24.2: HalfLife
Lesson Objectives
 Define halflife as it relates to radioactive nuclides and solve halflife 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
 halflife
 radioactive dating
Check Your Understanding
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 halflives of radioactive nuclei.
HalfLife
The rate of radioactive decay is often characterized by the halflife of a radioisotope. Halflife (t_{½}) is the time required for one half of the nuclei in a sample of radioactive material to decay. After each halflife has passed, one half of the radioactive nuclei will have transformed into a new nuclide (Table below). The rate of decay and the halflife 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 HalfLives Passed  Percentage of Radioisotope Remaining 

1  50 
2  25 
3  12.5 
4  6.25 
5  3.125 
As an example, iodine131 is a radioisotope with a halflife of 8 days. It decays by beta particle emission into xenon131.



\begin{align*}\mathrm{ ^{131}_{53} I \rightarrow ^{131}_{54} Xe + ^{\ 0}_{1} e}\end{align*}
13153I→13154Xe+ 0−1e

\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 iodine131 will have decayed, and the sample will now be 50% iodine131 and 50% xenon131. After another eight days pass (a total of 16 days), the sample will be 25% iodine131 and 75% xenon131. This continues until the entire sample of iodine131 has completely decayed (Figure below).
The halflife of iodine131 is eight days. Half of a given sample of iodine131 decays after each eightday time period elapses.
Halflives have a very wide range, from billions of years to fractions of a second. Listed below (Table below) are the halflives of some common and important radioisotopes.
Nuclide  HalfLife (t_{½})  Decay mode 

Carbon14  5730 years  β^{−} 
Cobalt60  5.27 years  β^{−} 
Francium220  27.5 seconds  α 
Hydrogen3  12.26 years  β^{−} 
Cobalt60  5.27 years  β^{−} 
Iodine131  8.07 days  β^{−} 
Nitrogen16  7.2 seconds  β^{−} 
Phosphorus32  14.3 days  β^{−} 
Plutonium239  24,100 years  α 
Potassium40  1.28 × 10^{9} years  β^{−} and e^{} capture 
Radium226  1600 years  α 
Radon222  3.82 days  α 
Strontium90  28.1 days  β^{−} 
Technetium99  2.13 × 10^{5} years  β^{−} 
Thorium234  24.1 days  β^{−} 
Uranium235  7.04 × 10^{8} years  α 
Uranium238  4.47 × 10^{9} years  α 
Sample Problem 24.1 illustrates how to use the halflife of a sample to determine the amount of radioisotope that remains after a certain period of time has passed.
Sample Problem 24.1: HalfLife Calculation
Strontium90 has a halflife 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
 t_{½} = 28.1 days
 time elapsed = 140.5 days
Unknown
 final mass of Sr90 = ? mg
First, find the number of halflives that have passed by dividing the time elapsed by the halflife. Then, reduce the amount of Sr90 by half, once for each halflife.
Step 2: Solve.



number of half lives = \begin{align*}\mathrm{140.5 \ days \times \dfrac{1 \ halflife}{28.1 \ days}}\end{align*}
140.5 days×1 half−life28.1 days = 5 halflives

number of half lives = \begin{align*}\mathrm{140.5 \ days \times \dfrac{1 \ halflife}{28.1 \ days}}\end{align*}



 mass of Sr90 = 5.00 mg × ½ × ½ × ½ × ½ × ½ = 0.156 mg

Step 3: Think about your result.
According to the table above (Table above), the passage of 5 halflives means that 3.125% of the original Sr90 remains, and 5.00 mg × 0.03125 = 0.156 mg. The remaining 4.844 mg has decayed by beta particle emission to yttrium90.
 The halflife of polonium218 is 3.0 min. How much of a 0.540 mg sample would remain after 9.0 minutes have passed?
 The halflife of hydrogen3, 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
Radioactive dating is a process by which the approximate age of an object is determined through the use of certain radioactive nuclides. For example, carbon14 has a halflife of 5,730 years and is used to measure the age of organic material. The ratio of carbon14 to carbon12 in living things remains constant while the organism is alive because fresh carbon14 is entering the organism whenever it consumes nutrients. When the organism dies, this consumption stops, and no new carbon14 is added to the organism. As time goes by, the ratio of carbon14 to carbon12 in the organism gradually declines, because carbon14 radioactively decays while carbon12 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. Uraniumcontaining 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 uranium238.
The decay of uranium238 proceeds along many steps until a stable nuclide, lead206, is reached. Each decay has its own characteristic halflife.
In the first step, uranium238 decays by alpha emission to thorium234 with a halflife of 4.5 × 10^{9} years. This decreases its atomic number by two. The thorium234 rapidly decays by beta emission to protactinium234 (t_{½} = 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, lead206.
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 fluorine18 isotope.



\begin{align*}\mathrm{ ^{14}_{7}N + ^{4}_{2}He \rightarrow ^{18}_{9}F }\end{align*}
147N+42He→189F

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

Fluorine18 quickly decays to the stable nuclide oxygen17 by releasing a proton.



\begin{align*}\mathrm{ ^{18}_{9}F \rightarrow ^{17}_{8}O+ ^{1}_{1}H } \end{align*}
189F→178O+11H

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

When beryllium9 is bombarded with alpha particles, carbon12 is produced with the release of a neutron.



\begin{align*}\mathrm{ ^{9}_{4}Be + ^{4}_{2}He \rightarrow ^{12}_{6}C+ ^{1}_{0}n }\end{align*}
94Be+42He→126C+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:



\begin{align*}\mathrm{ ^{9}_{4}Be(\alpha , n) ^{12}_{6}C}\end{align*}
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 halflives 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 
\begin{align*}\mathrm{^{238}_{92} U + ^{1}_{0}n \rightarrow ^{239}_{93} Np + ^{\ 0}_{1} \beta}\end{align*} 
94  Plutonium  Pu 
\begin{align*}\mathrm{^{239}_{93} Np \rightarrow ^{239}_{94} Pu + ^{\ 0}_{1} \beta}\end{align*} 
95  Americium  Am 
\begin{align*}\mathrm{^{239}_{94} Pu + ^{1}_{0}n \rightarrow ^{240}_{95} Am + ^{\ 0}_{1} \beta}\end{align*} 
96  Curium  Cm 
\begin{align*}\mathrm{^{239}_{94} Pu + ^{4}_{2}\alpha \rightarrow ^{242}_{96} Cm + ^{1}_{0}n }\end{align*} 
97  Berkelium  Bk 
\begin{align*}\mathrm{^{241}_{94} Am + ^{4}_{2}\alpha \rightarrow ^{243}_{97} Bk + 2^{1}_{0}n}\end{align*} 
98  Californium  Cf 
\begin{align*}\mathrm{^{242}_{96} Cm + ^{4}_{2}\alpha \rightarrow ^{245}_{98} Cf + ^{1}_{0}n }\end{align*} 
99  Einsteinium  Es 
\begin{align*}\mathrm{^{238}_{92} U + 15^{1}_{0}n \rightarrow ^{253}_{99} Es + 7^{\ 0}_{1} \beta}\end{align*} 
100  Fermium  Fm 
\begin{align*}\mathrm{^{238}_{92} U + 17^{1}_{0}n \rightarrow ^{253}_{100} Fm + 8^{\ 0}_{1} \beta}\end{align*} 
101  Mendelevium  Md 
\begin{align*}\mathrm{^{253}_{99} Es + ^{4}_{2}\alpha \rightarrow ^{256}_{101} Md + ^{1}_{0}n }\end{align*} 
102  Nobelium  No 
\begin{align*}\mathrm{^{246}_{96} Cm + ^{12}_{6}C \rightarrow ^{254}_{102} No+ 4^{1}_{0}n }\end{align*} 
103  Lawrencium  Lr 
\begin{align*}\mathrm{^{252}_{98} Cm + ^{10}_{5}B \rightarrow ^{257}_{103} Lr + 5^{1}_{0}n }\end{align*} 
104  Rutherfordium  Rf 
\begin{align*}\mathrm{^{249}_{98} Cf + ^{12}_{6}C \rightarrow ^{257}_{104} Rf+ 4^{1}_{0}n }\end{align*} 
105  Dubnium  Db 
\begin{align*}\mathrm{^{249}_{98} Cf + ^{15}_{7}N \rightarrow ^{260}_{105} Db+ 4^{1}_{0}n }\end{align*} 
106  Seaborgium  Sg 
\begin{align*}\mathrm{^{249}_{98} Cf + ^{18}_{8}O \rightarrow ^{263}_{106} Sg+ 4^{1}_{0}n }\end{align*} 
107  Bohrium  Bh 
\begin{align*}\mathrm{^{209}_{83} Bi+ ^{54}_{24}Cr \rightarrow ^{262}_{107} Bh+ ^{1}_{0}n }\end{align*} 
108  Hassium  Hs 
\begin{align*}\mathrm{^{208}_{82} Pb+ ^{58}_{26}Fe \rightarrow ^{265}_{108} Hs+ ^{1}_{0}n }\end{align*} 
109  Meitnerium  Mt 
\begin{align*}\mathrm{^{209}_{83} Bi+ ^{58}_{26}Fe \rightarrow ^{266}_{109} Mt+ ^{1}_{0}n }\end{align*} 
Lesson Summary
 A halflife is the time it takes for half of a given sample of a radioactive nuclide to decay. Scientists use the halflives 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
 What fraction of a radioactive isotope remains after one halflife? Two halflives? Five halflives?
 When does a decay series end?
 What is the difference between natural radioactive decay and artificial transmutation?
 Why is an electric field unable to accelerate a neutron?
Problems
 The halflife of protactinium234 is 6.69 hours. If a 0.812 mg sample of Pa239 decays for 40.1 hours, what mass of the isotope remains?
 2.86 g of a certain radioisotope decays to 0.358 g over a period of 22.8 minutes. What is the halflife of the radioisotope?
 Use the table above (Table above) to determine the time it takes for 100 mg of carbon14 to decay to 6.25 mg.
 Fill in the blanks in the following radioactive decay series.

\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*}
23290Th→α−−−−→β−−−−→β22890Th 
\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*}
23592U→α−−−−→β−−−−→α22789Ac 
\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*}
−−−−→α23391Pa→β−−−−→α−−−−

\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*}
 Fill in the blanks in the following artificial transmutation reactions.

\begin{align*}\mathrm{ ^{80}_{34}Se + ^2_1H \rightarrow \underline{\hspace{1cm}} + ^1_1H}\end{align*}
8034Se+21H→−−−−+11H 
\begin{align*}\mathrm{ ^{6}_{3}Li + ^1_0n \rightarrow \: ^{3}_{1}H + \underline{\hspace{1cm}}}\end{align*}
63Li+10n→31H+−−−− 
\begin{align*}\mathrm{ \underline{\hspace{1cm}} + ^4_2\alpha \rightarrow \: ^{30}_{15}P + ^1_0n}\end{align*}
−−−−+42α→3015P+10n

\begin{align*}\mathrm{ ^{80}_{34}Se + ^2_1H \rightarrow \underline{\hspace{1cm}} + ^1_1H}\end{align*}
 Write balanced nuclear equations for the following reactions and identify X.
 \begin{align*}\mathrm{ X(p, \alpha) ^{12}_{6}C}\end{align*}
 \begin{align*}\mathrm{ ^{10}_{5}B(n , \alpha) X}\end{align*}
 \begin{align*}\mathrm{ X(\alpha , p) ^{109}_{47}Ag}\end{align*}
Further Reading / Supplemental Links
 HalfLife, http://www.kentchemistry.com/links/Nuclear/halflife.htm
 Artificial Transmutation, http://www.kentchemistry.com/links/Nuclear/ArtificialTrans.htm
 Particle Accelerators, http://www.kentchemistry.com/links/Nuclear/accelerators.htm
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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