10.4
Half-Life
Figure 1 Pitchblende, the major
uranium ore, is a heavy mineral that
contains uranium oxides, lead, and trace
amounts of other radioactive elements.
Pierre and Marie Curie found radium
and polonium in pitchblende residues.
Investigation
10B
The Half-Life of Popcorn
To perform this investigation, turn to
page 300.
In this investigation, you will simulate
the radioactive decay using popcorn
kernels.
A sample of radioactive material, such as uranium ore (Figure 1), contains
an immense number of radioactive atoms, any of which can undergo
radioactive decay. The decay of a nucleus is an individual random event.
The rate of radioactive decay of a sample is not affected by physical or
chemical changes, including temperature and pressure. In addition, the age
of a nucleus does not affect the probability that it will decay. Although there
is no way of determining when an individual nucleus will decay, we can
predict the average rate of decay for a large number of nuclei. In the
beginning, there are a large number of radioactive parent nuclei and,
therefore, there will be a high rate of decay per second. As time passes and
parent nuclei decay, there will be fewer and fewer parent nuclei, and more
and more daughter nuclei. Over time, both the number of parent nuclei
present and the rate of decay will decrease.
The number of decays per second of a sample is known as the activity of
the sample and is measured in becquerels (Bq). A becquerel is equal to one
decay per second. The average length of time for half of the parent nuclei in
a sample to decay is called the half-life. The half-life is different for different
isotopes, but is a constant number for a given isotope. 10B Investigation
The activity of a sample depends on the size of the sample (how many
radioactive nuclei were present initially) and the age of the sample (how
many radioactive nuclei are left). However, for any sample, the number of
parent nuclei left in the sample and the activity level of the sample always
follow the curves shown in Figures 2 and 3. These curves are for a fictitious
(made up) radioactive source.
Activity Level versus Time
Number of Parent Nuclei versus Time
350
100
Activity level (Bq)
Number of parent nuclei
300
80
60
40
20
250
200
150
100
50
0
0
0
5
10
15
20
25
0
Time
Figure 2 Parent nuclei decay curve
290
Unit C Radioactivity
5
10
Time
15
20
Figure 3 Activity curve
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LEARNING TIP
Both curves have an identical shape. You can see from the figures that the
sample has a half-life of 5 units of time. The number of parent nuclei goes
from 100 to 50 to 25 to 12.5 to 6.25 at times of 0, 5, 10, 15, and 20 time
units. Similarly, the activity level of the sample goes from 320 to 160 to 80 to
40 to 20 at times of 0, 5, 10, 15, and 20 time units.
Some radioactive isotopes are used in medicine. For example, the
radioactive isotope thallium-201 can be injected into a patient’s bloodstream
where it is carried to the patient’s heart. A camera detects the radiation given
off from the decay of thallium-201 and produces an image of the heart
(Figure 4). Comparison of scans made during exercise and at rest may show
areas of the heart not receiving adequate blood flow. Figure 5 shows how the
activity level of an injection of thallium-201 decreases.
A line graph can be used to show a
trend over a period of time. Ask
yourself, “What information is
presented on the left side and along
the bottom of Figure 5? What has
happened to thallium-201 over a
period of time?”
Activity Level of Thallium-201 versus Time
120
Activity level (MBq)
100
80
60
40
20
0
0
1
2
3
Time (min)
4
5
Figure 5 Activity of thallium-201
Figure 4 A thallium scan of a normal heart
We can use the graph to determine the half-life of thallium-201. The
initial activity of 120 MBq is reduced to 60 MBq after 1.3 min. This means
that the half-life is 1.3 min. Note that the half-life is so short that most of the
thallium will decay quickly and not stay in the blood for much time.
As every half-life passes, the number of parent nuclei present and the
activity level decreases by half. To find out how many half-lives have passed,
divide the time by the half-life of the isotope. Table 1 shows how these
fractions can be expressed.
Table 1
Calculating Half-Lives Using Fractions
Number of
half-lives
1
2
3
4
5
Fraction
remaining
1
2
1
1
1
2
2
4
1
1
1
4
2
8
1
1
1
8
2
16
1
1
1
16
2
32
1
1
1 2
2
1
1
2 2
4
1
1
3 2
8
1
1
4 2
16
1
1
5 2
32
Exponential
notation
NEL
10.4 Half-Life
n
1
n
2
291
Another way to calculate the amount of the parent nuclei remaining is to
use percentages. The original amount is 100 %. Therefore, we can calculate
the amount left after every half-life by dividing the previous amount by two.
Table 2 shows the percentage left after the first five half-lives. This table can
be used for problems calculating the amount of parent nuclei left. Can you
determine the percentage that would be left after six half-lives?
Table 2
Calculating Half-Lives Using Percentages
Number of half-lives
Percent remaining
1
2
3
4
5
50 %
25 %
12.5 %
6.25 %
3.25 %
SAMPLE PROBLEM 1
Use Half-life to Determine the Time Passed
Cesium-124 has a half-life of 31 s. A sample of cesium-124 in a laboratory has an initial
mass of 20 mg.
(a) Calculate the amount of time it will take for the sample to decay to 5 mg.
(b) Calculate how much cesium-124 will remain after 93 s.
Solutions
Mass of Cesium-124
versus Time
Mass of cesium-124 (mg)
20
15
(a) First determine how many half-lives have passed. This can be done using either the
fraction or percentage method.
Fraction Method
Percentage Method
The fraction left is
5
1
1
2
20
4
2
The percent left is
5
100 % 25 %
20
By using either method, we can see that two half-lives have passed.
Now calculate the total amount of time that has passed.
10
Since two half-lives have passed, the total time that has passed will be 2 31 s 62 s.
Figure 6 shows a graph of the mass–time. From the graph, we can see that at about
62 s, the mass is reduced to 5 mg. This is in agreement with the calculated solution.
5
(b) Since the half-life of cesium-124 is 31 s, we can determine the number of half-lives:
0
0
20
40 60
Time (s)
80 100
Figure 6 Decay curve for cesium-124
total time
93 s
number of half-lives 3 half-lives
half-life
31 s
Now calculate the mass (m) remaining. This can be done using either the fraction or
percentage method.
Fraction Method
1
m 3 (20 mg)
2
Percentage Method
After three half-lives, there is 12.5 % remaining.
m 2.5 mg
12.5 %
m 20 mg
100 %
m 2.5 mg
The mass remaining is 2.5 mg. We can also see on the graph that the approximate mass
remaining is about 2.5 mg.
292
Unit C Radioactivity
NEL
Mass of Fluorine-18
versus Time
Practice
50
(a) Calculate the amount of time it will take for the initial mass of fluorine-18 to be
reduced from 50 mg to 12.5 mg. You can use the graph to confirm your answer.
40
(b) Calculate what mass of fluorine-18 remains after 5.4 h. You can use the graph to
confirm your answer.
Mass of fluorine-18 (mg)
A sample of fluorine-18 in a laboratory has an initial mass of 50 mg. Fluorine-18 has a
half-life of 1.8 h. Figure 7 shows the decay curve for fluorine-18.
30
20
10
0
0
1
2
3
4
5
6
Time (h)
Figure 7 Decay curve for fluorine-18
SAMPLE PROBLEM 2
Determine the Activity Level Using Half-life
Radium-226 has a half-life of 1600 years. A material containing radium-226 has an activity of
500 MBq.
(a) Determine what the activity level will be in the material after 8000 years.
(b) How many years earlier was the activity level in the material 2000 MBq?
Solutions
(a) First, determine the number of half-lives.
8000 years
years 5 half-lives
1600 half-life
Now determine the activity level.
1
activity 500 MBq 5 15.625 MBq
2
After 8000 years, the activity level will be 16 MBq.
(b) The activity level of 500 MBq is one-quarter the activity level of 2000 MBq.
1
1
2 This represents a period of two half-lives.
4
2
We can calculate the amount of time as
t 2 1600 years 3200 years
The activity level was 2000 MBq 3200 years earlier.
Practice
Silicon-32 has a half-life of 160 years. A material containing silicon-32 has an activity of
80 MBq.
(a) Determine what the activity level of the material will be after 320 years.
(b) How many years earlier was the activity level of the material 640 MBq?
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10.4 Half-Life
293
Decay Series
Table 3
Decay Series of Uranium-238
Decay
Half-life
238U
92
→
234Th
90
42He
4.5 109
years
10e
24 d
234Th
90
→
234Pa
91
234Pa
91
→
234U
92
10e
6.7 h
230Th
90
42He
2.5 105
years
234U
92
→
In biological families, a parent can have a daughter. After time passes, the
daughter becomes a parent and produces another generation. In a similar
way, a radioactive parent nucleus produces a daughter nucleus, which can
also be radioactively unstable. In turn, the daughter nucleus becomes a
parent nucleus, which continues the sequence of events. When radioactive
nuclei form such a chain, it is called a decay series. The decay series always
ends in the formation of a stable isotope.
For example, uranium-238 forms a decay series ending with the stable
isotope lead-206 as shown in Table 3. Note that some of the isotopes have
very short half-lives. However, uranium-238 has a half-life of about
4.5 billion years. The decay series of uranium-238 provides some isotopes
that would not otherwise be present on Earth. The decay series of
uranium-238 can be graphed as shown in Figure 8.
230Th
90
→
226Ra
88
42He
7.5 104
years
226Ra
88
→
222Rn
86
42He
1600 years
222Rn
86
→
218Po
84
42He
3.8 d
218Po
84
→
214Pb
82
42He
3.1 min
214Pb
82
→
214Bi
83
0
1e
27 min
10e
20 min
230
228
→
214Po
84
→
210Pb
82
42He
1.6 104 s
210Pb
82
→
210Bi
83
22 years
210Bi
83
210Po
84
→
→
210Po
84
206Pb
82
10e
42He
5d
138 d
238
236
Th-
234
234
232
214Po
84
0
1e
U-
238
Mass number
214Bi
83
Decay Series of Uranium-238
240
Pa- U234
234
Th230
Ra-
226
226
224
Rn-
222
222
220
Po-
218
216
Pb-
214
214
Pb-
210
210
208
Bi-
214
210
212
= decay
218
= decay
Po214
Bi- Po210
Pb-
206
206
204
81
82
83
84
85
86
87
88
Atomic number
89
90
91
92
93
Figure 8 Uranium-238 is changed into lead-206 in a decay series of 14 steps.
Radioactive Dating
Figure 9 The remains of a human were
found in glacial ice in the Alps. Scientists
used carbon-14 dating to determine
that he lived about 5300 years ago.
294
Unit C Radioactivity
Since radioactive isotopes decay according to their half-lives, it is possible to
date materials using appropriate isotopes. Carbon-14 is a radioactive isotope
that can be used to date material that was once alive (Figure 9). Almost all
naturally occurring carbon is carbon-12. However, an extremely small
fraction of carbon (about one atom in a trillion) is carbon-14. The half-life
of carbon-14 is 5730 years. With this half-life, there should be no carbon-14
NEL
14N
7
10n →
14C
6
11p
This process keeps the level of carbon-14 constant on Earth and in living
organisms.
When an organism dies, the amount of carbon-14 in the organism starts
to decrease as it radioactively decays, and no new carbon-14 enters the
organism through eating or respiration. Carbon-14 has a half-life of 5730
years, which means that the ratio of carbon-14 in an organism decreases by
half every 5730 years. Figure 10 shows the decay curve for carbon-14. It is
clear from the graph that carbon-14 can only be used to date objects less
than 40 000 years old. With a more accurate graph (or by calculation), the
useful time range can be extended to about 60 000 years. Note that
carbon-14 dating will only date things that were once alive. GO
Other isotopes can be used to date things that are more than 60 000 years
old or that were never alive. For example, uranium-235 decays to lead-207
with a half-life of 704 million years, and uranium-238 decays to lead-206
with a half-life of 4.46 billion years. Dating materials using two different
isotopes make the age estimates very accurate. Uranium-238 has been used
to determine that the oldest rocks that have been dated on Earth are about
4 billion years old.
Percentage of Carbon-14
Left versus Time
100
Amount of carbon-14 left (%)
left on Earth, which is about 4.5 billion years old. However, our Sun and all
the stars in the universe produce cosmic radiation. Energetic neutrons are
part of cosmic radiation, and the neutrons combine with nitrogen in the
upper atmosphere to form carbon-14 and a proton according to the
following nuclear equation:
80
60
40
20
0
0 10 20 30 40 50 60
Time (thousands of years)
Figure 10 Decay curve for carbon-14
To find out more about
carbon-14 dating go to
www.science.nelson.com
GO
To test your skills on half-life
and radioactive dating, go to
www.science.nelson.com
SAMPLE PROBLEM 3
Use Radioactive Dating to Determine the Age of a Sample
A piece of leather was found to have 12.5 % of its original carbon-14 present. Determine the
age of the leather using Figure 10 and by calculation.
Solution
From the graph, we can see that the time is approximately 17 000 years.
1
1
12.5 %
The decrease from 100 % to 12.5 % is a ratio of 3
8
2
100 %
years
Therefore, the time taken is 5730 3 half-lives 17 190 years
half-life
The piece of leather is approximately 17 200 years old.
Practice
A bone fragment was found to have 25 % of its original carbon-14 present. Determine the
age of the bone fragment using Figure 10 and by calculation.
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10.4 Half-Life
295
10.4
CHECK YOUR Understanding
1. How are the terms “activity” and “becquerel”
related?
4. A radioactive isotope source has a mass of
120 µg. If the isotope had a half-life of 20 s,
what would be the mass of the isotope after
2 min?
2. What is the activity level of the following
samples?
(a) 3600 decays in 42 s
(b) 35 decays in 35 min
(c) 45 000 decays in 7.5 min
(d) 1200 decays in 3 h
(e) 250 000 decays in 55 min and 17 s
Number of nitrogen-13 atoms (thousands)
3. Nitrogen-13 decays to produce carbon-13. A
laboratory sample contains 500 000 nitrogen-13
atoms. Use the decay curve for the sample over
time shown in Figure 11 to answer the following
questions.
Decay of Nitrogen-13 Atoms versus Time
500
5. Beryllium-7 has a half-life of 53 d. A sample
was observed for 1 min and there were
26 880 decays.
(a) What is the activity level of the sample?
(b) What will the activity level of the sample be
after 265 d?
(c) After how many days will the activity level
of the sample be 112 Bq?
(d) What was the activity level 106 d before the
sample was observed?
(e) How many days earlier was the activity level
eight times greater than the observed level?
6. A granite rock is thought to be about two
billion years old. Why is it not possible to
determine the age of the rock using carbon-14
dating?
400
7. A hair sample has 80 % of its original
carbon-14 present. What is the age of the
sample?
300
200
8. A bone fragment has lost 75 % of its original
carbon-14. What is the age of the bone
fragment?
100
0
0
5
10
15
20
25
Time (min)
30
35
40
9. An organic sample is 28 650 years old. What
percentage of the original carbon-14 is still
present in the sample?
Figure 11 Decay curve for nitrogen-13
(a) How many nitrogen-13 atoms will be left
after 16 min?
(b) How many carbon-13 atoms will be present
after 25 min?
(c) What is the half-life of nitrogen-13?
(d) How many nitrogen-13 atoms will be
present after 40 min?
296
Unit C Radioactivity
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