A radioactive sample decays by two modes: alp | vedclass

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

Vedclass · Accepted Answer

$90$

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(B) Let the decay constants for the two modes be $\lambda_{\alpha}$ and $\lambda_{\beta}$ respectively.Given that the probability of $\alpha$-decay is $66.6\%$ $(2/3)$ and $\beta$-decay is $33.3\%$ $(1/3)$,the ratio of decay constants is $\frac{\lambda_{\alpha}}{\lambda_{\beta}} = \frac{2/3}{1/3} = 2$,so $\lambda_{\beta} = \frac{\lambda_{\alpha}}{2}$.The effective decay constant $\lambda_{eff} = \lambda_{\alpha} + \lambda_{\beta}$.The effective half-life $T_{eff} = \frac{\ln 2}{\lambda_{eff}} = 60 	ext{ years}$.Substituting $\lambda_{\beta} = \frac{\lambda_{\alpha}}{2}$ into the equation: $\lambda_{eff} = \lambda_{\alpha} + \frac{\lambda_{\alpha}}{2} = \frac{3\lambda_{\alpha}}{2}$.Thus,$\frac{\ln 2}{\frac{3\lambda_{\alpha}}{2}} = 60 \Rightarrow \frac{\ln 2}{\lambda_{\alpha}} = 60 	imes \frac{3}{2} = 90 	ext{ years}$.The half-life for $\alpha$-decay only is $T_{\alpha} = \frac{\ln 2}{\lambda_{\alpha}} = 90 	ext{ years}$.

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