Radioactive Source Label, Background and Storage Decision
A school laboratory stores a sealed radioactive demonstration source labelled half-life 6.0 h and initial activity 8.0 kBq. After 18.0 h, a detector placed at a fixed distance reads 1.4 kBq. The background count rate in the room is 0.4 kBq. A teacher plans to use the source after 72 h and wants the net activity to be below 5 Bq before returning it to the cupboard. The detector efficiency and geometry are assumed unchanged, and the source is treated as a single nuclide. A student says the label must be wrong because 1.4 kBq is not one-eighth of 8.0 kBq. The teacher asks the student to correct the reading for background, decide whether the label is consistent, and estimate the net activity at 72 h.
1. Correct the 18.0 h reading for background and decide whether the half-life label is consistent.
Read the worked answer
After 18.0 h, the elapsed time is 18.0 / 6.0 = 3 half-lives.
The expected net activity isA = 8.0 × (1/2)^3 = 1.0 kBq.
The measured gross reading is 1.4 kBq, so the net reading is 1.4 - 0.4 = 1.0 kBq.
This agrees exactly with the label. The student compared the gross reading with the expected net activity and forgot that the detector also records background. The label is consistent with the measurement.
2. Estimate the net activity after 72 h. Will it meet the teacher's below-5-Bq requirement?
Read the worked answer
At 72 h, the number of half-lives is 72 / 6.0 = 12.
The net activity isA = 8.0 kBq / 2^12 = 8000 / 4096 = 1.95 Bq.
This is below 5 Bq, so the source meets the teacher's net-activity requirement. The calculation must use the net activity, not the gross detector reading, because the background is about 400 Bq and would completely dominate the gross reading at this stage.
3. Explain why the gross detector reading cannot be used directly for the final safety decision.
Read the worked answer
The detector reading is source counts plus background. At 18 h the background was 0.4 kBq, which was a large fraction of the 1.4 kBq gross reading. After 72 h the source itself is only about 2 Bq, while the background is still about 400 Bq. A gross reading of roughly 402 Bq would look alarming if background were ignored, but the net source activity is only about 2 Bq. Background must be subtracted, and the detector geometry and efficiency must remain unchanged for the comparison to be valid.