Radioactive Decay

Each dot is an unstable nucleus. Every nucleus has the same chance of decaying in each moment, regardless of its age. Run the sample and compare the random count with the smooth exponential curve, and see why small samples are noisy and large ones are predictable.

Speed

How the model works

Decay is random for each nucleus but statistically predictable for many. If the probability of decay per unit time is λ (the decay constant), the expected number left after time t is

N(t) = N₀ e^(−λt) = N₀ (½)^(t / t½), t½ = ln 2 / λ

The half-life t½ is the time for half of any sample to decay, whatever its size. The activity A = λN (decays per second) falls in the same way.

In each time step Δt the simulator gives every surviving nucleus an independent chance p = 1 − e^(−λΔt) of decaying, using a seeded random number generator. Repeat runs with the same seed replay exactly; change the seed to see a different random history. The relative scatter is about 1/√N, so 100 nuclei fluctuate by about 10%, while 10 000 fluctuate by about 1%.

Single-step decay to a stable daughter; no decay chains. Time is in arbitrary units, with half-lives scaled for viewing.

Guided experiments

One half-life at a time. 400 nuclei, t½ = 2 s. Prediction: about 200 remain at 2 s, 100 at 4 s, 50 at 6 s, never exactly, but close.
Small sample. Only 20 nuclei. Prediction: the count wobbles far from the curve, and the last few may linger. With so few, chance dominates.
Memoryless. Restart with the survivors after one half-life. Prediction: the survivors have exactly the same half-life as fresh nuclei. They are not “due” to decay.

Challenge

Carbon-14 has a half-life of 5730 years. A bone has 12.5% of the C-14 found in living tissue. How old is it?

Reveal the explanation

12.5% = (½)³, so three half-lives have passed: 3 × 5730 ≈ 17 190 years. In general t = t½ · log₂(N₀/N).

References

OpenStax, University Physics Volume 3, §10.5 “Radioactive Decay”.
NIST, “Radionuclide Half-Life Measurements”.