Kinetics consecutive first-order reactions

In the presence of excess bromine, the first-order rate coefficient was 10.3 x 10" 5, but kinetic studies here were complicated due to rapid reaction of bromine with the 2,4,6-tribromophenol to give the intermediate (LXII), with SO3 replaced by Br) which slowly decomposed with a rate coefficient k2 to give two products. By analysis in terms of two consecutive first-order reactions, the values of k2 and k2 were determined as 9.2 x 10"5 and 3.75 x 10"4 and the latter rate being the faster means that two moles of bromine were consumed for every mole of sulphonate undergoing substitution in fact, more than two moles were consumed, the reason for this being undetermined. 

M. Gui and S.C. Rutan, Determination of initial concentration of analyte by kinetic detection of the intermediate product in consecutive first-order reactions using an extended Kalman filter. Anal. Chim. Acta, 66 (1994) 1513-1519. 

First we construct a kinetic measurement which we then analyse in both of the above ways. The reaction is the set of two consecutive first order reactions with rate constants ki and fe ... 

If each type of sulfur compound is removed by a reaction that was first-order with respect to sulfur concentration, the first-order reaction rate would gradually, and continually, decrease as the more reactive sulfur compounds in the mix became depleted. The more stable sulfur species would remain and the residuum would contain the more difficult-to-remove sulfur compounds. This sequence of events will, presumably lead to an apparent second-order rate equation which is, in fact, a compilation of many consecutive first-order reactions of continually decreasing rate constant. Indeed, the desulfurization of model sulfur-containing compounds exhibits first-order kinetics, and the concept that the residuum consists of a series of first-order reactions of decreasing rate constant leading to an overall second-order effect has been found to be acceptable. 

A catalytic cycle is a sequence of steps. When one step is much slower than the others, we say that this step is rate determining and we ignore, for kinetics purposes, the other (fast) steps. Nevertheless, sometimes there are two slow steps with similar rates, and sometimes the rate of a specific step changes in the course of the reaction or under different conditions. One common situation is that of two consecutive first-order reactions, as in Eq. (2.44). 

Kinetics of Consecutive Reactions Reactions which take place in two or more steps, one after the other are called consecutive reactions. Their characteristic features are illustrated with an example consider two consecutive first order reaction. Thus, the sequence... 

The following example illustrates the ease with which the Solver can be used to perform non-linear least-squares curve fitting. Here we analyze kinetics data (absorbance vs. time) from a biphasic reaction involving two consecutive first-order reactions (A =— B =— C) to obtain two rate constants and the molar absorptivity of the intermediate species B. 

The equations for the concentrations of the species A, B and C in a reaction sequence of two consecutive first-order reactions can be found in almost any kinetics text. The expression for [B]f is... 

A complete pathway for the mineralization of 4-chlorophenol can be described by the hydroxylation reaction through dechlorination, ring cleavage, and mineralization. Reaction kinetics of the 4-chlorophenol and its intermediates can be reasonably well approximated by using a complex parallel and consecutive first-order reaction mechanism. 

Actually, by the equation immediately preceding Eq. (F). The same result is also given by Eq. (2-106), which was developed for consecutive first-order reactions. As shown in Example 2-8, in this case the results are the same for first- or second-order kinetics. 

A simplified model for evaluating the kinetic of the CL reaction has been developed in terms of three pool of substances as in the consecutive first-order reactions ... 

The kinetics of SO oxidation by [PtClg] have been studied over a wide range of experimental conditions.On the basis of the experimental evidence, a 2e reduction mechanism was proposed. The reduction of Cu(II) by SOl proceeds via Cu"SOf and Cu2S030H intermediates. " The rate law is consistent with consecutive first-order reactions leading to a mechanism involving complexation followed by a rate-determining electron transfer. Inner-sphere electron transfer is also assumed for the electron transfer between Fe(III) and The reduction... 

The kinetics observed are typical of two consecutive first-order reactions. The following data represent the progress of the reaction as reported in Swain s (7) analysis of Kaufler s work. 

It was previously thought that the first step in aquation of [Cr(malonate)3] was much faster than the subsequent aquation of the cw-[Cr(malonate)2(OH2)2] formed. Further studies have shown that the kinetic pattern is more complicated and that results need to be analysed in terms of consecutive first-order reactions. The complex cw-[Cr(oxalate)2(OH2)2] disproportionates very slowly to give [Cr(oxalate)(OH2)4l+ and [Cr(oxalate)3] . ... 

The kinetics of the decomposition of the transient nitroxide radicals derived from aceto-HX and SAHA are similar RC(0)N0 decays via a second-order reaction, which is followed by two consecutive first-order reactions. Typical kinetic traces are shown in Figure 2. 

Therefore we attempted to simulate advanced pyrolysis using a multi-step model (MSM). This model was developed using TGA- and DSC-derived kinetic coefficients, determined for chemically and thermally treated oil shale samples by modelling particular reaction steps. The MSM is based on the reaction scheme shown in Fig. 4-116 which displays a series of parallel and consecutive first order reactions. K and B denote the kerogen and bitumen originally present in the oil shale B, B, and to /Jj are non-volatilized intermediates and products (solids and liquids) to are volatilized products (gases and vapors) and/j to/jg are the stoichiometric coefficients that fulfil the condition ... 

Aluminum hydroxide was used as an adsorbent in a number of publications. Coprecipitation of Cd(OH)2 and Al(OH)3 was measured by Simon et al. in the absence [69] and presence [70] of NH3. Shiao et al. [50] demonstrate that cadmium distribution coefficients between liquid and surface are practically uninfluenced by the NaNOs concentration, while NaCl has a remarkable influence. According to Shimada et al. [71] and Packter and Derry [1], coprecipitation leads to the formation of Zn aluminates. Besides investigating various ions [62], Mustafa and coworkers describe sorption on a-Al(OH)3 [72,72], especially that of Zn, at different temperatures. They observed Langmuir isotherms and derived thermodynamic data. Kinetic studies were carried out by Simon et al. [74] using polaro-graphic techniques. In their experiments equilibrium was reached after 3h with two consecutive first-order reactions, in contrast to the findings of Lo and Leckie... 

A simple kinetic analysis of consecutive first-order reactions [410] may illustrate the problem. The data in Figure 1.5 show that the higher the ratio k2/ki, the lower the yield of the intermediate B. 

Bisset and Benson,at DuPont, studied the partial oxidation of n-butane using a cobalt molybdate catalyst. The MA formation kinetics suggested that there were two consecutive first-order reactions ... 

Consecutive reactions involving one first-order reaction and one second-order reaction, or two second-order reactions, are very difficult problems. Chien has obtained closed-form integral solutions for many of the possible kinetic schemes, but the results are too complex for straightforward application of the equations. Chien recommends that the kineticist follow the concentration of the initial reactant A, and from this information rate constant k, can be estimated. Then families of curves plotted for the various kinetic schemes, making use of an abscissa scale that is a function of c kit, are compared with concentration-time data for an intermediate or product, seeking a match that will identify the kinetic scheme and possibly lead to additional rate constant estimates. 

It should be noted that the kinetics were first-order over at least three half-lives (with the exception of the dicyclopropylcarbonium ion), but the reaction products were not well defined in some cases— probably due to relatively fast consecutive reactions of the unsatmated oxocarbonium ions formed. In the case of the oxocarbonium ions formed from the allyl cations a novel quantitative eyclization to give cyclopentenone derivatives was observed (Hogeveen and Gaasbeek, 1970) ... 

Even though the governing phenomena of coupled reaction and mass transfer in porous media are principally known since the days of Thiele (1) and Frank-Kamenetskii (2), they are still not frequently used in the modeling of complex organic systems, involving sequences of parallel and consecutive reactions. Simple ad hoc methods, such as evaluation of Thiele modulus and Biot number for first-order reactions are not sufficient for such a network comprising slow and rapid steps with non-linear reaction kinetics. 

These authors used 6 1 and 30 1 alcohol-to-oil molar ratios for both methanol and butanol. As expected, a pseudo first-order reaction was found at large excess of alcohol for both alcohols. At low excess alcohol, however, the butanolysis reaction (30°C) was second-order, but the methanolysis reaction (40°C) was reported to be a combination of a second-order consecutive reaction and a fourth-order shunt reaction. The shunt reaction, in which three methanol molecules simultaneously attack a TG molecule, was adopted to better fit the kinetic data. However, such a reaction is highly unlikely. Nureddini et al. later found that the inclusion of a shunt mechanism was not necessary to fit the kinetic data of the transesterification reaction, and Boocock et al showed that the shunt reaction assumption came as a misinterpretation of the observed kinetics. At low temperatures (20 0°C) the multiphase methanolysis reaction... 

We take the simplest kinetic scheme for our requirements two consecutive first-order chemical reactions converting an initial reactant P into a final product B through a single intermediate species A... 

A major problem in the chemistry of the cobalt(III) species has been the fact that, although reaction Eq. (56) has to proceed in two steps, only one reaction stage is normally observed. The species which apparently follow first-order kinetics are L4Co(OH)2CoL44+ for L4 = (NH3)4, (NH3)3(H20), (en)2, and (dien)(H20) (132,186,187, 365, 365 a so-called induction period has usually been observed for these reactions. In a few other cases, however, hydrolysis has been reported to follow consecutive first-order kinetics (177,359). In none of the systems studied so far has it been possible to isolate the monohydroxo-bridged intermediate, and in the absence of additional information on the properties of this intermediate, unambiguous interpretation of the kinetic data is not possible. The following discussion should be read with this in mind. 

As discussed in Section IV, Agrawal and Wei (1984) and Ware and Wei (1985b) have successfully modeled experimental deposit profiles by using the theory of coupled, multicomponent first-order reaction and diffusion. Wei and Wei (1982) employed this theory to evaluate the influence of catalyst properties on the shape of the deposit profile. Agrawal (1980) developed a model for the deactivation of unimodal and bimodal catalysts based on the consecutive reaction path. These approaches represent a more realistic consideration of the HDM reaction mechanism than first-order kinetics and will, accordingly, be discussed in more detail. 

The disappearance of melphalan from aqueous solutions has been shown to follow first-order kinetics [62,82,85] and the concentration profile during hydrolysis is consistent with a mechanism consisting of two consecutive pseudo first-order reactions (Scheme III A) [53]. 

By means of stopped-flow spectrophotometry, the reaction of flavopapain IV withN-benzyl-l,4-dihydronicotinamide (NBzNH) has been studied anaerobically. Using conditions of excess substrate and following the decrease in absorbance of Eox (at 427 nm) with time, we observed biphasic kinetic behavior. The experimental data could be fitted to a scheme using two consecutive first-order processes, and rate constants could be determined for the two phases. A reasonable interpretation of our results is that a labile intermediate is formed in the reaction, as illustrated in Equation 4, where ES represents the intermediate. In this interpretation, the second, substrate-independent, slower phase of the reaction corresponds to the breakdown of the intermediate ES (fc3 step in Equation 4). The initial, faster phase of the reaction corresponds to the formation of the intermediate from Eox and N-benzyl-l,4-dihydronicotinamide. Using Equation 5, the calculated rate constants for this phase, fcf, can be related to Ks and k2- Values of kCat and Km were calculated from the relationships of Equations 6 and 7, using the measured values of the kinetic parameters k2, 3, and Ks, and the numbers obtained were in reasonable agreement with the values obtained aerobically. 

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