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 ${}^{131}I$ is injected into the blood of a person. The activity of the amount of ${}^{131}I$ injected was $2.4\times {10}^5$ Becquerel (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 $115$ 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 $ln2\approx 0.7$).

Answer 1

$t_{\frac{1}{2}}=8days=8\times 24hr$

$R_0=2.4\times {10}^{5}Bq$

$Using\frac{0.691}{t_{\frac{1}{2}}}\times t=ln\frac{R_0}{R}$

$\frac{0.692}{8\times 24}\times 11.5=ln\frac{2.4\times {10}^5}{R}$

$\frac{2.4\times {10}^5}{R}=e^{0.041}=1+0.041$

$R=\frac{2.4\times {10}^5}{1.041}=2.3\times {10}^5$

$115Bqisinvolume=2.5ml$

$2.3\times {10}^{5}Bqisinvolume=\frac{2.5}{115}\times 2.3\times {10}^5$

$=0.05\times {10}^5$

$=5\times {10}^{3}ml$

$=5litres$