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•• A radioactive nucleus A has decay constant rA. It decays to an offspring nucleus B, which decays much more rapidly, with decay constant rB ≫ rA. The population of A decreases exponentially as usual [NA(t) = N0 exp(−rAt)]. Because B decays so rapidly, each B decays almost as soon as it is produced. Therefore, the rate of production of the offspring B (by decay of the parents A) is approximately equal to the rate of loss (by decay of the B nuclei themselves). Use this condition to find an expression for NB(t) in terms of N0,rA, rB, and t. Show in particular that NB remains proportional to NA.
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