A radioactive element ThA (_{84}Po^{216}) can | vedclass

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(D) The decay constant for $\alpha$-decay is $\lambda_1 = \frac{\ln 2}{T_1}$.The decay constant for $\beta$-decay is $\lambda_2 = \frac{\ln 2}{T_2}$.W

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$\frac{T_1 T_2}{T_1 + T_2}$

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(D) The decay constant for $\alpha$-decay is $\lambda_1 = \frac{\ln 2}{T_1}$.The decay constant for $\beta$-decay is $\lambda_2 = \frac{\ln 2}{T_2}$.When a radioactive nucleus can decay by two different processes,the total decay constant is the sum of the individual decay constants: $\lambda_{eq} = \lambda_1 + \lambda_2$.Since $\lambda_{eq} = \frac{\ln 2}{T_{eq}}$,we have $\frac{\ln 2}{T_{eq}} = \frac{\ln 2}{T_1} + \frac{\ln 2}{T_2}$.Dividing by $\ln 2$,we get $\frac{1}{T_{eq}} = \frac{1}{T_1} + \frac{1}{T_2} = \frac{T_1 + T_2}{T_1 T_2}$.Therefore,the effective half-life is $T_{eq} = \frac{T_1 T_2}{T_1 + T_2}$.

$A$ radioactive element $ThA$ $(_{84}Po^{216})$ can undergo $\alpha$ and $\beta$ types of disintegrations with half-lives $T_1$ and $T_2$ respectively. Then the effective half-life of $ThA$ is:

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