A \(0.20-\mathrm{mL}\) sample of a solution containing \({ }_{1}^{3} \mathrm{H}\) that produces \(3.7 \times\) \(10^{3}\) cps is injected into the bloodstream of an animal. After allowing circulatory equilibrium to be established, a \(0.20-\mathrm{mL}\) sample of blood is found to have an activity of 20 . cps. Calculate the blood volume of the animal.

Understanding the concentration of radioactive isotopes in a solution is crucial in various scientific calculations and experiments, especially in the field of nuclear chemistry. Radioactive isotope concentration refers to the amount of a radioactive substance within a given volume. This is commonly measured in counts per second (cps), indicating how many atoms are decaying and thereby emitting radiation each second.

In our exercise scenario, the initial concentration is provided, and the concentration after dilution through the animal's bloodstream is measured. To calculate the change in concentration, one must understand the principle of radioactive dilution. This principle states that when a solution containing radioactive isotopes is diluted, the total activity (the product of concentration and volume) remains unchanged if there is no decay over the course of the experiment.

Therefore, by using the equation: \[A_1 \times V_1 = A_2 \times V_2\] we can relate the initial concentration (activity) of the radioactive substance to its concentration after dilution. This equation is pivotal in calculating parameters such as the blood volume of an organism.

The calculation of an organism's blood volume is a practical application of the aforementioned concepts of radioactive isotope concentration. The concept revolves around injecting a known volume of a radioactive tracer and then, after dilution within the bloodstream, measuring the activity in a blood sample.

The blood volume is crucial for diagnostic and research purposes in both human medicine and veterinary fields. To calculate blood volume accurately, scientists use the formula derived from the conservation of activity during the dilution process: \[V_{blood} = \frac{A_1}{A_2} \times V_{sample}\] where:

• \(V_{blood}\) is the total blood volume,
• \(A_1\) is the initial activity of the injected solution,
• \(A_2\) is the activity measured in the blood sample after equilibrium, and
• \(V_{sample}\) is the volume of the blood sample.

By applying this formula, we can determine the circulatory blood volume using a small sample post-injection, which is a method known for its accuracy and minimal invasiveness.

The use of radioactive materials and the principles of nuclear chemistry have widespread applications in modern science and industry. These applications range from medical diagnostics and treatments, such as the use of radioactive tracers in PET scans and the treatment of cancer with radiotherapy, to environmental tracking where isotopes can trace the path of pollutants.

Moreover, in research, radioactive isotopes are invaluable tools for understanding complex biochemical pathways and reactions by labeling specific molecules and tracking their progress through these systems. Nuclear chemistry also plays a significant role in energy production through nuclear reactors and, increasingly, in the study of sustainability and long-term disposal of nuclear waste.

With every application, the fundamental principles that govern the behavior of radioactive substances, such as decay rates, isotope concentration, and the conservation of activity, are utilized to ensure precise and reliable results in scientific studies.