Problem 33 - Chemistry An Atoms First Approach

Question

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

Step-by-Step Solution

Answer

After 2 days (48 hours), approximately \(3.18\%\) of the administered dose of \(100 \: \mu g\) of Technetium-99 remains in the patient's body.

Step 1: Convert the given time to hours

Since we are given the half-life in hours, we need to convert the given time of 2.0 days to hours. There are 24 hours in a day, so 2.0 days are equal to \( 2 \times 24 \) hours. 2 days = \(2 \times 24 \) hours = 48 hours.

Step 2: Calculate the decay constant

The decay constant, denoted as λ, can be found using the formula: λ = \( \frac{log(2)}{t_{1/2}} \), where \(t_{1/2}\) is the half-life of the radioactive isotope. Using the half-life of 6.0 hours, we get: λ = \( \frac{log(2)}{6} \) ≈ 0.1155 h⁻¹.

Step 3: Apply the decay formula

Now, we will use the decay formula to find the remaining amount of Technetium-99 after 48 hours: \(N_t = N_0 \times e^{-\lambda t}\), where: - \(N_t\) is the remaining amount of Technetium-99 after time \(t\), - \(N_0\) is the initial amount of Technetium-99, - \(\lambda\) is the decay constant, and - \(t\) is the time in hours. We use \(N_0 = 100\) µg, \(\lambda = 0.1155\) h⁻¹, and \(t = 48\) h: \(N_t = 100\times e^{-0.1155 \times 48}\) ≈ 3.18 µg.

Step 4: Calculate the fraction of the remaining dose

Next, we calculate the fraction of the remaining dose by dividing the remaining amount of Technetium-99 by the initial administered dose: Fraction = \( \frac{N_t}{N_0} \) = \( \frac{3.18}{100} \) = 0.0318. To express this as a percentage, multiply by 100: Percentage = 0.0318 × 100 ≈ 3.18%. Therefore, after 2 days (48 hours), about 3.18% of the administered dose of Technetium-99 remains in the patient's body.

Key Concepts

Technetium-99

Technetium-99, often represented as Tc-99, is a crucial element in the field of nuclear medicine. It is a radioactive isotope that belongs to the chemical element technetium. Technetium-99 is widely used in medical imaging, specifically for bone scans.
This isotope is absorbed by bones, allowing doctors to see detailed images of bone structure and identify any abnormalities or diseases.
One reason Tc-99 is preferred in medical diagnostics is due to its properties:

• It emits gamma rays, which are ideal for imaging purposes.
• It has a relatively short half-life of 6 hours, which minimizes radiation exposure to patients.

With its widespread use, Tc-99 helps in diagnosing a variety of conditions, from fractures to cancers, making it a vital tool in healthcare diagnostics.

Half-life Calculation

Half-life is a critical concept in understanding radioactive decay. It represents the time taken for half of a radioactive substance to disintegrate. In essence, it measures how quickly a substance loses its radioactivity.

In the context of Technetium-99, the half-life is 6.0 hours. This means every 6 hours, the amount of Tc-99 reduces to half. Calculating how much radioactive material remains after a certain time is a common practice in medical and scientific calculations.
To determine how much of a substance remains, following these steps is typical:

• Convert time to consistent units, often hours, as half-life is given in hours.
• Use the decay formula: \[N_t = N_0 imes e^{-rac{log(2)}{t_{1/2}} imes t}\]Where:
  • \( N_0 \) is the initial amount.
  • \( N_t \) is the remaining amount.
  • \( t_{1/2} \) is the half-life.
  • \( t \) is the elapsed time.

This approach allows researchers and medical professionals to calculate the decay and plan accordingly.

Radiographic Agents

Radiographic agents are substances used in medical imaging to enhance the visibility of internal bodily structures. These agents, like Technetium-99, play a significant role in diagnostic imaging by providing contrast in scans.
This increased contrast helps doctors see clearly defined images, aiding in the accurate diagnosis of various conditions. Radiographic agents may be:

• Injected into the body.
• Ingested orally.
• Administered rectally.

When using Technetium-99, it is typically injected, as it is absorbed by target tissues like bones, providing a clear view during imaging processes like bone scans. Understanding the role of radiographic agents is crucial for analyzing their interactions with specific isotopes, ensuring successful and safe diagnostic procedures.

Decay Constant

The decay constant, denoted as \( \lambda \), is an important parameter in the study of radioactive decay. It is a measure of the probability per unit time that a given atom will decay. Essentially, it aids in quantifying the rate at which a radioactive substance loses its activity.
To compute the decay constant for a substance like Technetium-99, the following formula can be used:\[\lambda = \frac{log(2)}{t_{1/2}}\]Where:

• \( log(2) \) is the natural logarithm of 2, approximately 0.693.
• \( t_{1/2} \) is the half-life of the substance, for Tc-99 it's 6 hours.

Once calculated, the decay constant provides insights into how quickly the isotope decays, essential for scheduling diagnostic scans and managing patient care, ensuring optimal use with minimal exposure.

Radiopharmaceutical

Radiopharmaceuticals are a class of pharmaceuticals that contain radioactive isotopes, such as Technetium-99, used for diagnostic and therapeutic purposes in medicine. Technetium-99 plays a pivotal role in radiopharmaceuticals due to its effective imaging properties and minimal patient radiation dose.

These compounds exhibit several characteristics:

• They are designed to target specific organs, tissues, or cellular receptors within the body.
• They provide detailed images to observe physiological processes or locate diseases.
• Their short half-lives ensure patient safety by minimizing radiation exposure.

In diagnostics, Tc-99 radiopharmaceuticals help visualize bones, organs, and other physiological processes, making them invaluable in detecting cancer, heart diseases, and more. Safe preparation and administration are crucial to maximize their benefits while minimizing risks.