Question

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 After an adequate period of time to allow for the complete circulation of the tritium, a 5.0-mL sample of blood was withdrawn and found to have \(1.3 \times 10^{3} \mathrm{cpm}\) on the scintillation counter.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

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

Step-by-step solution

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. \(\frac{cpm_{solution}}{cpm_{blood}} = \frac{1.7 \times 10^{5}}{1.3 \times 10^{3}}\)

Simplify the cpm ratio

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

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 \(V_{solution} = 5.0 \, \mathrm{mL}\).

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: \(\frac{V_{solution}}{V_{circulation}} = 130.77\)

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.

Key concepts

Tritium Decay

Tritium is a radioactive isotope of hydrogen. It contains one proton and two neutrons in its nucleus, giving it a mass number of three. In the context of the experiment discussed, tritium serves as a radioactive label to understand how substances circulate within a body. Although tritium is radioactive, it decays very slowly over time.

The decay process of tritium is known as beta decay. During beta decay, a neutron in the nucleus transforms into a proton, releasing a beta particle, which is essentially an electron, along with an anti-neutrino. This process effectively changes the element, converting tritium (hydrogen) into helium-3. The half-life of tritium is approximately 12.3 years, meaning that in this amount of time, half of any amount of tritium will have decayed.

Understanding tritium decay is crucial as it allows scientists to estimate the duration of experiments involving tritium accurately. In the exercise, however, it was assumed that a negligible amount of tritium decayed during the experiment. This implies that the experiment was conducted in a timeframe much shorter than the half-life, ensuring the readings of counts per minute (cpm) remained reliable.

Circulatory System

The circulatory system serves a vital role in transporting substances throughout an organism's body. It includes the heart, blood vessels, and blood, which work together to deliver oxygen and nutrients to cells while removing waste products. In this context, the circulatory system's volume refers to the total amount of blood contained within it.

The experiment aimed to measure this volume by using a tracer, which in this case was tritium. Once injected, the tritium disperses uniformly in the blood, circulating throughout the body. By comparing the concentration of tritium in the injected solution to that in a withdrawn blood sample, the overall blood volume can be estimated.

This method highlights the remarkable aspect of the circulatory system's efficiency, showcasing its ability to distribute substances evenly throughout, which is crucial for maintaining homeostasis and ensuring that physiological functions proceed without interruption.

Radioactive Tracer

Radioactive tracers are substances containing a small amount of radioactive material, used to track the movement of substances within a system. They play a significant role in medical imaging, research, and various industrial applications.

In the given exercise, tritium acted as a radioactive tracer. Once injected into the bloodstream, it mixed with the blood, allowing scientists to trace and measure its distribution within the circulatory system. The choice of tritium was likely due to its ability to emit detectable radiation without significantly affecting the normal biological processes due to its relatively low energy particles.

Using a radioactive tracer like tritium provides several advantages, such as precise measurement and minimal disturbance to the system being studied. This is imperative in both medical diagnostics and in biological research within controlled experiments. Through techniques like scintillation counting, which measures the radiation emitted by the tracer, researchers can gain precise insights into the volumes and flows within complex systems like the circulatory system in this experiment.