is an isotope of Iodine that B decays to an isotope of Xenon with a half-life of 8 days. A small amount of a serum labelled with is injected into the blood of a person. The activity of the amount of injected was . It is known that the injected serum wi

Question

${}^{131}I$ is an isotope of Iodine that B 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 $2.4 \times 10^{5} Bq$ . It is known that the injected serum will get distributed uniformly in the bloodstream 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 litres, is approximately (you may use $e^{x} \approx 1 + x$ for $| x |$ $< < 1$ and $\ln 2 \approx 0.7$ ). [2017]

Smart Achievers · Accepted Answer

(5)

$According to question,$

${}^{131}I$

$A_{0} = 2.4 \times 10^{5} Bq = λ N_{0}$

$Let the volume be V$

$Given:$ $At$ $t = 0$ , $A_{0} = λ N_{0} = 2.4 \times 10^{5} Bq$

$t = 11.5 hrs,$ $A = λ N$

$After$ $t = 11.5 h$ , $2.5$ $ml of blood is drawn from the person's body and gives an activity of$ $115$ $Bq.$

$∴$ $115 = λ (\frac{N}{V} \times 2.5);$ $115 = \frac{λ}{V} \times 2.5 \times N_{0} e^{- λ t}$

$\Rightarrow 115 = \frac{λ N_{0}}{V} \times 2.5 \times e^{- \frac{\ln 2}{8 days} (11.5 hrs)}$

$\Rightarrow V = \frac{2.4 \times 10^{5}}{115} \times 2.5 (1 - \frac{1}{24})$ $[from approximation e^{x} \approx 1 + x]$

$\Rightarrow V = \frac{2.4 \times 10^{5}}{115} \times 2.5 \times \frac{23}{24} = 5 \times 10^{3} ml = 5 litres$

Solution

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