Some populations grow (Section 23.1), others are eaten away through use (Section 23.2), and some just decay away on their own. In fact, there are some “resources” that we want to become exhausted, such as nuclear waste.
“Decaying” resources can have enormous economic consequences, which could involve the following:
Radioactive materials are characterized by exponential decay, which we encountered in Section 21.5 (page 890) as geometric growth with a negative rate of growth, in connection with depreciation and inflation. Here, we see it in a different light, with a different way of measuring it. Unlike the purchasing power of the dollar, which declines at an irregular rate as inflation varies, a radioactive substance emits particles and decreases in quantity at a predictable continuous rate. The negative growth rate is usually written instead as , where the (positive) quantity is called the decay constant.
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Decay Constant DEFINITION
The decay constant for a substance decaying exponentially is the fraction of the substance that decays per unit of time.
The amount of radioactive substance remaining is given by the continuous interest formula of Section 21.4 (page 882): For an original amount , the amount remaining after time units is
However, the rate of decay of the substance is often described instead in terms of the half-life. A substance decaying geometrically never completely vanishes, even after millions of years. Since there is no time until it is all gone, we settle for measuring how long it takes until half of it is gone.
Half-life DEFINITION
The half-life of a substance decaying exponentially is the time that it takes for one-half of a quantity of the substance to decay.
Most chemical elements can occur in several isotopes (versions with the same atomic number but different atomic weights), some of which may be radioactive.
Isotope DEFINITION
An isotope of a chemical element is a form whose atomic nucleus contains the same number of protons (the atomic number for the element) as other forms but a different number of neutrons (giving it a different atomic weight).
For instance, iodine with atomic weight 131, iodine-131, is produced in nuclear reactors and nuclear explosions; it is a radioisotope (also called radionuclide), meaning that it is radioactive. There was great concern about the release of iodine-131 across Japan and the Pacific after the earthquake, tsunami, and subsequent nuclear reactor disasters in Japan in early 2011. Iodine in the atmosphere falls to the ground and gets into water supplies, milk, and other foods. Iodine is absorbed into the human body from food and concentrated in the thyroid gland; radioactive iodine can cause thyroid cancer.
Iodine-131 from the earlier reactor disaster at Chernobyl in Ukraine in 1986 has caused thousands of cancers over the years. Iodine-131 was also released from 1945 to 1963 as a result of atmospheric testing of nuclear weapons by the United States, the Soviet Union, and Great Britain, by France until 1974, and by China until 1980. (Humans need “good” iodine in their diet, iodine-127; but iodine-131 can replace it and cause cancer.) The half-life of iodine-131 is 8 days. This means that of 1 gram (g) of iodine-131 now, in 8 days only 0.5 g will remain (the rest will decay into nonradioactive xenon gas). Further, in days, only 0.25 g will remain; and in days, only 0.125 g (one-eighth of the original amount) will remain.
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We can determine the half-life from the decay constant. Let be the half-life, which means that the amount remaining of a quantity after time is . Hence, we have
To analyze this further and solve for , we need to take the natural logarithm of each side:
where we have made use of the general fact that . You can check on your calculator that ln .
Thus, we have the folllowing fundamental relationship.
Decay Constant and Half-Life Relationship RULE
EXAMPLE 5 Radioactivity in Your Home
The major radioactivity exposure for most Americans is from radon, which is formed from the decay of uranium. Radon-222 is a gas that enters many U.S. homes from the underlying soil and rock. It causes 10% to 15% of lung cancers in the United States (the rest are due to smoking). The half-life of radon-222 is 3.82 days. How long does it take for a quantity of radon-222 to decay to one-thousandth (0.1%) of the original amount?
For radon-222, we have days and want to determine the when :
Dividing both sides by gives
and taking the natural logarithm of each side gives
where the values and you get from your calculator. Solving for , we find
Would it take only one-tenth as long for the radon to decay to only 1% of the original amount?
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Carbon-14 Dating
Carbon-14 dating is a method of determining the age of organic materials, including mummies, charcoal from ancient fires, parchment, and cloth.
The element carbon, which is present in the food that we eat and in all living things, always has small traces of a radioactive form, called carbon-14. Plants and animals continually absorb carbon-14 during their lives, from the air (for plants) and from food (for animals), so that the concentration in their bodies stays the same while they are alive. Once they die, however, no new carbon-14 gets absorbed, and the carbon-14 already present decays.
Because we know the concentration of carbon-14 in living things, and we know how long it takes carbon-14 to decay, we can calculate from a sample how long ago a plant or animal was living.
The half-life of carbon-14 is 5730 years; its decay constant is
In other words, about 12 in 100,000 carbon-14 atoms decay each year. In each gram of carbon, approximately 814 carbon-14 atoms decay each hour. An approximate age of a sample can be determined by working backwards by half-lives. Suppose that a sample is decaying at 26 atoms per hour per gram of carbon. Table 23.1 shows that the 814 atoms per hour per gram of carbon would decrease to approximately 26 atoms per hour per gram of carbon in approximately 29,000 years, so that is the approximate age of the sample. (An age of 0 for the sample denotes the time of death of the living body.)
| Age in Half-Lives | Age in Years | Decays per Hour per Gram of Carbon |
|---|---|---|
| 0 | 0 | |
| 1 | 5,730 | |
| 2 | 11,460 | |
| 3 | 17,190 | |
| 4 | 22,920 | |
| 5 | 28,650 |
EXAMPLE 6 Carbon-14 Dating
How much of the original carbon-14 would be left after 50,000 years? (This is roughly the practical age limit for carbon-14 dating of the typically small samples available.) How many atoms would be decaying per hour per gram of carbon?
We have
so only about 0.24% of the original amount remains. This remaining amount would be decaying at a rate of atoms per hour per gram of carbon. An accurate estimate for the age of a sample of 1 milligram (mg) of carbon might take weeks or months of tallying counts on a decay counter. (The analysis is now often done by much faster atomic mass spectrometry rather than by counting decays, and calibration adjustments must be made for varying amounts of atmospheric carbon-14 over the years. Our calculations do not take such calibration, much of which is done from tree rings, into account.)
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For a 1-mg sample of carbon 50,000 years old, how many decays would you expect over a one-month period?
EXAMPLE 7 How Old?
We turn the question around to determine the age from the observed number of decays: How old is a sample that is decaying at a rate of 105 grams per hour per gram of carbon?
The formula that relates , the age of the sample in years, and , the number of carbon-14 atoms disintegrating per gram per hour, is
Solving for gives
Using natural logarithms, we can solve for as
Thus, a sample decaying at a rate of 105 atoms per hour per gram of carbon would be years old.
How much difference would it make if the decay rate was instead 110 atoms per hour per gram of carbon?
Although we are interested in the exhaustion and decay of nuclear waste, we are also interested in maintaining supplies of some radioactive materials:
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