14.3: Radioactivity and Half-Life

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Learning Objectives

Define half-life.
Determine the amount of radioactive substance remaining after a given number of half-lives.
Describe common radiometric carbon-14 dating technique.

Whether or not a given isotope is radioactive is a characteristic of that particular isotope. Some isotopes are stable indefinitely, while others are radioactive and decay through a characteristic form of emission. As time passes, less and less of the radioactive isotope will be present, and the level of radioactivity decreases. An interesting and useful aspect of radioactive decay is half life (t_1/2), which is the amount of time it takes for one-half of a radioactive isotope to decay. The half-life of a specific radioactive isotope is constant; it is unaffected by conditions and is independent of the initial amount of that isotope.

Each radioactive nuclide has a characteristic, constant half-life (t_1/2), the time required for half of the atoms in a sample to decay. An isotope’s half-life allows us to determine how long a sample of a useful isotope will be available, and how long a sample of an undesirable or dangerous isotope must be stored before it decays to a low-enough radiation level that is no longer a problem.

For example, cobalt-60 source, since half of the \(\ce{^{60}_{27}Co}\) nuclei decay every 5.27 years, both the amount of material and the intensity of the radiation emitted is cut in half every 5.27 years. (Note that for a given substance, the intensity of radiation that it produces is directly proportional to the rate of decay of the substance and the amount of the substance.) Thus, a cobalt-60 source that is used for cancer treatment must be replaced regularly to continue to be effective.

A graph, titled “C o dash 60 Decay,” is shown where the x-axis is labeled “C o dash 60 remaining, open parenthesis, percent sign, close parenthesis” and has values of 0 to 100 in increments of 25. The y-axis is labeled “Number of half dash lives” and has values of 0 to 5 in increments of 1. The first point, at “0, 100” has a circle filled with tiny dots drawn near it labeled “10 g.” The second point, at “1, 50” has a smaller circle filled with tiny dots drawn near it labeled “5 g.” The third point, at “2, 25” has a small circle filled with tiny dots drawn near it labeled “2.5 g.” The fourth point, at “3, 12.5” has a very small circle filled with tiny dots drawn near it labeled “1.25 g.” The last point, at “4, 6.35” has a tiny circle filled with tiny dots drawn near it labeled.”625 g.” — Figure \(\PageIndex{1}\): For cobalt-60, which has a half-life of 5.27 years, 50% remains after 5.27 years (one half-life), 25% remains after 10.54 years (two half-lives), 12.5% remains after 15.81 years (three half-lives), and so on. (CC BY 4.0; OpenStax)

We can determine the amount of a radioactive isotope remaining after a given number half-lives by using the following expression:

\[\text{amount remaining} = \text{initial amount} \times \left ( \frac{1}{2} \right )^{n}\]

where \(n\) is the number of half-lives. This expression works even if the number of half-lives is not a whole number.

Example \(\PageIndex{1}\): Fluorine-20

The half-life of fluorine-20 is 11.0 s. If a sample initially contains 5.00 g of fluorine-20, how much remains after 44.0 s?

Solution

If we compare the time that has passed to the isotope's half-life, we note that 44.0 s is exactly 4 half-lives, so using the previous expression, n = 4. Substituting and solving results in the following:

\[\begin{align*} \text{amount remaining} &= 5.00\,g \times \left ( \frac{1}{2} \right )^{4} \\[4pt] & =\: 5.00\,g\times \left ( \frac{1}{16} \right ) \\[4pt] &= 0.313\,g \end{align*}\]

Less than one-third of a gram of fluorine-20 remains.

Exercise \(\PageIndex{1}\): Titanium-44

The half-life of titanium-44 is 60.0 y. A sample of titanium contains 0.600 g of titanium-44. How much remains after 240.0 y?

Answer: 0.0375 g

Half-lives of isotopes range from fractions of a microsecond to billions of years. Table \(\PageIndex{1}\) - Half-Lives of Various Isotopes, lists the half-lives of some isotopes.

Table \(\PageIndex{1}\) Half-Lives of Various Isotopes
Isotope	Half-Life
³H	12.3 y
¹⁴C	5730 y
⁴⁰K	1.26 × 10⁹ y
⁵¹Cr	27.70 d
⁹⁰Sr	29.1 y
¹³¹I	8.04 d
²²²Rn	3.823 d
²³⁵U	7.04 × 10⁸ y
²³⁸U	4.47 × 10⁹ y
²⁴¹Am	432.7 y
²⁴⁸Bk	23.7 h
²⁶⁰Sg	4 ms

Chemistry Is Everywhere: Radioactive Elements in the Body

You may not think of yourself as radioactive, but you are. A small portion of certain elements in the human body are radioactive and constantly undergo decay. Most of the radioactivity in the human body comes from potassium-40 and carbon-14. Potassium and carbon are two elements that we absolutely cannot live without, so unless we can remove all the radioactive isotopes of these elements, there is no way to escape at least some radioactivity. There is debate about which radioactive element is more problematic. There is more potassium-40 in the body than carbon-14, and it has a much longer half-life. Potassium-40 also decays with about 10 times more energy than carbon-14, making each decay potentially more problematic. However, carbon is the element that makes up the backbone of most living molecules, making carbon-14 more likely to be present around important molecules, such as proteins and DNA molecules. Most experts agree that while it is foolhardy to expect absolutely no exposure to radioactivity, we can and should minimize exposure to excess radioactivity.

Key Takeaways

Natural radioactive processes are characterized by a half-life, the time it takes for half of the material to decay radioactively.
The amount of material left over after a certain number of half-lives can be easily calculated.