# Derive the Mathematical Expression for Law of Radioactive Decay for a Sample of a Radioactive Nucleus - Physics

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

#### 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.