Radioactive Decay Calculator

Enter initial amount and half-life to visualise how a radioactive sample decays over time.

Starting values — Radioactive Decay Calculator

Half-life t½	5730
Initial atoms N	0
Time span (half-lives)	5
Log	off

Change any control above and the results recompute from these.

Loading interactive simulation...

U-238 takes 14 steps to reach lead 🖖

Real isotopes rarely decay directly to something stable — U-238 passes through about 14 sequential decays (through thorium, radium, radon, polonium and more) before finally reaching stable Pb-206, with half-lives ranging from billions of years down to microseconds. When an intermediate step is much shorter-lived than its parent, the chain reaches secular equilibrium: every daughter's activity locks onto the parent's decay rate, which is why a single exponential like the one graphed here still accurately describes the parent's decline even though the full chain has over a dozen steps.

Random atoms, predictable crowds 🖖

Each unstable nucleus decays at a moment no one can predict — there is no countdown ticking inside it, only a fixed probability per second. Yet with trillions of atoms, that randomness averages into the smooth curve you see here. The half-life is simply the time for a sample to lose half its atoms, so after 3 half-lives only (1/2)³ = 1/8 remains, no matter how much you started with.

The 'constant' that isn't quite constant 🖖

Textbooks say decay rates ignore temperature, pressure and chemistry — and for alpha and beta decay that's almost perfectly true. But isotopes that decay by electron capture, like beryllium-7, borrow an electron from their own atom, so their rate depends on the electron density at the nucleus. Chemists have shifted Be-7's half-life (~53 days) by around 1% just by changing its chemical compound or squeezing it under pressure.

Problem solved in full

A Carbon-14 sample sitting for five 5730 years half-life 5 steps
Carbon-14 has a half-life of 5730 years; a sample has been sitting for five of them. Find what is left, and find where this smooth exponential stops describing anything real.
1. The decay constant is not the half-life — it is the probability per year that any one nucleus decays, and the two are related by ln 2 because that is the factor a half-life removes.
  
  $λ = \frac{ln 2}{t _{1/2}} = \frac{0.693147}{5730} = 1.21 \times 1 0^{- 4} y r^{- 1}$
2. Five half-lives is five halvings. The fraction left is a power of a half, and the exponential form gives the same number.
  
  $\frac{N}{N _{0}} = (\frac{1}{2})^{5} = 0.03125 = 3.125%$
3. From a thousand nuclei, 31.25 remain. And there is the problem: a quarter of a nucleus does not exist, and the smooth curve has quietly stopped being a description of the sample.
  
  $N = 1000 \times 0.03125 = 31.25$
4. What actually remains is a random number with a mean near 31. Decay is a Poisson process, so the spread is the square root of the count — about ±18% on a single measurement, and there is no averaging it away without more sample.
  
  $σ = 31 = 5.6 ⟹ \pm 18% on a single count$
5. The elapsed time is five half-lives, or 28,650 years. Double that and only one nucleus in a thousand is left, which is why radiocarbon dating runs out around fifty thousand years — not because the physics changes, but because the signal falls below what can be distinguished from contamination.
  
  $t = 5 \times 5730 = 28650 yr; (\frac{1}{2})^{10} = 0.098% at 57300 yr$
Answer

The tool prints λ = 1.21 × 10⁻⁴ per year, 3.125% remaining, 31.25 nuclei and an elapsed time of 28,650 years. Steps 3 and 4 are where a physics answer becomes a laboratory one. Exponential decay is an average over a population, and its accuracy comes from the population being astronomically large — a gram of carbon holds about 5 × 10²² atoms, so the fluctuations are invisible. Run the same law down to tens of nuclei and the noise is the answer, which is the reason a dating laboratory quotes an uncertainty in years rather than a date.

$N (t) = N_{0} e^{- λ t} : a smooth law over a countable population$

Learning path

Clocks in the rock

Leads to Radiometric dating

References (2)

The decay rate that is not quite constant, in the third block: T. Ohtsuki, H. Yuki, M. Muto, J. Kasagi and K. Ohno, "Enhanced Electron-Capture Decay Rate of ⁷Be Encapsulated in C₆₀ Cages." Physical Review Letters 93(11), 112501, 2004.
The U-238 chain, its 14 steps and their half-lives: National Nuclear Data Center, Brookhaven National Laboratory — Evaluated Nuclear Structure Data File (ENSDF).

By the 42Math team · Updated 2026-08-17

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

All Bio, Geo, Chem lessons →

Example problems

Carbon-14 - Carbon-14: t½ = 5,730 years. Used for dating organic materials.
Iodine-131 - Iodine-131: t½ = 8.02 days. A common medical isotope used in thyroid treatment.
Uranium-238 - Uranium-238: t½ = 4.47 Gyr. Extremely long-lived, used for age of Earth dating.
Radium-226 - Radium-226: t½ = 1,600 years. Discovered by Marie and Pierre Curie.