# Nuclear and particle physics: Unit 4.4(a)

> define activity decay constant half-life and use A equals lambda N and N equals N0 exp minus lambda t

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Specification statement (Unit 4.4(a)): define activity decay constant half-life and use A equals lambda N and N equals N0 exp minus lambda t

## Explanation

1. **Activity** A is the number of decays per second (becquerel, Bq). The **decay constant** λ is the probability per second that a nucleus decays: A = λN.
2. Decay is random and exponential: N = N₀e^(−λt), and likewise A = A₀e^(−λt).
3. **Half-life** t½ is the time for N (or A) to halve: t½ = ln 2 / λ. After n half-lives, a fraction (½)ⁿ remains.

## Worked example

Technetium-99m has a half-life of 6.0 h. A sample's activity is 400 MBq. Find λ, the activity after 24 h and the number of nuclei at the start.

1. λ = ln 2 / (6.0 × 3600) = 3.2 × 10⁻⁵ s⁻¹.
2. 24 h = 4 half-lives, so A = 400 / 2⁴ = 25 MBq.
3. N₀ = A₀/λ = 4.0 × 10⁸ / 3.2 × 10⁻⁵ = 1.2 × 10¹³ nuclei.

Answer: 3.2 × 10⁻⁵ s⁻¹; 25 MBq; 1.2 × 10¹³ nuclei.

## Check

A source's activity falls from 800 Bq to 100 Bq in 30 minutes. What is its half-life?

- 3.75 min
- 10 min
- 15 min
- 30 min

The answer and its explanation are shown in the app once you choose an option.

## Apply it

Explain what it means to say radioactive decay is random and spontaneous, and why the half-life is still predictable.

The model answer is shown in the app after you have tried it.

## Related

- [Nuclear and particle physics lesson](https://revise-gray.vercel.app/lesson?subject=wjec-alevel-physics&topic=wjec-alevel-physics.nuclear)
- [Nuclear and particle physics in the library](https://revise-gray.vercel.app/library?topic=wjec-alevel-physics.nuclear)