\({ }^{131} I\) is an isotope of Iodine that \(\beta\) decays to an isotope of Xenon with a half-life of 8…

\({ }^{131} I\) 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 \({ }^{131} I\) is injected into the blood of a person. The activity of the amount of \({ }^{131} I\) 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 litres is approximately (you may use $e^x \approx 1+x$ for \(|x|<<1\) and \(\ln 2=0.7\).

Solution

t12=8 days

A=A0 e-λt

eλt 115=A0

A0=1151+λt=115 1+ln2t12×11.5

A0=115×1.042=119.82

A0 120 Bq

120 Bq is the activity of 2.5 ml

   2.4×105 Bq  is the activity of 2.5×10-3120×2.4×105

    Total volume of blood = 5 litres

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Asked in: JEE Advanced 2017 (Paper 1)

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