Second-Order Reactions with One Reactant

The simplest example of a second-order reaction has one type of molecule reacting with itself  

A gas phase reaction believed to be elementary and second order is 

collisions between two HI molecules supply energy and also supply the reactants needed to satisfy the observed stoichiometry. 

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For a second-order reaction with one reactant, SI a = —ka. Equation (1.49) becomes a quadratic in The solution is... 

For a second-order reaction with one reactant, we have... 

The type of chemical system that has received the most attention is the one in which the dissolved gas (component A) undergoes an irreversible second-order reaction with a reactant (component B) dissolved in the liquid. For the present, the gas will be taken as consisting of pure A, so that complications arising from gas film resistance can be avoided. The stoichiometry of the reaction is represented by... 

For a second-order reaction involving one reactant, we would determine the rate constant k by plotting 1/[A] versus f to yield a straight line with a slope of k in accordance with Eq. 2-11. 

An irreversible, elementary reaction must have Equation (1.20) as its rate expression. A complex reaction may have an empirical rate equation with the form of Equation (1.20) and with integral values for n and w, without being elementary. The classic example of this statement is a second-order reaction where one of the reactants is present in great excess. Consider the slow hydrolysis of an organic compound in water. A rate expression of the form... 

Second-Order Batch Reactions with One Reactant. We choose to write the stoichiometric equation as... 

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... 

In this chapter we report on an investigation of the kinetics of this reaction with several monomers. Bunnett and Levitt have studied the kinetics of nucleophilic substitution between p-substituted bromobenzenes and sodium methoxide (1) and found these reactions to be second order. Therefore, for a polymer-forming reaction between difunctional reactants, one would expect a second-order reaction with respect to the concentra-... 

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). 

The reactivity of the model phenols and benzyl alcohols with phenyl isocyanate was determined in the presence of a tertiary amine (DMCHA) and a tin catalyst (DBTDL) by measurement of the reaction kinetics. The experimental results based on initial equal concentrations of phenyl isocyanate and protic reactants showed that the catalyzed reactions followed second order reaction with respect to the disappearance of isocyanate groups (see Figure 1). It was also found that a linear relationship exists between the experimental rate constant kexp, and the initial concentration of the amine catalyst (see Figure 2). In the case of the tin catalyst, the reaction with respect to catalyst concentration was found to be one-half order (see Figures 3-4). A similar relationship for the tin catalyzed urethane reaction was found by Borkent... 

Second-Order Batch Reactions with One Reactant... 

EXAMPLE 11.11 The following is a fictitious set of data for the concentration of the reactant in a chemical reaction with one reactant. Determine whether the reaction is first, second, or third order. Find the rate constant and the initial concentration. 

Note that here the half-life is inversely proportional to the initial reactant concentration. This relationship means that a second-order reaction with a high initial reactant concentration has a shorter half-life, and one with a low initial reactant concentration has a longer half-life. Therefore, as a second-order reaction proceeds, the half-life increases. 

Finally, let s increase the complexity just one step further. Consider a mechanism in which a reactive intermediate is formed in a first order reaction along with a stable product (Pi), followed by a second order reaction with a second reactant that converts the intermediate to product P2 (Eq. 7.53). We go through the same mathematical process of analyzing the rate (Eq. 7.54), applying the SSA (Eq. 7.55), and performing algebraic manipulation (Eq. 7.56), to arrive at the result (Eq. 7.57). The prediction is that the reaction is first order in A, less than first order in B, and is retarded by P] (since its concentration is only in the denominator). Such predictions are important in guiding the experiments used to test if a reaction fits this mechanistic scenario. 

Solvated electrons can be obtained in concentrations up to 10" moldm . Reactions with solvated electrons are very fast second-order reactions, with rate constants between 10 to 5x10 mor dm s", which means that their rates are close to the diffusion control. A reaction is diffusion-controlled when the reaction rate is dependent upon the rate at which reactants diffuse toward one another. A diffusion-controlled reaction must have a small activation energy, because if is high (EJRT 1), then the reaction rate is controlled by the number of molecules with energy higher than the activation energy, not by the diffusion rate. Reactions with high E are activation-controlled. 

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. 

Use of the isolation or pseudo-order technique. This approach is discussed in Chapter 2, where it was shown how a second-order reaction could be converted to a pseudo-first-order reaction by maintaining one of the reactant concentrations at an essentially eonstant level. The same method may be usefully applied to eomplex reactions. In this way, for example. Scheme XI can be studied under conditions such that it functions as Scheme IX. A corollary that must be kept in mind is that a reaction system that is observed to behave in accordance with (as an example) Scheme IX may actually be more complex than it appears to be, if an unsuspected reactant is present under pseudo-order conditions. 

Second law of thermodynamics A basic law of nature, one form of which states that all spontaneous processes occur with an increase in entropy, 457 Second order reaction A reaction whose rate depends on the second power of reactant concentration, 289,317q gas-phase, 300t... 

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. 

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. 

One way to ensure that back reactions are not important is to measure initial rates. The initial rate is the limit of the reaction rate as time reaches zero. With an initial rate method, one plots the concentration of a reactant or product over a short reaction time period during which the concentrations of the reactants change so little that the instantaneous rate is hardly affected. Thus,by measuring initial rates, one can assume that only the forward reaction in Eq. (35) predominates. This would simplify the rate law to that given in Eq. (36) which as written would be a second-order reaction, first-order in reactant A and first-order in reactant B. Equation (35), under these conditions, would represent a second-order irreversible elementary reaction. 

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