Reactions integrated rate laws

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... 

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... 

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. 

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. 

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... 

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). 

Where does the equation for half-life come from Each rate law has an associated integrated rate law. (A calculus teacher may be able to show you howto arrive at the integrated rate law.) For first-order reactions, the integrated rate law is... 

Robinson, M. K., Markinkus, K., Kennelly, P.J., and Timkovich, R. (1979). Implications of the integrated rate law for the reactions of Paracoccus denitrificans nitrite reductase. Biochemistry 18, 3921-3926. 

Besides its qualitative description, radioactive decay has an important quantitative description. Radioactive decay can be described as a first-order reaction, that is, the number of decays is proportional to the number of decaying nuclei present. It is described by the integrated rate law... 

One of the simplest integrated rate laws is for first-order reactions. We want to use the rate law to find the concentration of a reactant A at a time t, given that the initial molar concentration of A is [A]0. 

Equations 13a and 13b are two forms of the integrated rate law for a first-order reaction. The variation of concentration with time predicted by Eq. 13b is shown in Fig. 13.9. This behavior is called an exponential decay. The change in concentration is initially rapid, but it changes more slowly as time goes on and the reactant is used up. 

To relate the reaction s half-life to the rate constant, let s begin with the integrated rate law ... 

Integrated Rate Law, Second-Order Reaction activity... 

Since the integrated rate law has the form y = mx + b, a graph of 1 / [A] versus time is a straight line if the reaction is second order ... 

Throughout the course of the reaction, the rate remains constant (= k), independent of the concentration of the reactant. The integrated rate law is... 

Beginning with the integrated rate law, derive a general equation for the half-life of a zeroth-order reaction of the type A — Products. How does the length of each half-life compare with the length of the previous one Make the same comparison for first-order and second-order reactions. 

The above expression is the first-order differential rate law for the conversion of A to B. The change in concentration of A over the complete course of the reaction is given by the integrated rate law, which is found by solving the differential rate law ... 

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