Question

Technetium- \(99 \mathrm{~m}\) (gamma emitter, \(t_{1 / 2}=6.02 \mathrm{hr}\) ) is widely used for diagnosis in medicine. A sample prepared in the early morning for use that day had an activity of \(4.52 \times 10^{-6} \mathrm{Ci}\). What will its activity be at the end of the day- that is, after \(8.00 \mathrm{hr}\) ?

Short answer

The activity of Technetium-99m at the end of 8.00 hours will be 2.95 x 10^-6 Ci.

Step-by-step solution

Identify the Known Values

The half-life of Technetium-99m is 6.02 hours, the initial activity is 4.52 x 10^-6 Ci, and the time that has passed is 8.00 hours.

Use the Half-Life Formula

The activity of a radioactive isotope after a certain amount of time can be calculated using the formula \( A = A_0 (1/2)^{t/t_{1/2}} \), where \( A \) is the final activity, \( A_0 \) is the initial activity, \( t \) is the time elapsed, and \( t_{1/2} \) is the half-life period of the substance.

Calculate the Number of Half-Lives

First, determine the number of half-lives that have passed by dividing the elapsed time by the half-life: \( n = \frac{t}{t_{1/2}} = \frac{8.00}{6.02} \).

Compute Final Activity

Use the value of \( n \) from Step 3 to determine the final activity using the formula from Step 2: \( A = 4.52 \times 10^{-6} (1/2)^{\frac{8.00}{6.02}} \).

Perform the Calculations

Calculate the result using a scientific calculator or equivalent software to find the final activity at the end of 8.00 hours.

Key concepts

Technetium-99m

Technetium-99m is a radioactive isotope with a wide array of applications, particularly known for its role in medical diagnostic procedures. As a metastable nuclear isomer of technetium-99, it decays by emitting gamma rays, which can be detected to provide detailed images of internal body structures, such as the brain, bones, and heart. This imaging technique is essential for non-invasive diagnoses and is commonly used in the form of a radioisotope scan. What makes Technetium-99m highly valuable is its short half-life of approximately 6.02 hours, which means it decays quickly, minimizing radiation exposure to patients.

Despite its many benefits, handling Technetium-99m requires careful consideration due to its radioactive nature. Healthcare professionals must calculate its activity accurately to ensure that a sufficient amount is available for diagnostic tests during its peak effectiveness, while also ensuring safety standards are met.

Radioactive Isotope Activity

Radioactive isotope activity, often measured in curies (Ci), represents the rate at which the atoms of a radioactive isotope decay. One curie is defined as the amount of any radioactive substance that undergoes 37 billion decay events per second. The activity of a radioactive isotope is a crucial concept in understanding how it behaves over time, and it diminishes as the atoms decay into a more stable form.

The activity level at any given time can indicate the amount of the isotope present and its potential for emitting radiation. This concept is pivotal in disciplines such as nuclear medicine, where accurately measuring the activity of isotopes like Technetium-99m ensures the safety and effectiveness of diagnostic imaging.

Half-Life Formula

The half-life formula is a fundamental equation used in nuclear chemistry to calculate the time required for half of the radioactive atoms in a sample to decay. The formula is expressed as \(A = A_0 (1\/2)^{t\/t_{1/2}}\), where \(A\) is the activity at time \(t\), \(A_0\) is the initial activity, \(t\) is the elapsed time, and \(t_{1/2}\) is the half-life of the substance.

The real-world application of this formula is critical in various fields such as medicinal drug dosing, waste management of radioactive materials, and dating archaeological findings. Understanding and applying the half-life formula allows professionals to predict how quickly a substance will reduce to a safe or effective level.

Nuclear Chemistry

Nuclear chemistry is the branch of chemistry that deals with radioactivity, nuclear processes, and the changes in the structure of atomic nuclei. This field encompasses the study of both naturally occurring and artificially created radioactive isotopes, like Technetium-99m. It plays a significant role in energy production, medicine, and various industrial applications.

In nuclear chemistry, understanding the principles behind radioactive decay, such as alpha and beta particles, gamma rays, and half-life, is essential for harnessing the benefits of radioactivity while managing its potential hazards. This knowledge is utilized in developing nuclear power, treating cancer with radiation therapy, and conducting research within many scientific disciplines.