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Question:

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

Solution:

a) According to radioactive decay law, the rate of decay is proportional to the amount of nuclei, N. Thus,
−dN/dt ∝ N (negative sign comes because the amount of nuclei decreases with time)
So, dN/dt = −λN
or dN/N = −λdt
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 = e⁻λt
Hence, N = N₀e⁻λt, this is the mathematical expression of radioactive decay of nuclei.
b) The relation between mean life (τavg) and decay constant (λ):
τavg = 1/λ