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Difficulty: Very hardRadioactive Decay Law and Half-life

A Geiger-Müller counter records a total count rate of 340 counts per minute340\text{ counts per minute} near a radioactive source. The background radiation in the laboratory produces a steady count rate of 20 counts per minute20\text{ counts per minute}. If the total count rate recorded by the counter drops to 60 counts per minute60\text{ counts per minute} after an elapsed time of 15 minutes15\text{ minutes}, what is the half-life of the radioactive source in minutes?

Answer: 5 minutes

Answer

The half-life of the radioactive source is 5 minutes5\text{ minutes}.
To find the true activity of the radioactive source, the constant background radiation of 20 cpm20\text{ cpm} must be subtracted from all detector readings. The initial source activity is 340−20=320 cpm340 - 20 = 320\text{ cpm} and the activity after 15 minutes15\text{ minutes} is 60−20=40 cpm60 - 20 = 40\text{ cpm}. The fraction of source activity remaining is 40/320=1/8=(1/2)340 / 320 = 1/8 = (1/2)^3, which means 33 half-lives have elapsed in 15 minutes15\text{ minutes}. Dividing the total time by the number of half-lives (15/315 / 3) gives a half-life of 5 minutes5\text{ minutes}.

Step-by-Step Solution

1
Calculate the initial activity of the radioactive source by subtracting the background count rate.
A0=340 cpm−20 cpm=320 cpmA_0 = 340\text{ cpm} - 20\text{ cpm} = 320\text{ cpm}
Background radiation contributes to the detector reading and must be isolated from the source activity.
2
Calculate the activity of the source after 15 minutes by subtracting the background count rate.
A(t)=60 cpm−20 cpm=40 cpmA(t) = 60\text{ cpm} - 20\text{ cpm} = 40\text{ cpm}
The background count remains constant at 20 cpm20\text{ cpm}, so the actual count due to the source is 40 cpm40\text{ cpm}.
3
Determine the remaining fraction of the radioactive source.
A(t)A0=40320=18\frac{A(t)}{A_0} = \frac{40}{320} = \frac{1}{8}
Radioactive decay follows an exponential decay law based on the fraction of initial undecayed nuclei.
4
Calculate the number of elapsed half-lives nn.
\left(\frac{1}{2}\right)^n = \frac{1}{8} = \left(\frac{1}{2}\right)^3 \implies n = 3
The remaining fraction equals (1/2)n(1/2)^n where nn is the number of half-lives.
5
Compute the half-life T1/2T_{1/2}.
T_{1/2} = \frac{t}{n} = \frac{15\text{ minutes}}{3} = 5\text{ minutes}
The total elapsed time is the product of the number of half-lives and the duration of one half-life.

Key Concept

Radioactive Decay Law and Half-life with Background Radiation Correction
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Topics

Radioactive Decay Law and Half-life

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