Question

The mean lives of a radioactive substance are 1620 and 405 years for $\alpha$-emission and $\beta$-emission respectively. Find out the time during which three fourth of a sample will decay if it is decaying both by $\alpha$-emission and $\beta$-emission simultaneously.

• A. 449 years
• B. 440 years
• C. 520 years
• D. 549 years

Answer 1

The correct option is A

449 years

When a substance decays by $\alpha$ and $\beta$ emission simultaneously, the average rate of disintegration ${\lambda }_{av}$ is given by
${\lambda }_{av}={\lambda }_{\alpha }+{\lambda }_{\beta }$ when ${\lambda }_{\alpha }$ = disintegration constant for $\alpha$-emission only ${\lambda }_{\beta }$ = disintegration constant for $\beta$-emission only
Mean life is given by $T_m=\frac{1}{\lambda },{\lambda }_{av}={\lambda }_{\alpha }+{\lambda }_{\beta }\Rightarrow \frac{1}{T_m}=\frac{1}{T_{\alpha }}+\frac{1}{T_{\beta }}=\frac{1}{1620}+\frac{1}{405}=\frac{1}{324}$
${\lambda }_{av}\times t=2.303log\frac{N_0}{N_t},\frac{1}{324}t=2.303log\frac{100}{25}\Rightarrow t=2.303\times 324log4=449years$