Hong Life CalculatorsUnderstanding the radioactive decay calculator — concept and principle
Radioactive nuclei decay at random, but a large sample follows a precise rule: in every half-life, half of what remains decays. This calculator gives the amount left after a chosen time, the fraction remaining, the decay constant and the mean life. It is used for nuclear medicine doses, storage of radioactive waste, radiometric dating and physics homework.
How it is calculated
- N = N₀ × (½)^(t / t½)
- Fraction remaining = (½)^(t / t½) × 100%
- Decay constant λ = ln 2 / t½
- Mean life τ = t½ / ln 2 = 1/λ
N₀ can be in any unit that is proportional to the number of nuclei: atoms, grams, becquerels or counts per minute. The half-life and elapsed time must use the same time unit; λ is then per that unit and τ is in that unit. The default half-life, 5730, is that of carbon-14 in years.
The principle
Each nucleus has the same chance λ of decaying per unit time, independent of its age and of its neighbours. The number that decays in a short interval is therefore proportional to how many are left: dN/dt = −λN. The solution is N = N₀e^(−λt), and setting N = N₀/2 gives t½ = ln 2/λ. Writing the exponential with base ½ gives the half-life form used here. The mean life τ is the average time a nucleus survives; it is about 1.443 times the half-life.
Worked example
- Inputs: N₀ = 100, t½ = 5730, t = 10000 (same units, years).
- t / t½ = 10000 ÷ 5730 = 1.74520.
- (½)^1.74520 = 0.29829, so N = 100 × 0.29829 = 29.83 and the fraction remaining is 29.83%.
- λ = 0.693147 ÷ 5730 = 0.00012097 per year.
- τ = 5730 ÷ 0.693147 = 8266.64 years.
| Elapsed time (same units) | Remaining |
|---|---|
| 2,500 | 73.90274338 |
| 5,000 | 54.61615479 |
| 7,500 | 40.36283672 |
| 10,000 | 29.82924364 |
| 12,500 | 22.04462938 |
| 15,000 | 16.29158588 |
| 20,000 | 8.897837763 |
How to read the result
After 1, 2, 3, … half-lives, 50%, 25%, 12.5%, … remain; after 10 half-lives about 0.1% remains. Activity in becquerels follows the same curve, because activity equals λN. A short half-life means intense but brief radioactivity; a long one means weak but persistent radioactivity.
Common mistakes
- Mixing time units, such as a half-life in days with elapsed time in hours.
- Confusing half-life with mean life (they differ by a factor of ln 2).
- Forgetting that decay products may themselves be radioactive, so total activity can behave differently.
- Applying the law to very small numbers of atoms, where random fluctuations dominate.
Frequently asked questions
How long until only 1% remains?
Solve (½)^(t/t½) = 0.01: t = t½ × log₂ 100 ≈ 6.64 half-lives.
Can temperature or pressure change a half-life?
For almost all decay modes, no measurable effect occurs under ordinary conditions; electron-capture decays can shift very slightly with chemical environment.
How do I find a half-life from two measurements?
t½ = t × ln 2 ÷ ln(N₀/N), where t is the time between the measurements.
Sources: IAEA Nuclear Data Services, Live Chart of Nuclides · K. S. Krane, Introductory Nuclear Physics, Wiley