Question

A $0.10-{\mathrm{cm}}^3$ sample of a solution containing a radioactive nuclide $(5.0\times {10}^3$ counts per minute per milliliter) is injected into a rat. Several minutes later $1.0{\mathrm{cm}}^3$ blood is removed. The blood shows 48 counts per minute of radioactivity. Calculate the volume of blood in the rat. What assumptions must be made in performing this calculation?

Short answer

The volume of blood in the rat is approximately $10.42{\text{cm}}^3$. The assumptions made during this calculation include that the radioactive nuclide is uniformly distributed within the blood, and there is no loss of radioactivity in the rat.

Step-by-step solution

Determine the initial radioactivity of the injected solution

The concentration of radioactive nuclide in the injected solution is given as $5.0\times {10}^3$ counts per minute per milliliter (or per cm³, since 1 mL = 1 cm³). The volume of the injected solution is 0.10 cm³, so the total radioactive nuclide count in this solution can be calculated as: Radioactivity_count = concentration × volume Radioactivity_count = $(5.0\times {10}^3)\frac{\text{counts}}{\text{minute}\cdot {\text{cm}}^3}\cdot 0.10{\text{cm}}^3$

Calculate the total radioactive nuclide count in the injected solution

Now we multiply the concentration and volume to find the total radioactive nuclide count: Radioactivity_count = $(5.0\times {10}^3)\cdot 0.10=500\frac{\text{counts}}{\text{minute}}$ This means that 500 counts per minute of radioactive nuclide were injected into the rat.

Calculate the concentration of radioactive nuclide in the removed blood sample

We are given that the removed blood sample shows 48 counts per minute of radioactivity and has a volume of 1.0 cm³. Using this, we can find the concentration of radioactive nuclide in the blood: Blood_concentration = $\frac{\text{Radioactivity\_count}}{\text{Volume}}$ Blood_concentration = $\frac{48\frac{\text{counts}}{\text{minute}}}{1.0{\text{cm}}^3}$

Calculate the volume of blood in the rat using mass balance

We assume that the radioactive nuclide is uniformly distributed within the blood. Therefore, the total radioactive nuclide count in the blood remains constant at 500 counts per minute. Let V be the volume of blood in the rat. Total radioactive nuclide count = Blood_concentration × Volume of blood in the rat 500 ~\frac{\text{counts}}{\text{minute}} = 48 ~\frac{\text{counts}}{\text{minute} \cdot \text{cm}^3} \cdot V~\text{cm}^3 Now we solve for V: V = $\frac{500\frac{\text{counts}}{\text{minute}}}{48\frac{\text{counts}}{\text{minute}\cdot {\text{cm}}^3}}$

Calculate the volume of blood in the rat

Now we perform the calculation to find the volume of blood in the rat: V = $\frac{500}{48}=10.42{\text{cm}}^3$ So, the volume of blood in the rat is approximately 10.42 cm³. Assumptions made while performing this calculation include: 1. The radioactive nuclide is uniformly distributed within the blood. 2. There is no loss of radioactivity in the rat, and the radioactive nuclide count remains constant.