Question

State the law of radioactive decay. Arrive at the expression for the number of radioactive atoms in a sample at an instant.

Answer 1

$\underset{–}{\text{Law of radioactive decay:}}$

The law states that the rate of decay of a radioactive substance at any instant is directly proportional to the number of parent atoms present in the sample at that instant.

i.e., rate of disintegration $N_t$

(or)$\frac{-d{N}_t}{dt}\propto N_t$

$\Rightarrow \frac{d{N}_t}{dt}=–\lambda N_t$

$\lambda \to$ decay constant (constant of proportionality) also called disintegration constant.

$\frac{-d{N}_t}{N_t}=-\lambda dt$

Integrating on both sides,

${\int }_N^{N_t}\frac{d{N}_t}{N_t}={\int }_0^t-\lambda dt,weget,$

$ln\left[\frac{N_t}{N_0}\right]=-\lambda (t-0)\Rightarrow \frac{N_t}{N_0}=e^{-\lambda t}$

(or) $N=N_0{e}^{-\lambda }$