Problem 31 - Chemistry and Chemical Reactivity

Question

Gold-198 is used in the diagnosis of liver problems. The half-life of \(^{198} \mathrm{Au}\) is 2.69 days. If you begin with \(2.8 \mu \mathrm{g}\) of this gold isotope, what mass remains after 10.8 days?

Step-by-Step Solution

Answer

After 10.8 days, approximately 0.164 µg of Gold-198 remains.

Step 1: Understanding Half-Life

To solve this problem, we need to understand that half-life is the time it takes for half of a substance to decay. The half-life of Gold-198 is given as 2.69 days.

Step 2: Calculate Number of Half-Lives

Determine how many half-lives have passed in the given time period. Use the formula: \( \text{Number of half-lives} = \frac{\text{Total time elapsed}}{\text{Half-life period}} \). Here, it is \( \frac{10.8 \text{ days}}{2.69 \text{ days}} \approx 4.02 \).

Step 3: Calculate Remaining Mass

The remaining mass after a certain number of half-lives can be found using the formula: \( \text{Remaining mass} = \text{Initial mass} \times \left(\frac{1}{2}\right)^{\text{Number of half-lives}} \). Substituting the values, we get \( 2.8 \times \left(\frac{1}{2}\right)^{4.02} \approx 0.164 \mu g \).

Key Concepts

Half-Life Calculation

Half-life is a fundamental concept in nuclear chemistry. It describes the time it takes for half of the atoms in a radioactive substance to decay. This is a crucial characteristic of each radioactive isotope. For instance, Gold-198 has a half-life of 2.69 days. This means that every 2.69 days, half of any amount of Gold-198 will have decayed into another element.

To find out how much of a sample remains after a certain period, you need to calculate how many half-lives have passed. This is done using the formula:

• \(\text{Number of half-lives} = \frac{\text{Total time elapsed}}{\text{Half-life period}}\)

Once you know the number of half-lives, you can determine the remaining mass:

• \(\text{Remaining mass} = \text{Initial mass} \times \left(\frac{1}{2}\right)^{\text{Number of half-lives}}\)

This mathematical approach provides an efficient way to track how radioactive decay reduces the quantity of a material over time.

Isotopes

Isotopes are different forms of the same element. They have the same number of protons but differ in the number of neutrons. This difference affects their atomic masses but not their chemical properties.

Gold-198 is an isotope of gold. Its atomic number is 79, which tells us it has 79 protons. However, because it is an isotope, the neutron count doesn't match that of the most stable gold isotope.

The unique combination of protons and neutrons in Gold-198 makes it radioactive, meaning it can emit radiation as it decays. This property is why it's useful in medical applications, like diagnosing liver problems. Understanding isotopes helps scientists and medical professionals leverage specific characteristics of certain elements to solve real-world issues.

Nuclear Chemistry

Nuclear chemistry involves the study of changes in atomic nuclei, which includes reactions that involve the core of the atom. Many of these reactions concern the forces that hold the nucleus together and the energy changes accompanying any alterations in these forces.

Radioactive decay is a primary focus of nuclear chemistry. It involves the transformation of an unstable nucleus into a more stable arrangement by releasing particles and energy. In the case of radioactive isotopes like Gold-198, nuclear reactions dictate how the material changes over time.

• Energy released during these reactions can be harnessed for practical applications, such as electricity generation or medical treatment.
• Understanding these chemical processes can also help in managing nuclear waste and improving radiation safety protocols.
  

Nuclear chemistry fundamentally enriches our understanding of both everyday and extraordinary phenomena, explaining the unseen changes happening within atomic nuclei.