State the law of radioactive decay. Hence derive the expression N=N0e−λt where symbols have their usual meanings.
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Solution
Law of radioactive decay : At any instant, the rate of radioactive disintegration is directly proportional to the number of nuclei of the radioactive element present at that instant. We know by radioactive decay law dNdt∝N or dNdt=−λN where λ is the constant of proportionality called radioactive decay constant. Let N0 be the number of nuclei present at time t=0, and N the number of nuclei present at time t. ∫NN0dNN=−∫t0λdt=−λ∫t0dt logeN−logeN0=−λt loge(NN0)=−λt NN0=e−λt N=N0e−λt