Question

A radioactive nucleus can decay by two different processes. The half life for the first process is $t_1$ and that for the second process is $t_2$. If effective half life is $t$, then

• A. $t=t_1+t_2$
• B. $\frac{1}{t}=\frac{1}{t_1}+\frac{1}{t_2}$
• C. $t=\frac{2t_1{t}_2}{t_1+t_2}$
• D. $t=\frac{t_1+t_2}{2}$

Answer 1

The correct option is B $\frac{1}{t}=\frac{1}{t_1}+\frac{1}{t_2}$

$T_{1/2}=\frac{0.693}{\lambda }$
$\therefore \lambda =\frac{0.693}{t}$
According to law of radioactivity,
$\frac{dN}{dt}=-N\lambda$ ... (1)
Now, the total rate of decay is
$\frac{dN}{dt}=-N({\lambda }_1+{\lambda }_2)$ ... (2)
from the equation (1) and (2),
$N({\lambda }_1+{\lambda }_2)=N\lambda$
$\Rightarrow \lambda ={\lambda }_1+{\lambda }_2$
$\frac{1}{t}=\frac{1}{t_1}+\frac{1}{t_2}$