Question

For a substance, the average life for $\alpha$-emission is $3240$ years and for $\beta$ emission is $810$ years. After how much time (in years) the one-fourth of the material remains by simultaneous emission? ( $\ln 2=0.693$ )
(Round of the answer to nearest integer.)

Answer 1

Answer: 898

${\lambda }_{\alpha }=\frac{1}{3240}$ per year and ${\lambda }_{\beta }=\frac{1}{810}$ per year
Fraction of the remained activity, $\frac{A}{A_0}=\frac{1}{4}$
Total decay constant,
$\lambda ={\lambda }_{\alpha }+{\lambda }_{\beta }=\frac{1}{3240}+\frac{1}{810}=\frac{1+4}{3240}$ per year
$=\frac{1}{648}$ per year
We know that, $A=A_0{e}^{-\lambda t}$
$\Rightarrow t=\frac{1}{\lambda }{\log }_e\frac{A_0}{A}$
$\Rightarrow t=\frac{1}{\lambda }{\log }_{e}4=\frac{2}{\lambda }{\log }_{e}2$
$t=648\times 2\times 0.693=898$ years