Write and derive the law of radioactivity decay. 

"In any radioactive sample, which undergoes $\alpha, \beta$ or $\gamma$-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 $\mathrm{N}$ is the number of nuclei in the sample and $\Delta \mathrm{N}$ undergo decay in time $\Delta t$ then $\frac{\Delta \mathrm{N}}{\Delta t} \propto \mathrm{N}$

The number $\Delta \mathrm{N}$ of the decaying nucleus is always positive, $\therefore \frac{\Delta \mathrm{N}}{\Delta t}=\lambda \mathrm{N}$

where $\lambda$ is called decay constant or disintegration constant. If the period $\Delta t$ corresponds to zero,

$\lim _{\Delta t \rightarrow 0} \frac{\Delta \mathrm{N}}{\Delta t}=-\lambda \mathrm{N}$

$\therefore-\frac{d \mathrm{~N}}{d t}=\lambda \mathrm{N}$$...(1)$

where $d \mathrm{~N}$ is the change in $\mathrm{N}$, which may be positive or negative. Here it is negative because as time goes by the number of nucleus remaining will decreases.

Writing equation $(1)$ as follows,

$\frac{d \mathrm{~N}}{\mathrm{~N}}=-\lambda d t$

“In any radioactive sample, which undergoes $\alpha, \beta$ or $\gamma$-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 $\mathrm{N}$ is the number of nuclei in the sample and $\Delta \mathrm{N}$ undergo decay in time $\Delta t$ then $\frac{\Delta \mathrm{N}}{\Delta t} \propto \mathrm{N}$

The number $\Delta \mathrm{N}$ of the decaying nucleus is always positive,

$\therefore \frac{\Delta \mathrm{N}}{\Delta t}=\lambda \mathrm{N}$

where $\lambda$ is called decay constant or disintegration constant.

If the period $\Delta t$ corresponds to zero,

$\lim _{\Delta t \rightarrow 0} \frac{\Delta \mathrm{N}}{\Delta t}=-\lambda \mathrm{N}$

$\therefore-\frac{d \mathrm{~N}}{d t}=\lambda \mathrm{N}$

where $d \mathrm{~N}$ is the change in $\mathrm{N}$, which may be positive or negative. Here it is negative because as time goes by the number of nucleus remaining will decreases.

Writing equation $(1)$ as follows,

$\frac{d \mathrm{~N}}{\mathrm{~N}}=-\lambda d t$