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5: Calculations Involving a Limiting Reactant
616 Chapter 19 Radioactivity and Nuclear Energy
19.1 Radioactive Decay
To learn the types of radioactive decay. • To learn to write nuclear
equations that describe radioactive decay.
Many nuclei are radioactive; that is, they spontaneously decompose,
forming a different nucleus and producing one or more particles. An example is carbon-14, which decays as shown in the equation
S 147N ϩ Ϫ10e
where Ϫ10e represents an electron, which in nuclear terminology is called a
beta particle, or B particle. This nuclear equation, which is typical of
those representing radioactive decay, is quite different from the chemical
equations we have written before. Recall that in a balanced chemical equation the atoms must be conserved. In a nuclear equation both the atomic number (Z) and the mass number (A) must be conserved. That is, the sums of the Z
values on both sides of the arrow must be equal, and the same restriction applies to the A values. For example, in the above equation, the sum of the Z
values is 6 on both sides of the arrow (6 and 7 Ϫ 1), and the sum of the A values is 14 on both sides of the arrow (14 and 14 ϩ 0). Notice that the mass
number for the ␤ particle is zero; the mass of the electron is so small that it
can be neglected here. Of the approximately 2000 known nuclides, only 279
do not undergo radioactive decay. Tin has the largest number of nonradioactive isotopes—ten.
Over 85% of all known nuclides
Types of Radioactive Decay
There are several different types of radioactive decay. One frequently observed decay process involves production of an alpha (A) particle, which
is a helium nucleus (42He). Alpha-particle production is a very common
mode of decay for heavy radioactive nuclides. For example, 222
88Ra, radium222, decays by ␣-particle production to give radon-218.
S 42He ϩ 218
Notice in this equation that the mass number is conserved (222 ϭ 4 ϩ 218)
and the atomic number is conserved (88 ϭ 2 ϩ 86). Another ␣-particle producer is 230
S 42He ϩ 226
Notice that the production of an a particle results in a loss of 4 in mass number (A) and a loss of 2 in atomic number (Z).
B-particle production is another common decay process. For example, the thorium-234 nuclide produces a ␤ particle as it changes to protactinium-234.
Notice that both Z and A
balance in each of these nuclear
91Pa ϩ Ϫ1e
Iodine-131 is also a ␤-particle producer:
S Ϫ10e ϩ 131
Recall that the ␤ particle is assigned a mass number of 0 because its mass is
tiny compared with that of a proton or neutron. The value of Z is Ϫ1 for the
␤ particle, so the atomic number for the new nuclide is greater by 1 than the
atomic number for the original nuclide. Therefore, the net effect of ␤-particle
production is to change a neutron to a proton.
Production of a ␤ particle results in no change in mass number (A) and
an increase of 1 in atomic number (Z).
19.1 Radioactive Decay
Table 19.1 Various Types of Radioactive Processes
7N S 6C ϩ 1e
33As ϩ Ϫ1e S 32Ge
84Po S 82Pb ϩ 2He
A gamma ray is a high-energy
photon produced in connection
with nuclear decay.
excited nucleus S ground-state nucleus ϩ 00␥
A gamma ray, or ␥ ray, is a high-energy photon of light. A nuclide in
an excited nuclear energy state can release excess energy by producing a
gamma ray, and ␥-ray production often accompanies nuclear decays of various types. For example, in the ␣-particle decay of 238
The 00␥ notation indicates Z ϭ 0
and A ϭ 0 for a ␥ ray. A
gamma ray is often simply
indicated by ␥.
S 42He ϩ 234
90Th ϩ 20␥
two ␥ rays of different energies are produced in addition to the ␣ particle
(42He). Gamma rays are photons of light and so have zero charge and zero
Production of a ␥ ray results in no change in mass number (A) and no
change in atomic number (Z).
The positron is a particle with the same mass as the electron but opposite charge. An example of a nuclide that decays by positron production is sodium-22:
S 01e ϩ 22
Note that the production of a positron appears to change a proton into a neutron.
Production of a positron results in no change in mass number (A) and
a decrease of 1 in atomic number (Z).
Electron capture is a process in which one of the inner-orbital electrons is captured by the nucleus, as illustrated by the process
ϩ Ϫ10e S 201
79Au ϩ 0␥
Kopal/Mediamed Publiphoto/Photo Researchers, Inc.
This reaction would have been of great interest to the alchemists, but unfortunately, it does not occur often enough to make it a practical means of
changing mercury to gold. Gamma rays are always produced along with electron capture.
Table 19.1 lists the common types of radioactive decay, with examples.
Often a radioactive nucleus cannot achieve a stable (nonradioactive)
state through a single decay process. In such a case, a decay series occurs
until a stable nuclide is formed. A well-known example is the decay series
that starts with 238
92U and ends with 82Pb, as shown in Figure 19.1. Similar se235
ries exist for 92U:
Bone scintigraph of a patient’s
cranium following administration
of the radiopharmaceutical
and for 232
618 Chapter 19 Radioactivity and Nuclear Energy
The decay series from 238
Each nuclide in the series
82Pb is radioactive, and
the successive transformations
(shown by the arrows) continue
82Pb is finally formed. The
horizontal red arrows indicate
␤-particle production (Z increases
by 1 and A is unchanged). The
diagonal blue arrows signify
␣-particle production (both A
and Z decrease).
= α-particle production
= β-particle production
Writing Nuclear Equations, I
Write balanced nuclear equations for each of the following processes.
produces a positron.
produces a ␤ particle.
produces an ␣ particle.
a. We must find the product nuclide represented by AZX in the
S 01e ϩ AZX
The key to solving this problem is to recognize that both A and Z
must be conserved. That is, we can find the identity of AZX by
recognizing that the sums of the Z and A values must be the same
on both sides of the equation. Thus, for X, Z must be 5 because Z ϩ
1 ϭ 6. A must be 11 because 11 ϩ 0 ϭ 11. Therefore, AZX is 115B. (The
fact that Z is 5 tells us that the nuclide is boron. See the periodic
table on the inside front cover of the book.) So the balanced
equation is 116C S 01e ϩ 115B.
A ϭ 11
A ϭ 11 ϩ 0 ϭ 11
19.1 Radioactive Decay
b. Knowing that a ␤ particle is represented by
we can write
where Z Ϫ 1 ϭ 83 and A ϩ 0 ϭ 214. This means that Z ϭ 84 and
A ϭ 214. We can now write
S Ϫ10e ϩ 214
Using the periodic table, we find that Z ϭ 84 for the element
polonium, so 214
84X must be 84Po.
Z ϭ 83
A ϭ 214
Z ϭ 84 Ϫ 1 ϭ 83
A ϭ 214 ϩ 0 ϭ 214
c. Because an ␣ particle is represented by 42He, we can write
S 42He ϩ AZX
where A ϩ 4 ϭ 237 or A ϭ 237 Ϫ 4 ϭ 233, and Z ϩ 2 ϭ 93 or
Z ϭ 93 Ϫ 2 ϭ 91. Thus A ϭ 233, Z ϭ 91, and the balanced equation
Z ϭ 93
A ϭ 237
S 42He ϩ 233
Z ϭ 91 ϩ 2 ϭ 93
A ϭ 233 ϩ 4 ϭ 237
Self-Check EXERCISE 19.1 The decay series for 238
92U is represented in Figure 19.1. Write the balanced nuclear equation for each of the following radioactive decays.
a. Alpha-particle production by
b. Beta-particle production by
See Problems 19.25 through 19.28. ■
Writing Nuclear Equations, II
In each of the following nuclear reactions, supply the missing particle.
a. A does not change and Z for Pt is 1 lower than Z for Au, so the
missing particle must be an electron.
ϩ Ϫ10e S 195
Z ϭ 79 Ϫ 1 ϭ 78
Z ϭ 78
A ϭ 195 ϩ 0 ϭ 195
A ϭ 195
This is an example of electron capture.
b. For Z and A to be conserved, the missing particle must be a
18Ar ϩ 1e
620 Chapter 19 Radioactivity and Nuclear Energy
Z ϭ 19
A ϭ 38
Z ϭ 18 ϩ 1 ϭ 19
A ϭ 38 ϩ 0 ϭ 38
Potassium-38 decays by positron production.
Self-Check EXERCISE 19.2 Supply the missing species in each of the following nuclear equations.
See Problems 19.21 through 19.24. ■
19.2 Nuclear Transformations
To learn how one element may be changed into another by particle
In 1919, Lord Rutherford observed the first nuclear
transformation, the change of one element into another. He found that bombarding 147N with ␣ particles
produced the nuclide 178O:
ϩ 42He S 178O ϩ 11H
with a proton (11H) as another product. Fourteen years
later, Irene Curie and her husband Frederick Joliot observed a similar transformation from aluminum to
Culver Pictures/The Granger Collection
Irene Curie and Frederick Joliot.
ϩ 42He S 30
15P ϩ 0n
where 10n represents a neutron that is produced in the
Notice that in both these cases the bombarding
particle is a helium nucleus (an ␣ particle). Other
small nuclei, such as 126C and 157N, also can be used to
bombard heavier nuclei and cause transformations.
However, because these positive bombarding ions are
repelled by the positive charge of the target nucleus,
the bombarding particle must be moving at a very
high speed to penetrate the target. These high speeds
are achieved in various types of particle accelerators.
Neutrons are also employed as bombarding particles to effect nuclear
transformations. However, because neutrons are uncharged (and thus not repelled by a target nucleus), they are readily absorbed by many nuclei, producing new nuclides. The most common source of neutrons for this purpose
is a fission reactor (see Section 19.8).
By using neutron and positive-ion bombardment, scientists have been
able to extend the periodic table—that is, to produce chemical elements that
are not present naturally. Prior to 1940, the heaviest known element was
uranium (Z ϭ 92), but in 1940, neptunium (Z ϭ 93) was produced by neu239
tron bombardment of 238
92U. The process initially gives 92U, which decays to
93Np by ␤-particle production:
ϩ 10n S 239
92U S 93Np ϩ Ϫ1e
19.3 Detection of Radioactivity and the Concept of Half-life
Table 19.2 Syntheses of Some of the Transuranium Elements
neptunium (Z ϭ 93)
92U ϩ 0n S 92U S 93Np ϩ Ϫ1e
94Pu ϩ 2 0n S 94Pu S 95Am ϩ Ϫ1e
americium (Z ϭ 95)
curium (Z ϭ 96)
californium (Z ϭ 98)
rutherfordium (Z ϭ 104)
dubnium (Z ϭ 105)
seaborgium (Z ϭ 106)
94Pu ϩ 2He S 96Cm ϩ 0n
96Cm ϩ 2He S 98 Cf ϩ 0n or
92U ϩ 6C S 98Cf ϩ 4 0n
98Cf ϩ 6C S 104Rf ϩ 4 0n
98Cf ϩ 7N S 105Db ϩ 4 0n
98Cf ϩ 8O S 106Sg ϩ 4 0n
In the years since 1940, the elements with atomic numbers 93 through
112, called the transuranium elements, have been synthesized. In addition, the production of element 114 (in 1999), elements 113 and 115 (in
2004), and element 118 (in 2006) have been reported. Table 19.2 gives some
examples of these processes.
19.3 Detection of Radioactivity
and the Concept of Half-life
Geiger counters are commonly
called survey meters.
To learn about radiation detection instruments. • To understand half-life.
The most familiar instrument for measuring radioactivity levels is the
Geiger–Müller counter, or Geiger counter (Figure 19.2). High-energy
particles from radioactive decay produce ions when they travel through matter. The probe of the Geiger counter contains argon gas. The argon atoms
have no charge, but they can be ionized by a rapidly moving particle.
Arϩ(g) ϩ eϪ
That is, the fast-moving particle “knocks” electrons off some of the argon
atoms. Although a sample of uncharged argon atoms does not conduct a current, the ions and electrons formed by the high-energy particle allow a current to flow momentarily, so a “pulse” of current flows every time a particle
enters the probe. The Geiger counter detects each pulse of current, and these
events are counted.
A scintillation counter is another instrument often employed to detect radioactivity. This device uses a substance, such as sodium iodide, that
A schematic representation of
a Geiger–Müller counter. The
high-speed particle knocks
electrons off argon atoms to
Arϩ ϩ eϪ
and a pulse of current flows.
622 Chapter 19 Radioactivity and Nuclear Energy
gives off light when it is struck by a high-energy particle. A detector senses
the flashes of light and thus counts the decay events.
One important characteristic of a given type of radioactive nuclide is its
half-life. The half-life is the time required for half the original sample of nuclei
to decay. For example, if a certain radioactive sample contains 1000 nuclei at
a given time and 500 nuclei (half of the original number) 7.5 days later, this
radioactive nuclide has a half-life of 7.5 days.
A given type of radioactive nuclide always has the same half-life. However, the various radioactive nuclides have half-lives that cover a tremendous
range. For example, 234
91Pa, protactinium-234, has a half-life of 1.2 minutes,
92U, uranium-238, has a half-life of 4.5 ϫ 10 (4.5 billion) years. This
means that a sample containing 100 million 91Pa nuclei will have only 50
91Pa nuclei in it (half of 100 million) after 1.2 minutes have passed.
In another 1.2 minutes, the number of nuclei will decrease to half of 50 million, or 25 million nuclei.
100 million 234
50 million 234
(50 million decays)
25 million 234
(25 million decays)
This means that a sample of 234
91Pa with 100 million nuclei will show 50 million decay events (50 million 234
91Pa nuclei will decay) over a time of 1.2 minutes. By contrast, a sample containing 100 million 238
92U nuclei will undergo
50 million decay events over 4.5 billion years. Therefore, 234
91Pa shows much
greater activity than 238
Thus, at a given moment, a radioactive nucleus with a short half-life is
much more likely to decay than one with a long half-life.
Table 19.3 The Half-lives
for Some of the Radioactive
Nuclides of Radium
Table 19.3 lists various radioactive nuclides of radium.
a. Order these nuclides in terms of activity (from most decays per day
b. How long will it take for a sample containing 1.00 mole of
reach a point where it contains only 0.25 mole of 223
a. The shortest half-life indicates the greatest activity (the most decays
over a given period of time). Therefore, the order is
b. In one half-life (12 days), the sample will decay from 1.00 mole of
88Ra to 0.50 mole of 88Ra. In the next half-life (another 12 days),
it will decay from 0.50 mole of 223
88Ra to 0.25 mole of 88Ra.
1.00 mol 223
0.50 mol 223
0.25 mol 223
Therefore, it will take 24 days (two half-lives) for the sample to
change from 1.00 mole of 223
88Ra to 0.25 mole of 88Ra.
19.4 Dating by Radioactivity
Self-Check EXERCISE 19.3 Watches with numerals that “glow in the dark” formerly were made by including radioactive radium in the paint used to letter the watch faces. Assume that to make the numeral 3 on a given watch, a sample of paint containing 8.0 ϫ 10Ϫ7 mole of 228
88Ra was used. This watch was then put in a
drawer and forgotten. Many years later someone finds the watch and
wishes to know when it was made. Analyzing the paint, this person finds
1.0 ϫ 10Ϫ7 moles of 228
88Ra in the numeral 3. How much time elapsed between the making of the watch and the finding of the watch?
Use the half-life of 228
88Ra from Table 19.3.
See Problems 19.37 through 19.42. ■
A watch dial with radium paint.
19.4 Dating by Radioactivity
To learn how objects can be dated by radioactivity.
Archaeologists, geologists, and others involved in reconstructing the ancient
history of the earth rely heavily on the half-lives of radioactive nuclei to provide accurate dates for artifacts and rocks. A method for dating ancient articles made from wood or cloth is radiocarbon dating, or carbon-14 dating, a technique originated in the 1940s by Willard Libby, an American
chemist who received the Nobel Prize for his efforts.
Radiocarbon dating is based on the radioactivity of 146C, which decays
by ␤-particle production.
S Ϫ10e ϩ 147N
Carbon-14 is continuously produced in the atmosphere when high-energy
neutrons from space collide with nitrogen-14.
Mark W. Philbrick/BYU
Brigham Young researcher Scott
Woodward taking a bone sample
for carbon-14 dating at an
archaeological site in Egypt.
ϩ 10n S 146C ϩ 11H
Just as carbon-14 is produced continuously by this process, it decomposes
continuously through ␤-particle production. Over the years, these two opposing processes have come into balance, causing the amount of 146C present
in the atmosphere to remain approximately constant.
Carbon-14 can be used to date wood and cloth artifacts because the
6C, along with the other carbon isotopes in the atmosphere, reacts with
oxygen to form carbon dioxide. A living plant consumes this carbon dioxide
in the photosynthesis process and incorporates the carbon, including 146C,
into its molecules. As long as the plant lives, the 146C content in its molecules
remains the same as in the atmosphere because of the plant’s continuous uptake of carbon. However, as soon as a tree is cut to make a wooden bowl or a
flax plant is harvested to make linen, it stops taking in carbon. There is no
longer a source of 146C to replace that lost to radioactive decay, so the material’s 146C content begins to decrease.
Because the half-life of 146C is known to be 5730 years, a wooden bowl
found in an archaeological dig that shows a 146C content of half that found
in currently living trees is approximately 5730 years old. That is, because half
the 146C present when the tree was cut has disappeared, the tree must have
been cut one half-life of 146C ago.
C H E M I S T R Y I N F OCUS
While connoisseurs of gems value the clearest
possible diamonds, geologists learn the most from
impure diamonds. Diamonds are formed in the
earth’s crust at depths of about 200 kilometers,
where the high pressures and temperatures favor
the most dense form of carbon. As the diamond is
formed, impurities are sometimes trapped, and
these can be used to determine the diamond’s
date of “birth.” One valuable dating impurity is
92U, which is radioactive and decays in a series of
steps to 206
82Pb, which is stable (nonradioactive). Because the rate at which 238
92U decays is known, de238
termining how much 92U has been converted to
82Pb tells scientists the amount of time that has
elapsed since the 238
92U was trapped in the diamond
as it was formed.
Using these dating techniques, Peter D.
Kinney of Curtin University of Technology in
Perth, Australia, and Henry O. A. Meyer of Purdue University in West Lafayette, Indiana, have
identified the youngest diamond ever found.
Discovered in Mbuji Mayi, Zaire, the diamond is
628 million years old, far younger than all previously dated diamonds, which range from 2.4 to
3.2 billion years old.
The great age of all previously dated diamonds had caused some geologists to speculate
that all diamond formation occurred billions of
years ago. However, this “youngster” suggests
that diamonds have formed throughout geologic
time and are probably being formed right now in
the earth’s crust. We won’t see these diamonds for
a long time, because diamonds typically remain
deeply buried in the earth’s crust for millions of
years until they are brought to the surface by volcanic blasts called kimberlite eruptions.
It’s good to know that eons from now there
will be plenty of diamonds to mark the engagements of future couples.
Smithsonian Institution, Natural History Museum, Department of
The Hope diamond.
19.5 Medical Applications of Radioactivity
Nuclides used as radiotracers
have short half-lives so that
they disappear rapidly from the
To discuss the use of radiotracers in medicine.
Although we owe the rapid advances of the medical sciences in recent
decades to many causes, one of the most important has been the discovery
and use of radiotracers—radioactive nuclides that can be introduced into
organisms in food or drugs and subsequently traced by monitoring their radioactivity. For example, the incorporation of nuclides such as 146C and 32
into nutrients has yielded important information about how these nutrients
are used to provide energy for the body.
Iodine-131 has proved very useful in the diagnosis and treatment of illnesses of the thyroid gland. Patients drink a solution containing a small
amount of NaI that includes131I, and the uptake of the iodine by the thyroid
gland is monitored with a scanner (Figure 19.3).
Thallium-201 can be used to assess the damage to the heart muscle in a
person who has suffered a heart attack because thallium becomes concentrated
C H E M I S T R Y I N F OCUS
A positron emission tomography (PET) scanner.
ne of the most valuable applications of radioactivity is in the use of radiotracers for medical diagnosis. Radiotracers are radioactive atoms
that are attached to biologically active molecules. The resultant radioactivity is monitored to
check on the functioning of organs such as the
heart or to trace the path and final destination
of a drug.
One particularly valuable radiotracer technique is called positron emission tomography
(PET). As its name suggests, PET uses positron
producers, such as 18F and 11C, as “labels” on biologic molecules. PET is especially useful for brain
scans. For example, a modified form of glucose
with 18F attached is commonly used to map glucose metabolism. Areas of the brain where glucose is being rapidly consumed “light up” on the
PET screen. The brain of a patient who has a tumor or who has Alzheimer’s disease will show a
much different picture than will a brain of a patient without Alzheimer’s disease. Another application of PET is in seeing how much of a particular labeled drug reaches the intended target.
This enables pharmaceutical companies to check
the effectiveness of a drug and to set dosages.
One of the challenges of using PET is the
speed required to synthesize the labeled molecule.
For example, 18F has a half-life of 110 minutes.
Thus, in a little less than 2 hours after the 18F has
been produced in a particle accelerator, half of it
has already decayed. Also, because of the dangers
of handling radioactive 18F, synthesis operations
must be carried out by robotic manipulations inside a lead-lined box. The good news is that PET is
incredibly sensitive—it can detect amounts of 18F
as small as 10Ϫ12 mol. The use of 11C is even more
challenging synthetically than the use of 18F because 11C has a half-life of only 20 minutes.
PET is a rapidly growing technology. In particular, more chemists are needed in this field to
improve synthetic methods and to develop new
radioactive tracers. If this is of interest to you, do
some exploring to see how to prepare yourself
for a job in this field.
Pascal Goetgheluck/Photo Researchers, Inc.
PET, the Brain’s Best Friend
Scan of radioactive iodine in a
Scan of an enlarged thyroid.
After consumption of Na131I, the patient’s thyroid is scanned for radioactivity
levels to determine the efficiency of iodine absorption.
626 Chapter 19 Radioactivity and Nuclear Energy
Table 19.4 Some Radioactive Nuclides, Their Half-lives, and Their Medical Applications
Area of the Body Studied
red blood cells
eyes, liver, tumors
red blood cells
heart, bones, liver, lungs
*Z is sometimes not written when listing nuclides.
in healthy muscle tissue. Technetium-99, which is also taken up by normal
heart tissue, is used for damage assessment in a similar way.
Radiotracers provide sensitive and nonsurgical methods for learning
about biologic systems, for detecting disease, and for monitoring the action
and effectiveness of drugs. Some useful radiotracers are listed in Table 19.4.
19.6 Nuclear Energy
To introduce fusion and fission as producers of nuclear energy.
The protons and the neutrons in atomic nuclei are bound together with
forces that are much greater than the forces that bind atoms together to form
molecules. In fact, the energies associated with nuclear processes are more
than a million times those associated with chemical reactions. This potentially makes the nucleus a very attractive source of energy.
Because medium-sized nuclei contain the strongest binding forces (56
has the strongest binding forces of all), there are two types of nuclear
processes that produce energy:
1. Combining two light nuclei to form a heavier nucleus. This process
is called fusion.
2. Splitting a heavy nucleus into two nuclei with smaller mass
numbers. This process is called fission.
As we will see in the next several sections, these two processes can supply
amazing quantities of energy with relatively small masses of materials
19.7 Nuclear Fission
To learn about nuclear fission.
Nuclear fission was discovered in the late 1930s when 235
92U nuclides bombarded with neutrons were observed to split into two lighter elements.
92U S 56Ba ϩ 36Kr ϩ 3 0n