Half-life is defined as the amount of time it takes a given quantity to decrease to half of its initial value. The term is most commonly used in relation to atoms undergoing radioactive decay, but can be used to describe other types of decay, whether exponential or not. One of the most well-known applications of half-life is carbon-14 dating. The half-life of carbon-14 is approximately 5,730 years, and it can be reliably used to measure dates up to around 50,000 years ago. The process of carbon-14 dating was developed by William Libby, and is based on the fact that carbon-14 is constantly being made in the atmosphere. It is incorporated into plants through photosynthesis, and then into animals when they consume plants. The carbon-14 undergoes radioactive decay once the plant or animal dies, and measuring the amount of carbon-14 in a sample conveys information about when the plant or animal died.
If an archaeologist found a fossil sample that contained 25% carbon-14 in comparison to a living sample, the time of the fossil sample's death could be determined by rearranging equation 1, since Nt, N₀, and t₁/₂ are known.
For example, if the fossil sample contains 25% carbon-14, you can calculate its age:
t = (t₁/₂) * ln(2) * ln(N₀ / Nt)
t = (5,730 years) * ln(2) * ln(1 / 0.25) ≈ 11,460 years old.
A relationship exists between the half-life (t₁/₂), mean lifetime (τ), and decay constant (λ):
τ = 1 / λ
This relationship allows you to determine any of these values if you know the other two. It's a fundamental aspect of understanding decay processes and is crucial in fields such as nuclear physics and radiocarbon dating.