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, $2.5ml$ of blood is drawn from the person's body, and gives an activity of $115Bq$. 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 $2\approx 0.7)$

Answer 1

Answer: 5

$I^{131}\stackrel{\lambda }{\to }{X}_e$
$T_{1/2}=8$ days
$\lambda =\frac{\mathrm{\ell n}2}{8}$
$A_0=2.4\times 10^5$
After time $t$
$A=A_0{e}^{-\lambda t}$
$A=2.4\times 10^5{e}^{-0.7}\times \frac{11.5}{24}$
$A=2.4\times 10^5{e}^{-\frac{1}{24}}$
$A=2.4\times 10^5\left(1-\frac{1}{24}\right)$
$A=2.4\times 10^5\times \frac{23}{24}=\frac{115}{2.5}\times V$
$V=\frac{10^5}{20}ml=5lt$