Integrated rate laws are determined by integration of the corresponding differential rate laws. Rate constants for those rate laws are determined from measurements of concentration at various times during a reaction.
The half-life of a reaction is the time required to decrease the amount of a given reactant by one-half. The half-life of a zero-order reaction decreases as the initial concentration of the reactant in the reaction decreases. The half-life of a first-order reaction is independent of concentration, and the half-life of a second-order reaction decreases as the concentration increases.
What is the half-life of this reaction? What fraction of the cyclopropane remains after 0. Fluorine is a radioactive isotope that decays by positron emission to form oxygen with a half-life of Physicians use 18 F to study the brain by injecting a quantity of fluoro-substituted glucose into the blood of a patient.
The glucose accumulates in the regions where the brain is active and needs nourishment. The isomerization of cyclobutene to butadiene is first-order and the rate constant has been measured as 2. Determine the partial pressure of cyclobutene and its concentration after The plot is nicely linear, so the reaction is second order. The reaction is first order. Skip to content Chapter Learning Objectives By the end of this section, you will be able to:.
Answer: Figure 1. The linear relationship between the ln[H 2 O 2 ] and time shows that the decomposition of hydrogen peroxide is a first-order reaction. Answer: The plot of ln[ A ] vs. Answer: 0. Example 4 Determination of Reaction Order by Graphing Test the data given to show whether the dimerization of C 4 H 6 is a first- or a second-order reaction. Figure 2. These two graphs show first- and second-order plots for the dimerization of C 4 H 6.
Since the first-order plot left is not linear, we know that the reaction is not first order. The linear trend in the second-order plot right indicates that the reaction follows second-order kinetics. Answer: Yes. Figure 4. The intensity of the color symbolizes the concentration of H 2 O 2 at the indicated times; H 2 O 2 is actually colorless. Answer: 5. Chemistry End of Chapter Exercises Describe how graphical methods can be used to determine the order of a reaction and its rate constant from a series of data that includes the concentration of A at varying times.
Use the data provided in a graphical method and determine the order and rate constant of the reaction. Time h 0 2. What is the half-life for the first-order decay of phosphorus? The rate constant for the decay is 4. What is the half-life for the first-order decay of carbon? The rate constant for this second-order reaction is 8. What is the half-life for the decomposition of O 3 when the concentration of O 3 is 2. The rate constant for this second-order reaction is The reaction of compound A to give compounds C and D was found to be second-order in A.
The rate constant for the reaction was determined to be 2. If the initial concentration is 0. The half-life of a reaction of compound A to give compounds D and E is 8. How long will it take for the concentration to drop to 0. The integrated rate law for second-order reactions has the form of the equation of a straight line:. A plot of 1 [ A ] t 1 [ A ] t versus t for a second-order reaction is a straight line with a slope of k and a y -intercept of 1 [ A ] 0.
If the plot is not a straight line, then the reaction is not second order. In order to distinguish a first-order reaction from a second-order reaction, prepare a plot of ln[C 4 H 6 ] t versus t and compare it to a plot of 1 [ C 4 H 6 ] t 1 [ C 4 H 6 ] t versus t. The values needed for these plots follow. The plots are shown in Figure The plot of 1 [ C 4 H 6 ] t 1 [ C 4 H 6 ] t versus t is linear, indicating that the reaction is second order.
According to the second-order integrated rate law, the rate constant is equal to the slope of the 1 [ A ] t 1 [ A ] t versus t plot. A zero-order reaction thus exhibits a constant reaction rate, regardless of the concentration of its reactant s. For purposes of this introductory text, it will suffice to note that zero-order kinetics are observed for some reactions only under certain specific conditions.
Figure The decomposition reaction exhibits first-order behavior at a quartz SiO 2 surface, as suggested by the exponentially decaying plot of concentration versus time. On a tungsten surface, however, the plot is linear, indicating zero-order kinetics. In each succeeding half-life, half of the remaining concentration of the reactant is consumed. Using the decomposition of hydrogen peroxide Figure During the second half-life from 6. The concentration of H 2 O 2 decreases by half during each successive period of 6.
The decomposition of hydrogen peroxide is a first-order reaction, and, as can be shown, the half-life of a first-order reaction is independent of the concentration of the reactant. However, half-lives of reactions with other orders depend on the concentrations of the reactants.
An equation relating the half-life of a first-order reaction to its rate constant may be derived from the integrated rate law as follows:.
Substituting these terms into the rearranged integrated rate law and simplifying yields the equation for half-life:. This equation describes an expected inverse relation between the half-life of the reaction and its rate constant, k. Faster reactions exhibit larger rate constants and correspondingly shorter half-lives. Slower reactions exhibit smaller rate constants and longer half-lives. Following the same approach as used for first-order reactions, an equation relating the half-life of a second-order reaction to its rate constant and initial concentration may be derived from its integrated rate law:.
We also see that the reaction rate is not proportional to the monomer concentration, so the reaction is not first order. Comparing the data in the second and fourth rows shows that the reaction rate decreases by a factor of 2.
Because 1. This means that the reaction is second order in the monomer. Substituting values at time 10 min, for example, gives the following:. We can also determine the reaction order using the integrated rate law.
The measurements show that the concentration of the monomer initially 5. This graph also shows that the reaction rate decreases smoothly with increasing time.
Any pair of points on the line can be used to calculate the slope, which is the second-order rate constant.
Although in this example the stoichiometric coefficient is the same as the reaction order, this is not always the case. The reaction order must always be determined experimentally. For two or more reactions of the same order, the reaction with the largest rate constant is the fastest.
Because the units of the rate constants for zeroth-, first-, and second-order reactions are different, however, we cannot compare the magnitudes of rate constants for reactions that have different orders. At high temperatures, nitrogen dioxide decomposes to nitric oxide and oxygen. Given: balanced chemical equation, initial concentrations, and initial rates.
A We can determine the reaction order with respect to nitrogen dioxide by comparing the changes in NO 2 concentrations with the corresponding reaction rates. Comparing Experiments 2 and 4, for example, shows that doubling the concentration quadruples the reaction rate [ 5. Similarly, comparing Experiments 1 and 4 shows that tripling the concentration increases the reaction rate by a factor of 9, again indicating that the reaction rate is proportional to [NO 2 ] 2.
This behavior is characteristic of a second-order reaction. We can calculate the rate constant k using data from any experiment in the table. Selecting Experiment 2, for example, gives the following:. When the highly reactive species HO 2 forms in the atmosphere, one important reaction that then removes it from the atmosphere is as follows:.
If a flask that initially contains 0. Given: balanced chemical equation, rate constant, time interval, and initial concentration. Asked for: final concentration and time required to reach specified concentration. This high rate constant means that HO 2 decomposes rapidly under the reaction conditions given in the problem. In fact, the HO 2 molecule is so reactive that it is virtually impossible to obtain in high concentrations. Given a 0. The differential rate law for this reaction is as follows:.
Because the reaction is first order both in A and in B, it has an overall reaction order of 2. The integrated rate law for this reaction is rather complex, so we will not describe it.
We can recognize second-order reactions of this sort because the reaction rate is proportional to the concentrations of each reactant. The reaction rate of a zeroth-order reaction is independent of the concentration of the reactants. The reaction rate of a first-order reaction is directly proportional to the concentration of one reactant.
The reaction rate of a simple second-order reaction is proportional to the square of the concentration of one reactant. Knowing the rate law of a reaction gives clues to the reaction mechanism. Learning Objectives To apply rate laws to zeroth, first and second order reactions.
Units In a zeroth-order reaction, the rate constant must have the same units as the reaction rate, typically moles per liter per second. First-Order Reactions In a first-order reaction , the reaction rate is directly proportional to the concentration of one of the reactants.
The expected shapes of the curves for plots of reactant concentration versus time top and the natural logarithm of reactant concentration versus time bottom for a first-order reaction. Compare the observed effect with behaviors characteristic of zeroth- and first-order reactions to determine the reaction order.
Write the rate law for the reaction. Solution The reaction order with respect to ethyl chloride is determined by examining the effect of changes in the ethyl chloride concentration on the reaction rate. Selecting Experiment 1 gives the following: 1.
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