Second-Order Dependence

Second-order kinetics play an important role in the reactions of complex ions. Two identical reactants may be 

Second-order reactions between two dissimilar molecules A and B are invariably studied under pseudo first-order conditions (Sec. 1.4.4) because this is by far the simpler procedure. If however this condition cannot be used because, for example, both reactants absorb heavily 

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The kinetics of the nitration of benzene, toluene and mesitylene in mixtures prepared from nitric acid and acetic anhydride have been studied by Hartshorn and Thompson. Under zeroth order conditions, the dependence of the rate of nitration of mesitylene on the stoichiometric concentrations of nitric acid, acetic acid and lithium nitrate were found to be as described in section 5.3.5. When the conditions were such that the rate depended upon the first power of the concentration of the aromatic substrate, the first order rate constant was found to vary with the stoichiometric concentration of nitric acid as shown on the graph below. An approximately third order dependence on this quantity was found with mesitylene and toluene, but with benzene, increasing the stoichiometric concentration of nitric acid caused a change to an approximately second order dependence. Relative reactivities, however, were found to be insensitive... 

The kinetics of hydrolysis reactions maybe first-order or second-order, depending on the reaction mechanism. However, second-order reactions may appear to be first-order, ie, pseudo-first-order, if one of the reactants is not consumed in the reaction, eg, OH , or if the concentration of active catalyst, eg, reduced transition metal, is a small fraction of the total catalyst concentration. 

The solutions are most stable above pH 11 where the decomposition rate is nearly independent of pH. In this region, the decomposition rate has a second-order dependence on the concentration of hypochlorite. It also increases with increa sing ionic strength. Thus concentrated solutions decompose much faster than dilute solutions. Because of an unusually high activation energy, the decomposition rate increases greatiy with temperature. Nevertheless, solutions with less than about 6% available chlorine and a pH above 11 have acceptable long-term stabiUty below about 30°C. 

Rates that are independent of aromatic substrate concentration have been found for reaction of benzyl chloride catalyzed by TiCl4 or SbFj in nitromethane. This can be interpreted as resulting from rate-determining formation of the electrophile, presumably a benzyl cation. The reaction of benzyl chloride and toluene shows a second-order dependence on titanium tetrachloride concentration under conditions where there is a large excess of hydrocarbon. ... 

Detailed investigations indicated an interesting mechanism for azide openings catalyzed by 2 [6]. Chloride-epoxide addition products were observed in the initial stages of the ARO reaction with 2 in amounts commensurate with the catalyst loading. Azide complex 3, characterized as the TH F adduct, was isolated from the reaction mixture and proved to be an active and recyclable catalyst for the ARO, pointing to the role of 2 as that of a precatalyst. Kinetic experiments revealed a second-order dependence on the concentration of 3, a zero-order dependence on azide source, and inverse-order dependence on epoxide concentration. The sue-... 

Jacobsen developed a method employing (pybox)YbCl3 for TMSCN addition to meso-epoxides (Scheme 7.22) [46] with enantioselectivities as high as 92%. Unfortunately, the practical utility of this method is limited because low temperatures must be maintained for very long reaction times (up to seven days). This reaction displayed a second-order dependence on catalyst concentration and a positive nonlinear effect, suggesting a cooperative bimetallic mechanism analogous to that proposed for (salen)Cr-catalyzed ARO reactions (Scheme 7.5). 

The kinetic data based on the demonstration of specific acid catalysis in buffers, solvent isotope effects and acidity functions all support mechanisms where the proton-transfers are fast. It is possible to write equations which accommodate these facts together with the first-order dependence on hydrazo-compound and the concurrent first and second-order dependence on acidity. These are... 

For the thermally-initiated case, the initiation rate has a second-order dependence on monomer concentration as suggested by FloryLS.] instead of a third-order dependence as suggested by Hui and Hamielec[ ]. 

Kinetic studies of the hydride cluster [W3S4H3(dmpe)3] with acids in a non-coordinating solvent, i.e., dichloromethane, under the pseudo-first-order condition of acid excess, show a completely different mechanism with three kineti-cally distinguishable steps associated to the successive formal substitution of the coordinated hydrides by the anion of the acid, i.e., Ch in HCl [37]. The first two kinetic steps show a second-order dependence with the acid concentration. 

This is the first example of a proton transfer process to a hydride complex with a second-order dependence. Theoretical calculations indicate that the role of the HX molecules is the formation of W-H H-Cl- H-Cl adducts that convert into W-Cl, H2 and HCl2 in the rate-determining state through hydrogen complexes as transition states. 

A second-order dependence is consistent with a mechanism in which the first, rate-determining step is the collisional interaction of two molecules of butadiene. 

Alternative (ii) corresponds to the [Re(CO)4X]2 case, equations (41) and (42) above. However, it was here favoured largely because no second-order term was observed for the Re(CO)5X and Re(CO)4LX substitution. In the case of Mo(CO)5Py, expected to be closely similar to Mo (CO)4dipy, a second-order dependence has been observed. 

Both reaction paths are acid-catalysed and are subject to retardation by specific ions probably by removal of free Br . The second-order dependence with respect to reductant has several precedents, e.g. Fe(iri) oxidation of 1 and Mn(III) oxidation of HN3. The acid catalysis results from suppression of the hydrolysis to MnOH which is ineffective in this oxidation. 

The V(V) sulphate oxidation of arsenite shows an interesting second-order dependence on oxidant concentration, the rate expression being... 

The enoHsation may be rate-determining (to afford the zero-order dependence on oxidant concentration) or the oxidation step may be slower (to give the first-order dependence). The second-order dependence on oxidant concentration for acetone and nitroethane cannot involve slow oxidation of a free radical and no ready alternative explanation is available. Maltz showed that the rate of oxidation of isobutanal equals the rate of enolisation, and that two main paths of oxidation are followed subsequent to enolisation leading either to tetramethyldihydropyrazine and a poly-aquocyanoiron(II) species or to isobutyric acid. 

The tendency for N-nitrosamides to undergo hydrolysis by a nucleophilic catalysed pathway has been confirmed by studies of N-alkylnitroso acetamides (22) Results summarised in Table I for N -n-butyl-JJ -nitroso acetamide show that its decomposition is also subject to steric constraints (2,6-lutidinestrong nucleophiles (eg. imidazole, thiols) irrespective of their base strength (pK ). Further, the second order dependence on [Imidazole] is more clearly defined for the decomposit-... 

The temporal reaction heat flow data may be graphically manipulated to reveal the overall second order dependence in a quantitative manner. Reaction heat flow is converted to reaction rate using eq. (1), and the concentration of the limiting substrate 5 may be calculated according to eq. (3). From these calculations we may constract the plot in Figure 50.2b of reaction rate vs. [5]. The reaction is known to be first order in both [5] and [6] these plots reveal the curvature typical of overall second order kinetics. 

Interestingly, Hoveyda and coworkers observed a second-order dependence of the reaction rate on the concentration of zirconium in these reactions, suggesting that the zirconacyclopentane is formed from a bimetallic alkene-zirconate complex such as A in Fig. 1 [21]. This finding suggests that olefin alkylations and substitutions occur via reaction of a nucleophilic alkene unit [23]. 

Using l,8-diphenyloctatetra-l,3,5,7-ene, (DOT), as a model compound either in dilute, ( 10-5m), hexane or ethanol solutions or incorporated into a film of undegraded PVC confirmed that in the presence of HC1 it underwent a photochemical reaction which resembled that of the polyenes in thermally degraded PVC. The results indicated that the initial rates of reactions proceeding in either solvent showed a second order dependence on HC1 pressure and that the reaction was considerably slower in ethanol than in hexane. Further, when cast in PVC films, the characteristic absorption maxima of DOT were shifted about 16nm to longer wavelengths compared with their absorption in hexane and there... 

Analysis of the second semicircle can also be carried out. In essence, this depends on the second step of the reaction mechanism and a second order dependence on [CP] is found which rules out mechanisms (3)and (4)above. 

It is of interest to consider the Cu(n) reaction in more detail. The dissociation of this complex was performed in HC104 over the concentration range 0.19-0.58 mol dm-3. The reaction was observed to be first-order in complex and a plot of kobs versus [H+] confirmed the second-order dependence on [H+], The reaction has been described in terms of the following steps ... 

The experimentally observed second-order dependence on L for lm implies that 1 1Tl[L] and kohs = 2i i 2l[L]2. 

The monomeric cw-diaqua Cu(H) complex 32 has been shown to promote efficiently the transesterification of HPNP (pH = 8 and T = 298 K) with second-order dependence on complex concentration [51]. In this... 

The second-order dependence on PyO implicates that it is also involved in a second stage. One can write this reaction step, the significance of which will be considered in the following section. 

Provided that manganese is present predominantly as manganous ion, the kinetic model predicts a second-order dependence on [Mn(II)]... 

King and coworkers107 reported results for the hydroformylation reaction of ethylene using the Fe(CO)5 catalyst in the temperature range of 110-140 °C. The conclusions from kinetics testing were (1) the reaction rate was inhibited by Pco in the range 10-25 atm (2) the rate was found to exhibit second order dependency on... 

For several of these ligands the reaction is first order in ethene and thus it was suggested that the insertion of ethene in the titanacyclopentane ring is the rate-determining step, as was also found for the triazacyclohexane catalyst [13], Turnover numbers of the titanium catalyst are very high, but since some of the chromium catalysts have a second order dependency in ethene a comparison cannot be made at higher pressure these chromium catalysts will be more effective [7,17],... 

It should be noted, and will be further discussed later, that the definition of kuni in Equation 14.13 does not mean that the reaction is a first order reaction. A first order reaction would imply that kunj is independent of concentration. As indicated by Equation 14.13, kunj depends on the concentration of the third body M with which A collides for activation and deactivation. For k2 [M] much larger than ka, which means high pressure, kunj does indeed become independent of concentration. At low pressure, however, kunj depends on [M] and the overall rate of making products becomes second order, depending linearly on both [M] and [A],... 

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