The Clock of the Elements
Half-life ($t_{1/2}$) is the time required for a quantity to reduce to half of its initial value. The term is most commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay, but it is also used in biology (how fast a drug leaves your system) and finance.
The remarkable thing about half-life is that it is constant over time.
- After 1 half-life, 50% of the material remains.
- After 2 half-lives, 25% remains.
- After 3 half-lives, 12.5% remains.
The Relationship to Decay
The half-life of a substance is inversely proportional to its Decay Constant ($\lambda$). A large decay constant means a short half-life (it decays rapidly). The mathematical link between the two is the natural logarithm of 2 ($\approx 0.693$).
The Formula
Example Calculation
You are studying an unknown isotope. You determine its decay constant is $0.05 , \text{s}^{-1}$.
- Natural Log of 2: $\ln(2) \approx 0.693$.
- Divide by Decay Constant: $0.693 / 0.05 = 13.86 , \text{seconds}$.
The half-life of the isotope is roughly $13.9 , \text{seconds}$. Every $13.9$ seconds, exactly half of the remaining substance will vanish.