Modern Physics 3 Question 41
40. A small quantity of solution containing $Na^{24}$ radio nuclide (half-life $=15 h$ ) of activity 1.0 microcurie is injected into the blood of a person. A sample of the blood of volume $1 cm^{3}$ taken after $5 h$ shows an activity of 296 disintegrations per minute. Determine the total volume of the blood in the body of the person. Assume that the radioactive solution mixes uniformly in the blood of the person.
$(1994,6$ M)
$\left(1\right.$ curie $=3.7 \times 10^{10}$ disintegrations per second $)$
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Solution:
- $\lambda=$ Disintegration constant
$$ \frac{0.693}{t _{1 / 2}}=\frac{0.693}{15} h^{-1}=0.0462 h^{-1} $$
Let $R _0=$ initial activity $=1$ microcurie
$=3.7 \times 10^{4}$ disintegrations per second.
$r=$ Activity in $1 cm^{3}$ of blood at $t=5 h$
$=\frac{296}{60}$ disintegration per second
$=4.93$ disintegration per second, and
$R=$ Activity of whole blood at time $t=5 h$
Total volume of blood should be
$$ V=\frac{R}{r}=\frac{R _0 e^{-\lambda t}}{r} $$
Substituting the values, we have
$$ \begin{aligned} V & =\frac{3.7 \times 10^{4}}{4.93} e^{-(0.0462)(5)} cm^{3} \\ V & =5.95 \times 10^{3} cm^{3} \quad \text { or } \quad V=5.95 L \end{aligned} $$
