Second-order reactions examples

Diffusion with a second order reaction (example 3.2.1) is considered here again. The dimensionless concentration is governed by ... 

Example 5 Application of Effectiveness For a second-order reaction in a plug flow reactor the Thiele modulus is ( ) = SVQ, and inlet concentration is C50 = 1.0. The equation will he integrated for 80 percent conversion with Simpsons rule. Values of T) are... 

The classical example of a second-order reaction is the formation of HI(g) which was discussed above for which the reaction rate is given by... 

A second order reaction is performed adiabatically in a CESTR. Use die data in Example 6-11 to plot bodi conversions for die mass and heat balance equations. The second order rate constant k is... 

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. 

Choice of initial conditions. To give a very obvious example, in Chapter 2 we saw that a second-order reaction A -I- B —> products could be run with the initial conditions Ca = cb, thus permitting a very simple plotting form to be used. For complex reactions, it may be possible to obtain a usable integrated rate equation if the initial concentrations are in their stoichiometric ratio. 

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. 

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

The characteristics of zero-, first-, and second-order reactions are summarized in Table 11.2. To determine reaction order, the properties in either of the last two columns at the right of the table can be used (Example 11.5). 

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... 

Example 1.6 Apply Equation (1.54) to calculate the mean residence time needed to achieve 90% conversion in a CSTR for (a) a first-order reaction, (b) a second-order reaction of the type A - - B — Products. The rate constant... 

The examples in this section have treated a single, second-order reaction, although the approach can be generalized to multiple reactions with arbitrary... 

Some progress has been made in developing theoretical expressions for rj(6) for deactivation processes such as coking. Deactivation by loss of active sites can be modeled as a chemical reaction proceeding in parallel with the main reaction. It may be substantially independent of the main reaction. Site sintering, for example, will depend mainly on the reaction temperature. It is normally modeled as a second-order reaction ... 

Example 15.12 Find the outlet concentration from a completely segregated stirred tank for a first-order reaction. Repeat for a second-order reaction with = —kcp-. 

Example 15.15 Calculate limits on the fraction unreacted for a second-order reaction with = 5. Consider the following states of knowledge ... 

Part (c) in Example 15.15 illustrates an interesting point. It may not be possible to achieve maximum mixedness in a particular physical system. Two tanks in series—even though they are perfectly mixed individually—cannot achieve the maximum mixedness limit that is possible with the residence time distribution of two tanks in series. There exists a reactor (albeit semi-hypothetical) that has the same residence time distribution but that gives lower conversion for a second-order reaction than two perfectly mixed CSTRs in series. The next section describes such a reactor. When the physical configuration is known, as in part (c) above, it may provide a closer bound on conversion than provided by the maximum mixed reactor described in the next section. 

Parameter setup for Example 8.1. Irreversible second-order reaction, small scale... 

The variables are defined as previously. Thus if, for example, experimental data is to be tested for second-order reaction behaviour, then data plotted as Cai versus t should be examined from experiments, during which (k x Cao) is kept constant. 

In a reaction of either zero-order or second-order, the time to reach a certain fraction of the initial concentration, for example, t /2 or /yo% [the time required for the drug concentration to decrease to 90% of its original value (i.e., 10% degradation)] is dependent on to- This is illustrated in Fig. 1, in which a zero-order reaction (Fig. la) and a second-order reaction (Fig. lb) are plotted with two initial concentrations. It is readily seen that for a zero-order reaction, the ti/2 increases... 

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. 

Integrals involving partial fractions occur often in chemical kinetics. For example, the differential equation which represents a second-order reaction is... 

The use of blocked isocyanates to cure hydroxyl containing coatings is an example of a complex system having many practical applications. The chemistry of blocked isocyanates has been reviewed previously by Wicks (J.,.2). The cure reaction proceeds via consecutive first and second order reactions ... 

If the two competing reactions have the same concentration dependence, then the catalyst pore structure does not influence the selectivity because at each point within the pore structure the two reactions will proceed at the same relative rate, independent of the reactant concentration. However, if the two competing reactions differ in the concentration dependence of their rate expressions, the pore structure may have a significant effect on the product distribution. For example, if V is formed by a first-order reaction and IF by a second-order reaction, the observed yield of V will increase as the catalyst effectiveness factor decreases. At low effectiveness factors there will be a significant gradient in the reactant concentration as one moves radially inward. The lower reactant concentration within the pore structure would then... 

As a 3-step mechanism, the electron-transfer paradigm provides a pair of discrete intermediates [D, A] and D+, A for the prior organization and the activation, respectively, of the donor and the acceptor. The quantitative evaluation of these intermediates would allow the overall second-order reaction (k2) to be determined. Although the presence of [D, A] does not necessarily imply its transformation to D+, A-, a large number and variety of donor/ acceptor couples showing transient charge-transfer absorptions associated with [D, A] have now been identified. In each case, the product can be predicted from the expected behavior of the individual ion radicals D+ and A-. Consider for example, the labile 1 1 benzene complex with bromine that has been isolated at low temperatures and characterized crystallographically (Chart 9).256... 

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. 

It can be shown that the experimental data given do not conform to the hypothesis of a first-order reaction, by the test corresponding to that in Example 4-3. We then consider the possibility of a second-order reaction. From equation 4.2-6, we write the combined assumed form of the rate law and the material balance equation (for constant volume), in terms of CH3CHO (A), as... 

For the gas-phase, second-order reaction C2H4 + C4H6 - CgHio (or A + B - C) carried out adiabatically in a 2-liter experimental CSTR at steady-state, what should the temperature (T/K) be to achieve 40% conversion, if the (total) pressure (P) is 1.2 bar (assume constant), the feed rate (q0) is 20 cm3 s-1, and.. the reactants are equimolar in the feed. The Arrhenius parameters are EA = 115,000 J mol-1 and A =3.0x 107L mol-1s-1 (Rowley and Steiner, 1951 see Example 4-8). Thermochemical data are as follows (from Stull et al., 1969) ... 

The reactivity of NO with O2 is dramatically affected upon coordination of one of the diatomic components to a metal center. For example, the second-order reactions of NO with oxyhemoglobin, Hb(02) and oxymyoglobin, Mb(02) (e.g. Eq. (47)) are quite fast and have been used as colorimetric tests for NO (105). The nitrogen product is NO3 rather than N02 that is the product of aqueous autoxidation (106). While the reaction of 02 with nitrosyl myoglobin Mb(NO) (Eq. (48)) might superficially appear similar it is much slower and follows a different rate law (107). Possible mechanisms will be discussed below. 

An example of the effect of temperature on selectivity (yield) for the case of two reactions where A goes to product P by a first-order reaction, and P goes to impurity X by a second-order reaction is shown in Figure 3.9. Say that the undesired reaction is highly exothermic. If the product P is removed as soon as it is formed, the second (undesired) reaction will not occur. It is evident that the overall reaction would be more hazardous and the yield of product P less if an incorrect reactor type is selected. From Figure 3.9, it can be seen that the higher the temperature, the greater the decrease in selectivity. At low... 

Numerical example. Take a second order reaction with = 0.8 and y = 5.0, and find the other variables as functions of Da. 

Worked Example 8.17 The following kinetic data were obtained for the second-order reaction between osmium tetroxide and an alkene, to yield a 1,2-diol. Values of k are pseudo-order rate constants because the 0s04 was always in a tiny minority. Determine the second-order rate constant k2 from the data in the following table ... 

For example, for the second-order reaction in (3.142) the Jacobian matrix is given by... 

For non-linear chemical reactions, this term leads to new unclosed terms that are difficult to model. For example, even the isothermal second-order reaction, (3.142), where the joint dissipation chemical source term is given by... 

The preceding approach applies to all linear systems that is, those involving mechanisms in which only first-order or pseudo-first-order homogeneous reactions are coupled with the heterogeneous electron transfer steps. As seen, for example, in Section 2.2.5, it also applies to higher-order systems, involving second-order reactions, when they obey pure kinetic conditions (i.e., when the kinetic dimensionless parameters are large). If this is not the case, nonlinear partial derivative equations of the type... 

When a product, formed in a second order reaction, acts as a catalyst or effects the rate of reaction, the reaction is known as autocatalytic reaction. For example, the acid catalysed hydrolysis of various esters and similar compounds and various biochemical processes. 

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