A radioactive isotope is continuously produced at a rate such that its saturation activity would be 3200 Bq. Production is continued for exactly three half-lives and is then stopped. The sample is subsequently allowed to decay freely for another two half-lives. If the final activity is N Bq, find N.
Physics · Nuclear physics · radioactivity and law of decay
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Question

A radioactive isotope is continuously produced at a rate such that its saturation activity would be 3200 Bq. Production is continued for exactly three half-lives and is then stopped. The sample is subsequently allowed to decay freely for another two half-lives. If the final activity is N Bq, find N.

Physics · Nuclear physics · radioactivity and law of decay

G1 Physics · Accepted Answer

During constant production, activity approaches saturation according to A(t) = A(sat)[1−exp(−λt)]. After three half-lives: exp(−λt) = 1 ÷ 2³ = 1 ÷ 8. Activity when production stops: A(stop) = 3200[1−1 ÷ 8] = 3200×7 ÷ 8 = 2800 Bq. After production stops, the sample decays normally. Two more half-lives reduce activity by a factor of 4: A(final) = 2800 ÷ 4 = 700 Bq. Hence N = 700.