A given number of atoms, `N_(0)` of a radioactive element with a half life T is uniformly distributed in the blood stream of a
(i) Normal person A having total volume V of blood in the body.
(ii) Person B in need of blood transfusion having a volume V' of blood in the body.
The number of radioactive atoms per unit volume in the blood streams of the two persons after a time `nT` are found to be `N_1 and N_2`.
Prove mathematically that the additional volume of blood that needs to be transfused in the body of person B equals `((N_2 - N_1) /(N_2) )` V.

Explanation

Initial number of radioactive atoms, per unit volume, in the blood streams of persons A and B are `(N_(0) //V) and (N_(0) //V.)` respectively.
After a time `nT (T = "Half life")`, these numbers would get reduced by a factor `2^n`.
Hence` N_(1) = ((N_0) /( V) ) . (1)/ (2^n) `
and `N_2 = ((N_0) /( V.) ) . (1)/(2^n)`
`(N_1) /( N_2) = (V.)/( V)`
or `V.=V. (N_1) /( N_2)`
`therefore` Additional volume of blood needed by person B is
`V-V. = V - V (N_1)/( N_2)`
`= ((N_2 - N_1) /( N_2) )` V