To determine how much of a 20-gram sample of radioactive Carbon-14 will remain after 2,375 years, we follow these steps:
1. Understand the concept of half-life: The half-life of a radioactive substance is the time it takes for half of its mass to decay. For Carbon-14, the half-life is 5,730 years.
2. Identify the initial mass and the time elapsed:
- Initial mass = 20 grams
- Time elapsed = 2,375 years
3. Calculate the number of half-lives:
- The number of half-lives is calculated by dividing the time elapsed by the half-life of the substance.
- Number of half-lives = 2,375 years / 5,730 years ≈ 0.4145 half-lives
4. Determine the remaining mass:
- Every half-life, the mass of the substance is halved. Thus, we use the formula:
[tex]\[
\text{Remaining mass} = \text{Initial mass} \times \left(\frac{1}{2}\right)^{\text{Number of half-lives}}
\][/tex]
- Plugging in the values:
[tex]\[
\text{Remaining mass} = 20 \text{ grams} \times \left(0.5\right)^{0.4145}
\][/tex]
- This calculation yields:
[tex]\[
\text{Remaining mass} ≈ 15 \text{ grams}
\][/tex]
Thus, after 2,375 years, approximately 15 grams of the initial 20-gram sample of Carbon-14 will remain.
