Question

Mean life of a sample is 200 years for $\alpha -$decay and 300 years for $\beta -$ decay. If the sample decays by both $\alpha$ and $\beta$ emission simultaneously, then time for the sample to reduce to half is ($\ln 2=0.693$)

• A. 83 years
• B. 500 years
• C. 250 years
• D. 67 years

Answer 1

The correct option is A 83 years

${\lambda }_{\alpha }=\frac{1}{200}\&{\lambda }_{\beta }=\frac{1}{300}\Rightarrow \lambda ={\lambda }_{\alpha }+{\lambda }_{\beta }=\frac{1}{200}+\frac{1}{300}$

$\lambda =\frac{1}{120}$, By $N=N_0{e}^{-\lambda t}$
$e^{\lambda t}=2$ $t=\frac{ln2}{\lambda }=0.693\times 120=83years$