Question

${ }^{131} I$ 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 ${ }^{181} I$ is injected into the blood of a person. The activity of the amount of ${ }^{131} I$ injected was $2.4 \times 10^{5}$ Becquerel $( Bq ) .$ 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, $2.5 ml$ of blood is drawn from the person's body, and gives an activity of $115 Bq$. The total volume of blood in the person's body, in liters is approximately (you. may use $e ^{x} \approx 1+x$ for $|x|<<1$ and In $\left.2 \approx 0.7\right)$

Answer 1

$I^{131} \ce{->[\lambda]} X_e$
$ T _{1 / 2}=8$ days
$\lambda=\frac{\operatorname{\ell n} 2}{8} $
$A _{0}=2.4 \times 10^{5} $
After time $t$
$ A = A _{0} e ^{-\lambda t } $
$ A =2.4 \times 10^{5} e ^{-0.7} \times \frac{11.5}{24} $
$ A =2.4 \times 10^{5} e ^{-\frac{1}{24}} $
$ A =2.4 \times 10^{5}\left(1-\frac{1}{24}\right)$
$ A =2.4 \times 10^{5} \times \frac{23}{24}=\frac{115}{2.5} \times V $
$ V =\frac{10^{5}}{20} ml =5\, lt$