One of the interesting uses for half-life calculations involves radiocarbon
dating, where the content of carbon-14 in organic (formally living matter) is
used to calculate the age of a sample. The process begins in the upper
atmosphere, where nitrogen is bombarded constantly by high-energy neutrons
from the sun. Occasionally, one of these neutrons collides with a nitrogen
nucleus and the isotope that is formed undergoes the following nuclear
equation:
$$
{ }_{0}^{1} n+{ }_{7}^{14} N \rightarrow{ }_{1}^{1} \rho+{ }_{6}^{14} C
$$
Plants take up atmospheric carbon dioxide by photosynthesis, and are ingested
by animals, so every living thing is constantly exchanging carbon-14 with its
environment as long as it lives. Once it dies, however, this exchange stops,
and the amount of carbon-14 gradually decreases through radioactive decay with
a half-life of about 5,730 years, following the nuclear equation shown below:
$$
{ }_{6}^{14} C \rightarrow_{-1}^{0} \beta+{ }_{7}^{14} N
$$
Thus, by measuring the carbon-14/carbon-12 ratio in a sample and comparing it
to the ratio observed in living things, the number of half-lives that have
passed since new carbon-14 was absorbed by the object can be calculated.