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${ }^{131} I$ is an isotope of Iodine that $\beta$ decays to an isotope of Xenon with a half-life of $8$ days. A small amount of a serum labelled with ${ }^{131} \mathrm{I}$ is injected into the blood of a person. The activity of the amount of ${ }^{131} \mathrm{I}$ injected was $2.4 \times 10^5$ Becquerel $(\mathrm{Bq})$. It is known that the injected serum will get distributed uniformly in the blood stream in less than half an hour. After $11.5$ hours, $2.5 \mathrm{ml}$ of blood is drawn from the person's body, and gives an activity of $115 \mathrm{~Bq}$. The total volume of blood in the person's body, in liters is approximately (you may use $\mathrm{e}^{\mathrm{x}} \approx 1+\mathrm{x}$ for $|\mathrm{x}| \ll 1$ and $\ln 2 \approx 0.7$ ).
$2$
$3$
$4$
$5$
Solution
Final activity,
$\mathrm{A}_{\mathrm{f}}=\frac{\mathrm{v}}{\mathrm{v}_{\text {body }}} \times \mathrm{A}_0 \times \mathrm{e}^{-\hat{\mathrm{z} . t}}$
$\Rightarrow \mathrm{v}_{\text {body }}=\frac{\mathrm{v}}{\mathrm{A}_{\mathrm{f}}} \times \mathrm{A}_0 \mathrm{e}^{-\frac{\ln (2) \times t}{192}}$
$=4.998 \approx 5 \text { litres }$
