Question

The mean lives of a radioactive substance for $\alpha$ and $\beta$ emissions are 1620 years and 405 years respectively. After how much time will the activity be reduced to one fourth

• A. 405 year
• B. 1620 year
• C. 449 year
• D. None of these

Answer 1

The correct option is C 449 year

${\lambda }_{\alpha }=\frac{1}{1620}$ per year and ${\lambda }_{\beta }=\frac{1}{405}$ per year and it is given that the fraction of the remained activity
$\frac{A}{A_0}=\frac{1}{4}$
Total decay constant $\lambda ={\lambda }_{\alpha }+{\lambda }_{\beta }=\frac{1}{1620}+\frac{1}{405}=\frac{1}{324}$ per year
We know that $A=A_0{e}^{-\lambda t}\Rightarrow t=\frac{1}{\lambda }lo{g}_e\frac{A_0}{A}\Rightarrow t=\frac{1}{\lambda }lo{g}_{e}4=\frac{2}{\lambda }lo{g}_{e}2=324\times 2\times 0.693=449years$.