A small amount of a solution containing the $Na^{24}$ radionuclide with an initial activity of $A_0

Question

A small amount of a solution containing the $Na^{24}$ radionuclide with an initial activity of $A_0 = 2000 	ext{ disintegrations per second (Bq)}$ was injected into the bloodstream of a patient. After exactly 15 hours, a $1 	ext{ cm}^3$ blood sample was drawn, and its activity was measured to be $12 	ext{ disintegrations per minute}$. Assuming the injected serum gets distributed uniformly in the bloodstream quickly, what is the approximate total volume of blood in the patient's body? (The half-life of $Na^{24}$ is 15 hours).

Professor PC Thomas Classes · Answer

The time elapsed is exactly one half-life ($t = 15 	ext{ hours} = T_{1/2}$).
The total activity of the $Na^{24}$ in the body after one half-life will be:
$A_t = A_0 / 2 = 1000 	ext{ disintegrations per second}$.
To match units, convert the total activity into disintegrations per minute (dpm):
$A_t = 1000 	imes 60 = 60000 	ext{ dpm}$.
The activity of $1 	ext{ cm}^3$ of blood is $12 	ext{ dpm}$.
Let the total volume of blood be $V 	ext{ cm}^3$. Since the activity is uniformly distributed:
$\frac{60000}{V} = 12 \implies V = \frac{60000}{12} = 5000 	ext{ cm}^3$.
Since $1000 	ext{ cm}^3 = 1 	ext{ Liter}$, the total blood volume is $5.0 	ext{ Liters}$.