Question

A small quantity of solution containing ${}_{24}$Na radionuclide (half life 15 hours) of activity 1.0 microcuric is injected into the blood of a person.A sample of the blood of volume $1c{m}^3$ taken after 5 hours shows an activity of 296 disintegrations per minute.Determine the total volume of blood in litre in the body of the person.Assume that the radioactivity solution mixes uniformly in the blood of the person.$(1Curie=3.7\ast {10}^{10}$ disintegrations per second).

Answer 1

$\lambda =\frac{0.693}{T_{1/3}}=\frac{0.693}{15\times 3600}=1.283\times {10}^{-5}/sec$
Also, activity is given by $A=\frac{dN}{dt}=\lambda N_0$
$3.7\times {10}^4=1.283\times {10}^{-5}\times N_0$
$N_0=\frac{3.7\times {10}^4}{1.283\times {10}^{-5}}=2.883\times {10}^9$
Let the number of radioactive nuclei present after 5 hrs be $N_1$ in $1㎤$ sample of blood ,
Then, $\frac{dN}{dt}=\lambda N_1$
$\frac{296}{60}=\frac{0.693}{15\times 3600}{N}_1$
$N_1=3.844\times {10}^5$
Let $N_0^1$ be the no. of radioactive nuclei in per $㎤$of sample, then
$\frac{N_1}{N_0}={\left(\frac{1}{2}\right)}^{t/T_{1/2}};N_0^1={\left(2\right)}^{5/15}\times N_1{\left(2\right)}^{1/3}\times 3.844\times {10}^5$
$=4.878\times {10}^5$
Volume of blood $V=\frac{N_0}{N_0^1}=\frac{2.883\times {10}^9}{4.878\times {10}^5}=5.91\ell \approx 6L$