Second-Order Kinetics Overall

For the reaction of A and B to produce C and D, it is more likely that the kinetics would be second order overall, with first order in the concentration of both A and B rather than just first order in A. If this were the case, then the solution would be different than that which we foxmd in the foregoing and would be derived as follows  

This cannot be solved as written we need an expression for Cb in terms of Ca in order to substitute and do the integration. This can be obtained by going back to the stoichiometric statement  

we can substitute this expression for Cb in terms of Ca into the differential equation describing the change in Ca, make it autonomous, and derive an expression for the time dependence of Ca  

Using Mathematica we have two primary choices on how to proceed with the solution to this equation—we can rearrange it into its separable components and then integrate both sides of the equation or we can solve it directly we will do the latter. 

Chapter 7 Reacting Systems—Kinetics and Batch Reactors 

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The characteristics of this mechanism are that (i) the attacking base and the substrate both take part in the rate-determining step, which has second-order kinetics overall (first-order in base and first-order in substrate) (ii) a large primary isotope effect is usually observed (iii) since the mechanisms of 5n2 and E2 differ much more than those of 5n1 and 1 reactions, the substitution/elimination ratio can be controlled in most cases by choosing appropriate conditions (iv) no rearrangement reactions are observed (v) the rate of elimination depends upon the strength of the base (vi) the stereospecificity of an E2 reaction depends on the conditions (see Section 5.1.2.4). 

Most inner-sphere processes exhibit second order kinetics overall, and interpreting the data is seldom simple. Any one of bridge formation, electron transfer or bridge cleavage can be rate-determining. In the reaction between... 

Overall the reaction exhibits second order kinetics Both the ester and the base are involved m the rate determining step or m a rapid step that precedes it... 

All these facts—the observation of second order kinetics nucleophilic attack at the carbonyl group and the involvement of a tetrahedral intermediate—are accommodated by the reaction mechanism shown m Figure 20 5 Like the acid catalyzed mechanism it has two distinct stages namely formation of the tetrahedral intermediate and its subsequent dissociation All the steps are reversible except the last one The equilibrium constant for proton abstraction from the carboxylic acid by hydroxide is so large that step 4 is for all intents and purposes irreversible and this makes the overall reaction irreversible... 

This bimolecular process is called the S/ 2 mechanism. It yields overall second-order kinetics (unless the nucleophile is the solvent, in which case apparent first-order kinetics are seen). 

The simplest overall interpretation of these data is in terms of a rate-determining dissociation. Entropies of activation are positive and the solvent-dependence for a better leaving group (Cl) is less marked than for a worse one (Br) in the case of reaction (38) . For the dimeric carbonyls, [M(CO)4X]2, bridge-breaking, essentially the same dissociation, could result in a rapid pre-equilibrium. If this were followed by a second dissociative step, then the kinetics could be first-order (as for Mn), while a rate-determining entry of L could produce second-order kinetics (as for Re). 

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. 

Jacobsen et al. reported enhanced catalytic activity by cooperative effects in the asymmetric ring opening (ARO) of epoxides.[38] Chiral Co-salen complexes (Figure 4.27) were used, which were bound to different generations of commercial PAMAM dendrimers. As a direct consequence of the second-order kinetic dependence on the [Co(salen)] complex concentration of the hydrolytic kinetic resolution (HKR), reduction of the catalyst loading using monomeric catalyst leads to a sharp decrease in overall reaction rate. 

The types of systems we deal with are primarily gas-solid (Section 9.1) and gas-liquid (Section 9.2). In these cases, we assume first- or second-order kinetics for the intrinsic reaction rate. This enables analytical expressions to be developed in some situations for the overall rate with transport processes taken into account. Such reaction models are incorporated in reactor models in Chapters 22 and 24. 

The two limiting cases of nucleophilic displacement reactions are designated as SnI or Sn2 to indicate those mechanisms that respectively display overall first-order or second-order kinetics. These mechanisms are illustrated by the classical case of the reaction of hydroxide ion with chloromethane, and they differ with respect to the timing of the bond-breaking step relative to the bondmaking step. 

On the basis of overall second-order kinetics of hydroboration with disiamylbor-ane, a two-step mechanism was suggested involving the interaction between the... 

Overall second-order kinetics have been observed for catalysis by bases, alcohols or acids, and in the base-catalysed reactions the formation of ether groups is relatively insignificant46147). The base catalysis can be further activated by an acid co-catalyst, HA. For example any resident hydroxyl groups can act as internal co-catalysts. Tanaka and Kakiuchi48) proposed the following scheme for the reaction catalysed by base (B) with acid co-catalyst ... 

The reaction is said to follow second-order kinetics because its rate is proportional to a product of two concentrations. The kinetics are first order in either [A] or [B] alone, but second order overall. The rate constant for a second-order reaction has the dimensions of m-1s . 

Kinetics of symmetrisation were followed using a nephelometric method, normally with the ammonia present in a large excess under these conditions, reaction (17) was shown to follow second-order kinetics with respect to the alkyl mercuric bromide38. Variation of the concentration of the excess ammonia showed38 that reaction (17) was also second-order in ammonia, and hence38 that the reaction is of the fourth-order overall, viz. 

Scheme 9.4 Possible mechanisms involving intermediates and second-order kinetic terms for the same overall transformation, X + R -+ P + Y.

Nevertheless, the development of general kinetic data for the hydrodesulfurization of different feedstocks is complicated by the presence of a large number of sulfur compounds each of which may react at a different rate because of structural differences as well as differences in molecular weight. This may be reflected in the appearance of a complicated kinetic picture for hydrodesulfurization in which the kinetics is not, apparently, first order (Scott and Bridge, 1971). The overall desulfurization reaction may be satisfied by a second-order kinetic expression when it can, in fact, also be considered as two competing first-order reactions. These reactions are (1) the removal of nonasphaltene sulfur and (2) the removal of asphaltene sulfur. It is the sum of these reactions that gives the second-order kinetic relationship. 

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