A patient is given 0.050 mg of technetium-99m (where m means metastable—an unstable but long-lived state), a radioactive isotope with a half-life of about 6.0 hours. How long until the radioactive isotope decays to 6.3 * 10-3 mg?

In nuclear physics, the half-life of a radioactive isotope is a constant that represents the time required for half of the isotope's unstable nuclei to decay. This concept is crucial as it helps us predict how quickly a substance loses its radioactivity. For example, technetium-99m, used in medical diagnostics, has a half-life of 6 hours, meaning that every 6 hours, the amount of this isotope in a sample is reduced by half.

When a patient receives a dose, the healthcare professionals can estimate how long the radioactivity will be at certain levels within the body. This is important for both the effectiveness of the diagnostic procedure and the safety of the patient.

To calculate the decay of a substance, you need to know the initial quantity, the half-life of the substance, and the desired final quantity. Once you have this information, you can work out the number of half-lives that have passed to find the time elapsed or the remaining quantity after a specific time.

Technetium-99m is an unstable, yet commonly used isotope in medical imaging due to its short half-life and the clear images it provides. Over time, technetium-99m decays into technetium-99, which is less harmful and can be eliminated from the body.

Understanding its decay is essential for several reasons:

• Safety: Ensuring that patients and healthcare workers are exposed to the minimal necessary radiation.
• Effectiveness: Calculating the optimal timing for imaging to get the best results.
• Logistics: Managing the supply of technetium-99m, as it needs to be used shortly after production due to its rapid decay.

In our example, a patient received 0.050 mg of technetium-99m. To find out how long it takes to decay to 6.3 x 10^-3 mg, we applied the half-life formula. This practical application illustrates the importance of half-life knowledge in the real-world scenario of nuclear medicine.

Exponential decay problems are mathematical models that describe how the quantity of a substance decreases over time. These problems follow a specific exponential decay formula:

N = N₀e^-λt

where:

• N is the final quantity,
• N₀ is the initial quantity,
• λ (lambda) is the decay constant, related to the half-life,
• t is the time elapsed.

However, to solve these problems hands-on, like the decay of technetium-99m, we often use the half-life formula:

N = N₀(1/2)^k

where k is the number of half-lives that have passed. This formula simplifies the calculations and allows you to solve without knowing the decay constant.

You can rearrange this formula to find the number of half-lives (k) if you know the initial and final quantities. Then, using logarithms to solve for k, you can determine either the time elapsed or the remaining quantity of the substance. Mastery of these formulas is essential for students tackling nuclear chemistry, environmental science, and health-related fields.