Question

A \(0.10-\mathrm{cm}^{3}\) sample of a solution containing a radioactive nuclide \(\left(5.0 \times 10^{3} \text { counts per minute per milliter) is injected }\right.\) into a rat. Several minutes later 1.0 \(\mathrm{cm}^{3}\) of 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 cm³, assuming that the radioactive nuclide is evenly distributed in the rat's blood and the concentration of radioactivity in the blood sample is representative of the concentration in the rat's entire blood volume.

Step-by-step solution

Calculate the total counts of radioactivity

The problem states that the injected radioactive nuclide concentration is 5.0x10^3 counts per minute per milliliter (cpm/ml) and the sample volume is 0.10 cm³. To find the total counts of radioactivity, we'll multiply both values: Total counts of radioactivity \(= \text{nuclide concentration} \times \text{sample volume}\) Total counts of radioactivity \(= (5.0 \times 10^{3} \,\text{cpm/ml}) \times 0.10\, \mathrm{cm}^{3}\) #Step 2: Calculate the total counts of radioactivity injected into the rat#

Calculate the total counts

Now, we can calculate the total counts of radioactivity: Total counts of radioactivity \(= (5.0 \times 10^{3} \,\text{cpm/ml}) \times 0.10\, \mathrm{cm}^{3} = 500\, \mathrm{cpm}\) #Step 3: Find the concentration of the radioactive nuclide in the rat's blood#

Determine the concentration in the rat's blood

The blood sample has a volume of 1.0 cm³ and shows 48 counts per minute of radioactivity. Thus, the concentration of the radioactive nuclide in the rat's blood can be calculated as: Blood concentration \(= \cfrac{\text{counts per minute in blood sample}}{\text{volume of blood sample}}\) Blood concentration \(= \cfrac{48\, \mathrm{cpm}}{1.0\,\mathrm{cm^{3}}}\) Blood concentration \(= 48\, \mathrm{cpm/cm^{3}}\) #Step 4: Calculate the volume of blood in the rat#

Calculate the blood volume

Finally, to find the volume of blood in the rat, we'll use the total counts of radioactivity and the concentration of the radioactive nuclide in the rat's blood: Blood volume \(= \cfrac{\text{total counts of radioactivity}}{\text{blood concentration}}\) Blood volume \(= \cfrac{500\, \mathrm{cpm}}{48\, \mathrm{cpm/cm^{3}}}\) Blood volume \(= 10.42\, \mathrm{cm^{3}}\) #Step 5: State the assumptions made in this calculation#

List the assumptions

In performing this calculation, we made the following assumptions: 1. The radioactive nuclide is evenly distributed in the rat's blood. 2. The concentration of radioactivity in the blood sample is representative of the concentration in the rat's entire blood volume. The volume of blood in the rat is approximately 10.42 cm³.

Key concepts

Understanding Radioactivity Measurement

Radioactivity measurement involves quantifying how many nuclear disintegrations occur in a radioactive substance in a given time. In our scenario, this is counted using a unit called "counts per minute" (cpm). This unit tells us how often radioactive particles are emitted by the nuclide every minute.
Radioactive tracers, like the one used in our rat, help measure biological processes because their emission can be detected precisely.

• "Counts" refers to detecting signals from radioactive decay.
• "Per minute" means we measure how many of these decays happen every minute.

In our exercise, the injected solution has a concentration of 5000 cpm/ml. This value signifies that each milliliter of the solution emits 5000 decay signals every minute, and it's crucial to track how radioactivity decreases in sample analysis.

Calculating Blood Volume Using Tracers

Blood volume calculation using radioactive tracers is a common method in various scientific and medical applications. Here's how it's done. First, a known quantity of radioactive material with a measurable radioactivity level is injected into the bloodstream.
To calculate total blood volume:

• Determine the injected radioactivity: For instance, a 0.10 cm³ sample at 5000 cpm/ml leads to 500 cpm being injected.
• Detect the radioactivity in a blood sample: Measuring a sample of the rat's blood shows 48 cpm in 1.0 cm³.
• Measure the concentration in the entire blood: By dividing 500 cpm by 48 cpm/cm³, we find that the rat's blood volume is approximately 10.42 cm³.

Using this method allows scientists to understand and work with biological systems with minimal invasiveness, crucial in medical diagnostics and treatment monitoring.

Assumptions in Blood Volume Calculations

Scientific calculations, especially those involving living organisms, often rely on assumptions to simplify complex reality. Let's delve into the assumptions needed in the blood volume calculation using radioactive tracers:

• Uniform Distribution: It's assumed the radioactive substance disperses evenly throughout the entire bloodstream. Without this assumption, readings could vastly under or overestimate actual blood volume.
• Representativeness of Sample: We assume that the radioactivity concentration in the blood sample matches that of the whole blood system. This means the sample accurately reflects the conditions elsewhere in the rat's bloodstream.

Both assumptions help ensure simplicity and feasibility in calculations, but it's important to realize they can introduce errors if violated.
For instance, if the nuclide doesn't mix uniformly, the blood volume could be calculated amiss. However, the method offers a good estimate in controlled settings where assumptions likely hold true.