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Chapter 19: Problem 56

To measure the volume of the blood in an animal's circulatory system, the following experiment was performed. A 5.0-mL sample of an aqueous solution containing \(1.7 \times 10^{5}\) counts per minute (cpm) of tritium was injected into the bloodstream. After an adequate period of time to allow for the complete circulation of the tritium, a 5.0-mL sample of blood was withdrawn and found to have \(1.3 \times 10^{3} \mathrm{cpm}\) on the scintillation counter. Assuming that only a negligible amount of tritium has decayed during the experiment, what is the volume of the animal's circulatory system?

Short Answer

Expert verified
Answer: The approximate volume of the animal's circulatory system is 0.038 mL.

Step by step solution

01

Determine the ratio of the cpm of tritium in the aqueous solution to the cpm in the blood sample

First, we find the ratio of the counts per minute (cpm) of tritium in the aqueous solution and the cpm in the blood sample. \(\frac{cpm_{solution}}{cpm_{blood}} = \frac{1.7 \times 10^{5}}{1.3 \times 10^{3}}\)
02

Simplify the cpm ratio

Next, we need to simplify the given cpm ratio. Calculate the resulting ratio. \(\frac{1.7 \times 10^{5}}{1.3 \times 10^{3}} = \frac{170,000}{1,300} \approx 130.77\)
03

Set up the volume ratio

Now, we'll set up the ratio for the volumes. We know the volume of the injected solution, but we want to find the volume of the animal's circulatory system. Let the volume of the circulatory system be denoted by \(V_{circulation}\). Then the ratio of the volumes can be expressed as: \(\frac{V_{solution}}{V_{circulation}}\) Where \(V_{solution} = 5.0 \, \mathrm{mL}\).
04

Equate the cpm ratio with the volume ratio

Now, we know that the ratio of volumes should be equal to the ratio of the cpm of tritium in the aqueous solution and the cpm of tritium in the blood sample. Thus: \(\frac{V_{solution}}{V_{circulation}} = 130.77\)
05

Solve for the unknown volume of the circulatory system

We now need to solve for \(V_{circulation}\). We can do this by cross-multiplying: \(V_{circulation} = \frac{V_{solution}}{130.77}\) Plug in the value for \(V_{solution}\): \(V_{circulation} = \frac{5.0}{130.77}\) Finally, calculate the value for \(V_{circulation}\): \(V_{circulation} \approx 0.038 \, \mathrm{mL}\) So, the volume of the animal's circulatory system is approximately 0.038 mL.

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