Question

State radioactive decay law. Derive $N=N_o{e}^{-\lambda t}$ for a radioactive element.

Answer 1

Radioactive Decay law :

This law states that the number of nuclei undergoing decay per unit time is directly proportional to the total number of nuclei present in the sample at that instant of time.

Mathematically,

$\frac{dN}{dt}\propto N$

$\frac{dN}{dt}=-\lambda N$

Here, $\lambda =$ decay constant

$-$ve sign shows that the number of radioactive nuclei is decreasing with time

$\frac{dN}{N}=-\lambda dt$

On integrating the above equation,

${\int }_{N_o}^N\frac{dN}{N}=-{\int }_0^t\lambda dt$

$ln\frac{N}{N_o}=-\lambda t$

$N=N_o{e}^{-\lambda t}$

Here, $N_o$ represent the number of nuclei present at t= 0