Modern Physics Ques 141
- 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. $\left(1\right.$ curie $=3.7 \times 10^{10}$ disintegrations per second $)$
$(1994,6$ M)
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Answer:
Correct Answer: 141.$(5.95$ $L)$
Solution:
Formula:
- $\lambda=$ 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} $
BETA
