Second-Order Reactions with Two Reactants

Second-order reactions with two reactants are common  

C2H5OH + CH3COH C2H5OCCH3 + H2O typically follow second-order kinetics. 

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Solution This can be done by substituting the various rate equations into Equation (1.36), integrating, and applying the initial condition of Equation (1.37). Two versions of these equations can be used for a second-order reaction with two reactants. Another way is to use the previous results for... 

Ex. 4 A second order reaction with two reactants is started with 0.1M concentrations of each reactant. It is 20% completed in 500 seconds. How long will it take the reaction to go to 70% completion ... 

Fig. 12.1. Dependence of conversion on Damkohler number for a) a first-order reaction (Eq. (13)) b) a second-order reaction (Eq. (14)) and c) a second-order reaction with two reactants and a nonstoichiometric feed composition (Eqs. (15) and (16), here for X = 0.5).

For a second-order reaction with two reactants, 01 = = —kab. Write two versions... 

Table 1.2 summarizes the design equations for elementary reactions in ideal reactors. Note that component A is the only component or else is the stoichiometrically limiting component. Thus a = alao for batch reactions and a = afor flow reactors and Ya = a in both cases. For the case of a second-order reaction with two reactants, the stoichiometric ratio is also needed ... 

Let us now consider the important case of a second order reaction with two reactants, with kinetics described by eq. (3.4). We can write the mass balances, combined with the rate equation, as follows (assuming constant density)... 

Not many redox reactions on mineral surfaces follow the two basic assumptions of Eq. 18 outer-sphere transfer and regeneration of the surface during the reaction. Fast reductive dissolutions with powerful reducing agents may potentially lead to an additional test of Eq. 18. In many cases the surface species arc consumed during the reaction. Astumian and Schelly (1984) developed a theory to compare a second-order reaction of two reactants in solution (Eq. 14) with u heterogeneous reaction of a dissolved species A and an adsorbed species B. The two different reaction environments are outlined in Figure 5. Here we denote the surface complex as =MO-B ... 

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. 

Second-Order Batch Reactions with Two Reactants. The batch reaction is now... 

Occasionally it is as useful to obtain relative constants for a series of reactants acting on a common substrate, as it is to have actual rate values. Relative rate constants are obtained by competition methods, which avoid the kinetic approach entirely. The method is well illustrated by considering the second-order reactions of two Co(III) complexes Co" and Cob" (which might, for example, be Co(NH3)5CP and Co(NH3)5Br +), with a common reductant (Cr(II) (leading in this case to CrCU and CrBr + respectively) ... 

The simplest case of parallel second-order steps is that of formation of two different dimers of a reactant A, corresponding to the network 5.23 and rate equations 5.24 (see next page). At all times, both products are formed in the same ratio rP rQ = kAP kAQ, so that the decay of A is an ordinary second-order reaction with rate coefficient k = kAP + kAQ. Likewise, the product formations are ordinary second-order reactions. (One could think of the initial amount of A as divided into two portions in the ratio kAV kAQ that react independently of one another and at the same rate, one to P and the other to Q.) All equations and plots for irreversible second-order reactions thus are valid (see Section 3.3.1). 

Micellar rate enhancements of bimolecular, non-solvolytic reactions are due largely to increased reactant concentrations at the micellar surface, and micelles should favor third- over second-order reactions. The benzidine rearrangement typically proceeds through a two-proton transition state (Shine, 1967 Banthorpe, 1979). The first step is a reversible pre-equilibrium and in the second step proton transfer may be concerted with N—N bond breaking (17) (Bunton and Rubin, 1976 Shine et al., 1982). Electron-donating substituents permit incursion of a one-proton mechanism, probably involving a pre-equilibrium step. 

A second-order reaction may typically involve one reactant (A -> products, ( -rA) = kAc ) or two reactants ( pa A + vb B - products, ( rA) = kAcAcB). For one reactant, the integrated form for constant density, applicable to a BR or a PFR, is contained in equation 3.4-9, with n = 2. In contrast to a first-order reaction, the half-life of a reactant, f1/2 from equation 3.4-16, is proportional to cA (if there are two reactants, both ty2 and fractional conversion refer to the limiting reactant). For two reactants, the integrated form for constant density, applicable to a BR and a PFR, is given by equation 3.4-13 (see Example 3-5). In this case, the reaction stoichiometry must be taken into account in relating concentrations, or in switching rate or rate constant from one reactant to the other. 

Two stirred tanks are available, one 100 m3 in volume, the other 30 m3 in volume. It is suggested that these tanks be used as a two-stage CSTR for carrying out a liquid phase reaction A + B product. The two reactants will be present in the feed stream in equimolar proportions, the concentration of each being 1.5 kmol/m3. The volumetric flowrate of the feed stream will be 0.3 x 10-3 m3/s. The reaction is irreversible and is of first order with respect to each of the reactants A and B, i.e. second order overall, with a rate constant 1.8 x 10-4 m3/kmols. 

Reactant A decomposes according to a second order reaction. Two streams are to be processed, the first with Ca0 = 1 and = 1 the other with Ca0 - 2... 

A distinction between "molecularity" and "kinetic order" was deliberately made, "Mechanism" of reaction was said to be a matter at the molecular level. In contrast, kinetic order is calculated from macroscopic quantities "which depend in part on mechanism and in part on circumstances other than mechanism."81 The kinetic rate of a first-order reaction is proportional to the concentration of just one reactant the rate of a second-order reaction is proportional to the product of two concentrations. In a substitution of RY by X, if the reagent X is in constant excess, the reaction is (pseudo) unimolecular with respect to its kinetic order but bimolecular with respect to mechanism, since two distinct chemical entities form new bonds or break old bonds during the rate-determining step. 

Reaction rates are influenced not only by the activation energy and the temperature, but also by the concentrations of the reactants. When there is only one educt, A (1), v is proportional to the concentration [A] of this substance, and a first-order reaction is involved. When two educts, A and B, react with one another (2), it is a second order reaction (shown on the right). In this case, the rate v is proportional to the product of the educt concentrations (12 mM at the top, 24 mM in the middle, and 36 mM at the bottom). The proportionality factors k and k are the rate constants of the reaction. They are not dependent on the reaction concentrations, but depend on the external conditions for the reaction, such as temperature. 

To consider the early and late mixing of a microfluid, consider the two flow patterns shown in Fig. 11.17 for a reactor processing a second-order reaction. In (a) the reactant starts at high concentration and reacts away rapidly because n> l. n b) the fluid drops immediately to a low concentration. Since the rate of reaction drops more rapidly than does the concentration you will end up with a lower conversion. Thus, for microfluids... 

Reactant A decomposes according to a second order reaction. Two streams are to be processed, the first with Ca0 = 1 and = 1 the other with Ca0 = 2 and Vg = 2. For the first stream alone in a PFR the volume needed for 75% conversion is Vrl. What arrangement of streams and reactors will require the least total volume for conversion down to Ca = 0.25 ... 

The same general conclusions apply since backmixing of products with reactants should be avoided, a tubular plug-flow reactor or a batch reactor is preferred. However, there is one respect in which a series reaction involving a second reactant B does differ from simple series reaction with one reactant, even when the orders are the same. This is in the stoichiometry of the reaction the reaction cannot proceed completely to the product Q, even in infinite time, if less than two moles... 

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