Question

Carbon-11 has a half-life of about 20 minutes. If we start with 200 grams, how much will be left after 3 hours?

A. 25 grams

B. 2.5 grams

C. 0.39 grams

D. 0.039 grams

Answer 1

To determine how much Carbon-11 will remain after 3 hours, we first need to calculate how many half-lives will occur in that time period.

1. **Convert hours to minutes**:
\[
3 \text{ hours} = 3 \times 60 = 180 \text{ minutes}
\]

2. **Determine the number of half-lives**:
The half-life of Carbon-11 is 20 minutes. Thus, the number of half-lives in 180 minutes is:
\[
\text{Number of half-lives} = \frac{180 \text{ minutes}}{20 \text{ minutes/half-life}} = 9 \text{ half-lives}
\]

3. **Calculate the remaining amount**:
Initially, we have 200 grams of Carbon-11. After each half-life, the amount remaining is half of the previous amount. The formula for the remaining amount after \( n \) half-lives is:
\[
\text{Remaining amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^n
\]
where \( n \) is the number of half-lives.

Substituting the values:
\[
\text{Remaining amount} = 200 \times \left(\frac{1}{2}\right)^9
\]

4. **Calculating \(\left(\frac{1}{2}\right)^9\)**:
\[
\left(\frac{1}{2}\right)^9 = \frac{1}{512}
\]

5. **Now calculate the remaining amount**:
\[
\text{Remaining amount} = 200 \times \frac{1}{512} = \frac{200}{512} = \frac{25}{64} \approx 0.390625 \text{ grams}
\]

Thus, the amount of Carbon-11 remaining after 3 hours is approximately **0.39 grams**.

Therefore, the answer is **C. 0.39 grams**.