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Questions
Derive the mathematical expression for law of radioactive decay for a sample of a radioactive nucleus
Deduce the expression, N = N0 e−λt, for the law of radioactive decay.
Derive the expression N = Noe-λt where symbols have their usual meanings
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Solution
In any radioactive sample which undergoes α, β or γ-decay, it is found that the number of nuclei undergoing decay per unit time is proportional to the total number of nuclei in the sample.
If N is the number of nuclei in the sample and ΔN undergoes decay in time Δt, then we have
`(DeltaN)/(Deltat) propN`
`:.(DeltaN)/(Deltat)=lambdaN`
where λ is called the radioactive decay constant or disintegration constant.
The change in the number of nuclei in the sample is dN = −ΔN in time Δt, i.e. in the limit dt → 0. Thus, the rate of change of N is
`(dN)/dt= -lambdaN`
`:.(dN)/N = -lambdadt`
Now, integrating both the sides, we get
`int_(N_0)^N (dN)/N=-lambdaint_(t_0)^tdt`
∴ lnN - lnN0 = -λ(t-t0)
Here, N0 is the number of radioactive nuclei in the sample at some arbitrary time t0 and N is the number of radioactive nuclei at any subsequent time t. Setting t0 = 0 and rearranging, we get
`ln`
∴ N(t) = N0e-λt
This is the law of radioactive decay.
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