A radioactive source has a half-life of 2 min.At time t = 0 it is placed near a detector and the counting rate is observed to be 3000 counts/s.If the detection efficiency is 25%,how many radioactive nuclei are there at time t = 0? A)1.04 $\times$ 10⁶ B)1.67 $\times$ 10⁶ C)2.08 $\times$ 10⁶ D)3.41 $\times$ 10⁶ E)3.79 $\times$ 10⁶

Question

Quizplus · Accepted Answer

The number of radioactive nuclei can be calculated using the formula:

N=N0(1/2)^(t/T)

where N0 is the initial number of radioactive nuclei, t is the time elapsed, and T is the half-life.

At t = 0, N = N0. The counting rate observed by the detector is proportional to the number of radioactive decays per second:

C = ελN

where ε is the detection efficiency, λ is the decay constant (ln(2)/T), and C is the counting rate.

Substituting the values given, we get:

N0 = C/(ελ) = 3000/(0.25*0.693/120) = 16667

Therefore, the initial number of radioactive nuclei is 1.67x10^10. The answer is choice (C).

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A Radioactive Source Has a Half-Life of 2 Min $\times$ 10⁶
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