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The mean lives of a radio active substance are 1620 years and 405 years for $α -$ emissions and $β$ emissions respectively. Then the effective mean life of the substance for both $α and β$ emissions is

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324 years

382 years

364 years

298 years

answer is A.

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Detailed Solution

$\begin{array}{l} λ_{e} dt = λ_{1} dt + λ_{2} dt \\ \Rightarrow \frac{1}{T_{e}} = \frac{1}{T_{1}} + \frac{1}{T_{2}} \Rightarrow T_{e} = \frac{T_{1} T_{2}}{T_{1} + T_{2}} \end{array}$

$∴ T_{e} = (\frac{1620 \times 405}{1620 + 405}) years = 324 years$

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