Radioactive Source: Background and Half-Life
A student uses a Geiger–Müller tube to measure the activity of a radioactive sample. With no source nearby, the counter records a background count of 12 counts per minute. Placed 5 cm from the tube, the sample gives an initial count rate of 252 counts per minute. After 18 days, the count rate is 42 counts per minute. The sample is believed to contain one isotope with a half-life of about 6 days. The student says the sample will be safe to handle after 36 days because the activity will have fallen to zero. The student also notices that repeat readings vary by about ±5 counts per minute and suggests this means the half-life cannot be determined accurately. The sample is a sealed source, and the counter is used in the same position each time.
1. Correct the count rates for background and determine the half-life from the data.
Read the worked answer
Background = 12 counts/min. Initial net rate = 252 − 12 = 240 counts/min. After 18 days, net rate = 42 − 12 = 30 counts/min. The ratio is 30/240 = 1/8 = (1/2)^3, so 18 days corresponds to 3 half-lives. Half-life = 18/3 = 6.0 days. This agrees with the suggested value. The calculation assumes the background has not changed and that the source geometry is the same for each measurement.
2. Evaluate the student's claim that the sample will be safe after 36 days because the activity will be zero.
Read the worked answer
After 36 days, 36/6.0 = 6 half-lives have passed. The net activity would be 240 × (1/2)^6 = 240/64 = 3.75 counts/min, so the measured rate would be about 3.75 + 12 = 15.8 counts/min, close to background but not zero. Radioactive decay is exponential; the activity approaches zero but never reaches it. Also, 'safe' cannot be decided from count rate alone: it depends on the type of radiation (alpha, beta, gamma), its energy, and whether the source is sealed. A low count rate does not necessarily mean zero risk, especially if alpha or beta radiation is absorbed before reaching the detector. The claim is therefore wrong about zero activity and oversimplified about safety.
3. Evaluate whether the ±5 counts/min fluctuation prevents an accurate half-life determination.
Read the worked answer
Radioactive decay is random, so repeated counts fluctuate. For a count of about 42, a fluctuation of ±5 is roughly consistent with counting statistics; the standard deviation is about √42 ≈ 6.5 if only the source count is considered. The key comparison is 240 versus 30 net counts/min: the difference is large compared with the fluctuation. Averaging repeat readings or counting for longer periods would reduce the relative uncertainty. The half-life of 6 days is therefore well supported by these data; the fluctuation affects the precision but does not prevent the determination.