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A radioactive sample decays by two modes by $\alpha $ decay and by $\beta -decay$. $66.6 \%$ of times it decays by $\alpha -decay$ and $33.3 \%$ of times, it decays by $\beta -decay$. If half life of sample is $60$ years then what will be half life of sample, if it decays only by $\alpha - decay$. ............ $years$
$30$
$90$
$120$
$180$
Solution
Let decay constants for two modes are $\lambda_{\alpha}$ and $\lambda_{\mathrm{b}}$ respectively.
$\frac{\mathrm{d} \mathrm{N}_{\alpha}}{\mathrm{dt}}=-\lambda_{\alpha} \mathrm{N}$
$\frac{\mathrm{d} \mathrm{N}_{\beta}}{\mathrm{dt}}=-\lambda_{\beta} \mathrm{N}$
$\frac{\mathrm{d} \mathrm{N}_{\alpha} / \mathrm{dt}}{\mathrm{dN}_{\beta} / \mathrm{dt}}=2 \Rightarrow \frac{\lambda_{\alpha}}{\lambda_{\beta}}=2$ $(i)$
Also $\frac{\ln 2}{\lambda_{\alpha}+\lambda_{\beta}}=60$
$\frac{\ln 2}{\lambda_{\alpha}+\frac{\lambda_{\alpha}}{2}}=60$
$\frac{\ln 2}{\lambda_{\alpha}}=90$ years.
