Radioactivity is often used in determining how old something is; this is known as radioactive dating. When carbon-14 is used the process is called radiocarbon dating, but radioactive dating can involve other radioactive nuclei. The trick is to use a half-life which is of the order of, or somewhat smaller than, the age of the object.

Carbon-14 is used because all living things take up carbon from the atmosphere, so the proportion of carbon-14 in the carbon in a living organism is the same as the proportion in the carbon-14 in the carbon in the atmosphere. For many thousands of years this proportion has been about 1 atom of C-14 for every 8.3 x 10¹¹ atoms of carbon.

When an organism dies the carbon-14 slowly decays, so the proportion of C-14 is reduced over time. Carbon-14 has a half life of 5730 years, making it very useful for measuring ages of objects that are a few thousand to several tens of thousands of years old. To measure the age of something, then, you measure the activity of carbon-14, and compare it to the activity you'd expect it to have if it was brand new. Plugging these numbers into the decay equation along with the half-life, you can calculate the time period over which the nuclei decayed, which is the age of the object.

An example

A sample of wood has an activity of 0.22 Bq coming from carbon-14. An equivalent piece of wood cut from a growing tree would have an activity of 0.88 Bq from its carbon-14. How old is the sample?

N_o = 0.88 Bq       N = 0.22 Bq

l =

0.693 5730

= 1.21 x 10-4 yr-1

0.22 = 0.88 e^-lt

Divide both sides by 0.88: 0.25 = e^-lt

Take the natural log of both sides: ln(0.25) = -lt

The age of the sample is:

t = -ln(0.25)/l = 1.386/1.21 x 10^-4 = 11500 years.