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# (a) Derive the mathematical expression for law of radioactive decay for a sample of a radioactive nucleus.(b) How is the mean life of a given radioactive nucleus related to the decay constant?

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#### a) According to radioactive decay law, the rate of decay is proportional to amount of nuclei ,N. Thus, −dNdt∝N (negative sign come because the amount of nuclei decreases with time)So, dNdt=−λN or dNN=−λtIntegrating, lnN=−λt+c where C is the integrating constant.At t=0, N=N0 so lnN0=cNow, lnN=−λt+lnN0or ln(N/N0)=−λt or N/N0=e−λtHence, N=N0e−λt, this is the mathematical expression of radioactive decay of nuclei. b) The relation between mean life (tavg) and decay constant (λ) : tavg=1λ

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