Chemistry
Chemistry
4th Edition
ISBN: 9780078021527
Author: Julia Burdge
Publisher: McGraw-Hill Education
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Chapter 14, Problem 126AP

The activity of a radioactive sample is the number of nuclear disintegrations per second, which is equal to the first-order rate constant times the number of radioactive nuclei present. The fundamental unit of radioactivity is the curie (Ci). where 1 Ci corresponds to exactly 3.70   ×   10 10 disintegrations per second. This decay rate is equivalent to that of 1 g of radium-226. Calculate the rate constant and half-life for the radium decay. Starting with 1.0 g of the radium sample, what is the activity after 500 yr? The molar mass of Ra-226 is 226.03 g/mol.

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Interpretation Introduction

Interpretation:

The rate constant and half-life for radium decay is to be calculated. Also, the activity of the radium sample after 500 yr is to be determined.

Concept introduction:

Rate constant for a reaction is the proportionality constant, which relates the rate of reaction and the concentration of reactants in the reaction.

Half-life is the time required by a substance to reduce by half of its original quantity. Half-life for a substance can be calculated as follows:

t1/2=0.693k[A]0

Here, t1/2 is half-life of the substance, k is the rate constant for the decomposition reaction of the substance, and [A]0 is the initial concentration of reactant A.

Answer to Problem 126AP

Solution: Rate constant and half-life for radium decay is 1.4×1011 s1 and 5.0×1010 s, respectively.

3.035×1010nuclear disintegrations/s.

Explanation of Solution

Given information: A 1.0 g of radium-226 sample disintegrates and its molar mass is 226.03 g/mol.

To determine the rate constant for the radium decay, the number of radium nuclei in 1.0g of sample is determined as follows:

n=(1.0 g Ra)×(1molRa226.03g Ra)×(6.022×1023Ra nucleimolRa)=2.7×1021 Ra nuclei

Now, calculate rate constant from the activity and the number of nuclei as follows:

Activity =kN

Here, k is the rate constant and N is the number of nuclei.

Rearrange the equation to determine therate constant.

k=activityN

Substitute values of activity and number of nuclei in the above expression as,

k=(3.70×1010nuclear disintegrations2.7×1021 nuclei)=1.4×1011s1

The half-life of radium is determined as follows:

t12=0.693k

Substitute the value of rate constant in the above equation,

t1/2=(0.6931.4×1011)=5.0×1010s

The activity of radium after 500 yr is as follows:

(500yr)×(365days1yr)×(24h1day)×(3600s1h)=1.58×1010s

Now, by using the first-order integrated rate law, the number of nuclei remaining after 500 yearsis calculated as follows:

lnNtNo=kt

Here, Nt is the number of nuclei at the given time, No is the number of nuclei present in the element sample without any disintegration, k is the rate constant, and t is time for disintegration.

Substitute values of k, t, and No in the above expression,

ln(Nt2.7×1021)=(1.4×10111/s)(1.58×1010s)(Nt2.7×1021)=e0.22Nt=2.168×1021 Ranuclei

Now, the activity of radium sample after 500 yr is calculated as follows:

Activity=kN=(1.4×1011s1)(2.168×1021nuclei)=3.035×1010nuclear disintegrations/s

Conclusion

The rate constant and half-life for the radium decay is 1.4×1011 s1 and 5.0×1010 s, respectively. The activity of the radium sample after 500 yr is 3.035×1010nuclear disintegrations/s.

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Chapter 14 Solutions

Chemistry

Ch. 14.2 - Answer questions 14.2.1 through 14.2.4 using the...Ch. 14.2 - Answer questions 14.2.1 through 14.2.4 using the...Ch. 14.2 - 14.2.5 The diagrams represent three experiments in...Ch. 14.3 - Prob. 1PPACh. 14.3 - Practice Problem BUILD For the following general...Ch. 14.3 - Practice Problem CONCEPTUALIZE Three initial-rate...Ch. 14.3 - The first-order decomposition of dinitrogen...Ch. 14.3 - The first-order decomposition of dinitrogen...Ch. 14.3 - 14.3.3 Consider the first-order reaction in which...Ch. 14.3 - Which figure below represents the numbers of...Ch. 14.3 - 14.3.5 Of the plots shown here, ___________...Ch. 14.4 - Practice Problem ATTEMPT The rate constant for the...Ch. 14.4 - Practice Problem BUILD Refer again to the reaction...Ch. 14.4 - Practice Problem CONCEPTUALIZE The diagrams on...Ch. 14.4 - Use the table of data collected for a first-order...Ch. 14.4 - Prob. 2CPCh. 14.4 - Prob. 3CPCh. 14.5 - Practice Problem ATTEMPT Ethyl iodide ( C 2 H 5 I)...Ch. 14.5 - Practice Problem BUILD Use the calculated k from...Ch. 14.5 - Practice Problem CONCEPTUALIZE Use the graph in...Ch. 14.5 - Use the following information to answer questions...Ch. 14.5 - Use the following information to answer questions...Ch. 14.5 - Use the following information to answer questions...Ch. 14.5 - 14.5.4 A plausible mechanism for the reaction: Ch. 14.6 - Practice ProblemATTEMPT Calculate the half-life of...Ch. 14.6 - Practice ProblemBUILD Calculate the rate constant...Ch. 14.6 - Practice Problem CONCEPTUALIZE The diagrams show a...Ch. 14.7 - Practice Problem ATTEMPT The reaction is second...Ch. 14.7 - Practice Problem BUILD Determine the initial...Ch. 14.7 - Practice ProblemCONCEPTUALIZE The diagrams below...Ch. 14.8 - Practice ProblemATTEMPT The second-order rate...Ch. 14.8 - Practice Problem BUILD Use the graph to determine...Ch. 14.8 - Prob. 1PPCCh. 14.9 - Practice ProblemATTEMPT Use the data in the...Ch. 14.9 - Practice ProblemBUILD Based on the data shown in...Ch. 14.9 - Practice Problem CONCEPTUALIZE According to the...Ch. 14.10 - Practice ProblemATTEMPT Calculate the rate...Ch. 14.10 - Practice ProblemBUILD Calculate the rate constant...Ch. 14.10 - Practice ProblemCONCEPTUALIZE According to the...Ch. 14.11 - Practice Problem ATTEMPT The reaction between and...Ch. 14.11 - Practice ProblemBUILD Propose a plausible...Ch. 14.11 - Practice Problem CONCEPTUALIZE How many steps are...Ch. 14.12 - Practice Problem ATTEMPT Show that the following...Ch. 14.12 - Practice Problem BUILD The reaction proceeds via...Ch. 14.12 - Practice Problem CONCEPTUALIZE The reaction of is...Ch. 14 - Prob. 1KSPCh. 14 - Prob. 2KSPCh. 14 - Prob. 3KSPCh. 14 - Prob. 4KSPCh. 14 - 14.1 What is meant by the rate of a chemical...Ch. 14 - Distinguish between average rate and instantaneous...Ch. 14 - What are the advantages of measuring the initial...Ch. 14 - Identify two reactions that are very slow (take...Ch. 14 - Write the reaction rate expressions for the...Ch. 14 - Write the reaction rate expressions for the...Ch. 14 - Consider the reaction: 2NO ( g ) + O 2 ( g ) → 2NO...Ch. 14 - 14.8 Consider the reaction: Suppose that at a...Ch. 14 - 14.9 Explain what is meant by the rate law of a...Ch. 14 - Prob. 10QPCh. 14 - What are the units for the rate constants of...Ch. 14 - 14.12 Consider the zeroth-order reaction: a ...Ch. 14 - 14.13 The rate constant of a first-order reaction...Ch. 14 - Identify two reactions that are very slow (take...Ch. 14 - The rate law for the reaction: N H 4 + ( a q )+N O...Ch. 14 - Use the data in Table 14.2 to calculate the rate...Ch. 14 - 14.17 Consider the reaction: From the following...Ch. 14 - Consider the reaction: X + Y → Z From the...Ch. 14 - Determine the overall orders of the reactions to...Ch. 14 - 14.20 Consider the reaction: The rate of the...Ch. 14 - Cyclobutane decomposes to ethylene according to...Ch. 14 - The following gas-phase reaction was studied at...Ch. 14 - Write an equation relating the concentration of a...Ch. 14 - 14.24 Define half-life. 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What role does...Ch. 14 - Prob. 35QPCh. 14 - Prob. 36QPCh. 14 - The burning of methane in oxygen is a highly...Ch. 14 - Sketch a potential-energy versus reaction progress...Ch. 14 - The reaction H+H 2 → H 2 +H has been studied for...Ch. 14 - Over the range of about ±3°C from normal body...Ch. 14 - For the reaction: NO ( g ) + O 3 ( g ) → NO 2 ( g...Ch. 14 - The rate constant of a first-order reaction is 4...Ch. 14 - The rate constants of some reactions double with...Ch. 14 - 14.44 The rate at which tree crickets chirp is ...Ch. 14 - The rate of bacterial hydrolysis of fish muscle is...Ch. 14 - Prob. 46QPCh. 14 - Given the same reactant concentrations, the...Ch. 14 - 14.48 Variation of the rate constant with...Ch. 14 - 14.49 Diagram A describes the initial state of...Ch. 14 - 14 50 What do we mean by the mechanism of a...Ch. 14 - 14.51 What is an elementary step? What is the...Ch. 14 - 14.52 Classify the following elementary reactions...Ch. 14 - Reactions can be classified as unimolecular,...Ch. 14 - Determine the molecularity, and write the rate law...Ch. 14 - 14.55 What is the rate-determining step of a...Ch. 14 - 14.56 The equation for the combustion of ethane ...Ch. 14 - Specify which of the following species cannot be...Ch. 14 - Classify each of the following elementary steps as...Ch. 14 - 14.59 The rate law for the reaction: is given by...Ch. 14 - For the reaction x 2 + y + z → x y + x z , it is...Ch. 14 - The rate law for the reaction: 2H 2 ( g ) + 2NO (...Ch. 14 - 14.62 The rate law for the decomposition of ozone...Ch. 14 - 14.63 How does a catalyst increase the rate of a...Ch. 14 - 14.64 What are the characteristics of a...Ch. 14 - A certain reaction is known to proceed slowly at...Ch. 14 - Most reactions, including enzyme-catalyzed...Ch. 14 - 14.67 Are enzyme-catalyzed reactions examples of...Ch. 14 - The concentrations of enzymes in cells are usually...Ch. 14 - When fruits such as apples and pears are cut. the...Ch. 14 - The first-order rate constant for the dehydration...Ch. 14 - Which two potential-energy profiles represent the...Ch. 14 - Consider the following mechanism for the...Ch. 14 - List four factors that influence the rate of a...Ch. 14 - 14.71 Suggest experimental means by which the...Ch. 14 - 14.75 “The rate constant for the reaction: is .”...Ch. 14 - Prob. 76APCh. 14 - The following diagrams represent the progress of...Ch. 14 - The following diagrams show the progress of the...Ch. 14 - Prob. 79APCh. 14 - Prob. 80APCh. 14 - 14.81 When methyl phosphate is heated in acid...Ch. 14 - The rate of the reaction: CH 3 COOC 2 H 5 ( a q )...Ch. 14 - Explain why most metals used in catalysis are...Ch. 14 - Prob. 84APCh. 14 - The bromination of acetone is acid-catalyzed: CH 3...Ch. 14 - The decomposition of N 2 O to N 2 and O 2 is a...Ch. 14 - 14.87 The reaction proceeds slowly in aqueous...Ch. 14 - Prob. 88APCh. 14 - The integrated rate law for the zeroth-order...Ch. 14 - 14.90 A flask contains a mixture of compounds A...Ch. 14 - Prob. 91APCh. 14 - 14.92 The rate law for the reaction . Which of the...Ch. 14 - 14.93 The reaction of to form 2EG is exothermic,...Ch. 14 - 14.94 The activation energy for the decomposition...Ch. 14 - Prob. 95APCh. 14 - 14.96 When 6 g of granulated Zn is added to a...Ch. 14 - Prob. 97APCh. 14 - 14.98 A certain first-order reaction is 35.5...Ch. 14 - 14.99 The decomposition of dinitrogen pentoxide...Ch. 14 - 14.100 The thermal decomposition of obeys...Ch. 14 - 14.101 When a mixture of methane and bromine is...Ch. 14 - 14.102 The rate of the reaction between to form...Ch. 14 - The rate constant for the gaseous reaction: H 2 (...Ch. 14 - A gas mixture containing CH 3 fragments. C 2 H 6...Ch. 14 - Consider the following elementary step: X + 2Y →...Ch. 14 - 14.106 The following scheme in which A is...Ch. 14 - 14.107 (a) Consider two reactions, A and B. If the...Ch. 14 - The rate law for the following reaction: CO ( g )...Ch. 14 - Consider the following elementary steps for a...Ch. 14 - Prob. 110APCh. 14 - Consider the following potential-energy profile...Ch. 14 - The rate of a reaction was followed by the...Ch. 14 - 14.113 The first-order rate constant for the...Ch. 14 - 14.114 Many reactions involving heterogeneous...Ch. 14 - Thallium(I) is oxidized by cerium(IV) as follows:...Ch. 14 - The activation energy for the reaction: N 2 O ( g...Ch. 14 - Δ H ° for the reaction in Problem 14.116 is -164...Ch. 14 - 14.118 At a certain elevated temperature, ammonia...Ch. 14 - 14.119 The following expression shows the...Ch. 14 - In a certain industrial process involving a...Ch. 14 - Strontium-90, a radioactive isotope, is a major...Ch. 14 - Prob. 122APCh. 14 - Prob. 123APCh. 14 - A factory that specializes in the refinement of...Ch. 14 - 14.125 When the concentration of A in the reaction...Ch. 14 - 14.126 The activity of a radioactive sample is the...Ch. 14 - Prob. 127APCh. 14 - Prob. 128APCh. 14 - Prob. 129APCh. 14 - Prob. 130APCh. 14 - Prob. 131APCh. 14 - Prob. 132APCh. 14 - Prob. 133APCh. 14 - 14.134 At a certain elevated temperature, ammonia...Ch. 14 - Polyethylene is used in many items, including...Ch. 14 - In recent years, ozone in the stratosphere has...Ch. 14 - Metastron, an aqueous solution of 89 SrCl 2 , is a...Ch. 14 - Metastron, an aqueous solution of 89 SrCl 2 , is a...Ch. 14 - Metastron, an aqueous solution of 89 SrCl 2 , is a...Ch. 14 - Metastron, an aqueous solution of 89 SrCl 2 , is a...
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