Question

A \(120-\mathrm{mg}\) sample of technetium- \(99 \mathrm{~m}\) is used for a diagnostic test. If technetium- \(99 \mathrm{~m}\) has a half-life of \(6.0 \mathrm{~h}\), how much of the technetium- \(99 \mathrm{~m}\) sample remains \(24 \mathrm{~h}\) after the test?

Short answer

The remaining amount of technetium-99m after 24 hours is 7.5 mg.

Step-by-step solution

Understand the half-life concept

The half-life of a substance is the time it takes for half of the substance to decay. Technetium-99m has a half-life of 6.0 hours.

Calculate the number of half-life periods

Determine the number of half-lives in the given time period: Given time = 24 hours Half-life = 6 hours Number of half-lives = \( \frac{24}{6} = 4 \).

Apply the half-life formula

Use the formula to determine the remaining amount of the substance: \( \text{Remaining amount} = \text{Initial amount} \times (\frac{1}{2})^{\text{Number of half-lives}} \).Substitute the values: \( \text{Remaining amount} = 120 \times (\frac{1}{2})^{4} \).

Simplify the expression

Calculate the power of \( (\frac{1}{2})^{4} \): \( (\frac{1}{2})^{4} = \frac{1}{16} \). Thus, the remaining amount is: \( 120 \times \frac{1}{16} = 7.5 \mathrm{mg} \).

Conclusion

So, after 24 hours, only 7.5 mg of the 120 mg sample of technetium-99m will remain.

Key concepts

technetium-99m

Technetium-99m, often abbreviated as Tc-99m, is a radioactive isotope widely used in medical diagnostics.
Its unique properties make it highly useful for imaging and functional studies of tissues and organs.
Tc-99m emits gamma rays, which can be detected by a camera to produce detailed images of the inside of the body.

One of the reasons Tc-99m is favored in medicine is its relatively short half-life of 6 hours.
This short half-life means it decays quickly, minimizing radiation exposure to the patient.
It also allows for rapid imaging procedures.
This balance between adequate imaging time and patient safety makes Tc-99m an invaluable tool in nuclear medicine.

radioactive decay

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation.
Different types of radioactive decay include alpha, beta, and gamma decay.

In the context of Tc-99m, gamma decay is of particular interest.
During gamma decay, the nucleus of Tc-99m releases gamma radiation without changing the number of protons or neutrons.
This emission is what allows for medical imaging techniques to visualize internal structures.

Understanding radioactive decay is crucial because it directly impacts how we calculate the remaining amount of a radioactive substance over time.
The rate of decay follows a predictable pattern, known as the half-life, which is vital for applications like medical imaging and radiotherapy.

half-life formula

The half-life formula is used to determine how much of a radioactive substance remains after a given period.
The general formula is: \[ \text{Remaining amount} = \text{Initial amount} \times \bigg(\frac{1}{2}\bigg)^{\text{Number of half-lives}} \]

For example, in our exercise, we wanted to find out how much Tc-99m remains after 24 hours if its half-life is 6 hours.
First, we calculate the number of half-lives by dividing the total time by the half-life: \[ \frac{24 \text{ hours}}{6 \text{ hours}} = 4 \]

Then, we apply the half-life formula: \[ \text{Remaining amount} = 120 \text{ mg} \times \bigg(\frac{1}{2}\bigg)^{4} \]
This simplifies to: \[ 120 \text{ mg} \times \frac{1}{16} = 7.5 \text{ mg} \]

So, after 24 hours, only 7.5 mg of the initial 120 mg of Tc-99m remains.
Understanding the half-life formula is fundamental for calculating the decay of radioactive substances over time.