To solve the problem of determining how much of a 120-gram sample of Radium-226 will remain after [tex]\( t \)[/tex] years, we can use the decay formula. Let's go through the detailed steps:
### Step-by-Step Solution:
1. Understand the Half-Life Concept:
- Radium-226 has a half-life of 1,620 years. This means that after 1,620 years, half of the original sample will have decayed.
2. Decay Formula:
- The amount of substance remaining after a certain time can be calculated using the formula:
[tex]\[
P(t) = P_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{\text{half}}}}
\][/tex]
Where:
- [tex]\( P(t) \)[/tex] is the remaining mass after time [tex]\( t \)[/tex]
- [tex]\( P_0 \)[/tex] is the initial mass (120 grams in this case)
- [tex]\( t \)[/tex] is the elapsed time (in years)
- [tex]\( t_{\text{half}} \)[/tex] is the half-life (1,620 years for Radium-226)
3. Substitute the Given Values:
- Initial mass [tex]\( P_0 = 120 \)[/tex] grams
- Half-life [tex]\( t_{\text{half}} = 1,620 \)[/tex] years
- Time elapsed [tex]\( t = 100 \)[/tex] years
4. Apply the Decay Formula:
[tex]\[
P(100) = 120 \left( \frac{1}{2} \right)^{\frac{100}{1620}}
\][/tex]
5. Calculate the Exponent:
- First, compute the fraction:
[tex]\[
\frac{100}{1620} \approx 0.0617
\][/tex]
- Next, compute the power of [tex]\( \frac{1}{2} \)[/tex] raised to this fraction:
[tex]\(
\left( \frac{1}{2} \right)^{0.0617}
\)[/tex]
6. Determine Remaining Mass:
[tex]\(
P(100) \approx 120 \times 0.957
\)[/tex]
7. Final Result:
- The remaining mass after 100 years is approximately:
[tex]\[
P(100) \approx 114.97386938263115 \text{ grams}
\][/tex]
### Conclusion:
After 100 years, about 115 grams (rounded from 114.97386938263115 grams) of the 120-gram sample of Radium-226 will remain.
