Half-Life with a Contaminated Sample
A medical physics technician receives a sample labelled as containing a radioisotope with a half-life of 6.0 hours, to be used in a tracer study. A Geiger counter measures an initial count rate of 4800 counts per minute. After 18 hours the count rate is 640 counts per minute, but the technician expected 600. A background count of 40 counts per minute was not subtracted in either reading. The technician must decide whether the sample is pure or contaminated with a longer-lived isotope.
1. Correct for background and determine whether the decay is consistent with a single 6.0-hour half-life.
Read the worked answer
Corrected rates: Initial: 4800 − 40 = 4760 cpm. After 18 h: 640 − 40 = 600 cpm.
Expected after 3 half-lives (18 h): 4760/2³ = 4760/8 = 595 cpm.
Observed: 600 cpm. Difference is 5 cpm, about 0.8% - well within typical counting statistics (√N fluctuations). The apparent discrepancy of 40 cpm vanishes once background is subtracted.
Judgement: The sample is consistent with a single 6.0-hour half-life. The technician's concern was an artefact of failing to subtract background. This is a classic trap: background matters most at low count rates, exactly where the discrepancy appeared.
2. The technician needs the count rate to fall below 100 cpm (background-corrected) before disposal. Estimate the additional waiting time required and comment on the reliability of the estimate.
Read the worked answer
Target: 100 cpm corrected.
From 600 cpm: need factor 6 reduction. Number of half-lives n satisfies 2ⁿ = 6, so n = log₂6 = ln6/ln2 = 2.585 half-lives.
Time: 2.585 × 6.0 h = 15.5 h additional.
Reliability: The estimate assumes the single-isotope model holds indefinitely. If any long-lived contaminant were present, the rate would plateau above the prediction. The technician should re-measure after ~15 h and confirm the rate continues to halve every 6 h; a plateau signals contamination. Also, at 100 cpm the background (40 cpm) is 40% of the signal, so statistical uncertainty is large and longer counting windows are needed.