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The graph represents the decay of a newly-prepared sample of radioactive nuclide X to a stable nuclide Y. The half-life of X is $τ$ The growth curve for Y intersects the
decay curve for X after time T.What is the time T?

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$τ$

$\frac{τ}{2}$

$\ln (2 τ)$

$\ln (\frac{τ}{2})$

answer is A.

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Detailed Solution

If half-life of X is $τ$ then its initial number $N_{0}$ becomes $\frac{N_{0}}{2}$ in time $τ$ . Now $\frac{N_{0}}{2}$ decayed X produce $\frac{N_{0}}{2}$ number of Y nuclei. Thus, the numbers of X and Y will be
equal at half life time or ' $τ$ '

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