Question

The half-life of Radium-226 is 1590 years. If a sample contains 200 mg, how many mg will remain after 4000 years?

mg

Give your answer accurate to at least 2 decimal places.

Answer 1

The half-life of Radium-226 is 1590 years. To determine how much of the 200 mg sample will remain after 4000 years, we first need to calculate how many half-lives fit into 4000 years.

1. **Calculate the number of half-lives**:
\[
\text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{4000 \text{ years}}{1590 \text{ years}} \approx 2.51
\]

2. **Determine how much remains after each half-life**:
After each half-life, the amount of substance reduces to half its previous value. To find the amount remaining after 2.51 half-lives, we can use the formula:
\[
\text{Remaining amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^{n}
\]
where \( n \) is the number of half-lives.

3. **Calculate the remaining amount**:
\[
\text{Remaining amount} = 200 \text{ mg} \times \left(\frac{1}{2}\right)^{2.51}
\]

4. **Calculate \( \left(\frac{1}{2}\right)^{2.51} \)**:
\[
\left(\frac{1}{2}\right)^{2.51} \approx 0.1768 \quad (\text{using a calculator})
\]

5. **Final calculation**:
\[
\text{Remaining amount} \approx 200 \text{ mg} \times 0.1768 \approx 35.36 \text{ mg}
\]

Thus, after 4000 years, approximately **35.36 mg** of Radium-226 will remain.