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

Consider a `1 cm^(3)` sample of the blood. If its initial acitivity was `A_(o)`, then the activity A after time t will be `A=A_(0)e^(-lambda t)`. In terms of half life `T_(1//2)`, the expression can be written as `A=A_(0)((1)/(2))^(t//T_(1)//2)`,
i.e., `(296)/(60 s)=A_(0)((1)/(2))^(5 hr//15 hr)` or `A_(0)=((296)/(60s))=6.22 s^(-1)`.
Since the total initial activity is given to be `A_(0)=1.0xx10^(-6)` curie `=1.0xx10^(-6)xx(3.7xx10^(10) "disintergration"//"second")=3.7xx10^(4)s^(-1)`,
`:.` The total volume of the blood is `V=`
`("Total acitivity")/("activity per" cm^(3))=(3.7xx10^(4)s^(-1))/(6.22s^(-1)//cm^(3))`
`5.95xx10^(3)=5.95` litres.