Problem 48 - Chemical Principles

Question

Technetium-99 \(\mathrm{m}\) (the \(\mathrm{m}\) signifies a "metastable," or moderately stable, species) is generated in nuclear reactors and shipped to hospitals for use in medical imaging. The radioisotope has a half-life of \(6.01 \mathrm{~h}\). If a 165 -mg sample of technetium- \(99 \mathrm{~m}\) is shipped from a nuclear reactor to a hospital 125 kilometers away in a truck that averages \(50.0 \mathrm{~km} \cdot \mathrm{h}^{-1}\), what mass of technetium\(99 \mathrm{~m}\) will remain when it arrives at the hospital?

Step-by-Step Solution

Answer

Approximately 41.25 mg of technetium-99m will remain when it arrives at the hospital.

Step 1: Calculate the travel time

To determine the travel time to the hospital, divide the distance by the average speed of the truck. Travel time can be calculated by using the formula: \( \text{Travel Time} = \frac{\text{Distance}}{\text{Speed}} \).

Step 2: Determine the number of half-lives

To find out how many half-lives have passed during the travel time, divide the travel time by the half-life of technetium-99m. Use the formula: \( \text{Number of Half-lives} = \frac{\text{Travel Time}}{\text{Half-life}} \).

Step 3: Calculate the remaining mass

The remaining mass of the sample after reaching the hospital can be calculated by repeatedly halving the sample's mass for each half-life elapsed. This can be expressed by the formula: \( \text{Remaining Mass} = \text{Initial Mass} \times \left(\frac{1}{2}\right)^{\text{Number of Half-lives}} \).

Key Concepts

Understanding Half-Life of Isotopes

The half-life of an isotope is the time required for half of the radioactive substance to decay or transform into another element or isotope. This concept is crucial for many scientific and medical applications, particularly in understanding how quickly a radioisotope will lose its radioactivity.

For example, let's say a particular isotope has a half-life of 2 hours. If we start with a 100 gram sample, after 2 hours only 50 grams will remain unchanged. After another 2 hours (a total of 4 hours), 25 grams will remain, and so forth. This decay continues until the isotope is no longer detectable or has transformed completely. The decay process follows a logarithmic scale, meaning it slows down as the quantity of the substance decreases.

The exercise you are dealing with uses the concept of half-life to calculate how much of the technetium-99m is left after a certain period. Understanding this concept is crucial for accurately determining the dosage and effectiveness of the radioisotope when it arrives at the hospital for use in medical imaging.

Radioisotope Applications in Medical Imaging

Medical imaging is an essential tool in modern healthcare, allowing doctors to see inside the body without invasive surgery. Radioisotopes are frequently used in this field because they emit radiation that can be detected by special imaging devices. Technetium-99m, mentioned in the exercise, is extensively used due to its short half-life and the fact that it emits low-energy gamma rays well-suited for imaging.

In medical imaging procedures like single-photon emission computed tomography (SPECT), technetium-99m can be attached to various compounds suitable for targeting different organs or structures in the body. Once administered to a patient, doctors use gamma cameras to capture images from the gamma rays emitted by the radioactive decay of technetium-99m.

Understanding the decay and half-life of radioisotopes like technetium-99m is crucial for scheduling and executing imaging procedures. This ensures that there is a sufficient amount of the radioisotope present for effective imaging, but not so much that it poses an unnecessary radiation risk to the patient.

Nuclear Chemistry and its Importance

Nuclear chemistry deals with the reactions and processes of atomic nuclei, which includes the principles behind radioactive decay and the applications of radioisotopes. Not only does it underpin the calculations for half-life and medical imaging we've discussed, but it also includes the study of nuclear reactions, such as fission and fusion, and their applications in energy production and weaponry.

Nuclear chemistry instruments our understanding of how elements come into being, why radioisotopes emit radiation, and how this radiation can be used or controlled. The ability to harness nuclear reactions is what powers nuclear reactors, which are a source of energy around the world, and supports the development of new medical technologies and treatments.

In educational contexts, like your exercise, focusing on the key elements of nuclear chemistry, such as radioactive decay and half-life, develops a foundational understanding of how these larger processes and applications function and affect the world we live in.