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(D) When a radioactive substance decays by two different processes simultaneously,the total decay constant $\lambda_{eff}$ is the sum of the individua

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$\frac{T_{\alpha} T_{\beta}}{T_{\alpha} + T_{\beta}}$

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(D) When a radioactive substance decays by two different processes simultaneously,the total decay constant $\lambda_{eff}$ is the sum of the individual decay constants: $\lambda_{eff} = \lambda_{\alpha} + \lambda_{\beta}$.We know that the half-life $T$ is related to the decay constant $\lambda$ by the formula $T = \frac{\ln 2}{\lambda}$,which implies $\lambda = \frac{\ln 2}{T}$.Substituting this into the effective decay constant equation: $\frac{\ln 2}{T_{eff}} = \frac{\ln 2}{T_{\alpha}} + \frac{\ln 2}{T_{\beta}}$.Dividing both sides by $\ln 2$,we get: $\frac{1}{T_{eff}} = \frac{1}{T_{\alpha}} + \frac{1}{T_{\beta}}$.Solving for $T_{eff}$: $\frac{1}{T_{eff}} = \frac{T_{\beta} + T_{\alpha}}{T_{\alpha} T_{\beta}}$.Therefore,$T_{eff} = \frac{T_{\alpha} T_{\beta}}{T_{\alpha} + T_{\beta}}$.

$A$ radioactive substance has decay constants $\lambda_{\alpha}$ and $\lambda_{\beta}$ for $\alpha$ and $\beta$ emission,respectively. If the substance emits both $\alpha$ and $\beta$ particles,find the effective half-life of the substance.

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