Radioactive Decay & Half-Life
Drag along the decay curve and watch the count rate halve over each fixed half-life. Why decay is exponential — and what λ and half-life really mean.
You can never say whena single radioactive nucleus will decay — it might be in the next second or in a thousand years. Yet a lump of the same material fades away on a schedule so reliable we use it to date rocks and time medical scans. That tension between random and predictable is the whole story of half-life.
Random for one nucleus, precise for a trillion
Radioactive decay is spontaneous and random. Nothing you do — heating, squeezing, chemistry — changes when an individual nucleus will throw out its particle. Every undecayed nucleus has the same fixed probability of decaying in the next moment, and no memory of how long it has already waited.
The magic is in the numbers. A microgram of material holds billions upon billions of nuclei, and while you cannot predict any single one, the fractionthat decays each second is rock-steady. Randomness at the bottom becomes near-perfect regularity at the top — the same reason a casino always knows its monthly takings.
One nucleus is a coin you can’t time; a mole of them is a trillion coins flipped at once, and the average always lands where the maths says it will.
Read off the decay curve
Because a fixed fraction decays in each equal slice of time, the amount left follows a falling exponential. Pick the decay preset below and drag along the curve: over every fixed step the count rate drops to exactly half of what it was.
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An exponential curve for radioactive decay. Selecting the decay preset shows N falling towards zero; because decay is exponential, the quantity halves over each equal time interval — the constant half-life. Dragging along the curve reads off the remaining amount at a given time.
What the half-life measures
The half-life (t½) is the time taken for the quantity to fall to half its starting value. Its defining feature is that it is constant: whether you start with 8 grams or 8 milligrams, one half-life later exactly half remains, then half of that, and so on.
Half-life is tied to the decay constantλ — the probability per unit time that a given nucleus decays. The larger λ is, the faster the source drops and the shorter its half-life:
Worked example
A radioactive source has an activity of 800 Bq and a half-life of 6 hours. What is its activity after 18 hours?
18 hours is exactly three half-lives (18 ÷ 6 = 3). Each half-life halves the activity, so just follow the chain:
800 Bq → (6 h) → 400 Bq → (12 h) → 200 Bq → (18 h) → 100 Bq.
You could reach the same answer with A = A₀e^(−λt), using λ = ln 2 / 6 h ≈ 0.116 h⁻¹, but when the time is a whole number of half-lives, halving repeatedly is faster and less error-prone.
Common mistakes
Practice
A nuclide’s activity falls from 240 Bq to 30 Bq. How many half-lives have passed?
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Halve until you reach 30: 240 → 120 → 60 → 30. That is three halvings, so three half-lives have passed.
Where it shows up
The same maths dates archaeological finds by carbon-14, sets the dose schedule for medical tracers, and governs how long nuclear waste stays hazardous — anywhere a quantity decays by a fixed fraction each step.
Frequently asked questions
What is half-life?+
Why is radioactive decay exponential?+
What is the decay constant?+
How are activity and the number of nuclei related?+
The ScholarsGate Physics Team
Oxbridge & Russell Group physics tutors
Written and reviewed by ScholarsGate tutors who teach A-Level and undergraduate physics. Every explainer is checked against the AQA, Edexcel, OCR and CIE specifications.
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