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Question

(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?

Solution
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a) According to radioactive decay law, the rate of decay is proportional to amount of nuclei ,N. Thus, dNdtN (negative sign come because the amount of nuclei decreases with time)
So, dNdt=λN or dNN=λt
Integrating, lnN=λt+c where C is the integrating constant.
At t=0, N=N0 so lnN0=c
Now, lnN=λt+lnN0
or ln(N/N0)=λt or N/N0=eλt
Hence, N=N0eλt, 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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