Around the year 1900 Ernest Rutherford (a key figure in early
investigations into atomic structure) found that his sources of
radioactivity were losing their potency. Over time, they began
to emit less and less radiation. He decided to
measure this decrease in radiation and discovered a simple law of
exponential decay to describe the decrease. One way to write this law is in terms of something called the half-life of the decay.
Individual unstable atoms decay completely randomly. It is impossible to predict when a single unstable atom will decay. The radioactive decay of an atom cannot be hurried up or slowed down in any way. Even so, it is possible to describe the behavior of large numbers of atoms using a simple idea. Each unstable isotope has a characteristic half-life which is the amount of time that has passed when half of a group of unstable atoms has decayed. The symbol used for half-life is t_{1/2}. Say there is an unstable isotope with t_{1/2} = 1 hour. If you have a sample of 100 grams of the unstable isotope then after one hour, 50 g would have decayed and 50 g would remain. The fraction remaining is ^{1}/_{2} (0.5 or 50%). After two hours the fraction remaining would be ^{1}/_{4} (0.25 or 25%) and the amount decayed would be 75 g. After three hours, the fraction remaining would be ^{1}/_{8} (0.125 or 12.5%). Four hours: ^{1}/_{16} (0.0625 or 6.25%), etc.
The fraction of remaining un-decayed atoms can be calculated using a simple equation:
R = (1/2)^{n}
R is the fraction of the material that remains
after some amount of time has passed. |
n is the number of half-lives that have passed.
To use this equation just plug the number of half-lives in for n. The expression (1/2)^{n} can be understood to mean 1/2^{n} so for n = 1, R = 1/2 and for n = 2, R = 1/4, and so on.
If you know how much time has passed but you don’t know how many half-lives that is equal to you can still use the equation above. To do so, just use the following simple expression that relates the total time that has passed to the number of half-lives:
t_{total} = nt_{1/2}
t_{total} is the total time |
n is the number of half-lives that have passed |
t_{1/2} is the length of the half-life
When using this equation you may find that sometimes the value for n will not be a whole number of half-lives. If the half-life is 1 hour and 3.7 hours have passed then n = 3.7/1 or 3.7 half-lives. You can still use this in the previous equation as long as you have a calculator to do the math. R = (1/2)^{3.7} = 0.077. This makes sense because it is between the fraction remaining for three half-lives (0.125) and the fraction remaining for four half-lives (0.0625).
The characteristic decay rates of many isotopes have been measured. This information is useful in cases such as when radioactive isotopes are used in medicine. For example, Iodine-131 (^{131}_{53}I) is a beta-emitter and is used to diagnose disorders of the thyroid. The thyroid is an organ of the endocrine system and secretes hormones important for the regulation of appetite and metabolism. The element iodine is concentrated in the thyroid because it uses the element to build enzymes important in performing its bodily functions. When a patient ingests a medicine containing iodine-131 the material concentrates in the thyroid. By using a machine to record the radiation from the thyroid (found at the base of the front of the neck) or even to make an image in the way x-rays images are made, doctors can make a diagnosis. Iodine-131 has a half-life of 8.02 days. When doctors need to use it for a diagnosis it would do little or no good to use iodine-131 that had decayed so much that the fraction remaining was too small to produce enough radiation to be detected. Using the equations above it is possible to determine how long it will be until the material is useless as a medicine.
Radioactive half-lives are also useful for figuring out how old something is by
measuring the fraction remaining. For example, carbon-14
(^{14 }_{6}C
) is created in the atmosphere by the action of cosmic neutrons
on nitrogen nuclei. Living things take up this carbon-14 and make
it part of themselves. It is a beta-emitter and is constantly decaying back into nitrogen-14. While an animal or plant is alive, the carbon-14 in the organism’s tissues is constantly renewed as it eats or converts carbon dioxide into sugars. When the organism dies, the carbon-14 decays at its characteristic rate without any more being added. The half-life of
carbon-14 is 5,730 years. It is possible to measure the amount of
carbon-14 in an object and compare it to how much carbon-14 there
must have been to begin with. This data can be used to find the fraction of the original amount of carbon-14. You can use
the formula given above to find the number of half-lives which
have passed since the animal or plant stopped taking in fresh
carbon from the environment (that is, since it died). For example let’ say R is reported as 0.0625 for an artifact collected from an ancient tomb. When R = 0.0625 (which equals ^{1}/_{16}) n = 4 because (^{1}/_{2})^{n} = ^{1}/_{16} when n = 4. If four half-lives have passed for carbon-14 then 4(5,715 yr) = 22,860 yr have passed.
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Information about Several Radioactive Isotopes
Nuclide
Half-life
Decay Mode
Nuclide
Half-life
Decay Mode
^{14 }_{ 6}C
5,715 yr
β^{-}
^{40}_{19}K
1.248 × 10^{9} yr
β^{-}
^{90}_{38}Sr
28.9 yr
β^{-}
^{238}_{92}U
4.468 × 10^{9} yr
α
^{239}_{94}Pu
24,110 yr
α
^{26}_{13}Al
7.17 × 10^{5} yr
EC
Write a balanced nuclear equation for
the decay of each of the isotopes in the table given above.
Define the concept of radioactive half-life.
If you have just one atom of a radioactive isotope is it possible to use the half-life of the isotope to predict when it will decay? Why or why not?
Say you have a sample of 1,000 radioactive atoms. Why does the amount of radioactivity, measured as decays per minute, decrease as time passes?
Fill in the following table using the definition of half-life and the fact that the half-life of carbon-14 is 5,715 years.
n
the # of t_{1/2}
R
(as a fraction)
R
(as a decimal)
R
(as a percent)
Years
0
1
1
100%
0
1
1/2
0.5
50%
5,715
2
3
4
5
6
7
8
Make a graph on paper or using MS Excel or a similar graphing utility for the data above about carbon-14. Place R (as a decimal) on the y-axis and place n (the number of half-lives) on the x-axis, evenly spaced. Title it “Radioactive Decay of Carbon-14 by Number of Half-lives”
There is a sample set of graphs for Carbon-14 available in Excel format. There is also a graph that shows the decay of any radioactive isotope here.
According to your graph, how much carbon-14 remains after 22,860 years?
At what time do only 3.125% of the original carbon-14 atoms remain?
How much of the original carbon-14 in a sample remains after 3 half-lives? How much time must pass before this happens?
Consider the following data and the graph constructed from that data:
Magnesium-28 is a beta emitter. Write a balanced nuclear decay equation.
What is the half-life of magnesium-28?
How much magnesium-28 remains undecayed after 50 hours?
How much of the magnesium-28 has decayed at 70 hours?
At what time will the rate detected by a Geiger counter (in counts per minute) from magnesium-28 atoms be greater: at the start or after 63 hours? Why?
Now consider the isotope hydrogen-3, also called tritium. It is also a beta-emitter and has a half-life of 12.3 years. Hydrogen-3 has one proton and two neutrons and is a by-product of the nuclear power industry. It is also the radioactive material at the heart of the “always visible technology” used in Luminox™ watches.
Radioactive Decay of Hydrogen-3
Number of
Half-lives (n)
Fraction Remaining (R)
as a decimal
Time (t_{total}) (years)
0
1
0
1
0.5
12.3
2
0.25
24.6
Fill in the following data table and make a graph similar to the graph above of the radioactive decay of magnesium-28. Place the Fraction Remaining (R) on the y-axis and place time, in years, on the x-axis. Title the graph “Radioactive Decay of Hydrogen-3”.
Hydrogen-3 is a beta emitter. Write a balanced nuclear decay equation.
How much hydrogen-3 remains undecayed after 24.6 years?
How much of the hydrogen-3 is still un-decayed at 55 years?
At what time will more radiation be detected from hydrogen-3 atoms by a Geiger counter: at the start or after 50 years? Why?
Answer the following questions using the graphs, data tables, and formulas found in the introduction to this activity and in your work up to this point. Some examples have been provided. For all problems you need only identify the variables that are given (R, n, or t_{total}), plug them into the two equations you have and solve for a missing variable. Half-lives are given in the table at the top of the first page of exercises.
If you are given a number of half-lives (n) you can use that information to find the amount of time that has passed (t_{total}) as long as you know the length of a half-life (t_{1/2}).
Say n = 4 and t_{1/2} = 25 years.
t_{total} = nt_{1/2} so t_{total} = (4)(25 years) = 100 years
If you are given a number of half-lives you can use that information to find R.
Say n = 3.
R = (1/2)^{n} so R = (1/2)^{3} = 1/8 = 0.125
See the table you filled out on the first page of exercises for values of R that belong with the different values of n. You have already calculated them out to n = 8
If you are given the fraction remaining (R) you can figure out the number of half-lives and the total time (t_{total}).
Say R = 0.0625
According to your table, when R = 0.0625, n = 4
See the first example for how to find t_{total} for a given value of n. Just substitute in the value of t_{1/2} for the isotope you’re working on.
If you are given a value for t_{total} you can find the number of half-lives (n) and the fraction remaining (R).
Say t_{total} = 11,430 years and the isotope is carbon-14.
t_{1/2} = 5,715 years.
t_{total} = nt_{1/2} can be solved for n: n = t_{total}/t_{1/2}
In this example n = (11,430 years)/(5,715 years) = 2
And for two half-lives, R = (1/2)^{n} = (1/2)^{2} = 1/4
What percent of uranium-238 atoms remain un-decayed after 4 half-lives? How much time must pass before the radioactivity from a sample of uranium-238 drops to this level? You may use the data table and graph you made for carbon-14 to answer this and later questions because the graph was made in terms of half-lives and looks the same no matter how much time is represented by one half-life.
How many half-lives aluminum-26 have passed if the fraction remaining is 3.125%? How much time does it take for an original sample of this isotope to decay to this level?
A team of anthropologists uncovers the site of an ancient fire pit inside a cave that was inhabited by Neandertals (an ancient race of people closely related to modern humans). When they analyze the carbon-14 content of the charred wood they find that approximately seven half-lives have passed since the wood was a growing tree. How old is the wood in the fire pit?
Let’s say doctors require a sample of iodine-131 to have at least 12.5% of its original potency as an emitter of beta-rays in order to use it for thyroid diagnosis. If iodine-131 has a half-life of 8.02 days then how many days does a fresh sample of the isotope last?
Radioactive iodine-131 also has a darker side. After the 1986 accident at the Ukrainian Chernobyl nuclear power plant many people were exposed to very high levels of iodine-131. It was released from the power plant in a large plume that spread across the region. The level of exposure was much greater than the tiny doses used by doctors to diagnose illness. To protect against exposure, tablets of non-radioactive potassium iodide were distributed to large numbers of people. As long as a person has extra iodine in their bodies the radioactive iodine will not stay in their bodies long enough to do harm. How many days would someone have to continue taking supplemental iodine if it is only safe to do so when the radioactivity has dropped to less than 5%? Citation: http://en.wikipedia.org/wiki/Potassium_iodide#Thyroid_protection_due_to_nuclear_accidents_and_emergencies or http://en.wikipedia.org/wiki/Potassium_iodide
Strontium-90 has a half-life of 28.9 years. It is produced in relatively large amounts by the fission of uranium-235 as when a nuclear bomb is exploded. Strontium-90 is a particularly dangerous radioactive isotope for two reasons. First, it has a high activity level due to its relatively short half-life. Second, it is chemically similar to calcium (they are both in Group 2 on the Periodic Table) and when accidentally ingested it can be incorporated into bones. When radioactivity is inside the body it can do much more damage than when it is outside the body. For this reason, the site of nuclear of explosions are dangerous to occupy for many years afterward. How many years would it take for the radioactivity of strontium-90 to reach less than 1% of its original amount?
Plutonium-239 is produced by the bombardment of uranium-238 atoms with neutrons. The resulting uranium-239 atoms decay by two beta-decays to become plutonium-239. Since plutonium-239 can undergo fission and produce energy it is used as both a nuclear fuel and as material for making nuclear bombs. It has the smallest critical mass of all fissionable isotopes Citation: http://en.wikipedia.org/wiki/Plutonium-239 at only 11 kg. This makes it a desirable material for nations wishing to control a nuclear weapon. How many years would 44 kg of plutonium-239 need to be safely stored to prevent the material from being used to make a nuclear weapon requiring 11 kg?
Approximately how many half-lives of potassium-40 have passed since the formation of the Earth 4.5 × 10^{9} years ago? How much of the original amount of potassium-40 that was in the Earth at the time of its formation still remains? This is the naturally occurring isotope of potassium that makes potassium-rich bananas slightly (very slightly!) more radioactive than some other fruits. (Don’t worry, eating a banana is not dangerous due to the fact that the amount of radiation is actually extremely small).
In 1988, three teams of scientists found
that the Shroud of Turin, which was reputed to be the burial
cloth of Jesus, contained 91% of the amount of Carbon-14 contained in
freshly made cloth of the same material. This means that only 0.136 half-lives had passed. How long ago was the Shroud made according to the data?
See also the more advanced activity I have written which uses logarithms and the correct exponential decay equation: Half-lives Activity
Last updated:
Jan 04, 2019
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