Question

A \(0.10-\mathrm{cm}^{3}\) sample of a solution containing a radioactive nuclide \(\left(5.0 \times 10^{3}\) counts per minute per milliliter) is injected \right. 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 estimated volume of blood in the rat is approximately \(10.42\,\mathrm{mL}\), under the assumptions of complete mixing of the radioactive solution with the rat's blood, uniform distribution of the radioactive nuclide throughout the rat's blood, and no chemical reactions or decay occurring between the radioactive nuclide and the blood components.

Step-by-step solution

(Step 1: Variables Setup)

Let's define the variables: \(V_\mathrm{injected} = 0.10\,\mathrm{cm}^3\) - the volume of the injected solution \(\rho_\mathrm{injected} = 5.0\times10^3\,\mathrm{counts/min/mL}\) - radioactivity concentration of the injected solution \(V_\mathrm{blood\_ sample} = 1.0\,\mathrm{cm}^3\) - the volume of the removed blood sample \(\rho_\mathrm{blood\_ sample} = 48\,\mathrm{counts/min}\) - radioactivity concentration of the removed blood sample \(V_\mathrm{rat}\) - the total volume of blood in the rat, which we want to find.

(Step 2: Convert volume units)

We will convert the injected volume measurement to milliliters (mL): \(V_\mathrm{injected} = 0.10\,\mathrm{cm}^3 = 0.10\,\mathrm{mL}\)

(Step 3: Calculate the number of counts)

We will calculate the initial number of counts from the injected solution: \(C_\mathrm{injected} = V_\mathrm{injected} \times \rho_\mathrm{injected}\) \(C_\mathrm{injected} = 0.10\,\mathrm{mL} \times 5.0\times10^3\,\mathrm{counts/min/mL} = 500\,\mathrm{counts/min}\) When injecting the solution, all the counts (\(C_\mathrm{injected}\)) get mixed with the rat's blood, and the sample has 48 counts/min.

(Step 4: Solve the proportion)

We can set up a proportion to solve for the unknown total volume of blood in the rat: \(\frac{C_\mathrm{injected}}{C_\mathrm{blood\_ sample}} = \frac{V_\mathrm{rat}}{V_\mathrm{blood\_ sample}}\) Substitute the known values into the proportion: \(\frac{500\,\mathrm{counts/min}}{48\,\mathrm{counts/min}} = \frac{V_\mathrm{rat}}{1.0\,\mathrm{mL}}\) Now, we can solve for the volume of blood in the rat: \(V_\mathrm{rat} = \frac{500\,\mathrm{counts/min}}{48\,\mathrm{counts/min}} \times 1.0\,\mathrm{mL}\) \(V_\mathrm{rat} \approx 10.42\,\mathrm{mL}\)

(Step 5: State the assumptions)

The assumptions made in calculating the volume of blood in the rat are: 1. Complete mixing of the radioactive solution with the rat's blood. 2. Uniform distribution of the radioactive nuclide throughout the rat's blood. 3. No chemical reactions or decay occurring between the radioactive nuclide and the blood components.

(Conclusion)

Based on the given information and calculations, the estimated volume of blood in the rat is approximately \(10.42\,\mathrm{mL}\).

Key concepts

Radioactivity Calculation

When dealing with radioactive substances, calculations help quantify the amount of radioactivity present within different samples. In our exercise, the fundamental goal is to find out how much blood exists in a rat after a radioactive solution is injected. To calculate this, we first determine the level of radioactivity, expressed as counts per minute (CPM), in both the injected solution and the blood taken from the rat. The initial number of counts from the injected sample can be calculated using the formula:

\[ C_\mathrm{injected} = V_\mathrm{injected} \times \rho_\mathrm{injected} \]where:

• \( V_\mathrm{injected} \) is the volume of the injected solution in mL,
• \( \rho_\mathrm{injected} \) is the radioactivity concentration of the injected solution.

After the solution is injected, it mixes with the rat's blood. We also measure the radioactivity in the blood sample taken from the rat. Using these values, we can calculate the total volume of blood by setting up a proportion between the initial and sampled radioactivity. This allows us to relate the known values to the unknown blood volume. Understanding these steps carefully makes it easier not only to solve the exercise but also to apply similar concepts to other scientific problems involving radioactivity.

Proportion Method

The proportion method is a powerful tool in calculations involving comparisons between two ratios. In this problem, it serves to determine the rat's blood volume by comparing the counts of radioactivity before and after mixing with the blood. The principle behind this method is that the proportion of counts before mixing remains the same after mixing, allowing for a simple calculation.

Using the equation:
\[ \frac{C_\mathrm{injected}}{C_\mathrm{blood\_ sample}} = \frac{V_\mathrm{rat}}{V_\mathrm{blood\_ sample}} \]we set up a relationship between the injected counts and the sampled counts with the respective volumes.

• \( C_\mathrm{injected}\) and \( C_\mathrm{blood\_ sample}\) are the counts per minute of the solution injected and the blood sample taken, respectively.
• \( V_\mathrm{rat}\) is the unknown we want to find, representing the total volume of blood in the rat.
• \( V_\mathrm{blood\_ sample} \) is the volume of the blood sample taken.

By inserting the known values, the formula simplifies the process, making it straightforward to solve for \( V_\mathrm{rat} \). This method efficiently utilizes ratios to find unknown quantities, provided all conditions assumed remain valid.

Assumptions in Scientific Experiments

Any scientific experiment, especially involving biological systems, requires specific assumptions to ensure calculations and results are valid. In our radioactivity measurement, we make several key assumptions:

• **Complete mixing:** It's assumed that once the radioactive solution is injected into the rat, it thoroughly mixes with all of the rat's blood. This ensures the radioactivity concentration is uniform across all samples.
• **Uniform distribution:** The distribution of radioactive nuclides must be even throughout the blood. This prevents inaccuracies when measuring the concentration in the blood sample compared to the expected total blood volume.
• **No interfering reactions:** Finally, it's assumed that there are no reactions or decay processes affecting the radioactive nuclide within the blood. If reactions occurred, they might alter the counts per minute, introducing significant error in the calculations.

Assumptions like these are crucial because they lay the foundation for the model's accuracy used to solve the exercise. Without clearly defined assumptions, any change in these conditions could lead to different outcomes, thus questioning the validity of any experimental results.