Question

The mean lives of a radioactive substance are 1620 years and 405 years for $\alpha$ and $\beta -$emissions respectively. Then the time during which $\frac{1}{4}$th of the sample will decay if it is decaying both $\alpha$ and $\beta -$emissions simultaneously will be:

• A. 92.4 years
• B. 449.2 years
• C. 106.0 years
• D. 172.0 years

Answer 1

The correct option is A 92.4 years

This is an example of successive, emissions in parallel paths.
${\lambda }_{av}={\lambda }_{\alpha }+{\lambda }_{\beta }=\frac{1}{1620}+\frac{1}{405}=\frac{5}{1620}y{r}^{-1}$
As one fourth part decays, $t=t,N=\frac{3}{4}{N}_o$
Thus, $t=\frac{2.303}{{\lambda }_{av}}log\frac{N_o}{N}=\frac{2.303\times 1620}{5}log\frac{4}{3}=92.4yrs.$