Problem 39 - Chemistry

Question

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

Step-by-Step Solution

Answer

After converting 2 days to hours, we get 48 hours. Then, we calculate the number of half-lives by dividing the total time (48 hours) by the half-life period (6 hours), giving us 8 half-lives. To find the remaining fraction of Technetium-99, we use the formula \((\frac{1}{2})^n\), where n is the number of half-lives. In this case, the remaining fraction is \((\frac{1}{2})^8 = \frac{1}{256}\). Finally, to find the remaining dose, we multiply the initial dose (100 µg) by the remaining fraction: \(100 \times \frac{1}{256} = \frac{100}{256} \approx 0.391\, \mu g\). So, approximately 0.391 µg of Technetium-99 remains in the patient's body after 2 days.

Step 1: Convert days to hours

We need to convert the given days to hours since the half-life is in hours. 1 day = 24 hours 2 days = 2 × 24 hours

Step 2: Calculate the number of half-lives

Determine the number of half-lives that have passed in 2.0 days (48 hours). Number of half-lives = total time / half-life period \(n = \frac{t}{T_{1/2}}\) Where: n = number of half-lives t = total time = 48 hours \(T_{1/2}\) = half-life period = 6 hours

Step 3: Find the remaining fraction of Technetium-99

We use the formula: Remaining fraction = \((\frac{1}{2})^n\) Where n is the number of half-lives calculated in step 2.

Step 4: Calculate the final answer

Substitute the values obtained in steps 2 and 3 into the formula to find the remaining fraction of Technetium-99. Then multiply this fraction by the initial administered dose (100 µg) to calculate the final fraction remaining in a patient's body.

Key Concepts

Technetium-99

Technetium-99 (\(^{99m} ext{Tc}\)) is a radioactive isotope commonly used in medical imaging, particularly in bone scans. This isotope is chosen for medical purposes because it emits gamma rays, which can be detected by medical imaging devices. Another significant advantage is its relatively short half-life, which allows it to disappear quickly from the body, reducing radiation exposure.
Some key features of technetium-99 include:

• It is absorbed by bones, making it ideal for bone scans.
• The gamma-ray emissions provide clear images for diagnosis.
• Its short half-life minimizes prolonged radiation, making it safer for patients.

Doctors prefer using technetium-99 not only for its diagnostic power but also because it has been extensively studied and deemed safe for patients.

Half-Life

The concept of half-life is essential to understanding how radioactive substances behave over time. The half-life of a substance is the time it takes for half of a given amount of the radioactive material to decay. For technetium-99, the half-life is 6 hours.
This means:

• After 6 hours, only half of the original amount remains.
• After another 6 hours (12 hours total), one quarter of the initial quantity is left.
• This process continues, with each half-life reducing the amount by half.

Understanding half-life helps medical professionals determine how long a radioactive substance will remain active within the body, thus planning safe and effective treatment periods.

Bone Scans

Bone scans are diagnostic tests that use radioactive materials like technetium-99 to examine the bones. Technetium-99 quickly accumulates in the bone tissue, allowing it to highlight areas of increased bone activity, which may indicate conditions like fractures, infections, or cancer.
The bone scan process usually involves:

• Injecting a trace amount of technetium-99 into the patient's bloodstream.
• Waiting for the isotope to be absorbed by the bones, typically a few hours.
• Using a special camera to detect the gamma rays emitted by technetium-99, creating an image of the bones.

This method is invaluable for early diagnosis and treatment planning, helping doctors provide targeted and timely care for bone-related conditions.

Nuclear Medicine

Nuclear medicine is a field that uses radioactive substances for diagnosis and treatment. It plays a crucial role in modern healthcare by offering precise and effective methods for detecting and treating various diseases. Techniques like bone scans fall under this specialization.
In nuclear medicine:

• Radioisotopes, such as technetium-99, are used to image organs and systems within the body.
• These procedures are minimally invasive and offer quick results.
• It combines techniques from biology, chemistry, and physics to innovate medical diagnostics and treatments.

The use of nuclear medicine in diagnostics has significantly increased the ability of physicians to detect diseases at an early stage, often before symptoms appear, allowing for more efficient and effective treatment interventions.