Problem 30 - Chemistry and Chemical Reactivity

Question

Radioactive gold-198 is used in the diagnosis of liver problems. The half-life of this isotope is 2.7 days. If you begin with a 5.6-mg sample of the isotope, how much of this sample remains after 1.0 day?

Step-by-Step Solution

Answer

After 1.0 day, approximately 4.32 mg of gold-198 remains.

Step 1: Understand the Concept of Half-Life

The half-life of a radioactive isotope is the time required for half of the isotope in a sample to decay. For gold-198, this time is 2.7 days, meaning every 2.7 days, half of the gold-198 sample will remain.

Step 2: Identify Variables

We are given the initial mass of the sample (5.6 mg) and the time in days for which we need to determine how much remains (1.0 day). The half-life of the isotope is 2.7 days.

Step 3: Use the Decay Formula

We use the decay formula: \[ A = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \] Where: - \( A \) is the amount of substance left after time \( t \), - \( A_0 \) is the initial amount, - \( t \) is the time elapsed, - \( T_{1/2} \) is the half-life of the substance.

Step 4: Substitute the Values into the Formula

Plug the values into the formula: \[ A = 5.6 \left( \frac{1}{2} \right)^{\frac{1.0}{2.7}} \]

Step 5: Calculate the Exponent

Calculate the exponent: \[ \frac{1.0}{2.7} \approx 0.3704 \]

Step 6: Compute the Remaining Quantity

Now compute: \[ \left( \frac{1}{2} \right)^{0.3704} \approx 0.7711 \] Multiply this by the initial amount: \[ 5.6 \times 0.7711 \approx 4.32 \text{ mg} \]

Step 7: Conclusion

After 1.0 day, approximately 4.32 mg of gold-198 remains from the original 5.6 mg sample.

Key Concepts

Half-life

Half-life is a core concept in understanding radioactive decay. It refers to the time it takes for half of a sample of a radioactive substance to decay into another substance. Each isotope has its own unique half-life, like gold-198, which has a half-life of 2.7 days. This means that if you start with a certain amount of gold-198, only half of it will remain after 2.7 days.

Knowing this, you can predict how long it will take for a sample to substantially decay. The predictable nature of half-life makes it useful for practical applications:

• Medical diagnostics, like using isotopes to detect liver issues.
• Carbon dating, which uses the half-life of carbon-14 to determine the age of artifacts.
• Calculating dosage levels for certain treatments.

Understanding half-life can empower you to understand more complex decay processes.
It's a foundational piece of knowledge when studying nuclear physics or chemistry.

Gold-198

Gold-198 is a radioactive isotope of gold, frequently used in medicine. This isotope is particularly useful in the field of diagnostic imaging, often aiding in detecting liver issues. Its radioactivity emits gamma rays, which can be captured and used to create diagnostic images.

A unique property of gold-198 is its relatively short half-life of 2.7 days. This makes it ideal for medical applications as it decays quickly.

• The short half-life minimizes long-term radiation exposure risks to patients.
• It ensures that the isotope doesn't linger in the body longer than necessary.
• Its rapid decay also makes it safer to handle and store.

Gold-198 offers a practical application in the medical field by allowing non-invasive imaging solutions. Understanding the properties and uses of gold-198 can help appreciate the benefits and limitations of radioactive materials in medicine.

Decay Formula

The decay formula is a mathematical expression used to calculate the remaining amount of a substance over time, given its half-life. The formula is:\[ A = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]Here, \( A \) represents the amount remaining, \( A_0 \) is the initial amount, \( t \) is the time elapsed, and \( T_{1/2} \) is the half-life.

This formula relies on the principle that each half-life period will reduce the remaining amount of substance by 50%. As a result, calculating using this formula helps predict how much of the isotope will persist after a specified amount of time:

• The formula forms a foundation for applications across various scientific fields.
• It's used in nuclear chemistry to predict radioactive decay outcomes.
• Through examples like gold-198, you see its practical applications in medicine.

Utilizing the decay formula lets you model and understand the behavior of isotopes over time. It provides the mathematical insight needed to engage with real-world decay processes effectively.