Radioactive decay – decay law and half-life

PhysicsQuantum & Nuclear PhysicsAges 17–18

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Use with my class

N₀ parent nuclei (red) decay randomly into daughter nuclei (grey), each with the same decay probability per unit time. The graph plots the simulated number of remaining nuclei N(t) against the theoretical curve N = N₀·2^(−t/T); the decay constant λ = ln2/T and the activity A = λN update continuously. Drag the cursor along the graph to read N at any instant.

Lesson: Radioactive decay, decay law, half-life, activity

What it shows

Radioactive decay is random for each individual nucleus: it is impossible to predict which one will decay next, yet every nucleus has the same probability of decaying per unit time. For a large number of nuclei this produces a predictable pattern, N = N₀·2^(−t/T), where the half-life T is the time for half of the remaining nuclei to decay. The decay constant is λ = ln2/T, and the activity A = λN falls at the same rate as N. With small samples, random fluctuations make the simulated count stray noticeably from the curve.

How to use

Set Initial number of nuclei N₀ and Half-life T, press Run and watch red parent nuclei turn grey. Compare the simulated count with the theoretical curve after each half-life. Drag on the graph to read N at 2T or 3T. Repeat with N₀ = 50 and N₀ = 400 to see how sample size affects the agreement, and use Simulation speed when the half-life is long.

Parameters you can change

Initial number of nuclei N₀ 50–400
Half-life T 1–20 s
Simulation speed 0.5–4 ×
Show theoretical curve N₀·2^(−t/T)

Questions to explore

Can you predict which nucleus will decay next? What can you predict about the sample as a whole?
Why does the simulated curve follow the theoretical curve more closely when N₀ is large?
What fraction of the original nuclei remains after three half-lives?

Teaching ideas

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