A small quantity of solution containing `Na^(24)` radio nuclide of activity 1 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 disintegration per minute. What will be the total value of the blood in the person? Assum that the radioactive solution mixes uniformly in the blood of the person. (Take, 1 curie `= 3.7xx10^(10)` disintegration per second and `e^(-lamdat)=0.7927,` where lamda=` disintegration constant)

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`.