Kinetics integrated rate law

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. 

Kinetics is the study of the speed of reactions. The speed of reaction is affected by the nature of the reactants, the temperature, the concentration of reactants, the physical state of the reactants, and catalysts. A rate law relates the speed of reaction to the reactant concentrations and the orders of reaction. Integrated rate laws relate the rate of reaction to a change in reactant or product concentration over time. We may use the Arrhenius equation to calculate the activation... 

Our goal in this chapter is to help you learn about nuclear reactions, including nuclear decay as well as fission and fusion. If needed, review the section in Chapter 2 on isotopes and the section in Chapter 13 on integrated rate laws which discusses first-order kinetics. And just like the previous nineteen chapters, be sure to Practice, Practice, Practice. 

Again plotting concentration versus time using these integrated second-order rate laws gives linear plots only if the reaction is a second-order process. The rate constants can be determined from the slopes. If the concentration-time plots are not linear, then the second-order rate equations do not correctly describe the kinetic behavior. There are integrated rate laws for many different reaction orders. 

A second kind of rate law, the integrated rate law, will also be important in our study of kinetics. The integrated rate law expresses how the concentrations depend on time. As we will see, a given differential rate law is always related to a certain type of integrated rate law, and vice versa. That is, if we determine the differential rate law for a given reaction, we automatically know the form of the integrated rate law for the reaction. This means that once we determine either type of rate law for a reaction, we also know the other one. 

In this section and the previous two, we discussed a series of experimental and mathematical methods for the study of reaction kinetics. Figure 16.10 is a useful summary of this information. Note that the integrated rate law provides an alternative method for obtaining reaction orders and the rate constant. 

Figure 16.10 Information sequence to determine the kinetic parameters of a reaction. Note that the integrated rate law does not depend on the comparison of experimental initial rates and that it is also used to determine reaction orders and the rate constant.

In Chapter 2, several types of kinetic schemes were examined in detail. While the mathematical apparatus was developed to describe these cases, little was said about other methods used in kinetic studies or about experimental techniques. In this chapter, we will describe some of the methods employed in the study of kinetics that do not make use of the integrated rate laws. In some cases, the exact rate law may be unknown, and some of the experimental techniques do not make use of the classical determination of concentration as a function of time to get data to fit to a rate law. A few of the techniques described in this chapter are particularly useful in such cases. 

Since the general kinetic treatment of soHd state reactions cannot be made in terms of concentrations, we need to put the integrated rate law for the contracting sphere in a form containing a. In this case, the amount reacted is — V, so... 

Some of the early methods that were developed to analyze data to determine kinetic parameters were based on differential methods. This refers to the fact that the methods do not involve attempts to obtain an integrated rate law, but rather a differential form is used directly. Suppose a reaction follows a rate law that can be written as... 

Analyze We are given the rate constant for a reaction that obeys first-order kinetics, as well as information about concentrations and times, and asked to calculate how much reactant (insecticide) remains after one year. We must also determine the time interval needed to reach a particular insecticide concentration. Because the exercise gives time in (a) and asks for time in (b), we know that the integrated rate law. Equation 14.13, is required. 

The experimental rate law can be determined by monitoring the concentration of one of the reactants or products as a function of time using spectroscopic means. For instance, the Beer-Lambert law states that the absorbance of a colored compound is directly proportional to its concentration (for optically dilute solutions anyway), so that the absorbance can be measured as the course of the reaction proceeds. The data are then fit to a model, such as the function that results when integrating one of the differential rate law equations. The integrated rate laws for some commonly occurring kinetics are listed in Table 17.1. Half-life equations are also included for some of the reactions in this table, where the half-life ftyi) is defined as the length of time that it takes for half of the initial reactant concentration to disappear. 

TABLE 17.1 Differential and integrated rate laws and half-lives for some commonly occurring chemical kinetics. 

If you are familiar with integration, you should be able to follow the same procedure to derive the integrated rate laws for first-order and second-order kinetics. 

In such instances, the rate of the reaction will not change as the reactants are consumed. The integrated rate law for this type of kinetics is given by... 

At this point, we can see that (1) if a plot of [A] versus time is Unear, the reaction is zero order and (2) if a plot of ln[A] versus time is linear, the reaction is first order. From the form of the integrated rate law equations for these cases, you should also be able to see that the slope of the linear plot must be equal to -k. So we can find both the rate constant and the reaction order from our graph. Next we will examine the correct model for second-order kinetics. 

Because this is an introductory text, we have chosen our examples so that they will fit one of these three simple models. In some cases, however, reaction kinetics can be quite complicated and the appropriate model might be none of the three we tried here. If none of these three plots were linear, we would have to conclude that the reaction was not zero, first, or second order. Similar integrated rate law models can be derived for other cases, but this is beyond the scope of an introductory class. 

Problem 3.4. An exact treatment of equilibrium reaction kinetics for reactions that do not go to completion was discussed in a dialog box in the text. Expressions 3.68 and 3.69 were provided as integrated rate laws for a simple equilibrium first-order reaction between A and B where the forward rate constant is given by Iq and the backward rate constant is given by k. Prove that as t -> oo, these expressions yield the equilibrium concentrations of species A and B and... 

This is a method for determining the concentration dependence of a rate law that avoids the need for an integrated rate law or pseudo-first-order conditions. It is based on the assumption that the reactant concentrations are essentially constant during the initial 10% of reaction. The use of this method requires that observation can begin very soon after mixing the reactants and that the detection method is sensitive enough to provide precise data over the small extent of reaction. The latter condition usually means that the reaction half-time is about ten seconds or longer, so that this method is convenient and efficient for slow reactions. Observation over a short initial period may avoid, but also may hide, kinetic and chemical complications that only are clearly apparent later in the reaction. 

Sometimes, even under pseudo-first-order conditions, the kinetic observations do not obey the first-order integrated rate law. This may indicate a number of chemical problems, such as impurities, a nonlinear analytical method or precipitate formation. However, it is also possible that the system is more complex, with parallel and/or successive reactions, as shown in the following system ... 

The first serious challenge to the single-step mechanism for the reaction for the nitroalkanes in water appeared in 2001 for the reaction of 1-(4-nitrophenyl)-ethane (NNPEh) with sodium hydroxide in water/acetonitrile (50 50 vol. %). The kinetics of the reaction (Scheme 1.16) were studied under pseudo-first-order conditions using stopped-flow spectrophotometry and the data were analyzed as IRC-time profiles. Rate constants and KIE were obtained by fitting experimental to calculated data for the reversible consecutive mechanism (Scheme 1.17) which is the simplest bimolecular mechanism. The theoretical data were obtained using the integrated rate law for that mechanism ignoring B as appropriate for pseudo-first-order conditions. 

Now that we have looked at some of the most common scenarios, it is useful to tabulate these along with more complex ones. This should serve as a simple reference table for you to apply when implementing kinetic experiments. Table 7.2 shows several reaction stoichiometries along with the rate laws and the integrated rate laws. Almost all of these scenarios are amenable to reduction to simpler forms when a large excess of one reagent is used or an initial-rate kinetic treatment is applied. 

By varying the time delay between the pump and probe pulses, information about the time it takes to form the intermediate can be gained. The actual lifetime of the intermediate can then be obtained by continuously monitoring the reactive intermediate over its short lifetime. This leads to decay traces such as that shown in Figure 7.17 A. These traces can be fit to the standard integrated rate laws to extract the appropriate rate constants. Some decay traces have more than one component. Figure 7.17 B, for example, shows a decay trace that indicates both a short and a long component. Hence, fast kinetic techniques can often be used to analyze multiple reactions of reactive intermediates. It may seem that the need to ini-... 

Revision of Material on Kinetics In Chapter 14, we have included a more thorough discussion of second-order reactions, and we have expanded the coverage of integrated rate laws (concentration-time equations) and reaction half-life to include zero-order reactions. Several new end-of-chapter problems covering these topics have also been added. 

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