To solve part (a), we will use the formula for radioactive decay:

$N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}$

Where:

• $N(t)$ is the amount of substance remaining after time $t$,
• $N_0$ is the initial amount (in this case, 3 g),
• $T_{1/2}$ is the half-life (13 years for plutonium-241),
• $t$ is the time elapsed (80 years).

Let's calculate how much of the 3 g sample remains after 80 years using the provided values.

$N(80) = 3 \times \left( \frac{1}{2} \right)^{\frac{80}{13}}$

Now I will compute this.After 80 years, approximately $0.0421$ grams of the 3-gram sample of plutonium-241 will remain.

Do you need further details or have any questions? Here are 5 related questions to expand on this:

1. How does the half-life formula change if the substance decays at a different rate?
2. What is the total percentage of plutonium-241 that decayed after 80 years?
3. How many half-lives of plutonium-241 occur in 80 years?
4. What is the remaining mass after 40 years using the same formula?
5. How would you solve this problem if you didn't know the time but needed to find it?

Tip: The exponential decay formula is powerful for predicting remaining amounts of any decaying substance if you know the half-life and initial quantity.