A radioactive sample decays by two modes: alp | vedclass

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Question

(B) Let the decay constants for the two modes be $\lambda_{\alpha}$ and $\lambda_{\beta}$ respectively.Given that the sample decays by $\alpha$-decay

Vedclass · Accepted Answer

$90$

Vedclass · Answer

(B) Let the decay constants for the two modes be $\lambda_{\alpha}$ and $\lambda_{\beta}$ respectively.Given that the sample decays by $\alpha$-decay $66.6 \%$ of the time and by $\beta$-decay $33.3 \%$ of the time,the ratio of the decay rates is $\frac{\lambda_{\alpha}}{\lambda_{\beta}} = \frac{66.6}{33.3} = 2$.Thus,$\lambda_{\alpha} = 2\lambda_{\beta}$ or $\lambda_{\beta} = \frac{\lambda_{\alpha}}{2}$.The effective decay constant $\lambda_{eff}$ is given by $\lambda_{eff} = \lambda_{\alpha} + \lambda_{\beta}$.The effective half-life $T_{eff}$ is $60 	ext{ years}$,so $\lambda_{eff} = \frac{\ln 2}{60}$.Substituting the values: $\lambda_{\alpha} + \frac{\lambda_{\alpha}}{2} = \frac{\ln 2}{60} \Rightarrow \frac{3\lambda_{\alpha}}{2} = \frac{\ln 2}{60}$.Therefore,$\lambda_{\alpha} = \frac{2 \ln 2}{180} = \frac{\ln 2}{90}$.The half-life for $\alpha$-decay only is $T_{\alpha} = \frac{\ln 2}{\lambda_{\alpha}} = 90 	ext{ years}$.

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