Second-order reactions reactants

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. 

At high reactant concentrations, this intermediate is favored and rapidly formed. It then decomposes in a second order reaction to form chlorine dioxide and chlorine ... 

Tubular flow reaclors operate at nearly constant pressure. How the differential material balance is integrated for a number of second-order reactions will be explained. When n is the molal flow rate of reactant A the flow reactor equation is... 

Titrimetric analysis is a classical method for generating concentration-time data, especially in second-order reactions. We illustrate with data on the acetylation of isopropanol (reactant B) by acetic anhydride (reactant A), catalyzed by A-methyl-imidazole. The kinetics were followed by hydrolyzing 5.0-ml samples at known times and titrating with standard base. A blank is carried out with the reagents but no alcohol. The reaction is... 

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. 

Scheme XIII is a more complex example. Suppose that reactant B is consumed only in the second-order reaction.

Another means is available for studying the exchange kinetics of second-order reactions—we can adjust a reactant concentration. This may permit the study of reactions having very large second-order rate constants. Suppose the rate equation is V = A caCb = kobs A = t Ca, soAtcb = t For the experimental measurement let us say that we wish t to be about 10 s. We can achieve this by adjusting Cb so that the product kc 10 s for example, if A = 10 M s , we require Cb = 10 M. This method is possible, because there is no net reaction in the NMR study of chemical exchange. 

Second-order reaction (Section 11.2) A reaction whose rate-limiting step is bimolecular and whose kinetics are therefore dependent on the concentration of two reactants. 

For a second-order reaction involving a single reactant, such as acetaldehyde (recall Example 11.1),... 

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... 

However, in most cases, relation (48) does not account for results obtained under experimental conditions used in industry, i.e. high reactant concentrations. Othmer carried out a detailed study in this field and suggested second-order reactions for the esterifications of n-butanol with acetic acid245 and monobutyl terephthalate246 catalyzed by sulfuric acid. Since such relations cannot be established in all cases, no reaction order could be found for the esterification of 2,3-butanediol with acetic arid247 in the presence of sulfuric add. Moreover, Othmer s reaction orders were obtained for very concentrated media and in our opinion cannot be connected to a mechanism. In fact, this was not Othmer s objective who established these relations for practical use in industrial esterifications. 

FIGURE 13.14 The characteristic shapes of the time dependence of the concentration of a reactant during a second-order reaction. The larger the rate constant, k, the greater is the dependence of the rate on the concentration of the reactant. The lower gray lines are the curves for first-order reactions with the same initial rates as for the corresponding second-order reactions. Note how the concentrations for second-order reactions fall away much less rapidly at longer times than those for first-order reactions do. 

Equation 8b is plotted in Fig. 13.14. We see that the concentration of the reactant decreases rapidly at first but then changes more slowly than a first-order reaction with the same initial rate. This slowing down of second-order reactions has important environmental consequences because many pollutants degrade by second-order reactions, they remain at low concentration in the environment for long periods. Equation 8a can be written in the form of a linear equation ... 

A second-order reaction has a long tail of low concentration at long reaction times. The half-life of a second-order reaction is inversely proportional to the concentration of the reactant. 

The second order reactions in Table 13.4 show wide variations in activation energy. In an activated complex, the reactant bonds are lengthened while the product bonds are beginning to form. Consider what bonds need to be stretched to form the activated complex and use bond enthalpies (Sections 2.14, 2.15, and 6.20) to explain the differences in activation energies. 

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... 

The concentration of all molecules is normally much higher than the concentration of reactant molecules, so that it remains essentially constant during the course of the reaction. Thus, what is truly a second-order reaction appears to be hrst order. 

When depends on a alone, the ODE is variable-separable and can usually be solved analytically. If depends on the concentration of several components (e.g., a second-order reaction of the two reactants variety, a = —kab), versions of Equations (1.22) and (1.23) are written for each component and the resulting equations are solved simultaneously. 

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... 

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

Note that the rates of product formation and reactant conversion indeed have the dimensions of mol per unit of time, and that these rates are proportional to the number of sites, or, in fact, the amount of catalyst present in the reactor. Also, in the case of a second order reaction, e.g. betv een adsorbed species A and B, we write the rate in the form r = Nk0j 0 by applying the mean-field approximation. Here the rate is proportional to both the total number of sites on the surface and the probability of finding a species A adjacent to a species B on the surface, the latter being proportional to the coverages of A and B. In the mean-field approximation A and B are distributed randomly over the N available sites this only tends to be valid when the adsorbents repel each other. Thus the rate is not r= k(N0/ )(N02,) since the reactants need to be on adjacent sites. Another important consideration is that we want the rate to be linearly proportional to the amount of catalyst in the reactor, in accordance with r = Nk0A0B for a second order surface reaction. 

A second-order reaction takes place in a two-phase batch system. Reactant A is supplied by gas-liquid transfer and reactant B supplied by liquid feed. The model equations are... 

This type of reaction is referred to as a second-order reaction because here we have two reactant concentration terms (or in this case, the square of one reactant concentration term) in the rate equation. Similarly the combination of two reactants to form a bimolecular complex is a second-order reaction ... 

Most biological reactions fall into the categories of first-order or second-order reactions, and we will discuss these in more detail below. In certain situations the rate of reaction is independent of reaction concentration hence the rate equation is simply v = k. Such reactions are said to be zero order. Systems for which the reaction rate can reach a maximum value under saturating reactant conditions become zero ordered at high reactant concentrations. Examples of such systems include enzyme-catalyzed reactions, receptor-ligand induced signal transduction, and cellular activated transport systems. Recall from Chapter 2, for example, that when [S] Ku for an enzyme-catalyzed reaction, the velocity is essentially constant and close to the value of Vmax. Under these substrate concentration conditions the enzyme reaction will appear to be zero order in the substrate. 

In order for the overall rate expression to be 3/2 order in reactant for a first-order initiation process, the chain terminating step must involve a second-order reaction between two of the radicals responsible for the second-order propagation reactions. In terms of our generalized Rice-Herzfeld mechanistic equations, this means that reaction (4a) is the dominant chain breaking process. One may proceed as above to show that the mechanism leads to a 3/2 order rate expression. 

Although equation 4.3.29 refers to a second-order reaction between an atom X and a molecule YZ, the theory is readily generalized to other reaction stoichiometries. An expression characterizing the equilibrium between reactants and the activated complex is used to eliminate the latter from equation 4.3.27, and the desired result is obtained. 

From a mathematical standpoint the various second-order reversible reactions are quite similar, so we will consider only the most general case—a mixed second-order reaction in which the initial system contains both reactant and product species. 

If the surface is nearly covered (0A 1) the reaction will be first-order in the gas phase reactant and zero-order in the adsorbed reactant. On the other hand, if the surface is sparsely covered (0A KAPA) the reaction will be first-order in each species or second-order overall. Since adsorption is virtually always exothermic, the first condition will correspond to low temperature and the second condition to high temperatures. This mechanism thus offers a ready explanation of a transition from first-to second-order reaction with increasing temperature. 

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