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Chapter 18: Problem 81

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

Expert verified
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

01

(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.
02

(Step 2: Convert volume units)

We will convert the injected volume measurement to milliliters (mL): \(V_\mathrm{injected} = 0.10\,\mathrm{cm}^3 = 0.10\,\mathrm{mL}\)
03

(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.
04

(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: \(V_\mathrm{rat} = \frac{500\,\mathrm{counts/min}}{48\,\mathrm{counts/min}} \times 1.0\,\mathrm{mL}\) \(V_\mathrm{rat} \approx 10.42\,\mathrm{mL}\)
05

(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.
06

(Conclusion)

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

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Most popular questions from this chapter

Iodine-131 has a half-life of 8.0 days. How many days will it take for 174 g of \(^{131}\) I to decay to 83 g of \(^{131}\) I?

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