Question

A radioactive nucleus can decay by two different processes. The mean value period for the first process is $t_1$ and that for the second process is $t_2$. The effective mean value period for the two processes is

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

Answer 1

Answer: (D)

$\frac{t_1{t}_2}{t_1+t_2}$

Mean value:
${\lambda }_1=\frac{1}{t_1}$
${\lambda }_2=\frac{1}{t_2}$

First order decay:
$\frac{dN}{dt}=-({\lambda }_{1}N+{\lambda }_{2}N)$
$\frac{dN}{dt}=-({\lambda }_1+{\lambda }_2)$
$\frac{dN}{dt}=-(\frac{1}{t_1}+\frac{1}{t_2})N$
$\frac{dN}{dt}=-\left(\frac{t_1+t_2}{t_1{t}_2}\right)N$
$\frac{dN}{dt}=-{\lambda }_{eff}N$
$t_{eq}=\frac{1}{{\lambda }_{eff}}$