Rate laws integrated

The rate law determined by the method described in the previous section allows us to predict the rate of a reaction for a given set of concentrations. But because the concentrations of the reacting substances wiU change with time, the rate law does not let us easily predict the concentrations or rate at some later time. 

The form of the integrated rate law depends on the order of reaction. If we know the integrated rate laws for a few common reaction orders, we can use them as models for comparison with data. As well see below, this provides a useful alternative way of determining the rate law for a reaction. 

The differential and integrated forms of the rate law are both valid. The integrated rate law Is useful if we want to consider concentrations as a function of time. 

The rate laws summarize useful information about the progress of a reaction and 

Integrated rate laws have two principal uses. One is to predict the concentration of a species at any time after the start of the reaction. Another is to help find the rate constant and order of the reaction. Indeed, although we have introduced rate laws through a discussion of the determination of reaction rates, these rates are rarely measured directly because slopes are so difficult to determine accurately. Almost all experimental work in chemical kinetics deals with integrated rate laws their great advantage being that they are expressed in terms of the experimental observables of concentration and time. Computers can be used to find numerical solutions of even the most complex rate laws. However, we now see that in a number of simple cases, solutions can be expressed as relatively simple functions and prove to be very useful. 

Model 1 Integrated First- and Second-Order Rate Laws 

As we have seen, the concentration of a reactant decreases as a reaction proceeds. In a few situations, the concentration of a reactant is a (relatively) simple function of time. For example, for a reaction that is first order in a single reactant, R, the rate law is 

This equation can be rearranged and integrated to provide an explicit relationship between (R) and time. The integrated form of a first-order rate law is 

Similarly, for a second-order reaction with the rate law 

Note that the integrated rate laws contain four potential variables (R), (R)o, k, and t. Knowledge of any three of these variables permits the calculation of the fourth variable. 

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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. 

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... 

The integrated rate law for a zero-order reaction is easy to find. Because the rate is constant (at k), the difference in concentration of a reactant from its initial value, [A]0, is proportional to the time for which the reaction is in progress, and we can write... 

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... 

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. 

Calculate a concentration, time, or rate constant by using an integrated rate law (Examples 13.3, 13.4, and 13.5). 

Models of population growth are analogous to chemical reaction rate equations. In the model developed by Malthus in 1798, the rate of change of the population N of Earth is dN/dt = births — deaths. The numbers of births and deaths are proportional to the population, with proportionality constants b and d. Derive the integrated rate law for population change. How well does it fit the approximate data for the population of Earth over time given below ... 

Reactions for common minerals fall in both categories, but many important cases tend, except under acidic conditions, to be surface controlled (e.g., Aagaard and Helgeson, 1982 Stumm and Wollast, 1990). For this reason and because of their relative simplicity, we will consider in this chapter rate laws for surface-controlled reactions. The problem of integrating rate laws for transport-controlled reactions into reaction path calculations, nonetheless, is complex and interesting (Steefel and Lasaga, 1994), and warrants further attention. 

The single particle acts as a batch reactor in which conditions change with respect to time, This unsteady-state behavior for a reacting particle differs from the steady-state behavior of a catalyst particle in heterogeneous catalysis (Chapter 8). The treatment of it leads to the development of an integrated rate law in which, say, the fraction of B converted, /B, is a function oft, or the inverse. 

So far, we have used only instantaneous data in the rate expression. These expressions allow us to answer questions concerning the speed of the reaction at a particular moment, but not questions like about how long it might take to use up a certain reactant. However, if we take into account changes in the concentration of reactants or products over time, as expressed in the integrated rate laws, we can answer these types of questions. 

The order of reaction can be determined graphically by using the integrated rate law. If a plot of the ln[A] versus time yields a straight line, then the reaction is first order with respect to reactant A. If a plot of 1/[A] versus time yields a straight line, then the reaction is second order with respect to reactant A. 

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. 

Simple integrated rate laws for single reactants allow us to express the rate of reaction as a function of time. These are summarised in Table 10.2. 

Table 10.2 Integrated rate laws for first- and second-order reactions...

Order Reaction and Rate Law Integrated Rate Law Plot... 

Figure 10.13 Integrated rate-law plot of the decay of quinine triplet species...

When RH is in large excess, the triplet state undergoes pseudo first-order reaction to form QH and R, so an integrated rate-law plot of ln[3Q ] against t gives a straight line of slope -k (Figure 10.13). 

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