Derive the mathematical expression for law of radioactive decay for a sample of a radioactive nucleus

Deduce the expression, N = N_{0} e^{−λt}, for the law of radioactive decay.

Derive the expression N = N_{o}e^{-λt }where symbols have their usual meanings

#### 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 - lnN_{0} = -λ(t-t_{0})

Here, N_{0} is the number of radioactive nuclei in the sample at some arbitrary time t_{0} and N is the number of radioactive nuclei at any subsequent time t. Setting t_{0} = 0 and rearranging, we get

`ln`

∴ N(t) = N_{0}e^{-λt}

This is the law of radioactive decay.