State radioactive decay law. Derive `N=N_0e^(-lambdat)` for a radioactive element

Explanation

Statement : In any radioactive sample which undergoes `alpha m beta , gamma - ` decay it is found that the number of nuclei undergoing the 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 `Delta N` undergo decay in the time `Delta t` then
`(Delta N)/(Delta t) prop N rArr (Delta N)/(Delta t) = lamda N`
Where `lamda` is called radioactive decay constant (or) disintegration constant
The change in the number of nuclei in the sample is
`dN = -Delta N` in time `Delta t `
Thus the rate of change of N is (in the limit `Delta t to 0` )
`(dN)/(dt) = - lamda N rArr (dN)/(N) = - lamda dt`
Now integration on both sides
`int_(N_0)^(N) (dN)/(N) = - lamda int_(t_0)^(t) dt`
`ln N - ln N_0 = - lamda (t - t_0)`
Here `N_0` is the number of radioactive nuclei in the sample at some arbitarary time `t_0` and N is the number of radioactive nuclei at any subsequent time t, but `t_0 = 0` ,
then `ln((N)/(N_0)) = - lamda t`
`rArr (N)/(N_0) = e^(-lamda t)`
`N = N_0 e^(-lamda t)`