Kinetic analysis, reaction order

Figure 12-11. Self-heat rate analysis. ARC data are shown along with a fitted model obtained by assuming the following kinetic parameters reaction order = 1, activation energy = 31.08 kcal/mol, and frequency factor = 2.31 El 2 min ...

Table 3 summarizes the results of the parameter estimation. All values of the kinetic parameters - reaction orders, rate constants and activation energies - were estimated by nonlinear regression analysis based on numerous experiments. 

As with the case of energy input, detergency generally reaches a plateau after a certain wash time as would be expected from a kinetic analysis. In a practical system, each of its numerous components has a different rate constant, hence its rate behavior generally does not exhibit any simple pattern. Many attempts have been made to fit soil removal (50) rates in practical systems to the usual rate equations of physical chemistry. The rate of soil removal in the Launder-Ometer could be reasonably well described by the equation of a first-order chemical reaction, ie, the rate was proportional to the amount of removable soil remaining on the fabric (51,52). In a study of soil removal rates from artificially soiled fabrics in the Terg-O-Tometer, the percent soil removal increased linearly with the log of cumulative wash time. 

In this chapter we will discuss the results of the studies of the kinetics of some systems of consecutive, parallel or parallel-consecutive heterogeneous catalytic reactions performed in our laboratory. As the catalytic transformations of such types (and, in general, all the stoichiometrically not simple reactions) are frequently encountered in chemical practice, they were the subject of investigation from a variety of aspects. Many studies have not been aimed, however, at investigating the kinetics of these transformations at all, while a number of others present only the more or less accurately measured concentration-time or concentration-concentration curves, without any detailed analysis or quantitative kinetic interpretation. The major effort in the quantitative description of the kinetics of coupled catalytic reactions is associated with the pioneer work of Jungers and his school, based on their extensive experimental material 17-20, 87, 48, 59-61). At present, there are so many studies in the field of stoichiometrically not simple reactions that it is not possible, or even reasonable, to present their full account in this article. We will therefore mention only a limited number in order for the reader to obtain at least some brief information on the relevant literature. Some of these studies were already discussed in Section II from the point of view of the approach to kinetic analysis. Here we would like to present instead the types of reaction systems the kinetics of which were studied experimentally. 

Racemization constitutes a special case of opposing first-order reactions. The equilibrium constant is unity, and the opposing rate constants are equal to one another. Racemization can be followed by polarimetry (monitoring the angle of optical rotation) or by circular dichroism (monitoring the ellipticity). The kinetic analysis can be done by either Eq. (3-15) or (3-16). The rate constant for racemization is krac = ke/2. 

To summarize the analysis of pH profiles, even complex ones, is not an arcane or difficult art. Systematic analysis in terms of ionic equilibria, predominant species, and the reaction orders with respect to [H+] provides the solution. Kinetically indistinguishable alternatives can never, by definition, be distinguished from the kinetic data contained in the pH profile. Other measurements, including some alluded to earlier and others given in Chapter 10, may, however, allow these distinctions. 

The hydrolytic depolymerisation of PETP in stirred potassium hydroxide solution was investigated. It was found that the depolymerisation reaction rate in a KOH solution was much more rapid than that in a neutral water solution. The correlation between the yield of product and the conversion of PETP showed that the main alkaline hydrolysis of PETP linkages was through a mechanism of chain-end scission. The result of kinetic analysis showed that the reaction rate was first order with respect to the concentration of KOH and to the concentration of PETP solids, respectively. This indicated that the ester linkages in PETP were hydrolysed sequentially. The activation energy for the depolymerisation of solid PETP in a KOH solution was 69 kJ/mol and the Arrhenius constant was 419 L/min/sq cm. 21 refs. 

A full development of the rate law for the bimolecular reaction of MDI to yield carbodiimide and CO indicates that the reaction should truly be 2nd-order in MDI. This would be observed experimentally under conditions in which MDI is at limiting concentrations. This is not the case for these experimements MDI is present in considerable excess (usually 5.5-6 g of MDI (4.7-5.1 ml) are used in an 8.8 ml vessel). So at least at the early stages of reaction, the carbon dioxide evolution would be expected to display pseudo-zero order kinetics. As the amount of MDI is depleted, then 2nd-order kinetics should be observed. In fact, the asymptotic portion of the 225 C Isotherm can be fitted to a 2nd-order rate law. This kinetic analysis is consistent with a more detailed mechanism for the decomposition, in which 2 molecules of MDI form a cyclic intermediate through a thermally allowed [2+2] cycloaddition, which is formed at steady state concentrations and may then decompose to carbodiimide and carbon dioxide. Isocyanates and other related compounds have been reported to participate in [2 + 2] and [4 + 2] cycloaddition reactions (8.91. 

Preliminary kinetic analysis revealed that the reactions mentioned for various sugars were close to first order with respect to the organic reactant, while the reaction order with respect to hydrogen varied between 0.5 and 2.2, being 0.7 for hydrogenation of lactose on sponge nickel and about 2 for fructose hydrogenation on CuO/ZnO. 

Steady-state approximation. Fractional reaction orders may be obtained from kinetic data for complex reactions consisting of elementary steps, although none of these steps are of fractional order. The same applies to reactions taking place on a solid catalyst. The steady-state approximation is very useful for the analysis of the kinetics of such reactions and is illustrated by Example 5.4.2.2a for a solid-catalysed reaction. 

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. 

Reaction step 5 in Scheme 3.1 can be rnled ont becanse the flnoranil ketyl radical (FAH ) reaches a maximum concentration within 100 ns as the triplet state ( FA) decays by reaction step 2 while the fluoranil radical anion (FA ) takes more than 500 ns to reach a maximum concentration. This difference snggests that the flnoranil radical anion (FA ) is being produced from the fluoranil ketyl radical (FAH ). Reaction steps 1 and 2 are the most likely pathway for prodncing the flnoranil ketyl radical (FAH ) from the triplet state ( FA) and is consistent with the TR resnlts above and other experiments in the literatnre. The kinetic analysis of the TR experiments indicates the fluoranil radical anion (FA ) is being prodnced with a hrst order rate constant and not a second order rate constant. This can be nsed to rnle ont reaction step 4 and indicates that the flnoranil radical anion (FA ) is being prodnced by reaction step 3. Therefore, the reaction mechanism for the intermolecular hydrogen abstraction reaction of fluoranil with 2-propanol is likely to predominantly occur through reaction steps 1 to 3. 

The experimental method used for this kinetie study is reaetion ealorimetry. In the ealorimeter, the energy enthalpy balance is continuously monitored the heat signal can then be easily converted in the reaction rate (in the case of an isothermal batch reactor, the rate is proportional to the heat generated or consnmed by the reaction). The reaction orders and catalyst stabihty were determined with the methodology of reaction progress kinetic analysis (see refs. (8,9) for reviews). 

We first consider the simple example of an uncatalyzed Diels-Alder reaction shown in Scheme 50.2 in order to demonstrate the use of different excess experiments. The Diels-Alder reaction is known to exhibit second overall order kinetics, as shown in eq. (4). We demonstrate with this known case how reaction progress kinetic analysis may be used to extract the reaction orders in both substrate concentrations, [5] and [6]. ... 

This method is primarily concerned with the phenomena that occur at electrode surfaces (electrodics) in a solution from which, as an absolute method, through previous calibration a component concentration can be derived. If desirable the technique can be used to follow the progress of a chemical reaction, e.g., in kinetic analysis. Mostly, however, potentiometry is applied to reactions that go to completion (e.g. a titration) merely in order to indicate the end-point (a potentiometric titration in this instance) and so do not need calibration. The overwhelming importance of potentiometry in general and of potentiometric titration in particular is due to the selectivity of its indication, the simplicity of the technique and the ample choice of electrodes. 

In general an analysis of a system in which noncompetitive parallel reactions are taking place is considerably more difficult than analyses of the type discussed in Chapter 3. In dealing with parallel reactions one must deal with the problems of determining reaction orders and rate constants for each of the individual reactions. The chemical engineer must be careful both in planning the experiment and in analyzing the data so as to obtain values of the kinetic con-... 

Equations 12.4.22 and 12.4.24 indicate that the observed reaction order will differ from the intrinsic reaction order in the presence of intraparticle and/or external mass transfer limitations. To avoid drawing erroneous conclusions about intrinsic reaction kinetics, we must be careful to either eliminate these limitations by proper choice of experimental conditions or to properly take them into account in our data analysis. 

Careful kinetic analysis of this thermal reaction shows that the rate of disappearance of the CT band is identical to that of the adduct formation in equation (50). Most importantly, the relative reactivity of the metal hydrides in Table 8 decreases with the increasing ionization potential in the order Bu3SnH < Bu3GeH < Et3SiH. 

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