Integrated rate, first order

In Problem 1 the reader was called upon to integrate simple first order and second order rate laws. In this section we will develop techniques for dealing with more complicated differential equations linear and nonlinear, with two or more dependent variables. 

The half-life of a first-order rate expression can be obtained by integrating the first-order differential rate equation. 

FIGURE A10.3 Determination of the rate constant by the integral method first-order kinetics r = kc. ... 

The integrated form of the rate law for equation 13.4, however, is still too complicated to be analytically useful. We can simplify the kinetics, however, by carefully adjusting the reaction conditions. For example, pseudo-first-order kinetics can be achieved by using a large excess of R (i.e. [R]o >> [A]o), such that its concentration remains essentially constant. Under these conditions... 

In a curve-fitting method the concentration of a reactant or product is monitored continuously as a function of time, and a regression analysis is used to fit an appropriate differential or integral rate equation to the data. Eor example, the initial concentration of analyte for a pseudo-first-order reaction, in which the concentration of a product is followed as a function of time, can be determined by fitting a rearranged form of equation 13.12... 

Several additional approaches for analyzing mixtures have been developed that do not require such a large difference in rate constants.Because both A and B react at the same time, the integrated form of the first-order rate law becomes... 

Proceeding in the same manner as for a first-order reaction, the integrated form of the rate law is derived as follows... 

The deomposition of AIBN in xylene at 77°C was studiedt by measuring the volume of N2 evolved as a function of time. The volumes obtained at time t and t = 00, are and, respectively. Show that the manner of plotting used in Fig. 6.1 is consistent with the integrated first-order rate law and evaluate k j. 

Flooding and Pseudo-First-Order Conditions For an example, consider a reaction that is independent of product concentrations and has three reagents. If a large excess of [BJ and [CJ are used, and the disappearance of a lesser amount of A is measured, such flooding of the system with all components butM permits the rate law to be integrated with the assumption that all concentrations are constant except A. Consequentiy, simple expressions are derived for the time variation of A. Under flooding conditions and using equation 8, if x happens to be 1, the time-dependent concentration... 

Separating the variables and integrating between the limits shown below yields Eqs. (2-6), (2-7), and (2-8) as equivalent forms of the integrated first-order rate equation. 

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. 

Evidently simple first-order behavior is predicted, the reactant concentration decaying exponentially with time toward its equilibrium value. In this case a complicated differential rate equation leads to a simple integrated form. The experi-... 

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 only kinetic data reported are in a Ph.D. thesis (41). Integral order kinetics were usually not obtained for the reaction of a number of ketones with piperidine and a number of secondary amines with cyclohexanone. A few of the combinations studied (cyclopentanone plus piperidine, pyrrolidine, and 4-methylpiperidine, and N-methylpiperazine plus cyclohexanone) gave reactions which were close to first-order in each reactant. Relative rates were based on the time at which a 50% yield of water was evolved. For the cyclohexanone-piperidine system the half-time (txn) for the 3 1 ratio was 124 min and for the 1 3 ratio 121 min. It appears that an... 

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. 

Using calculus, it is possible to develop integrated rate equations relating reactant concentration to time. We now examine several such equations, starting with first-order reactions. 

It follows that the rate constant is 0.35/min the integrated first-order equation for the decomposition of N205 is... 

Our initial experimental results indicated that the kinetic model— first order in liquid phase CO concentration—was the leading candidate. We designed an experimental program specifically for this reaction model. The integrated rate expression (see Appendix for nomenclature) can be written as ... 

Wilkinson s method allows the evaluation of the reaction order from data taken during the first half-life. This, as we saw, was not possible from treatment by the integrated rate law. Note, however, that relatively small errors in [A] can lead to a larger error in E at small conversions.17... 

This peculiar form applies when a dimeric molecule dissociates to a reactive monomer that then undergoes a first-order or pseudo-first-order reaction. This scheme is considered in Section 4.3. Unless one can work at either of the limits, the form is such that a numerical solution or the method of initial rates will be needed, since the integrated equation has no solution for [A]r. 

First-order kinetics. Show that the first-order integrated rate expression can be written as... 

An important application of an integrated rate law is to confirm that a reaction is in fact first order and to measure its rate constant. From Eq. 5a, we can write... 

STRATEGY The level of mercury(II) in the urine can be predicted by using the integrated first-order rate law, Eq. 5b. To use this equation, we need the rate constant. Therefore, start by calculating the rate constant from the half-life (Eq. 7) and substitute the result into Eq. 5b. 

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. 

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