Question

To measure the volume of the blood system of an animal, this experiment was done. A 1.0 -mL sample of an aqueous solution containing tritium with an activity of \(2.0 \times 10^{6} \mathrm{~s}^{-1}\) was injected into the bloodstream. After time was allowed for complete circulatory mixing, a 1.0-mL blood sample was withdrawn and found to have an activity of \(1.5 \times 10^{4} \mathrm{~s}^{-1}\). Calculate the volume of the circulatory system. (The half- life of tritium is 12.3 years, so this experiment assumes that only a negligible quantity of tritium has decayed during the experiment.)

Short answer

The volume of the circulatory system is approximately 133.33 mL.

Step-by-step solution

Understand the Concept

The problem involves calculating the total volume of the animal's circulatory system using the dilution principle. A known activity level of tritium is injected, and after mixing, another sample is taken to determine the final activity. We assume negligible decay during this process.

Identify the Initial Conditions

The initial sample injected has a volume of 1.0 mL and an activity of \(2.0 \times 10^{6} \, \text{s}^{-1}\). This is the total activity before mixing.

Analyze the Final Sample Conditions

After mixing, a 1.0 mL blood sample is withdrawn, showing an activity of \(1.5 \times 10^{4} \, \text{s}^{-1}\). This is lower due to the dilution in the entire blood system.

Use the Dilution Formula

The relation between the initial activity, final activity, and volumes is given by the dilution principle. Here, we calculate the volume \(V\) of the circulatory system using:\[V = \frac{\text{Initial Activity} \times \text{Volume of Injection}}{\text{Final Activity}}\]Substitute the known values: \(V = \frac{2.0 \times 10^{6} \, \text{s}^{-1} \times 1.0 \, \text{mL}}{1.5 \times 10^{4} \, \text{s}^{-1}}\).

Calculate the Circulatory System Volume

Perform the calculation:\[V = \frac{2.0 \times 10^{6}}{1.5 \times 10^{4}} \, \text{mL} = \frac{2000000}{15000} \, \text{mL} = 133.33 \, \text{mL}\]

Interpret the Result

The calculated volume of the circulatory system is approximately 133.33 mL. This is the total volume in which the tritium has been diluted.

Key concepts

Circulatory System Volume

The circulatory system volume refers to the total amount of blood present within an organism's circulatory system. This system is responsible for transporting nutrients, oxygen, and waste products throughout the body. In scientific research, understanding the volume of this system is crucial, especially when studying how substances like medications or tracers, such as radioactive isotopes, distribute in the body.
This volume can be measured using various methodologies, one of which is the dilution principle. By injecting a known concentration of a substance with a measurable activity into the bloodstream, and later extracting a blood sample to analyze the decreased concentration, scientists can back-calculate the total blood volume.
The specifics of this calculation rely heavily on the assumption that the substance has thoroughly mixed throughout the entire circulatory system, allowing for accurate volume estimation.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This is a spontaneous event that leads to the transformation of the atom into a different element or isotope. Understanding radioactive decay is essential in experiments involving radioactive tracers, like tritium.
The decay rate of a radioactive material is characterized by its half-life — the time span in which half of the radioactive atoms decay. In the given exercise, tritium, a radioactive isotope of hydrogen, is used. With a half-life of 12.3 years, tritium decays very slowly.
Because of this long half-life, for short-term experiments, the decay of tritium is negligible, allowing it to be effectively used as a tracer without introducing significant errors. This is why, in the given problem, it is assumed that only a negligible amount of tritium decayed during the experiment.

Tritium Activity

Tritium activity refers to the rate at which tritium atoms in a sample undergo radioactive decay. This rate is typically measured in disintegrations per second (s⁻¹), often referred to as "activity." In the context of the exercise, knowing the activity of tritium both before and after mixing is crucial for determining how much it has been diluted in the blood.
Initially, a known amount of tritium activity is injected into the animal's bloodstream. After allowing sufficient time for mixing within the circulatory system, a blood sample is taken to measure the new activity, which is inevitably lower due to dilution.
This drop in activity directly correlates with how much the tritium has spread throughout the blood, and it helps in calculating the volume of the blood system using the dilution principle. By comparing the initial and final activity levels, the activity serves as a crucial indicator of substance distribution.

Dilution Formula

The dilution formula is a mathematical relationship used to determine concentrations in solutions after dilution. It is particularly useful in experiments involving the distribution of substances within a large volume, such as blood in the circulatory system.
For the exercise at hand, the dilution principle is applied via the formula:\[ V = \frac{\text{Initial Activity} \times \text{Volume of Injection}}{\text{Final Activity}} \]This equation expresses the relationship between initial and final measurements to find out the volume of the system into which the tracer has been dispersed.
Here, the initial activity and volume of injected substance are used alongside the diminished activity in the diluted sample to calculate the total circulatory volume. This computed volume is essential for applications and studies in medicine and physiology, where understanding the distribution of substances through the bloodstream is of primary importance.