The decay constants of a radioactive substanc | vedclass

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Question

(C) When a radioactive substance decays through multiple processes simultaneously,the total decay constant is the sum of the individual decay constant

Vedclass · Accepted Answer

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

Vedclass · Answer

(C) When a radioactive substance decays through multiple processes simultaneously,the total decay constant is the sum of the individual decay constants.Thus,the effective decay constant is $\lambda_{eff} = \lambda_{\alpha} + \lambda_{\beta}$.The half-life $T$ is related to the decay constant $\lambda$ by $T = \frac{\ln 2}{\lambda}$,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 by $\ln 2$ on both sides,we get $\frac{1}{T_{eff}} = \frac{1}{T_{\alpha}} + \frac{1}{T_{\beta}}$.Solving for $T_{eff}$,we obtain $T_{eff} = \frac{T_{\alpha}T_{\beta}}{T_{\alpha} + T_{\beta}}$.

The decay constants of a radioactive substance for $\alpha$ and $\beta$ emission are $\lambda_{\alpha}$ and $\lambda_{\beta}$ respectively. If the substance emits $\alpha$ and $\beta$ simultaneously,then the average half-life of the material will be:

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