Kinetics second-order rate laws

A simple kinetic order for the nitration of aromatic compounds was first established by Martinsen for nitration in sulphuric acid (Martin-sen also first observed the occurrence of a maximum in the rate of nitration, occurrii for nitration in sulphuric acid of 89-90 % concentration). The rate of nitration of nitrobenzene was found to obey a second-order rate law, first order in the concentration of the aromatic and of nitric acid. The same law certainly holds (and in many cases was explicitly demonstrated) for the compounds listed in table 2.3. 

First-order nitrations. The kinetics of nitrations in solutions of acetyl nitrate in acetic anhydride were first investigated by Wibaut. He obtained evidence for a second-order rate law, but this was subsequently disproved. A more detailed study was made using benzene, toluene, chloro- and bromo-benzene. The rate of nitration of benzene was found to be of the first order in the concentration of aromatic and third order in the concentration of acetyl nitrate the latter conclusion disagrees with later work (see below). Nitration in solutions containing similar concentrations of acetyl nitrate in acetic acid was too slow to measure, but was accelerated slightly by the addition of more acetic anhydride. Similar solutions in carbon tetrachloride nitrated benzene too quickly, and the concentration of acetyl nitrate had to be reduced from 0-7 to o-i mol 1 to permit the observation of a rate similar to that which the more concentrated solution yields in acetic anhydride. 

Kinetic studies of these reactions reveal that they follow a second order rate law Rate = [Aryl halide] [Nucleophile]... 

If the process of APIO is properly described by Equation (19), which infers the presence of a soluble Fe(III) intermediate species, it will be difficult to analyze this species directly, given the low levels that are expected. We must therefore develop mathematical approaches to estimating the isotopic composition of this component, as was done for DIR. The equations used in the previous chapter (Chapter lOA Beard and Johnson 2004) to describe abiotic Fe(II) oxidation are useful for illustrating possible isotopic fractionations that may occur during APIO. We will assume that the overall oxidation process occurs through a series of first-order rate equations, where relatively slow oxidation of FefTI) to a soluble Fe(III) component occurs, which we will denote as Fe(III)jq for simplicity. The oxidation step is followed by precipitation of Fe(III)jq to ferrihydrite at a much faster rate, which maintains a relatively low level of Fe(III)jq relative to Fe(II)jq. The assumption of first-order kinetics is not strictly valid for the experiments reported in Croal et al. (2004), where decreasing FefTI) contents with time do not closely follow either zeroth-, first-, or second-order rate laws. However, use of a first-order rate law allows us to directly compare calculations here with those that are appropriate for abiologic Fe(II) oxidation, where experimental data are well fit to a first-order rate law (Chapter lOA Beard and Johnson 2004). 

Kinetic data presented in Fig. 22.2 for the adsorption of metobromuron were fitted to the pseudo-first-order and the pseudo-second-order rate laws. Linear form of pseudo-first-order model can be formulated as... 

The kinetics of hydrogenation of/ra 5-[IrCl(CO)(PPh3)2] in toluene and other organic solvents as well as that of the hydrogenation of trans-[IrCl(CO)(TPPMS)2] [78, 79] in water were studied in detail by Atwood and co-workers [80,81], The rate of both reactions could be described by an overall second-order rate law ... 

An equilibrium and kinetic study of the iron(II) phthalocyanine/nitric oxide system in DMSO, at 293 K, showed that formation of [Fe(pc)(NO)] obeys a simple second-order rate law, like [Fe(pc)] plus CO but unlike [Fe(pc)] plus dioxygen. A rate constant for dissociation of [Fe(pc)(NO)] was derived from its formation rate and equilibrium constants. " ... 

Compared to the time scale of their formation, the carbonyl oxides are quite long lived (10 -10 s), and so their subsequent reactions can be monitored kinetically. For most of the carbonyl oxides, the decay is best fit to a second-order rate law, indicating a bimolecular decomposition pathway. For benzophenone oxide, the ketone is the major product at room temperature, and no dimer can be detected. A bimolecular process involving O2 extrusion from two molecules of the oxides is suggested under these conditions. 

Hence, sometimes phenomena associated with enzyme kinetics control the rate of biotransformations. If suitable enzymes are present in the microbial community, for example due to consumption of structurally related growth substrates, then we may see immediate degradation of compounds of interest like BQ when they are added to these metabolically competent microbial communities (Fig. 17.17). For such cases, if the abundance of the bacteria is varied, the rate of removal changes accordingly. Consequently, the removal of BQ could be described by a second-order rate law (Smith et al., 1978) ... 

Again plotting concentration versus time using these integrated second-order rate laws gives linear plots only if the reaction is a second-order process. The rate constants can be determined from the slopes. If the concentration-time plots are not linear, then the second-order rate equations do not correctly describe the kinetic behavior. There are integrated rate laws for many different reaction orders. 

The kinetics and the products of bromination of several substituted stilbenes with Bu4N+Br3 have been investigated in aprotic solvents at different temperatures and concentrations. Stilbenes bearing electron-withdrawing or moderately electron-donating substituents gave stereospecifically the anti addition products the reaction followed a second-order rate law and inverse kinetic isotope effect Ap/Ap = 0.85 ( 0.05) was... 

Accordingly, an overall second-order rate law is expected at constant pH (first order in phenol and in Cr(VT)), and there will be a complex dependence on acidity varying between kinetic order 0 and 2. In addition, the k2 (two-electron transfer) pathway should be the main route at higher acidities, with predominance of the products derived from the phenoxy cation, X2. 

It is convenient at this juncture to introduce a concept that, in electro analytical chemistry, sometimes is referred to as the reaction order approach. Consider first the half-life-time, t1/2> which in conventional homogeneous kinetics refers to the time for the conversion of half of the substrate into product(s). From basic kinetics, it is well known that t /2 is independent of the substrate concentration for a reaction that follows a first-order rate law and that 1/t j2 is proportional to the initial concentration of the substrate for a reaction that follows a second-order rate law. Similarly, in electro analytical chemistry it is convenient to introduce a parameter that reflects a certain constant conversion of the primary electrode intermediate. In DPSCA, it is customary to use ti/2 (or to.s), which is the value of (f required to keep the value of Ri equal to 0.5. The reaction orders (see Equation 6.30) are then given by Equations 6.35 and 6.36, where Ra/b = a + b, and Rx = x (in reversal techniques such as DPSCA, in which O and R are in equilibrium at the electrode surface, it is not possible to separate the... 

The molecular formula of A is C4H6. When 1 mole of A was mixed with 1 mol of Br2, the flask became hot. A colorless product B was observed by gas chromatography. In another experiment, 1 mol of A was mixed with 1.5 mol of Br2 the flask became hot but the color of the solution remained red. In a third experiment, UV light was shone on a flask containing A, and the product C was observed. Kinetic studies showed that the formation of B and C both fit to second-order rate laws. 

Understanding the polymerization kinetics is of fundamental importance not only for comparing different initiators but also understanding the catalytic cycle and preparing more active compounds. The polymerization generally obeys a second-order rate law, with first-order dependencies on both monomer (lactide) and... 

This last reaction, which involves a cyanide linkage isomerization, was inferred from the observation of a first order decay for an intermediate. The half life of this intermediate is 1.6 sec at 25°. A similar reaction, the oxidation of Cr+2 by Com(NH3)5(CN)+2, was investigated by Espenson and Birk (1965). Two steps were observed in the kinetics. The first obeys a second order rate law,... 

Kinetic analysis of the data revealed that the decomposition obeyed a second-order rate law in both solvent systems. A simple kinetic approach was used where... 

From a kinetic point of view, the single electron transfer reaction is described by the second-order rate law [i],... 

The reactions of [Co(CN)s(H20)]2 with Nf6768 and SCN-68 have been reinvestigated and important differences found which indicate that the evidence in favour of limiting kinetics is not as clear as originally supposed. In particular, deviations from the simple second-order rate law for anation, rate = /c[Co(CN)5(H20)2 ][Y ], appear to be small and the question of reaction mechanism remains somewhat open at present. The effect of solvent on reaction between [Co(CN)5-(H20)]2 and N 69 and the small positive activation volumes for anation by Br-, I- and SCN-70 are indicative of a dissociative mechanism, but the existence of a long-lived five-coordinate intermediate has not been definitively established. 

Rate Law and Rate Constants of Hydroxyl Radical Reactions 179 The reaction kinetics can be described by the second order rate law of Eq. 6-30. 

With 30-70 wt% PVC, the kinetic curves show an initial sharp decrease followed by a slow component that satisfactorily obeys a second-order rate law (Fig. 13.6). Levin et al. proposed that the initial change in the absorption corresponds to the in-cage combination of the radicals pairs, while the slow component represents the combination of the radicals that have diffused into the (polymer) bulk. Interestingly, at >70 wt% PVC, only the fast in-cage combination of the geminate radical pairs can be observed. From the relative amplitudes of the fast and slow components of the kinetic curves, approximated values for the cage factor Fcab were calculated. Fcab increase from zero in films with <30 wt% PVC to near unity (>0.95) in films with >70 wt % PVC. 

Kinetic studies on the imidoyl complexes reveal strict second-order rate laws where the observed rate is related to the donor properties of the incoming ligandJ ... 

Kinetics of e -h recombination may depend on its mode if one electron is excited and this is recombined with h , the recombination rate obeys the first-order rate law, while if multiple e -h+ appears at the same time within a photocatalyst particle, the rate obeys the second-order rate law. Actually, in a femtosecond pmnp-probe diffuse reflection spectroscopic analysis of tita-nia samples, photoabsorption at 620 nm by trapped electrons showed second-order decay with a component of baseline as follows ... 

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