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 `1cm^3` taken after `5h` 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 person.
(1 curie `=3.7 xx 10^10` disintegrations per second)

Explanation

initial activity of `.^(24)Na`,
`A= (dN)/(dt)=1.0 mu C`
`=1.0 xx 10^(-6) xx 3.7 xx 10^(10)`
`=3.7 xx 10^(4)` disintegrations `s^(-1)`
half - life, `T=15 h=15 xx 3600 s`
Initial activity,
`A=(dN)/(dt) =lambda N_(0)`
`3.7 xx 10^(4) =(0.693)/(15 xx 3600) (N_(0))`
`:. N_(0) =(3.7 xx 10^(4) xx 15 xx 3600)/(0.693)`
`=2.883 xx 10^(9)`
Let the number of radioactive nuclei present after `5 h` be`N`' in `1 cm^(3)` of sample of blood. Then,
`(N')/(n'_(0)) =((1)/(2))^(t//T)`
`N'_(0)=(2)^(t//T)N'`
`=(2)^(5//15)N'=(2)^(1//3)xx3.844xx10^(5)`
Let `y=(2)^(1//3)`
because
log `y=(1)/(3) log 2 (1)/(3) xx 0.3010=0.1003`
`rArr` Antilong `0.1003 =1.2598`
`because N'_(0)=1.2598xx 3. 844 x 10^(5)`
`=4. 843xx10^(5)`
Volume of blood `=(N_(0))/(N'_(0)=(2.883xx10^(9))/(4.843xx10^(5))=0.595 xx10^(4) cm^(3)`
`=5.95 L`.