The half-life formulas
- N = N₀ (½)t / t½, where n = t / t½ is the number of half-lives.
- t½ = 0.693 / k for any first-order process, including radioactive decay. k is the decay constant (λ in physics).
- t = (2.303 / k) log(N₀ / N), the integrated first-order rate law from chemical kinetics.
Quick check: after 1 half-life 50% is left, after 2 it's 25%, after 3 it's 12.5%, and after 10 about 0.1%. If the amount left is a neat fraction like ⅛, count the halvings instead of using logs.
Where it comes up
In physics, radioactive decay (Nuclei, Class 12). In chemistry, first-order reactions (Chemical Kinetics, Class 12). The maths is the same. The 2.303 comes from converting ln to log, so keep the log table handy for practice.
Questions students ask
How much is left after 3 half-lives?
One eighth, or 12.5%, because ½ × ½ × ½ = ⅛.
What is the relation between half-life and the decay constant?
t½ = ln 2 / λ = 0.693 / λ. The mean life is 1/λ, which is 1.44 times the half-life.
Does half-life depend on the starting amount?
Not for first-order reactions or radioactive decay. For a zero-order reaction, t½ = [A]₀ / 2k, so it does.
How long does it take for 99% to decay?
About 6.64 half-lives, since (½)ⁿ = 0.01 gives n = log 100 / log 2 = 6.64.