Half-Life: Perfect Clocks from Pure Chance
No nucleus knows when it will decay — yet a trillion of them keep time better than any watch. Randomness, aggregated, is how we date pharaohs and planets.
Builds on: 6.2 Quanta: Where the Smooth World Turns Grainy
The most honest randomness there is
Some nuclei are unstable: sooner or later each one spits out radiation and becomes something else. When? Genuinely unknowable. A given carbon-14 nucleus might decay in the next second or outlast civilisation — it has no age, no wear, no schedule; a five-thousand-year-old nucleus is exactly as likely to decay this second as a fresh one. This is not ignorance to be fixed by better instruments; as far as physics can tell, it is chance all the way down — the quantum graininess of the last lesson, running the show inside the nucleus.
Chance, aggregated, is a clock
Now take a trillion of them. Each has a fixed probability per unit time, so a fixed fraction of the population decays per unit time — and that yields an unmistakable pattern:
The half-life is the signature of the isotope: carbon-14, 5,730 years; uranium-238, 4.5 billion; some medical isotopes, hours. Individually lawless, collectively metronomic — the same statistical magic that lets casinos budget precisely on games of pure chance.
Reading the clock backwards
Living things constantly refresh their carbon, holding a known trace of C-14. Death stops the refresh, and the clock starts draining. Measure how far a sample’s C-14 has fallen and the formula runs in reverse: an eighth of the original means three half-lives — about 17,000 years. This is radiocarbon dating, and it put honest dates on Ötzi, the Dead Sea Scrolls and every charcoal hearth archaeology has ever argued about. For rocks and planets, slower isotopes take over: uranium-lead dating is how we know the Earth is 4.54 billion years old — a number this course can now defend, not just quote.
What decay is not
Half-life describes populations, not appointments: after one half-life your particular nucleus hasn’t “half-decayed” — it is either gone or untouched. And decay is a nuclear event: burning, freezing or crushing a sample changes its chemistry, never its half-life. That indifference is exactly what makes the clock trustworthy across billions of years.
Roll 100 dice; remove every six; repeat. Each die is lawless, yet the population falls exponentially with a “half-life” of about 3.8 rolls — and your curve will wobble around the ideal one exactly the way real decay counts wobble. The lab below throws thousands so your wrists don’t have to.
⚗️ Lab — The Decay Farm
Ten thousand unstable nuclei, one probability dial, and time. Watch law emerge from chaos.
- Run it and watch the count halve on schedule — while any single nucleus stays unpredictable.
- Shrink the population to 20 and re-run: the clean curve turns ragged. Statistics needs numbers.
- Load the ’ancient sample’ and read its age off the fraction remaining — dating in one division.
Work it by hand — 0 / 2
No multiple choice here: compute the value and type it. Suffixes like 2.5k or 20m are understood; answers within ±2% count.