Nuclear Decay Chain Calculator
This nuclear decay calculator finds how much of a radioactive sample remains after a given time, using the exponential decay law that governs every radioactive nuclide. Radioactive decay is a first-order process: the rate at which nuclei decay is proportional to how many remain, so the population falls off exponentially with a characteristic constant. That decay constant is the natural logarithm of two divided by the half-life, the time for half the sample to decay. The remaining amount is the starting amount multiplied by e raised to minus the decay constant times the elapsed time. These relations are the foundation of the nuclear data the US National Institute of Standards and Technology and national laboratories tabulate. Enter the initial number of atoms or activity, the half-life, and the elapsed time (in the same time unit as the half-life), and the calculator returns the decay constant, the number of half-lives elapsed, and the amount remaining. Use it to track an isotope in a sample, plan radioactive source handling, or check a physics problem. Every figure is computed deterministically from the exponential decay formula shown in full below, with a worked example that reconciles exactly to the calculator so you can follow each step yourself.
The decay law N = N0 e^(-lambda t) with lambda = ln2 / half-life gives the remainder: 1,000 atoms with an 8 day half-life after 24 days (three half-lives) leave 125.00. Each half-life halves the amount.
Radioactive decay formula
lambda = ln(2) / half-life
N = N0 x e ^ ( - lambda x t )
N0 = starting amount, N = amount remaining
t = elapsed time (same units as the half-life)
The decay constant lambda is the probability per unit time that a given nucleus decays. Because decay is proportional to the amount present, the population follows an exponential curve, and after each half-life exactly half of whatever remained is left.
Worked example
Start with 1,000 atoms of a nuclide with an 8 day half-life and find the amount after 24 days.
- lambda = ln(2) / 8 = 0.693147 / 8 = 0.086643 per day
- half-lives elapsed = 24 / 8 = 3
- N = 1,000 x e ^ (-0.086643 x 24) = 1,000 x e ^ (-2.079442)
- N = 1,000 x 0.125000 = 125.00
After 24 days, three half-lives, 125.00 atoms remain. These are the calculator's default inputs, so the result above matches the widget exactly.
Fraction remaining by half-lives
Each half-life leaves half of what was present, so the fraction halves at every step.
| Half-lives | Fraction remaining | Of 1,000 |
|---|---|---|
| 0 | 1.0000 | 1,000.00 |
| 1 | 0.5000 | 500.00 |
| 2 | 0.2500 | 250.00 |
| 3 | 0.1250 | 125.00 |
| 4 | 0.0625 | 62.50 |
Nuclear data and half-life reference: US National Institute of Standards and Technology (NIST).
Nuclear Decay Chain Calculator: frequently asked questions
What is the decay constant?
The decay constant lambda is the probability per unit time that any single nucleus will decay. It relates to the half-life by lambda equals the natural logarithm of two divided by the half-life. A larger decay constant means a shorter half-life and faster decay. It carries units of inverse time, matching whatever unit you use for the half-life and elapsed time.
Do the time units have to match?
Yes. The half-life and the elapsed time must be in the same unit, whether seconds, days or years, because the decay constant is computed from the half-life and then multiplied by the elapsed time. If your half-life is in days, enter the elapsed time in days too, or convert one of them first.
What does this report for a decay chain?
This calculator tracks the exponential decay of a single parent nuclide, the dominant step in many chains. A full chain where a daughter is itself radioactive requires the Bateman equations to follow the build-up and decay of each member. For the parent's remaining amount, the single exponential here is exact.
Can I enter activity instead of atom count?
Yes. The same exponential law governs activity, which is proportional to the number of atoms present. If you enter the starting activity, the result is the activity remaining after the elapsed time, in the same units. The fraction remaining is identical whether you track atoms or activity.
Why is the amount after three half-lives one eighth?
Each half-life multiplies the remaining amount by one half. After three half-lives the factor is one half cubed, which is one eighth, or 0.125. Starting from 1,000 that leaves 125, exactly as the exponential formula returns when the elapsed time equals three half-lives.
Official sources
- Atomic and nuclear data reference: US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.