Half-Life Lab: Random Decay, Predictable Law
A grid of up to 900 nuclei, each rolling the same dice every instant — the exact per-frame decay probability 1 − e^(−λ·dt), with nothing about "exponential decay" programmed in. Individual nuclei flash and die at genuinely unpredictable moments (one is ringed: try to guess when it goes), yet the population traces the theory curve N₀·2^(−t/T½), and every halving crossing is marked and timed so students can verify the definition of half-life by measurement. Activity (decays per second) is counted live and halves alongside N. Run small samples for ragged statistics and large ones for smooth law — chance in the small, law in the large.
Every brass nucleus rolls the same dice every instant — when one loses, it flashes and becomes a faint ring. No nucleus knows what the others are doing, yet the population traces the dashed theory curve, and every red dot marks a measured halving. Randomness up close, law at a distance.
Watch the activity fall with N: fewer undecayed nuclei means fewer decays per second. Both halve every half-life — which is why old radioactive sources grow quiet but never quite reach zero.
1. Before pressing Start, pick the ringed nucleus and guess when it will decay. You will be wrong — everyone is. Single decays are genuinely unpredictable.
2. Compare the measured halvings printed in red: first halving, second, third. All roughly equal — the definition of half-life, checked by experiment.
3. Run the same settings twice. The individual flashes are completely different; the curve is nearly the same. Chance in the small, law in the large.
4. Try 100 nuclei, then 900. The small sample gives a ragged staircase, the big one hugs the theory curve — this is why real labs measure with billions of atoms.
5. After two half-lives, is the sample half gone? No — three quarters gone. Each halving acts on what remains, not on the original amount.
What Each Variable Means
Half-life. The time taken for HALF the undecayed nuclei to decay — the same however many you start with, and the same for every successive halving.
Undecayed nuclei. The population still waiting to decay. After one half-life: N₀/2. After two: N₀/4. After three: N₀/8.
Activity. Decays per second (measured in becquerel, Bq). Proportional to N — so activity also halves every half-life.
Decay constant. The fixed probability per second that any one nucleus decays. Nuclei have no memory: an old nucleus is no more likely to decay than a fresh one.