Chapter 21 | Nuclear Chemistry 1193

(a) The value of the rate constant is given by:

λ = ln 2 0.693 −1
t 1/2 = 5.27 y = 0.132 y

60 Nt
(b) The fraction of 27 Co that is left after time t is given by
N0
. Rearranging the first-order relationship

Nt = N0e–λt to solve for this ratio yields:

Nt ⎛ ⎞⎛
− 0.132/y 15.0/y ⎞

= e −λt = e
⎝ ⎠⎝ ⎠
= 0.138
N0
60 60
The fraction of 27 Co that will remain after 15.0 years is 0.138. Or put another way, 13.8% of the 27 Co
originally present will remain after 15 years.
60
(c) 2.00% of the original amount of 27 Co is equal to 0.0200 × N0. Substituting this into the equation for
time for first-order kinetics, we have:
⎛ Nt ⎞ ⎛0.0200 × N 0 ⎞
t = − 1 ln 1
λ ⎝N 0 ⎠ ⎝ ⎠ = 29.6 y
= − ln
0.132 y −1 N0

Check Your Learning

222
Radon-222, 86 Rn, has a half-life of 3.823 days. How long will it take a sample of radon-222 with a mass
of 0.750 g to decay into other elements, leaving only 0.100 g of radon-222?
Answer: 11.1 days

Because each nuclide has a specific number of nucleons, a particular balance of repulsion and attraction, and its own
degree of stability, the half-lives of radioactive nuclides vary widely. For example: the half-life of 209
83 Bi is 1.9 ×
239 222
1019 years; 94 Ra is 24,000 years; 86 Rn is 3.82 days; and element-111 (Rg for roentgenium) is 1.5 × 10–3
seconds. The half-lives of a number of radioactive isotopes important to medicine are shown in Table 21.2, and
others are listed in Appendix M.

Half-lives of Radioactive Isotopes Important to Medicine

Type[1] Decay Mode Half-Life Uses

F-18 β+ decay 110. minutes PET scans

Co-60 β decay, γ decay 5.27 years cancer treatment

Tc-99m γ decay 8.01 hours scans of brain, lung, heart, bone

I-131 β decay 8.02 days thyroid scans and treatment

Tl-201 electron capture 73 hours heart and arteries scans; cardiac stress tests

Table 21.2

1. The “m” in Tc-99m stands for “metastable,” indicating that this is an unstable, high-energy state of Tc-99.
Metastable isotopes emit γ radiation to rid themselves of excess energy and become (more) stable.
1194 Chapter 21 | Nuclear Chemistry

Radiometric Dating
Several radioisotopes have half-lives and other properties that make them useful for purposes of “dating” the origin
of objects such as archaeological artifacts, formerly living organisms, or geological formations. This process is
radiometric dating and has been responsible for many breakthrough scientific discoveries about the geological
history of the earth, the evolution of life, and the history of human civilization. We will explore some of the most
common types of radioactive dating and how the particular isotopes work for each type.

Radioactive Dating Using Carbon-14

The radioactivity of carbon-14 provides a method for dating objects that were a part of a living organism. This
method of radiometric dating, which is also called radiocarbon dating or carbon-14 dating, is accurate for dating
carbon-containing substances that are up to about 30,000 years old, and can provide reasonably accurate dates up to
a maximum of about 50,000 years old.
12
Naturally occurring carbon consists of three isotopes: 6 C, which constitutes about 99% of the carbon on earth;
13 14
6 C, about 1% of the total; and trace amounts of 6 C. Carbon-14 forms in the upper atmosphere by the reaction of
nitrogen atoms with neutrons from cosmic rays in space:
14 1 14 1
7N + 0n ⟶ 6C + 1H

14 12
All isotopes of carbon react with oxygen to produce CO2 molecules. The ratio of 6 CO 2 to 6 CO 2 depends on the
14 12 14
ratio of 6 CO to 6 CO in the atmosphere. The natural abundance of 6 CO in the atmosphere is approximately 1
part per trillion; until recently, this has generally been constant over time, as seen is gas samples found trapped in ice.
The incorporation of 146 C 146 CO 2 and 126 CO 2 into plants is a regular part of the photosynthesis process, which means
14 12 14 12
that the 6 C: 6 C ratio found in a living plant is the same as the 6 C: 6 C ratio in the atmosphere. But when the
12
plant dies, it no longer traps carbon through photosynthesis. Because 6C is a stable isotope and does not undergo
radioactive decay, its concentration in the plant does not change. However, carbon-14 decays by β emission with a
half-life of 5730 years:
14 12 0
6C ⟶ 7 N + -1 e

14 12
Thus, the 6 C: 6 C ratio gradually decreases after the plant dies. The decrease in the ratio with time provides a
measure of the time that has elapsed since the death of the plant (or other organism that ate the plant). Figure 21.11
visually depicts this process.

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Chapter 21 | Nuclear Chemistry 1195

Figure 21.11 Along with stable carbon-12, radioactive carbon-14 is taken in by plants and animals, and remains at a
constant level within them while they are alive. After death, the C-14 decays and the C-14:C-12 ratio in the remains
decreases. Comparing this ratio to the C-14:C-12 ratio in living organisms allows us to determine how long ago the
organism lived (and died).

14 14 12
For example, with the half-life of 6C being 5730 years, if the 6 C: 6 C ratio in a wooden object found in an
archaeological dig is half what it is in a living tree, this indicates that the wooden object is 5730 years old. Highly
accurate determinations of 146 C: 126 C ratios can be obtained from very small samples (as little as a milligram) by the
use of a mass spectrometer.

Link to Learning

Visit this website (http://openstaxcollege.org/l/16phetradiom) to perform

simulations of radiometric dating.

Example 21.6

Radiocarbon Dating
1196 Chapter 21 | Nuclear Chemistry

A tiny piece of paper (produced from formerly living plant matter) taken from the Dead Sea Scrolls has
an activity of 10.8 disintegrations per minute per gram of carbon. If the initial C-14 activity was 13.6
disintegrations/min/g of C, estimate the age of the Dead Sea Scrolls.
Solution
The rate of decay (number of disintegrations/minute/gram of carbon) is proportional to the amount of
radioactive C-14 left in the paper, so we can substitute the rates for the amounts, N, in the relationship:
⎛ Nt ⎞ ⎛ Rate t ⎞
t = − 1 ln ⟶ t = − 1 ln
λ ⎝N 0 ⎠ λ ⎝Rate 0 ⎠

where the subscript 0 represents the time when the plants were cut to make the paper, and the subscript t
represents the current time.
The decay constant can be determined from the half-life of C-14, 5730 years:

λ = ln 2 0.693 −4 −1
t 1/2 = 5730 y = 1.21 × 10 y

Substituting and solving, we have:

⎛ Rate t ⎞ ⎛ 10.8 dis/min/g C ⎞
t = − 1 ln 1
λ ⎝ Rate 0 ⎠ 1.21 × 10 −4 y −1 ⎝ 13.6 dis/min/g C ⎠
= − ln = 1910 y

Therefore, the Dead Sea Scrolls are approximately 1900 years old (Figure 21.12).

Figure 21.12 Carbon-14 dating has shown that these pages from the Dead Sea Scrolls were written or
copied on paper made from plants that died between 100 BC and AD 50.

Check Your Learning

More accurate dates of the reigns of ancient Egyptian pharaohs have been determined recently using plants
that were preserved in their tombs. Samples of seeds and plant matter from King Tutankhamun’s tomb have
a C-14 decay rate of 9.07 disintegrations/min/g of C. How long ago did King Tut’s reign come to an end?
Answer: about 3350 years ago, or approximately 1340 BC

14 12
There have been some significant, well-documented changes to the 6 C: 6 C ratio. The accuracy of a
14 12
straightforward application of this technique depends on the 6 C: 6 C ratio in a living plant being the same now as
it was in an earlier era, but this is not always valid. Due to the increasing accumulation of CO2 molecules (largely
12 ⎞ 14
6 CO 2⎠ in the atmosphere caused by combustion of fossil fuels (in which essentially all of the 6C has decayed),
14 12 12
the ratio of 6 C: 6 C in the atmosphere may be changing. This manmade increase in 6 CO 2 in the atmosphere
14 12
causes the 6 C: 6 C ratio to decrease, and this in turn affects the ratio in currently living organisms on the earth.
Fortunately, however, we can use other data, such as tree dating via examination of annual growth rings, to calculate

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Chapter 21 | Nuclear Chemistry 1197

correction factors. With these correction factors, accurate dates can be determined. In general, radioactive dating only
works for about 10 half-lives; therefore, the limit for carbon-14 dating is about 57,000 years.

Radioactive Dating Using Nuclides Other than Carbon-14

Radioactive dating can also use other radioactive nuclides with longer half-lives to date older events. For example,
uranium-238 (which decays in a series of steps into lead-206) can be used for establishing the age of rocks (and the
approximate age of the oldest rocks on earth). Since U-238 has a half-life of 4.5 billion years, it takes that amount
of time for half of the original U-238 to decay into Pb-206. In a sample of rock that does not contain appreciable
amounts of Pb-208, the most abundant isotope of lead, we can assume that lead was not present when the rock was
formed. Therefore, by measuring and analyzing the ratio of U-238:Pb-206, we can determine the age of the rock.
This assumes that all of the lead-206 present came from the decay of uranium-238. If there is additional lead-206
present, which is indicated by the presence of other lead isotopes in the sample, it is necessary to make an adjustment.
Potassium-argon dating uses a similar method. K-40 decays by positron emission and electron capture to form Ar-40
with a half-life of 1.25 billion years. If a rock sample is crushed and the amount of Ar-40 gas that escapes is measured,
determination of the Ar-40:K-40 ratio yields the age of the rock. Other methods, such as rubidium-strontium dating
(Rb-87 decays into Sr-87 with a half-life of 48.8 billion years), operate on the same principle. To estimate the lower
limit for the earth’s age, scientists determine the age of various rocks and minerals, making the assumption that the
earth is older than the oldest rocks and minerals in its crust. As of 2014, the oldest known rocks on earth are the Jack
Hills zircons from Australia, found by uranium-lead dating to be almost 4.4 billion years old.

Example 21.7

Radioactive Dating of Rocks

An igneous rock contains 9.58 × 10–5 g of U-238 and 2.51 × 10–5 g of Pb-206, and much, much smaller
amounts of Pb-208. Determine the approximate time at which the rock formed.
Solution
The sample of rock contains very little Pb-208, the most common isotope of lead, so we can safely assume
that all the Pb-206 in the rock was produced by the radioactive decay of U-238. When the rock formed, it
contained all of the U-238 currently in it, plus some U-238 that has since undergone radioactive decay.
The amount of U-238 currently in the rock is:
⎛ ⎞
9.58 × 10 −5 g U × ⎜ 1 mol U ⎟ = 4.03 × 10 −7 mol U
⎝ 238 g U ⎠

Because when one mole of U-238 decays, it produces one mole of Pb-206, the amount of U-238 that has
undergone radioactive decay since the rock was formed is:
⎛ ⎞ ⎛ ⎞
2.51 × 10 −5 g Pb × ⎜ 1 mol Pb ⎟ × ⎝ 1 mol U ⎠ = 1.22 × 10 −7 mol U
⎝ 206 g Pb ⎠ 1 mol Pb

The total amount of U-238 originally present in the rock is therefore:

4.03 × 10 −7 mol + 1.22 × 10 −7 mol = 5.25 × 10 −7 mol U
The amount of time that has passed since the formation of the rock is given by:
⎛ Nt ⎞
t = − 1 ln
λ ⎝ N0 ⎠

with N0 representing the original amount of U-238 and Nt representing the present amount of U-238.
1198 Chapter 21 | Nuclear Chemistry

U-238 decays into Pb-206 with a half-life of 4.5 × 109 y, so the decay constant λ is:

λ = ln 2 0.693 −10 −1
t 1/2 = 4.5 × 10 9 y = 1.54 × 10 y

Substituting and solving, we have:

1 ⎛4.03 × 10 −7 mol U ⎞
1.54 × 10 −10 y −1 ⎝5.25 × 10 −7 mol U ⎠ = 1.7 × 10 y
9
t= − ln

Therefore, the rock is approximately 1.7 billion years old.

Check Your Learning

A sample of rock contains 6.14 × 10–4 g of Rb-87 and 3.51 × 10–5 g of Sr-87. Calculate the age of the
rock. (The half-life of the β decay of Rb-87 is 4.7 × 1010 y.)

Answer: 3.7 × 109 y

21.4 Transmutation and Nuclear Energy

By the end of this section, you will be able to:
• Describe the synthesis of transuranium nuclides
• Explain nuclear fission and fusion processes
• Relate the concepts of critical mass and nuclear chain reactions
• Summarize basic requirements for nuclear fission and fusion reactors
After the discovery of radioactivity, the field of nuclear chemistry was created and developed rapidly during the early
twentieth century. A slew of new discoveries in the 1930s and 1940s, along with World War II, combined to usher in
the Nuclear Age in the mid-twentieth century. Science learned how to create new substances, and certain isotopes of
certain elements were found to possess the capacity to produce unprecedented amounts of energy, with the potential
to cause tremendous damage during war, as well as produce enormous amounts of power for society’s needs during
peace.

Synthesis of Nuclides
Nuclear transmutation is the conversion of one nuclide into another. It can occur by the radioactive decay of
a nucleus, or the reaction of a nucleus with another particle. The first manmade nucleus was produced in Ernest
Rutherford’s laboratory in 1919 by a transmutation reaction, the bombardment of one type of nuclei with other
nuclei or with neutrons. Rutherford bombarded nitrogen atoms with high-speed α particles from a natural radioactive
isotope of radium and observed protons resulting from the reaction:
14 4 17 1
7 N + 2 He ⟶ 8O + 1H

17 1
The 8O and 1H nuclei that are produced are stable, so no further (nuclear) changes occur.

To reach the kinetic energies necessary to produce transmutation reactions, devices called particle accelerators are
used. These devices use magnetic and electric fields to increase the speeds of nuclear particles. In all accelerators,
the particles move in a vacuum to avoid collisions with gas molecules. When neutrons are required for transmutation
reactions, they are usually obtained from radioactive decay reactions or from various nuclear reactions occurring in

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Chapter 21 | Nuclear Chemistry 1199

nuclear reactors. The Chemistry in Everyday Life feature that follows discusses a famous particle accelerator that
made worldwide news.

Chemistry in Everyday Life

CERN Particle Accelerator

Located near Geneva, the CERN (“Conseil Européen pour la Recherche Nucléaire,” or European Council
for Nuclear Research) Laboratory is the world’s premier center for the investigations of the fundamental
particles that make up matter. It contains the 27-kilometer (17 mile) long, circular Large Hadron Collider
(LHC), the largest particle accelerator in the world (Figure 21.13). In the LHC, particles are boosted to high
energies and are then made to collide with each other or with stationary targets at nearly the speed of light.
Superconducting electromagnets are used to produce a strong magnetic field that guides the particles around
the ring. Specialized, purpose-built detectors observe and record the results of these collisions, which are then
analyzed by CERN scientists using powerful computers.

Figure 21.13 A small section of the LHC is shown with workers traveling along it. (credit: Christophe
Delaere)

In 2012, CERN announced that experiments at the LHC showed the first observations of the Higgs boson,
an elementary particle that helps explain the origin of mass in fundamental particles. This long-anticipated
discovery made worldwide news and resulted in the awarding of the 2103 Nobel Prize in Physics to François
Englert and Peter Higgs, who had predicted the existence of this particle almost 50 years previously.

Link to Learning

Famous physicist Brian Cox talks about his work on the Large Hadron Collider at
CERN, providing an entertaining and engaging tour (http://openstaxcollege.org/
l/16tedCERN) of this massive project and the physics behind it.
View a short video (http://openstaxcollege.org/l/16CERNvideo) from CERN,
describing the basics of how its particle accelerators work.
1200 Chapter 21 | Nuclear Chemistry

Prior to 1940, the heaviest-known element was uranium, whose atomic number is 92. Now, many artificial elements
have been synthesized and isolated, including several on such a large scale that they have had a profound effect
on society. One of these—element 93, neptunium (Np)—was first made in 1940 by McMillan and Abelson by
bombarding uranium-238 with neutrons. The reaction creates unstable uranium-239, with a half-life of 23.5 minutes,
which then decays into neptunium-239. Neptunium-239 is also radioactive, with a half-life of 2.36 days, and it decays
into plutonium-239. The nuclear reactions are:
238 1 239
92 U + 0 n ⟶ 92 U
239 239 0
92 U ⟶ 93 Np + −1 et 1/2 half-life = 23.5 min
239 239 0
93 Np ⟶ 94 Pu + −1 et 1/2 half-life = 2.36 days

Plutonium is now mostly formed in nuclear reactors as a byproduct during the decay of uranium. Some of the neutrons
that are released during U-235 decay combine with U-238 nuclei to form uranium-239; this undergoes β decay to
form neptunium-239, which in turn undergoes β decay to form plutonium-239 as illustrated in the preceding three
equations. It is possible to summarize these equations as:
238 1 239 β − 239 β − 239
92 U + 0 n ⟶ 92 U ⎯⎯⎯→ 93 Np ⎯⎯⎯→ 94 Pu

Heavier isotopes of plutonium—Pu-240, Pu-241, and Pu-242—are also produced when lighter plutonium nuclei
capture neutrons. Some of this highly radioactive plutonium is used to produce military weapons, and the rest presents
a serious storage problem because they have half-lives from thousands to hundreds of thousands of years.
Although they have not been prepared in the same quantity as plutonium, many other synthetic nuclei have been
produced. Nuclear medicine has developed from the ability to convert atoms of one type into other types of atoms.
Radioactive isotopes of several dozen elements are currently used for medical applications. The radiation produced
by their decay is used to image or treat various organs or portions of the body, among other uses.
The elements beyond element 92 (uranium) are called transuranium elements. As of this writing, 22 transuranium
elements have been produced and officially recognized by IUPAC; several other elements have formation claims that
are waiting for approval. Some of these elements are shown in Table 21.3.

Preparation of Some of the Transuranium Elements

Name Symbol Atomic Number Reaction

239 1 240 0
americium Am 95 94 Pu + 0 n ⟶ 95 Am + −1 e

239 4 242 1
curium Cm 96 94 Pu + 2 He ⟶ 96 Cm + 0 n

242 4 243 1
californium Cf 98 96 Cm + 2 He ⟶ 97 Bk + 2 0 n

238 1 253 0
einsteinium Es 99 92 U + 15 0 n ⟶ 99 Es + 7 −1 e

253 4 256 1
mendelevium Md 101 99 Es + 2 He ⟶ 101 Md + 0 n

246 12 254 1
nobelium No 102 96 Cm + 6 C ⟶ 102 No + 4 0 n

249 257
rutherfordium Rf 104 98 Cf + 126 C ⟶ 104 Rf + 4 10 n

Table 21.3

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Chapter 21 | Nuclear Chemistry 1201

Preparation of Some of the Transuranium Elements

Name Symbol Atomic Number Reaction

206 54 257 1
82 Pb + 24 Cr ⟶ 106 Sg + 3 0 n
seaborgium Sg 106 249 18 263 1
98 Cf + 8 O ⟶ 106 Sg + 4 0 n

209 58 266 1
meitnerium Mt 107 83 Bi + 26 Fe ⟶ 109 Mt + 0 n

Table 21.3

Nuclear Fission
Many heavier elements with smaller binding energies per nucleon can decompose into more stable elements that have
intermediate mass numbers and larger binding energies per nucleon—that is, mass numbers and binding energies per
nucleon that are closer to the “peak” of the binding energy graph near 56 (see Figure 21.3). Sometimes neutrons are
also produced. This decomposition is called fission, the breaking of a large nucleus into smaller pieces. The breaking
is rather random with the formation of a large number of different products. Fission usually does not occur naturally,
but is induced by bombardment with neutrons. The first reported nuclear fission occurred in 1939 when three German
scientists, Lise Meitner, Otto Hahn, and Fritz Strassman, bombarded uranium-235 atoms with slow-moving neutrons
that split the U-238 nuclei into smaller fragments that consisted of several neutrons and elements near the middle of
the periodic table. Since then, fission has been observed in many other isotopes, including most actinide isotopes that
have an odd number of neutrons. A typical nuclear fission reaction is shown in Figure 21.14.

Figure 21.14 When a slow neutron hits a fissionable U-235 nucleus, it is absorbed and forms an unstable U-236
nucleus. The U-236 nucleus then rapidly breaks apart into two smaller nuclei (in this case, Ba-141 and Kr-92) along
with several neutrons (usually two or three), and releases a very large amount of energy.

Among the products of Meitner, Hahn, and Strassman’s fission reaction were barium, krypton, lanthanum, and
cerium, all of which have nuclei that are more stable than uranium-235. Since then, hundreds of different isotopes
have been observed among the products of fissionable substances. A few of the many reactions that occur for U-235,
and a graph showing the distribution of its fission products and their yields, are shown in Figure 21.15. Similar
1202 Chapter 21 | Nuclear Chemistry

fission reactions have been observed with other uranium isotopes, as well as with a variety of other isotopes such as
those of plutonium.

Figure 21.15 (a) Nuclear fission of U-235 produces a range of fission products. (b) The larger fission products of
U-235 are typically one isotope with a mass number around 85–105, and another isotope with a mass number that is
about 50% larger, that is, about 130–150.

Link to Learning

View this link (http://openstaxcollege.org/l/16fission) to see a simulation of

nuclear fission.

A tremendous amount of energy is produced by the fission of heavy elements. For instance, when one mole of U-235
undergoes fission, the products weigh about 0.2 grams less than the reactants; this “lost” mass is converted into a very
large amount of energy, about 1.8 × 1010 kJ per mole of U-235. Nuclear fission reactions produce incredibly large
amounts of energy compared to chemical reactions. The fission of 1 kilogram of uranium-235, for example, produces
about 2.5 million times as much energy as is produced by burning 1 kilogram of coal.
As described earlier, when undergoing fission U-235 produces two “medium-sized” nuclei, and two or three neutrons.
These neutrons may then cause the fission of other uranium-235 atoms, which in turn provide more neutrons that can
cause fission of even more nuclei, and so on. If this occurs, we have a nuclear chain reaction (see Figure 21.16).
On the other hand, if too many neutrons escape the bulk material without interacting with a nucleus, then no chain
reaction will occur.

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