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24.2: Half-Life

Difficulty Level: At Grade Created by: CK-12

Lesson Objectives

• Define half-life as it relates to radioactive nuclides and solve half-life problems.
• Describe the general process by which radioactive dating is used to determine the age of various objects.
• Explain the mechanism of a decay series.
• Define and write equations for artificial transmutation processes.

Lesson Vocabulary

• artificial transmutation
• decay series
• half-life

Recalling Prior Knowledge

• Kinetics is the study of what aspect of chemical reactions?
• What must be balanced in an equation for a nuclear reaction?

The rate of radioactive decay is different for every radioisotope. Less stable nuclei decay at a faster rate than more stable nuclei. In this lesson, you will learn about the half-lives of radioactive nuclei.

Half-Life

The rate of radioactive decay is often characterized by the half-life of a radioisotope. Half-life (t½) is the time required for one half of the nuclei in a sample of radioactive material to decay. After each half-life has passed, one half of the radioactive nuclei will have transformed into a new nuclide (Table below). The rate of decay and the half-life does not depend on the original size of the sample. It also does not depend upon environmental factors such as temperature and pressure.

Number of Half-Lives Passed Percentage of Radioisotope Remaining
1 50
2 25
3 12.5
4 6.25
5 3.125

As an example, iodine-131 is a radioisotope with a half-life of 8 days. It decays by beta particle emission into xenon-131.

13153I13154Xe+ 01e\begin{align*}\mathrm{ ^{131}_{53} I \rightarrow ^{131}_{54} Xe + ^{\ 0}_{-1} e}\end{align*}

After eight days have passed, half of the atoms of any sample of iodine-131 will have decayed, and the sample will now be 50% iodine-131 and 50% xenon-131. After another eight days pass (a total of 16 days), the sample will be 25% iodine-131 and 75% xenon-131. This continues until the entire sample of iodine-131 has completely decayed (Figure below).

The half-life of iodine-131 is eight days. Half of a given sample of iodine-131 decays after each eight-day time period elapses.

Half-lives have a very wide range, from billions of years to fractions of a second. Listed below (Table below) are the half-lives of some common and important radioisotopes.

Nuclide Half-Life (t½) Decay mode
Carbon-14 5730 years β
Cobalt-60 5.27 years β
Francium-220 27.5 seconds α
Hydrogen-3 12.26 years β
Cobalt-60 5.27 years β
Iodine-131 8.07 days β
Nitrogen-16 7.2 seconds β
Phosphorus-32 14.3 days β
Plutonium-239 24,100 years α
Potassium-40 1.28 × 109 years β and e- capture
Strontium-90 28.1 days β
Technetium-99 2.13 × 105 years β
Thorium-234 24.1 days β
Uranium-235 7.04 × 108 years α
Uranium-238 4.47 × 109 years α

Sample Problem 24.1 illustrates how to use the half-life of a sample to determine the amount of radioisotope that remains after a certain period of time has passed.

Sample Problem 24.1: Half-Life Calculation

Strontium-90 has a half-life of 28.1 days. If you start with a 5.00 mg sample of the isotope, how much remains after 140.5 days have passed?

Step 1: List the known values and plan the problem.

Known

• original mass = 5.00 mg
• t½ = 28.1 days
• time elapsed = 140.5 days

Unknown

• final mass of Sr-90 = ? mg

First, find the number of half-lives that have passed by dividing the time elapsed by the half-life. Then, reduce the amount of Sr-90 by half, once for each half-life.

Step 2: Solve.

number of half lives = 140.5 days×1 halflife28.1 days\begin{align*}\mathrm{140.5 \ days \times \dfrac{1 \ half-life}{28.1 \ days}}\end{align*} = 5 half-lives
mass of Sr-90 = 5.00 mg × ½ × ½ × ½ × ½ × ½ = 0.156 mg

According to the table above (Table above), the passage of 5 half-lives means that 3.125% of the original Sr-90 remains, and 5.00 mg × 0.03125 = 0.156 mg. The remaining 4.844 mg has decayed by beta particle emission to yttrium-90.

Practice Problems
1. The half-life of polonium-218 is 3.0 min. How much of a 0.540 mg sample would remain after 9.0 minutes have passed?
2. The half-life of hydrogen-3, commonly known as tritium, is 12.26 years. If 4.48 mg of tritium has decayed to 0.280 mg, how much time has passed?

Radioactive dating is a process by which the approximate age of an object is determined through the use of certain radioactive nuclides. For example, carbon-14 has a half-life of 5,730 years and is used to measure the age of organic material. The ratio of carbon-14 to carbon-12 in living things remains constant while the organism is alive because fresh carbon-14 is entering the organism whenever it consumes nutrients. When the organism dies, this consumption stops, and no new carbon-14 is added to the organism. As time goes by, the ratio of carbon-14 to carbon-12 in the organism gradually declines, because carbon-14 radioactively decays while carbon-12 is stable. Analysis of this ratio allows archaeologists to estimate the age of organisms that were alive many thousands of years ago. The ages of many rocks and minerals are far greater than the ages of fossils. Uranium-containing minerals that have been analyzed in a similar way have allowed scientists to determine that the Earth is over 4 billion years old.

Decay Series

In many instances, the decay of an unstable radioactive nuclide simply produces another radioactive nuclide. It may take several successive steps to reach a nuclide that is stable. A decay series is a sequence of successive radioactive decays that proceeds until a stable nuclide is reached. The terms reactant and product are generally not used for nuclear reactions. Instead, the terms parent nuclide and daughter nuclide are used to refer to the starting and ending isotopes in a decay process. The figure below (Figure below) shows the decay series for uranium-238.

The decay of uranium-238 proceeds along many steps until a stable nuclide, lead-206, is reached. Each decay has its own characteristic half-life.

In the first step, uranium-238 decays by alpha emission to thorium-234 with a half-life of 4.5 × 109 years. This decreases its atomic number by two. The thorium-234 rapidly decays by beta emission to protactinium-234 (t½ = 24.1 days). The atomic number increases by one. This continues for many more steps until eventually the series ends with the formation of the stable isotope, lead-206.

Artificial Transmutation

As we have seen, transmutation occurs when atoms of one element spontaneously decay and are converted to atoms of another element. Artificial transmutation is the bombardment of stable nuclei with charged or uncharged particles in order to cause a nuclear reaction. The bombarding particles can be protons, neutrons, alpha particles, or larger atoms. Ernest Rutherford performed some of the earliest bombardments, including the bombardment of nitrogen gas with alpha particles to produce the unstable fluorine-18 isotope.

147N+42He189F\begin{align*}\mathrm{ ^{14}_{7}N + ^{4}_{2}He \rightarrow ^{18}_{9}F }\end{align*}

Fluorine-18 quickly decays to the stable nuclide oxygen-17 by releasing a proton.

189F178O+11H\begin{align*}\mathrm{ ^{18}_{9}F \rightarrow ^{17}_{8}O+ ^{1}_{1}H } \end{align*}

When beryllium-9 is bombarded with alpha particles, carbon-12 is produced with the release of a neutron.

94Be+42He126C+10n\begin{align*}\mathrm{ ^{9}_{4}Be + ^{4}_{2}He \rightarrow ^{12}_{6}C+ ^{1}_{0}n }\end{align*}

This nuclear reaction contributed to the discovery of the neutron in 1932 by James Chadwick. A shorthand notation for artificial transmutations can be used. The above reaction would be written as:

94Be(α,n)126C\begin{align*}\mathrm{ ^{9}_{4}Be(\alpha , n) ^{12}_{6}C}\end{align*}

The parent isotope is written first. In the parentheses is the bombarding particle followed by the ejected particle. The daughter isotope is written after the parentheses.

Positively charged particles need to be accelerated to high speeds before colliding with a nucleus in order to overcome the electrostatic repulsion. The necessary acceleration is provided by a combination of electric and magnetic fields. Shown below (Figure below) is an aerial view of the Fermi National Accelerator Laboratory in Illinois.

The Fermi National Accelerator Laboratory in Illinois.

Transuranium Elements

Many, many radioisotopes that do not occur naturally have been generated by artificial transmutation. The elements technetium and promethium have been produced, since these elements no longer occur in nature. All of their isotopes are radioactive and have half-lives short enough that any amount of the elements that once existed have long since disappeared through natural decay. The transuranium elements are elements with atomic numbers greater than 92. All isotopes of these elements are radioactive and none occur naturally. Listed below (Table below) are the transuranium elements up through meitnerium, and the reactions by which they were formed.

Atomic Number Name Symbol Preparation
93 Neptunium Np 23892U+10n23993Np+ 01β\begin{align*}\mathrm{^{238}_{92} U + ^{1}_{0}n \rightarrow ^{239}_{93} Np + ^{\ 0}_{-1} \beta}\end{align*}
94 Plutonium Pu 23993Np23994Pu+ 01β\begin{align*}\mathrm{^{239}_{93} Np \rightarrow ^{239}_{94} Pu + ^{\ 0}_{-1} \beta}\end{align*}
95 Americium Am 23994Pu+10n24095Am+ 01β\begin{align*}\mathrm{^{239}_{94} Pu + ^{1}_{0}n \rightarrow ^{240}_{95} Am + ^{\ 0}_{-1} \beta}\end{align*}
96 Curium Cm 23994Pu+42α24296Cm+10n\begin{align*}\mathrm{^{239}_{94} Pu + ^{4}_{2}\alpha \rightarrow ^{242}_{96} Cm + ^{1}_{0}n }\end{align*}
97 Berkelium Bk 24194Am+42α24397Bk+210n\begin{align*}\mathrm{^{241}_{94} Am + ^{4}_{2}\alpha \rightarrow ^{243}_{97} Bk + 2^{1}_{0}n}\end{align*}
98 Californium Cf 24296Cm+42α24598Cf+10n\begin{align*}\mathrm{^{242}_{96} Cm + ^{4}_{2}\alpha \rightarrow ^{245}_{98} Cf + ^{1}_{0}n }\end{align*}
99 Einsteinium Es 23892U+1510n25399Es+7 01β\begin{align*}\mathrm{^{238}_{92} U + 15^{1}_{0}n \rightarrow ^{253}_{99} Es + 7^{\ 0}_{-1} \beta}\end{align*}
100 Fermium Fm 23892U+1710n253100Fm+8 01β\begin{align*}\mathrm{^{238}_{92} U + 17^{1}_{0}n \rightarrow ^{253}_{100} Fm + 8^{\ 0}_{-1} \beta}\end{align*}
101 Mendelevium Md 25399Es+42α256101Md+10n\begin{align*}\mathrm{^{253}_{99} Es + ^{4}_{2}\alpha \rightarrow ^{256}_{101} Md + ^{1}_{0}n }\end{align*}
102 Nobelium No 24696Cm+126C254102No+410n\begin{align*}\mathrm{^{246}_{96} Cm + ^{12}_{6}C \rightarrow ^{254}_{102} No+ 4^{1}_{0}n }\end{align*}
103 Lawrencium Lr 25298Cm+105B257103Lr+510n\begin{align*}\mathrm{^{252}_{98} Cm + ^{10}_{5}B \rightarrow ^{257}_{103} Lr + 5^{1}_{0}n }\end{align*}
104 Rutherfordium Rf 24998Cf+126C257104Rf+410n\begin{align*}\mathrm{^{249}_{98} Cf + ^{12}_{6}C \rightarrow ^{257}_{104} Rf+ 4^{1}_{0}n }\end{align*}
105 Dubnium Db 24998Cf+157N260105Db+410n\begin{align*}\mathrm{^{249}_{98} Cf + ^{15}_{7}N \rightarrow ^{260}_{105} Db+ 4^{1}_{0}n }\end{align*}
106 Seaborgium Sg 24998Cf+188O263106Sg+410n\begin{align*}\mathrm{^{249}_{98} Cf + ^{18}_{8}O \rightarrow ^{263}_{106} Sg+ 4^{1}_{0}n }\end{align*}
107 Bohrium Bh 20983Bi+5424Cr262107Bh+10n\begin{align*}\mathrm{^{209}_{83} Bi+ ^{54}_{24}Cr \rightarrow ^{262}_{107} Bh+ ^{1}_{0}n }\end{align*}
108 Hassium Hs 20882Pb+5826Fe265108Hs+10n\begin{align*}\mathrm{^{208}_{82} Pb+ ^{58}_{26}Fe \rightarrow ^{265}_{108} Hs+ ^{1}_{0}n }\end{align*}
109 Meitnerium Mt 20983Bi+5826Fe266109Mt+10n\begin{align*}\mathrm{^{209}_{83} Bi+ ^{58}_{26}Fe \rightarrow ^{266}_{109} Mt+ ^{1}_{0}n }\end{align*}

Lesson Summary

• A half-life is the time it takes for half of a given sample of a radioactive nuclide to decay. Scientists use the half-lives of some naturally occurring radioisotopes to estimate the age of various objects.
• A decay series is a sequence of steps by which a radioactive nuclide decays to a stable nuclide.
• Artificial transmutation is used to produce other nuclides, including the transuranium elements.

Lesson Review Questions

Reviewing Concepts

1. What fraction of a radioactive isotope remains after one half-life? Two half-lives? Five half-lives?
2. When does a decay series end?
3. What is the difference between natural radioactive decay and artificial transmutation?
4. Why is an electric field unable to accelerate a neutron?

Problems

1. The half-life of protactinium-234 is 6.69 hours. If a 0.812 mg sample of Pa-239 decays for 40.1 hours, what mass of the isotope remains?
2. 2.86 g of a certain radioisotope decays to 0.358 g over a period of 22.8 minutes. What is the half-life of the radioisotope?
3. Use the table above (Table above) to determine the time it takes for 100 mg of carbon-14 to decay to 6.25 mg.
4. Fill in the blanks in the following radioactive decay series.
1. 23290Thαββ22890Th\begin{align*}\mathrm{ ^{232}_{90}Th \overset{\alpha}{\rightarrow} \underline{\hspace{1cm}} \overset{\beta}{\rightarrow} \underline{\hspace{1cm}} \overset{\beta}{\rightarrow} \: ^{228}_{90} Th}\end{align*}
2. 23592Uαβα22789Ac\begin{align*}\mathrm{ ^{235}_{92}U \overset{\alpha}{\rightarrow} \underline{\hspace{1cm}} \overset{\beta}{\rightarrow} \underline{\hspace{1cm}} \overset{\alpha}{\rightarrow} \: ^{227}_{89} Ac}\end{align*}
3. α23391Paβα\begin{align*}\mathrm{ \underline{\hspace{1cm}} \overset{\alpha}{\rightarrow} \: ^{233}_{91}Pa \overset{\beta}{\rightarrow} \underline{\hspace{1cm}} \overset{\alpha}{\rightarrow} \underline{\hspace{1cm}}}\end{align*}
5. Fill in the blanks in the following artificial transmutation reactions.
1. 8034Se+21H+11H\begin{align*}\mathrm{ ^{80}_{34}Se + ^2_1H \rightarrow \underline{\hspace{1cm}} + ^1_1H}\end{align*}
2. 63Li+10n31H+\begin{align*}\mathrm{ ^{6}_{3}Li + ^1_0n \rightarrow \: ^{3}_{1}H + \underline{\hspace{1cm}}}\end{align*}
3. +42α3015P+10n\begin{align*}\mathrm{ \underline{\hspace{1cm}} + ^4_2\alpha \rightarrow \: ^{30}_{15}P + ^1_0n}\end{align*}
6. Write balanced nuclear equations for the following reactions and identify X.
1. \begin{align*}\mathrm{ X(p, \alpha) ^{12}_{6}C}\end{align*}
2. \begin{align*}\mathrm{ ^{10}_{5}B(n , \alpha) X}\end{align*}
3. \begin{align*}\mathrm{ X(\alpha , p) ^{109}_{47}Ag}\end{align*}

Points to Consider

Nuclear fission and nuclear fusion are two processes that can occur naturally or through bombardment. Both release tremendous amounts of energy.

• What are two modern applications of nuclear fission?
• Where does nuclear fusion occur naturally?

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Date Created:
Mar 29, 2013