half-life of a reaction (t_l/2)

time required for half of a given amount of reactant to be consumed

equation that relates the concentration of a reactant to elapsed time of reaction

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Chemistry 2e · 12.4 Integrated Rate Laws

Integrated Rate Laws: practice problems

9 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 33
Use the data provided to graphically determine the order and rate constant of the following reaction: $SO_{2} Cl_{2} ⟶ SO_{2} + Cl_{2}$
Time (s) 0 5.00 $\times$ 10³ 1.00 $\times$ 10⁴ 1.50 $\times$ 10⁴
[SO₂Cl₂] (M) 0.100 0.0896 0.0802 0.0719
Time (s) 2.50 $\times$ 10⁴ 3.00 $\times$ 10⁴ 4.00 $\times$ 10⁴
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Exercises 34
Pure ozone decomposes slowly to oxygen, $2O_{3} (g) ⟶ 3O_{2} (g) .$ Use the data provided in a graphical method and determine the order and rate constant of the reaction.
Time (h) 0 2.0 $\times$ 10³ 7.6 $\times$ 10³
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Exercises 36
What is the half-life for the first-order decay of phosphorus-32? $(_{15}^{32} P ⟶_{16}^{32} S + e^{-})$ The rate constant for the decay is 4.85 10⁻² day⁻¹.
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Exercises 38
What is the half-life for the decomposition of NOCl when the concentration of NOCl is 0.15 M? The rate constant for this second-order reaction is 8.0 $\times$ 10⁻⁸ L mol⁻¹ s⁻¹.
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Exercises 40
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 2.42 L mol⁻¹ s⁻¹. If the initial concentration is 0.500 mol/L, what is the value of t1/2?
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Exercises 42
Some bacteria are resistant to the antibiotic penicillin because they produce penicillinase, an enzyme with a molecular weight of 3 $\times$ 10⁴ g/mol that converts penicillin into inactive molecules. Although the kinetics of enzyme-catalyzed reactions can be complex, at low concentrations this reaction can be described by a rate law that is first order in the catalyst (penicillinase) and that also involves the concentration of penicillin. From the following data: 1.0 L of a solution containing 0.15 µg (0.15 $\times$ 10⁻⁶ g) of penicillinase, determine the order of the reaction with respect to penicillin and the value of the rate constant.
[Penicillin] (M) Rate (mol L⁻¹ min⁻¹)
2.0 $\times$ 10⁻⁶ 1.0 $\times$ 10⁻¹⁰
3.0 $\times$ 10⁻⁶ 1.5 $\times$ 10⁻¹⁰
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Exercises 44
There are two molecules with the formula C₃H₆. Propene, $CH_{3} CH = CH_{2},$ is the monomer of the polymer polypropylene, which is used for indoor-outdoor carpets. Cyclopropane is used as an anesthetic:
When heated to 499 °C, cyclopropane rearranges (isomerizes) and forms propene with a rate constant of
5.95 $\times$ 10⁻⁴ s⁻¹. What is the half-life of this reaction? What fraction of the cyclopropane remains after 0.75 h at 499 °C?
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Exercises 46
Suppose that the half-life of steroids taken by an athlete is 42 days. Assuming that the steroids biodegrade by a first-order process, how long would it take for $\frac{1}{64}$ of the initial dose to remain in the athlete’s body?
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Exercises 48
Nitroglycerine is an extremely sensitive explosive. In a series of carefully controlled experiments, samples of the explosive were heated to 160 °C and their first-order decomposition studied. Determine the average rate constants for each experiment using the following data:
Initial [C₃H₅N₃O₉] (M) 4.88 3.52 2.29 1.81 5.33 4.05 2.95 1.72
t (s) 300 300 300 300 180 180 180 180
% Decomposed 52.0 52.9 53.2 53.9 34.6 35.9 36.0 35.4
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Key terms · 2

half-life of a reaction (t_l/2): time required for half of a given amount of reactant to be consumed
integrated rate law: equation that relates the concentration of a reactant to elapsed time of reaction

Learn 12.4 on the whiteboard

The tutor teaches the whole section, drawing it out step by step and answering your questions as you go.

Read 12.4 in the textbook · This section on openstax.org

Integrated Rate Laws: practice problems

9 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 33

Use the data provided to graphically determine the order and rate constant of the following reaction: $SO_{2} Cl_{2} ⟶ SO_{2} + Cl_{2}$

Time (s)	0	5.00 $\times$ 10³	1.00 $\times$ 10⁴	1.50 $\times$ 10⁴
[SO₂Cl₂] (M)	0.100	0.0896	0.0802	0.0719
Time (s)	2.50 $\times$ 10⁴	3.00 $\times$ 10⁴	4.00 $\times$ 10⁴

Exercises 34

Pure ozone decomposes slowly to oxygen, $2O_{3} (g) ⟶ 3O_{2} (g) .$ Use the data provided in a graphical method and determine the order and rate constant of the reaction.

Time (h)

2.0

\times

10³

7.6

\times

10³

Exercises 36

What is the half-life for the first-order decay of phosphorus-32? $(_{15}^{32} P ⟶_{16}^{32} S + e^{-})$ The rate constant for the decay is 4.85 10⁻² day⁻¹.

Exercises 38

What is the half-life for the decomposition of NOCl when the concentration of NOCl is 0.15 M? The rate constant for this second-order reaction is 8.0 $\times$ 10⁻⁸ L mol⁻¹ s⁻¹.

Exercises 40

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 2.42 L mol⁻¹ s⁻¹. If the initial concentration is 0.500 mol/L, what is the value of t1/2?

Exercises 42

Some bacteria are resistant to the antibiotic penicillin because they produce penicillinase, an enzyme with a molecular weight of 3 $\times$ 10⁴ g/mol that converts penicillin into inactive molecules. Although the kinetics of enzyme-catalyzed reactions can be complex, at low concentrations this reaction can be described by a rate law that is first order in the catalyst (penicillinase) and that also involves the concentration of penicillin. From the following data: 1.0 L of a solution containing 0.15 µg (0.15 $\times$ 10⁻⁶ g) of penicillinase, determine the order of the reaction with respect to penicillin and the value of the rate constant.

[Penicillin] (M)	Rate (mol L⁻¹ min⁻¹)
2.0 $\times$ 10⁻⁶	1.0 $\times$ 10⁻¹⁰
3.0 $\times$ 10⁻⁶	1.5 $\times$ 10⁻¹⁰

Exercises 44

There are two molecules with the formula C₃H₆. Propene, $CH_{3} CH = CH_{2},$ is the monomer of the polymer polypropylene, which is used for indoor-outdoor carpets. Cyclopropane is used as an anesthetic:

A structural formula for cyclopropane is shown. Three C H subscript 2 groups are positioned as vertices of an equilateral triangle connected with single bonds represented by line segments.

When heated to 499 °C, cyclopropane rearranges (isomerizes) and forms propene with a rate constant of

5.95 $\times$ 10⁻⁴ s⁻¹. What is the half-life of this reaction? What fraction of the cyclopropane remains after 0.75 h at 499 °C?

Exercises 46

Suppose that the half-life of steroids taken by an athlete is 42 days. Assuming that the steroids biodegrade by a first-order process, how long would it take for $\frac{1}{64}$ of the initial dose to remain in the athlete’s body?

Exercises 48

Nitroglycerine is an extremely sensitive explosive. In a series of carefully controlled experiments, samples of the explosive were heated to 160 °C and their first-order decomposition studied. Determine the average rate constants for each experiment using the following data:

Initial [C₃H₅N₃O₉] (M)	4.88	3.52	2.29	1.81	5.33	4.05	2.95	1.72
t (s)	300	300	300	300	180	180	180	180
% Decomposed	52.0	52.9	53.2	53.9	34.6	35.9	36.0	35.4

Key terms · 2

half-life of a reaction (t_l/2): time required for half of a given amount of reactant to be consumed
integrated rate law: equation that relates the concentration of a reactant to elapsed time of reaction

Learn 12.4 on the whiteboard

The tutor teaches the whole section, drawing it out step by step and answering your questions as you go.

Read 12.4 in the textbook · This section on openstax.org