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(A) The decay constant $\lambda$ is related to the half-life $T_{1/2}$ by the formula $\lambda = \frac{\ln 2}{T_{1/2}}$.For $\alpha$-emission,the deca

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$\frac{3}{2} \frac{\ln 2}{T}$

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(A) The decay constant $\lambda$ is related to the half-life $T_{1/2}$ by the formula $\lambda = \frac{\ln 2}{T_{1/2}}$.For $\alpha$-emission,the decay constant is $\lambda_{\alpha} = \frac{\ln 2}{T}$.For $\beta$-emission,the decay constant is $\lambda_{\beta} = \frac{\ln 2}{2T}$.Since the decays occur simultaneously,the total decay constant $\lambda_{total}$ is the sum of the individual decay constants:$\lambda_{total} = \lambda_{\alpha} + \lambda_{\beta} = \frac{\ln 2}{T} + \frac{\ln 2}{2T}$.Taking $\frac{\ln 2}{T}$ as a common factor:$\lambda_{total} = \frac{\ln 2}{T} (1 + \frac{1}{2}) = \frac{\ln 2}{T} (\frac{3}{2}) = \frac{3}{2} \frac{\ln 2}{T}$.

The half-lives of a radioactive substance are $T$ and $2T$ years for $\alpha$-emission and $\beta$-emission,respectively. The total decay constant for the simultaneous decay of the $\alpha$ and $\beta$ radioactive substance is:

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