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

$\lambda_{\alpha}=\frac{1}{3240}$ 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