introduced by Raicu (1999 see Equation [1.16] above). To this end, we use the data reported by Walther and coworkers (Walther et al. 2005) from pump-probe as well as the transient phase grating measurements on trehalose-embedded MbCO. Their pump-probe data have been used without modification herein, while the phase grating data (also reproduced in Figure 1.12) have been corrected for thermal diffusion of the grating using the relaxation time reported above, r,, and Equation (1.25). 

Because reactions in solids tend to be heterogeneous, they are generally described by rate laws that are quite different from those encountered in solution chemistry. Concentration has no meaning in a heterogeneous system. Consequently, rate laws for solid-phase reactions are described in terms of a, the fraction of reaction (a = quantity reacted -r- original quantity in sample). The most commonly encountered rate laws are given in Table 1. These rate laws and their application to solid-phase reactions are described elsewhere. 1 4 10-12 Unfortunately, it is often merely assumed that solid-phase reactions are first order. This uncritical analysis of kinetic data produces results that must be accepted only with great caution. 

Let us discuss what meaning should be put into the parameters Rt, ve, and a, obtained above for the tunneling reactions of etr with different acceptors from an analysis of kinetic data by means of eqns. (7) of Chap. 5. 

PtLX+(aq) + Y (aq) - analysis of kinetic data demonstrating the primary salt effect,... 

The C-H bond strength in the (=Si-)3C-H group is (100 + 2)kcal/mol. This estimate is based on the analysis of kinetic data on the formation and decay of these groups and also on the results of quantum-chemical calculations of model systems. 

Enzymes are biocatalysts, as such they facilitate rates of biochemical reactions. Some of the important characteristics of enzymes are summarized. Enzyme kinetics is a detailed stepwise study of enzyme catalysis as affected by enzyme concentration, substrate concentrations, and environmental factors such as temperature, pH, and so on. Two general approaches to treat initial rate enzyme kinetics, quasi-equilibrium and steady-state, are discussed. Cleland s nomenclature is presented. Computer search for enzyme data via the Internet and analysis of kinetic data with Leonora are described. 

The differential method of analysis of kinetic data deals directly with the differential rate of reaction. A mecha-... 

No precise information about the olefin polymerisation mechanism has been obtained from kinetic measurements in systems with heterogeneous catalysts analysis of kinetic data has not yet afforded consistent indications either concerning monomer adsorption on the catalyst surface or concerning the existence of two steps, i.e. monomer coordination and insertion of the coordinated monomer, in the polymerisation [scheme (2) in chapter 2], Note that, under suitable conditions, each step can be, in principle, the polymerisation rate determining step [241]. Furthermore, no % complexes have been directly identified in the polymerisation process. Indirect indications, however, may favour particular steps [242]. Actually, no general olefin polymerisation mechanism that may be operating in the presence of Ziegler-Natta catalysts exists, but rather the reaction pathway depends on the type of catalyst, the kind of monomer and the polymerisation conditions. 

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