Question

Technetium-99 has been used as a radiographic agent in bone scans ( \({ }_{43}^{99} \mathrm{Tc}\) is absorbed by bones). If \({ }_{43}^{99} \mathrm{Tc}\) has a half-life of 6.0 hours, what fraction of an administered dose of \(100 \mu \mathrm{g}\) of \({ }_{43}^{99} \mathrm{Tc}\) remains in a patient's body after 2.0 days?

Short answer

After 2 days, the fraction of the initial dose of Technetium-99 remaining in the patient's body is \( \frac{100}{256} \).

Step-by-step solution

Calculate the number of half lives

To calculate the total number of half-lives passed in 2 days, first convert the days into hours: 2 days × 24 hours/day = 48 hours Now, divide the time that has passed (48 hours) by the half-life (6 hours) to find the number of half lives: Number of half lives = 48 hours / 6 hours/half life = 8 half lives

Calculate the remaining fraction of the initial dose

To find the remaining fraction of the initial dose, we use the formula: Remaining fraction = \((1/2)^n\), where n is the number of half lives. In our case, n = 8. Therefore, the remaining fraction is: Remaining fraction = \((1/2)^8\) = 1/256

Calculate the remaining dose in micrograms

Now, to find the remaining dose in micrograms, we multiply the initial dose (100 µg) by the remaining fraction (1/256): Remaining dose = Initial dose × Remaining fraction Remaining dose = 100 µg × (1/256) = 100/256 µg

Convert the remaining dose to a fraction

Finally, we will convert the remaining dose in micrograms to a fraction of the initial dose: Fraction of initial dose = Remaining dose / Initial dose Fraction of initial dose = (100/256 µg) / (100 µg) = 100/256 So, after 2 days, the fraction of the initial dose of Technetium-99 remaining in the patient's body is 100/256.

Key concepts

Half-Life

Understanding the concept of half-life is essential for various applications, including nuclear medicine and radiometric dating. Half-life is the period it takes for half the atoms in a sample of a radioactive substance to decay. This decay is a random process at the level of single atoms, meaning it's impossible to predict exactly when a particular atom will decay. However, statistically, for a large number of atoms, the half-life is remarkably consistent.

This predictability allows scientists and doctors to calculate how much time will pass before a radioactive compound, like Technetium-99 used in medical imaging, will decrease to a desired level. The formula for calculating the remaining quantity of a substance after a certain number of half-lives is \( (1/2)^n \), where \( n \) represents the number of half-lives. To apply this to practical scenarios, it's often necessary to convert units (e.g., days to hours) to align with the given half-life unit.

Technetium-99

Technetium-99 (\( {}_{43}^{99}Tc \) is a radioactive isotope widely utilized in the field of nuclear medicine, most notably in diagnostic procedures like bone scans. Specifically, Technetium-99m, which is a metastable isotope of Technetium-99, is used because it emits gamma rays that can be detected to provide a clear image of the bone structure.

This isotope has a relatively short half-life – about 6 hours – which is long enough to conduct medical tests but short enough to minimize radiation exposure to the patient. As a chemically versatile element, Technetium-99 can bind to pharmaceuticals that target specific tissues or structures in the body, such as bones, allowing doctors to detect abnormalities such as fractures, cancer, or infection through the captured images.

The controlled use of Technetium-99 requires a deep understanding of its decay rate to ensure its effectiveness and safety for patients, thus reinforcing the importance of half-life calculations in medicine.

Bone Scans as a Radiographic Agent

Bone scans are a type of medical imaging that makes use of radiographic agents to diagnose and track various conditions related to the skeletal system. The patient is administered a radiographic agent, such as Technetium-99m, which has a penchant for being absorbed by bone tissue. As the radioactive material decays, it emits gamma rays that can be detected and imaged, thus highlighting areas of bone metabolism or abnormalities.

The use of Technetium-99 in bone scans is prevalent because its short half-life presents a balance between adequate diagnostic time and lower patient radiation exposure. After the substance is administered, its concentration in the bones is highest, and with time it decays, which means the ability to get a clear image diminishes. This decay also has to be accurately calculated to ensure the safety and health of the patient.

Determining the remaining amount of the substance after a certain period, as illustrated in the original exercise, is critical for understanding the patient's exposure to radiation and the optimal timing for the imaging.