Question

If the decay constant of a radioactive substance is $\lambda$, then its half-life and mean life are respectively

• A. $\frac{1}{\lambda }$ and $\frac{{\log }_{e}2}{\lambda }$
• B. $\frac{{\log }_{e}2}{\lambda }$ and $\frac{1}{\lambda }$
• C. $\lambda {\log }_{e}2$ and $\frac{1}{\lambda }$
• D. $\frac{\lambda }{{\log }_{e}2}$ and $\frac{1}{\lambda }$

Answer 1

Answer: (B)

$\frac{{\log }_{e}2}{\lambda }$ and $\frac{1}{\lambda }$

Half life of a radioactive substance is defined as the time upto which half of the initial number of nuclei presented will decay.
From !st order decay, $N=N_o{e}^{-\lambda t}$
At $t=T_{1/2}$, we have $N=\frac{N_o}{2}$
$\therefore$ $\frac{N}{2}=N_o{e}^{-\lambda T_{1/2}}$
Or $e^{-\lambda T_{1/2}}=\frac{1}{2}$
Or $-\lambda T_{1/2}=-{\log }_{e}2$
$⟹T_{1/2}=\frac{{\log }_{e}2}{\lambda }$
Mean life time of a radioactive substance is defined as the average time taken by a nuclei to decay.
Mean life time $t_m=\frac{1}{\lambda }$
Correct option is B.