A detector records total count rate from a radioactive sample: (0 min, 860 counts min⁻¹), (5 min, 510), (10 min, 300), (15 min, 195), and (20 min, 142.5). A separate long background measurement gives 90.0 counts min⁻¹. Determine the corrected half-life and decay constant, predict the total count rate at 30 min, and explain how Poisson counting uncertainty and an incorrect background estimate affect a logarithmic analysis.

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Accepted Answer

Corrected rates are 770, 420, 210, 105 and 52.5 counts min⁻¹; half-life from 10–20 min is 5.00 min, λ = 0.1386 min⁻¹, and predicted total at 30 min is about 103 counts min⁻¹. Subtract background from every total reading. Corrected rates are 770, 420, 210, 105 and 52.5 counts min⁻¹. The corrected rate halves from 210 to 105 in 5 min and again to 52.5 in the next 5 min, so the consistent later-data half-life is 5.00 min. Decay constant is λ = ln2/T_half = 0.6931/5.00 = 0.1386 min⁻¹. At 30 min, two more half-lives after 20 min, corrected rate is 52.5/4 = 13.1 counts min⁻¹. Add background to predict total 103 counts min⁻¹. Counts in a fixed interval have approximate standard uncertainty √N, so relative uncertainty grows as activity falls. A wrong background makes ln(corrected rate) curve, especially late when background is a large fraction; fit background and decay carefully rather than subtracting an arbitrary value.

Infer half-life and background from count data

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Measured total count rate

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Measured total count rate — supplied values
Time / min	Total count rate / counts min⁻¹
0	860
5	510
10	300
15	195
20	142.5