Half-Life With a Background Count: Subtract It First

The question

A student uses a detector and counter to investigate a radioactive sample.

With the sample removed, the counter records a background count rate of $20$ counts/min.

The sample is then placed in front of the detector. At time $t = 0$ , the measured count rate is $660$ counts/min. The half-life of the isotope in the sample is $6.0$ hours.

(a) Calculate the count rate the student will measure at $t = 18$ hours. [3 marks]

(b) Calculate the time at which the measured count rate will be $60$ counts/min. [3 marks]

Cambridge IGCSE note: correcting count rates for background radiation, and half-life calculations that include it, are Extended content in 0625.

The common wrong answer

Most students spot that $18$ hours is $3$ half-lives and halve the reading three times:

660 \rightarrow 330 \rightarrow 165 \rightarrow 82.5

“The measured count rate is $82.5$ counts/min.”

The mistake is halving $660$ . The detector is counting two things at once: radiation from the sample and background radiation from the surroundings (rocks, buildings, radon in the air, cosmic rays). Only the part from the sample halves every $6.0$ hours. The background stays at about $20$ counts/min whatever the sample does.

A second slip is to subtract the background at the start, halve correctly to get $80$ , and then stop. That is the count rate from the sample alone, but the question asks what the student will measure, and the detector still picks up the background. Cambridge examiners reported both of these errors on a multiple-choice question of this type in 2024.

Part (b) shows the problem even more clearly. Halving $660$ gives $330$ , $165$ , $82.5$ , $41.25$ , so it never lands on $60$ , and students end up guessing a time between $3$ and $4$ half-lives.

The correct working

(a) Subtract the background to get the count rate from the sample alone (the corrected count rate):

660 - 20 = 640\ \text{counts/min}

Number of half-lives in $18$ hours:

\frac{18}{6.0} = 3

Halve the corrected count rate three times:

640 \rightarrow 320 \rightarrow 160 \rightarrow 80\ \text{counts/min}

The detector also picks up the background, so add it back on:

80 + 20 = 100\ \text{counts/min}

The measured count rate at $t = 18$ hours is $100$ counts/min.

(b) A measured count rate of $60$ counts/min means the sample itself is giving

60 - 20 = 40\ \text{counts/min}

Now count the halvings from $640$ down to $40$ :

640 \rightarrow 320 \rightarrow 160 \rightarrow 80 \rightarrow 40

That is $4$ half-lives:

t = 4 \times 6.0 = 24\ \text{hours}

The measured count rate is $60$ counts/min at $t = 24$ hours.

Graph of measured count rate in counts per minute against time in hours. The curve falls from 660 at 0 hours through 340 at 6 hours, 180 at 12 hours, 100 at 18 hours and 60 at 24 hours, levelling off towards a dashed background line at 20 counts per minute. — The measured count rate levels off at the background of 20 counts/min, not at zero.

Why it works

Half-life is about the nuclei in the sample: every $6.0$ hours, half of the undecayed nuclei that are left decay, so the count rate from the sample halves. The background has nothing to do with the sample, so it doesn’t halve.

The graph makes this easy to see. The measured readings go $660$ , $340$ , $180$ , $100$ , $60$ . None of these is half of the one before, because each one has an extra $20$ on top. Take the $20$ off and you get $640$ , $320$ , $160$ , $80$ , $40$ , which halve perfectly. And however long you wait, the measured count rate never falls below about $20$ counts/min: it levels off at the background, not at zero.

So the method is always: subtract the background, halve, then add the background back if the question asks for what the detector reads.

How to spot it next time

If a question gives a background count rate, it is there for a reason. Write “corrected count rate = measured − background” as your first line.
Check the wording of the answer: “count rate from the sample” means corrected; “measured” or “recorded” count rate means add the background back on.
If halving the readings never lands on the number you need, that is a sign you have forgotten the background.