Half-Life Calculator
Calculate the half-life, elapsed time, initial quantity, or remaining quantity of a decaying substance. Pick what you want to solve for, then enter the known values. The decay rate can be given as a half-life, a decay constant (λ), or a mean lifetime (τ) — the calculator also reports the other two with every result.
What Is Half-Life?
Half-life is the time needed to reduce the quantity of a substance to half its initial amount. It is commonly used to describe the decay of radioactive material, where the half-life is the time needed for half the atoms of an isotope to decay.
After one half-life, half of the original amount remains; after two, a quarter; after three, an eighth, and so on. The remaining quantity follows exponential decay.
Decay Constant and Mean Lifetime
The half-life t½ is related to the decay constant λ — the probability that an isotope decays per unit time — by λ = ln(2) ÷ t½. The longer the half-life, the smaller the decay constant.
The mean lifetime τ is the average time an isotope exists before it decays, given by τ = t½ ÷ ln(2), which also equals 1 ÷ λ. All three describe the same decay rate, so entering any one of them is enough.
Calculating Elapsed Time (Carbon Dating)
Rearranging the half-life formula lets you solve for elapsed time by comparing the remaining quantity to the original: t = t½ × ln(N₀ ÷ Nₜ) ÷ ln(2).
For example, a sample with 15% of its original carbon-14 (t½ = 5730 years) gives t = 5730 × ln(100 ÷ 15) ÷ ln(2) ≈ 15,680 years old.
Formula
Nₜ = N₀ × (½) ^ (t ÷ t½)
where N₀ = initial quantity, Nₜ = remaining quantity, t = elapsed time, t½ = half-life
Example: Carbon-14 (t½ = 5730 yr): amount left after 11460 yr from 100 g:
Nₜ = 100 × (½) ^ (11460 ÷ 5730) = 100 × (½)² = 25 g
Half-Life Decay Table
The table below shows the remaining quantity after exponential decay over elapsed half-life time periods.
| Number of Half-Lives | Remaining (fraction) | Remaining (decimal) |
|---|---|---|
| 0 | 1 | 1.0 |
| 1 | 1/2 | 0.5 |
| 2 | 1/4 | 0.25 |
| 3 | 1/8 | 0.125 |
| 4 | 1/16 | 0.0625 |
| 5 | 1/32 | 0.03125 |
| 6 | 1/64 | 0.015625 |
| 7 | 1/128 | 0.0078125 |
| 8 | 1/256 | 0.00390625 |
| 9 | 1/512 | 0.001953125 |
| 10 | 1/1024 | 0.0009765625 |