Half-Life Calculator
Calculate radioactive decay, remaining amount, half-life, or elapsed time.
Note: Half-life (t½) and elapsed time (t) must use the same time units (e.g. seconds, years).
Remaining Amount
Additional Information
What is Half-Life?
The half-life (t½) of a radioactive isotope is the time required for half of the atoms in a sample to decay. After one half-life, 50% of the original material remains; after two half-lives, 25% remains; after three, 12.5%, and so on. Half-life is a constant for each isotope and is independent of the amount of material present, temperature, or pressure.
Radioactive decay is a first-order process, meaning the rate of decay is proportional to the number of undecayed atoms. This is why a fixed fraction (one half) decays in each half-life interval regardless of the starting quantity.
The Decay Formula
N = N₀ × (1/2)^(t / t½)
- • N = Remaining amount after time t
- • N₀ = Initial amount
- • t = Elapsed time
- • t½ = Half-life
Rearranging the formula lets you solve for any of the four quantities:
N = N₀ × 0.5^(t / t½)
N₀ = N / 0.5^(t / t½)
t½ = t × ln(0.5) / ln(N / N₀)
t = t½ × ln(N / N₀) / ln(0.5)
The decay constant is related to half-life by λ = ln(2) / t½, and the number of half-lives elapsed is simply t / t½.
Examples
Carbon-14 Dating
Carbon-14 has a half-life of about 5,730 years. If a sample originally contained 100% of its carbon-14 and now contains 25%, then two half-lives have elapsed (100% → 50% → 25%), meaning the sample is roughly 11,460 years old. This principle underlies radiocarbon dating of organic materials.
Medical Isotopes
Technetium-99m, widely used in medical imaging, has a half-life of about 6 hours. Starting from a 100 mCi dose, after 6 hours 50 mCi remains, and after 12 hours only 25 mCi remains. Short half-lives are desirable for diagnostic isotopes so that radiation exposure is minimized after the procedure.
Worked Example
With N₀ = 100, t½ = 10, and t = 20: the number of half-lives is t / t½ = 2, so N = 100 × 0.5² = 25 (25% remaining). The decay constant is λ = ln(2) / 10 ≈ 0.0693.
Note: This calculator assumes ideal first-order radioactive decay of a single isotope and the same time units for half-life and elapsed time. It does not account for decay chains, branching ratios, or daughter-product ingrowth. For precise nuclear or analytical work, consult specialized references.