Integrated second-order rate

When even second-order reactions are included in a group to be analyzed, individual integration methods maybe needed. Three cases of coupled first- and second-order reactions will be touched on. All of them are amenable only with difficulty to the evaluation of specific rates from kinetic data. Numerical integrations are often necessary. 

This is the equation for a plug flow reactor. It can be derived directly from the rate equations with the aid of Laplace transforms. The sequences of second-order reactions of Figs. 7-5n and 7-5c required numerical integrations. 

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... 

We can reach two useful conclusions from the forms of these equations First, the plots of these integrated equations can be made with data on concentration ratios rather than absolute concentrations second, a first-order (or pseudo-first-order) rate constant can be evaluated without knowing any absolute concentration, whereas zero-order and second-order rate constants require for their evaluation knowledge of an absolute concentration at some point in the data treatment process. This second conclusion is obviously related to the units of the rate constants of the several orders. 

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. 

The differential rate equation is - dCf /dt = kCffiR, and the mass balanee equation is Ca i A = r r- Eliminating Ca between these equations and integrating gives the usual second-order integrated equation, whieh can be written in this form ... 

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. 

The power to which the concentration of reactant A is raised in the rate expression is called the order of the reaction, m. If tn is 0, the reaction is said to be zero-order If m = 1, the reaction is first-order if mi = 2, it is second-order and so on. Ordinarily, the reaction order is integral (0,1,2,...), but fractional orders such as are possible. 

Kinetic data for the reaction between PuOi- and Fe2+, given in Table 2-4, are fitted to the integrated rate law for mixed second-order kinetics. The solid curve represents the least-squares fit to Eq. (2-34). left and (2-35). right. 

Now we derive the integrated rate law for second-order reactions with the rate law Rate of consumption of A = [A]2... 

To obtain the integrated rate law for a second-order reaction, we recognize that the rate law is a differential equation and write it as... 

As we have seen for first- and second-order rate laws, each integrated rate law can be rearranged into an equation that, when plotted, gives a straight line and the rate constant can then be obtained from the slope of the plot. Table 13.2 summarizes the relationships to use. 

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... 

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... 

Students who have taken calculus will recognize that Equation results from integration of the second-order rate law. 

Deriving the integrated rate equation of a second-order reaction is a little more complicated. Let us assume the second-order reaction... 

Since the measurements of conductance change are not directly related to the composition of the solution, as an alternative method numerical integration of the differential rate equations implied by the proposed mechanism was employed. The second order rate coefficients obtained by this method are... 

Figure 4.21 illustrates the validity of an integral form of the rate equation governing aggregation of silver atoms at several temperatures. The activation energy of surface aggregation (second order) was found... 

The order of the reaction, n, can be defined as n = a + b. Extended to the general case, the order of a reaction is the numerical sum of the exponents of the concentration terms in the rate expression. Thus if a = b = 1, the reaction just described is said to be second-order overall, first-order relative to A, and first-order relative to B. In principle, the numerical value of a or b can be integral or fractional. 

It is obvious that to quantify the rate expression, the magnitude of the rate constant k needs to be determined. Proper assignment of the reaction order and accurate determination of the rate constant is important when reaction mechanisms are to be deduced from the kinetic data. The integrated form of the reaction equation is easier to use in handling kinetic data. The integrated kinetic relationships commonly used for zero-, first-, and second-order reactions are summarized in Table 4. [The reader is advised that basic kinetic... 

The complexity of the integrated form of the second-order rate equation makes it difficult to apply in many practical applications. Nevertheless, one can combine this equation with modem computer-based curve-fitting programs to yield good estimates of reaction rate constants. Under some laboratory conditions, the form of Equation (A1.25) can be simplified in useful ways (Gutfreund, 1995). For example, this equation can be simplified considerably if the concentration of one of the reactants is held constant, as we will see below. 

Initially, it could be postulated that the reaction could be zero order, first order or second order in the concentration of A and B. However, given that all the reaction stoichiometric coefficients are unity, and the initial reaction mixture has equimolar amounts of A and B, it seems sensible to first try to model the kinetics in terms of the concentration of A. This is because, in this case, the reaction proceeds with the same rate of change of moles for the two reactants. Thus, it could be postulated that the reaction could be zero order, first order or second order in the concentration of A. In principle, there are many other possibilities. Substituting the appropriate kinetic expression into Equation 5.47 and integrating gives the expressions in Table 5.5 ... 

ILLUSTRATION 3.2 USE OF A GRAPHICAL INTEGRAL METHOD FOR DETERMINING THE RATE CONSTANT FOR A CLASS II SECOND-ORDER REACTION... 

The integrated rate equation for a reaction that is second order with respect to N02 as the only reactant ... 

Since rate = [N02], we must use the second order integrated rate equation - [no2]0 =... 

This problem may be solved by linear regression using equations 3.4-11 (n = 1) and 3.4-9 (with n = 2), which correspond to the relationships developed for first-order and second-order kinetics, respectively. However, here we illustrate the use of nonlinear regression applied directly to the differential equation 3.4-8 so as to avoid use of particular linearized integrated forms. The method employs user-defined functions within the E-Z Solve software. The rate constants estimated for the first-order and second-order cases are 0.0441 and 0.0504 (in appropriate units), respectively (file ex3-8.msp shows how this is done in E-Z Solve). As indicated in Figure 3.9, there is little difference between the experimental data and the predictions from either the first- or second-order rate expression. This lack of sensitivity to reaction order is common when fA < 0.5 (here, /A = 0.28). 

The values of a A, and EA must be determined from experimental data to establish the form of the rate law for a particular reaction. As far as possible, it is conventional to assign small, integral values to a2, etc., giving rise to expressions like first-order, second-order, etc. reactions. However, it may be necessary to assign zero, fractional and even negative values. For a zero-order reaction with respect to a particular substance, the rate is independent of the concentration of that substance. A negative order for a particular substance signifies that the rate decreases (is inhibited) as the concentration of that substance increases. 

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. 

Rate data for the condensation of formaldehyde (F) with sodium paraphenolsulfonate (M) were taken by Stults et al (CEP Symp Series 4 38, 1952) at 100°C and pH = 8.35. Equal quantities of the reactants were present initially. Check first and second order mechanisms with the tabulated data. Integrated rate equations are... 

A second order reaction is catalyzed in a PFR by a porous catalyst made up of small spheres. At the inlet the Thiele modulus is 0O = 10. Integrate the rate equation up to 90% conversion. 

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