# Some nuclei of a radioactive

Question:

Some nuclei of a radioactive material are undergoing radioactive decay. The time gap between the instances when a quarter of the nuclei have decayed and when half of the nuclei have decayed is given as :

(where $\lambda$ is the decay constant)

1. $\frac{1}{2} \frac{\ln 2}{\lambda}$

2. $\frac{\ln 2}{\lambda}$

3. $\frac{2 \ln 2}{\lambda}$

4. $\frac{\ln \frac{3}{2}}{\lambda}$

Correct Option: , 4

Solution:

$\frac{3 \mathrm{~N}_{0}}{4}=\mathrm{N}_{0} \mathrm{e}^{-\lambda \mathrm{t}_{1}}$

$\frac{\mathrm{N}_{0}}{2}=\mathrm{N}_{0} \mathrm{e}^{-\lambda \mathrm{t}_{2}}$

$\ln (3 / 4)=-\lambda t_{1} \ldots . .$ (i)

$\ln (1 / 2)=-\lambda t_{2} \quad \ldots .$ (i)

$\ln (3 / 4)-\ln (1 / 2)=\lambda\left(t_{2}-t_{1}\right) \quad \ldots .$ (i)

$\Delta \mathrm{t}=\frac{\ln (3 / 2)}{\lambda}$