Radioactive Half-Life

A hospital clock that keeps perfect time

In hospitals across Canada, patients receive tiny amounts of a radioisotope called technetium-99m so doctors can take pictures of bones, the heart or other organs with a special camera. Doctors choose it partly because of its half-life of about 6 hours. Long enough to take the pictures, short enough that a day later only a small fraction of it is left.

No one can say when any one nucleus of technetium-99m will decay. But for billions of them, the pattern is as reliable as a clock: every 6 hours, half of whatever is left decays. That reliable pattern is called half-life, and it is used everywhere from cancer treatment to dating ancient campfires.

Random for one, predictable for many

Radioactive decay is random. A single unstable nucleus might decay in the next second or in a thousand years; there is no way to know. Nothing outside the nucleus (heat, pressure, chemistry) speeds it up or slows it down.

But every nucleus of a given radioisotope has the same chance of decaying in a given time. With a huge number of nuclei, that chance turns into a predictable rule: in a fixed amount of time, the same fraction decays.

The half-life is the time it takes for half of the radioactive nuclei in a sample to decay. After one half-life, ½ remain; after two, ¼; after three, ⅛; and so on. Notice that the amount never quite reaches zero in this model; it just keeps halving.

The activity of a sample (decays per second, measured in becquerels, Bq) is proportional to the number of radioactive nuclei left, so it also halves every half-life.

Calculating with half-life

If N₀ is the starting amount (number of nuclei, mass, or activity), T is the half-life and t is the time that has passed, then the amount left is:

N = N₀ (½)^(t/T)

The exponent t/T is the number of half-lives that have passed. When it is a whole number, you can just halve repeatedly. When it is not, use the y^x key on a scientific calculator. (You study exponential functions like this in MCR3U, and solving for t when it is in the exponent uses logarithms in MHF4U.)

Applications and consequences

Medicine. Radioisotopes for imaging inside the body are chosen with short half-lives (hours to days), so the patient's dose drops quickly. Sources for radiation therapy and sterilizing equipment, like cobalt-60, have half-lives of years, so they stay useful for a long time. Canada is a major supplier of cobalt-60, which is produced in some Ontario CANDU reactors.

Dating. Carbon-14 dates once-living material up to about 50 000 years old (after that, too little carbon-14 is left to measure). Isotopes with much longer half-lives, like uranium-238, are used to date rocks billions of years old.

Smoke alarms. Americium-241's half-life of 432 years means the source in a smoke alarm hardly weakens during the alarm's life.

Nuclear waste. Used reactor fuel contains many radioisotopes. Short-lived ones become much less radioactive within years, but some, such as plutonium-239, have half-lives of thousands of years. Waste containing them must be isolated from people and the environment for a very long time. This is one of the biggest challenges for nuclear power, and you will look at how Canada plans to manage it in Nuclear and Geothermal Technology and Society.

Accidents and spills. When radioactive material is released, short-half-life isotopes such as iodine-131 are the main hazard in the first weeks, while longer-lived isotopes can contaminate land for decades.