Radioactivity and the Nucleus: Question 7

Worked solution

Part (a): Why the background must be subtracted

The GM tube detects radiation from any source reaching it, not just the source being tested. It also picks up the steady background radiation that is present in the room even with no source there. This background contributes a fixed number of counts per minute to every reading. To find the count rate due to the source alone, the constant background count rate must be subtracted from each total reading; otherwise the calculated half-life would be distorted by counts that have nothing to do with the source’s decay.

Part (b): Calculating the corrected count rate

Corrected count rate $=$ total count rate $-$ background count rate, where the background count rate is $20$ counts per minute at every time:

$420 - 20 = 400 \qquad 220 - 20 = 200 \qquad 120 - 20 = 100 \qquad 70 - 20 = 50$

Part (c): Finding the half-life

Half-life is the time taken for the count rate due to the source to fall to half its previous value. Reading down the corrected table:

$400 \to 200 \quad (5\text{ minutes}: \text{halved})$ $200 \to 100 \quad (5\text{ minutes}: \text{halved})$ $100 \to 50 \quad (5\text{ minutes}: \text{halved})$

The corrected count rate halves every $5$ minutes, consistently across the whole table, so:

$\text{half-life} = \boxed{5}\text{ minutes}$

Final answers

• (a) Background counts are present even without the source and must be removed so only the source’s own count rate remains.
• (b) Corrected count rate $=$ 400, 200, 100, 50 counts per minute at $0$, $5$, $10$, $15$ minutes.
• (c) Half-life $=$ 5 minutes