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Chapter 11: Problem 77

The half-life of technetium-99m (Problem 11.76) is 6.01 hours. If a sample with an initial activity of \(5.55 \times 10^{5} \mathrm{~Bq}\) is injected into a patient, what is the activity in 24 hours, assuming that none of the sample is excreted?

Short Answer

Expert verified
The activity after 24 hours is approximately 3.47 x 10^4 Bq.

Step by step solution

01

Understand the problem

We need to calculate the remaining activity of a radioactive substance (technetium-99m) after 24 hours, given its half-life and initial activity.
02

Identify the half-life equation

The equation for radioactive decay is \( A = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \), where \( A \) is the remaining activity, \( A_0 \) is the initial activity, \( t \) is the time elapsed, and \( T_{1/2} \) is the half-life.
03

Substitute known values into the equation

Here, \( A_0 = 5.55 \times 10^{5} \text{ Bq} \), \( t = 24 \text{ hours} \), and \( T_{1/2} = 6.01 \text{ hours} \). Substitute these values into the decay formula: \[ A = 5.55 \times 10^{5} \left( \frac{1}{2} \right)^{\frac{24}{6.01}} \]
04

Calculate the remaining activity

First, calculate \( \frac{24}{6.01} \approx 3.99 \). Next, calculate \( \left( \frac{1}{2} \right)^{3.99} \approx 0.0625 \). Finally, calculate \( A = 5.55 \times 10^{5} \times 0.0625 \approx 3.47 \times 10^{4} \text{ Bq} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Half-Life Calculation
The concept of half-life is essential in understanding radioactive decay. It refers to the amount of time it takes for half of the radioactive substance to decay. This constant rate of decay allows scientists to calculate how much of a substance remains after a certain period.

In the case of technetium-99m, its half-life is 6.01 hours. To find the remaining activity after a specific time, we use the formula:
  • \( A = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \)
  • Where \( A \) is the remaining activity, \( A_0 \) is the initial activity, \( t \) is the time elapsed, and \( T_{1/2} \) is the half-life.
This formula uses the half-life to determine how the activity decreases over time, given an initial amount of radioactive material.
Technetium-99m
Technetium-99m is a widely used radioactive isotope in the medical field. Its short half-life of 6.01 hours makes it ideal for diagnostic imaging without prolonged exposure to radiation.

It decays by emitting gamma rays, which can be captured to create detailed images of organs. This property makes technetium-99m especially valuable in nuclear medicine for tracking biological processes and diagnosing various conditions.
  • Its rapid decay ensures that its radioactivity diminishes quickly, reducing potential risks to patients.
  • The use of technetium-99m allows physicians to gain critical insights into a patient's health rapidly and safely.
Activity of Radioactive Substances
The activity of a radioactive substance is measured in becquerels (Bq), representing the number of decays per second. Higher activity means more frequent decay events.

In the original exercise, the initial activity of technetium-99m is given as \(5.55 \times 10^{5}\) Bq. Calculating its activity after 24 hours involves understanding how decay processes reduce activity over time.

Using the half-life formula, activity calculations provide insights into how a substance behaves in a given period:
  • As the substance decays, its activity will decrease exponentially.
  • This understanding allows for precise predictions on how much of a substance remains active after a certain time.
  • Such calculations are vital in fields like medicine and environmental science, where safe handling of radioactive substances is crucial.

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Most popular questions from this chapter

What are the main sources of background radiation?

The nuclear disasters at the Chernobyl nuclear power plant disaster in 1986 and in Fukushima in 2011 resulted in significant releases of radioactive nuclear materials into the environment. Perform a web search to find information about one or both of these disasters and answer the following questions: (a) What was the primary nuclear fuel? What other radioactive materials besides the fuel were released during the accident? (b) What is the estimated amount of radioactive material released into the environment, and in what form was it released? (c) How much additional radiation exposure would be expected for a person living near the power plant? How much additional radiation exposure would be expected for someone living at a considerable distance from the plant? How do these levels compare to the average background radiation dose for the average person? Express your answer as a percentage.

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