One of the hazards of nuclear explosion is the generation of $^{90} S r$ and its subsequent incorporation in bones. This nuclide has a half-life of 28 years. Suppose one microgram was absorbed by a newborn child, how much $^{90} S r$ will remain (in micrograms) in his bones on their $14^{t h}$ birthday? (take $\sqrt{2} = 1.4$ )

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Solution

We know, radioactive decay follows first order reaction.
$λ = \frac{l n 2}{t_{\frac{1}{2}}}$
$\Rightarrow 14 = \frac{t_{\frac{1}{2}}}{l n 2} l n \frac{w_{0}}{w_{t}} \Rightarrow 14 = \frac{28}{l n 2} l n \frac{w_{0}}{w_{t}}$
$\Rightarrow \frac{1}{2} \times l n 2 = l n \frac{w_{0}}{w_{t}}$
$\sqrt{2} = \frac{w_{0}}{w_{t}}$
$w_{t} = \frac{w_{0}}{\sqrt{2}}$
$w_{t} = 0.714 \times 10^{- 6}$ gram

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We know, radioactive decay follows first order reaction. λ=ln2t12 ⇒14=t12ln 2lnw0wt⇒14=28ln 2lnw0wt ⇒12×ln2=lnw0wt √2=w0wt wt=w0√2 wt=0.714×10−6 gram