Half-Life and Activity in a Sealed Sample
A sealed vial contains a radioactive isotope used in a school lab. A detector measures an initial count rate of 4800 counts per minute above background. After 36 hours the count rate is 600 counts per minute above background. The vial is then stored for another 24 hours. A technician wants to know whether the sample will be safe to handle at that time, defined by the school as a count rate below 200 counts per minute above background. The isotope emits gamma radiation that is not fully stopped by the thin plastic vial, and the detector has a dead time that makes it undercount at very high rates.
1. Determine the half-life of the isotope from the two measurements and predict the count rate after the additional 24 hours.
Read the worked answer
The count rate falls from 4800 to 600, a factor of 8, which is 2^3, so three half-lives have elapsed in 36 hours. The half-life is therefore 36/3 = 12 hours.
After another 24 hours, two more half-lives pass: 600 → 300 → 150 counts per minute above background. The predicted rate is 150 counts per minute, which is below the 200 threshold, so on this basis the sample would be considered safe.
The judgement to make is whether the exponential model is trustworthy here. Two points fix a half-life only if the decay is genuinely a single isotope and the detector is linear over the whole range. The stated dead-time problem means the initial 4800 may be an underestimate, which would make the true ratio larger than 8, implying more than three half-lives and a shorter half-life. That would make the final rate even lower, so the safety conclusion is robust in direction.
2. The technician worries the detector's dead time invalidates the result. Explain how this affects the conclusion and what measurement would resolve the concern.
Read the worked answer
Dead time causes the detector to miss counts when the rate is high, so the measured 4800 is lower than the true initial rate. The measured ratio of 8 is therefore smaller than the true ratio, meaning the true decay factor over 36 hours is greater than 8 and the true half-life is shorter than 12 hours.
A shorter half-life means that after the additional 24 hours the true rate is below the 150 predicted by the naive model, so the sample is even more clearly under the 200 threshold. The dead-time error therefore does not threaten the safety conclusion; it only makes the half-life estimate an upper bound.
To resolve the concern, the technician should repeat the measurement with a calibrated source of known activity to characterize the dead time, or use a detector with a shorter dead time, or measure at a larger distance with a collimator so the rate falls in the linear range. Re-measuring the initial rate under low-rate conditions (for example, far from the source) and scaling by distance would give a corrected ratio.