Radio activity

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Question

A radio active sample has a half life T. What fraction of initial number of active nuclei will decay in a time $t = \frac{T}{2}$ ?

Solution

After time $+ \frac{T}{2}$
$\frac{N}{N_{0}} = \frac{1}{{(2)}^{\frac{\frac{T}{2}}{T}}} = \frac{1}{\sqrt{2}}$
$∴$ Required fraction $= \frac{N_{0} - N}{N_{0}} = (1 - \frac{1}{\sqrt{2}})$

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