Question

Carbon-11 is a radioactive isotope used in positron emission tomography (PET) scans for medical diagnosis. Positron emission is another, though rare, type of radioactivity. The half-life of carbon-11 is 20.3 min. If \(4.23 \times 10^{-6} \mathrm{~g}\) of carbon- 11 is left in the body after \(4.00 \mathrm{~h}\), what mass of carbon-11 was present initially?

Short answer

The initial mass of carbon-11 was approximately 8.26 mg.

Step-by-step solution

Understand the Problem

We need to find the initial mass of Carbon-11, knowing the final mass after 4 hours and the half-life of Carbon-11. The half-life is the time taken for half of the substance to decay.

Convert Time

Convert the decay time from hours to minutes since the half-life is in minutes. \( 4 \text{ hours} = 4 \times 60 = 240 \text{ minutes} \).

Calculate the Number of Half-Lives

Determine how many half-lives have passed during the decay process. This can be calculated using the formula: \( n = \frac{\text{total time}}{\text{half-life}} \). This gives \( n = \frac{240}{20.3} \approx 11.82 \) half-lives.

Use the Decay Formula

Use the decay formula to relate initial and final quantities of the substance: \( \frac{N}{N_0} = 0.5^n \), where \( N \) is the final mass, \( N_0 \) is the initial mass, and \( n \) is the number of half-lives. Substituting the known values: \( \frac{4.23 \times 10^{-6}}{N_0} = 0.5^{11.82} \).

Solve for Initial Mass

Rearrange the decay formula to solve for \( N_0 \):\[ N_0 = \frac{4.23 \times 10^{-6}}{0.5^{11.82}} \]Calculate \( 0.5^{11.82} \) and then solve for \( N_0 \). After computing, \( N_0 \approx 8.26 \times 10^{-3} \) g.

Key concepts

Half-life calculation

The concept of half-life is a fundamental aspect of radioactive decay. It's used to determine how long it will take for a substance to reduce by half. In the case of Carbon-11, the half-life is 20.3 minutes. This means that every 20.3 minutes, only half of the Carbon-11 remains. Understanding how to calculate the half-life helps us find the initial amount of a substance if we know how much is left after a certain time period.
To calculate the number of half-lives that have passed, use the formula:

• \( n = \frac{\text{total time}}{\text{half-life}} \)

Substituting in the exercise's values, 240 minutes of decay occurred, allowing us to determine the number of half-lives passed is approximately 11.82.
The decay formula \( \frac{N}{N_0} = 0.5^n \) connects the initial and final quantities of Carbon-11, allowing us to determine the original mass if the final mass and number of half-lives are known.

Positron emission tomography (PET)

Positron Emission Tomography, or PET, is a type of imaging technology used to observe metabolic processes in the human body. This method is particularly useful in diagnosing diseases, such as cancer and brain disorders. It involves injecting a small amount of radioactive material, such as Carbon-11, into the body. As this isotope decays, it emits positrons, which collide with electrons producing gamma rays.
These gamma rays are detected by a PET scanner, allowing doctors to create a detailed image of internal processes. This method provides extremely valuable insights because it displays how organs and tissues are functioning in real-time. The precise tracking of isotopes like Carbon-11 thus plays a crucial role in understanding bodily functions and diagnosing potential medical conditions.
PET scans are non-invasive and provide meaningful information that often leads to significant medical breakthroughs in both research and treatment.

Carbon-11 isotope

Carbon-11 is a radioactive isotope frequently used in PET scans. It has a relatively short half-life of 20.3 minutes, which necessitates its quick synthesis and use in medical imaging. Despite its short-lived nature, it is highly valuable due to its ability to integrate into various biological molecules, enabling its use in a variety of medical studies and diagnostic tests.
This isotope facilitates the tracking of physiological changes and observes pathological processes in real-life scenarios. Because of its high energy decay and interaction dynamics, it can produce clear images crucial for diagnosing diseases.
Though its presence in the body is brief, the emitted positrons from Carbon-11 provide critical data that influences medical decisions and treatments, showcasing the importance of such isotopes in advancing medical science.

Medical diagnostic imaging

Medical diagnostic imaging encompasses various technologies utilized to observe the internal workings of the body. Techniques like PET scans, CT scans, MRI, and X-rays, provide non-invasive ways to view organs, tissues, and bones.
PET scans, specifically, rely on radioactive isotopes like Carbon-11, which help capture detailed images. This has revolutionized how practitioners diagnose, manage, and understand numerous diseases. These imaging forms offer insight into both structural and functional aspects of medical conditions.
The importance of diagnostic imaging lies in its ability to offer vivid pictures without physical intervention, aiding in early detection and treatment planning for patients. It is an integral part of modern medicine, ensuring health professionals have the necessary tools to deliver precise and effective care.