A given number of atoms No 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 N1 and N2.
Prove mathematically that the additional volume of blood that needs to be transfused in the body of person B equals
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Pre requisite:
Equation of Nuclear decay:
Where
The above relation can also be written as
where
Given
Person A Person B
Volume V V'
Initial
number of
atoms per 0
unit volume
before transfusion
Final
Number of
atoms per
unit volume
Consider the volume of blood transfused is x
Then the number of present in the transfused blood is
Then the
Number of atoms present in the blood of person B after transfusion is
Volume of the blood in person B is
Thus the number of atoms per unit volume is
After a time of ,
Number of atoms per unit volume in person A
Number of atoms per unit volume in person B
Hence
Now solve for x
Equation of Nuclear decay:
Where
The above relation can also be written as
where
Given
Person A Person B
Volume V V'
Initial
number of
atoms per 0
unit volume
before transfusion
Final
Number of
atoms per
unit volume
Consider the volume of blood transfused is x
Then the number of present in the transfused blood is
Then the
Number of atoms present in the blood of person B after transfusion is
Volume of the blood in person B is
Thus the number of atoms per unit volume is
After a time of ,
Number of atoms per unit volume in person A
Number of atoms per unit volume in person B
Hence
Now solve for x