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Answer :
To determine which radioisotope Rachel has, we need to calculate the half-life based on the given information:
1. Measure the Initial Mass and Final Mass:
- Initial mass of the radioisotope = 104.8 kg
- Final mass at a later time = 13.1 kg
2. Determine the Time Elapsed:
- Rachel takes her first measurement at 12:02 P.M. and the second one at 4:11 P.M.
- Convert these times to minutes since midnight:
- 12:02 P.M. is 12 hours and 2 minutes, which is [tex]\(12 \times 60 + 2 = 722\)[/tex] minutes.
- 4:11 P.M. is 16 hours and 11 minutes, which is [tex]\(16 \times 60 + 11 = 971\)[/tex] minutes.
- The time elapsed is therefore [tex]\(971 - 722 = 249\)[/tex] minutes.
- Convert minutes to hours: [tex]\(249 \div 60 \approx 4.15\)[/tex] hours.
3. Use the Decay Formula:
- The decay formula for radioactive substances is:
[tex]\[
N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{\text{half-life}}}
\][/tex]
- Rearrange to solve for the half-life ([tex]\(\text{half-life}\)[/tex]):
[tex]\[
\text{half-life} = \frac{t \times \log(2)}{\log\left(\frac{N_0}{N(t)}\right)}
\][/tex]
- Knowing [tex]\(N_0 = 104.8\)[/tex] kg and [tex]\(N(t) = 13.1\)[/tex] kg, and using the elapsed time [tex]\(t = 4.15\)[/tex] hours:
- Upon calculating, we find the half-life to be approximately 1.38 hours.
4. Identify the Isotope:
- Now, we compare the calculated half-life to the known half-lives of the isotopes listed:
- Potassium-42: 12.4 hours
- Nitrogen-13: 9.97 minutes
- Barium-139: 1.5 hours
- Radon-220: 55 seconds
- The calculated half-life of approximately 1.38 hours is closest to the half-life of Barium-139, which is 1.5 hours.
Therefore, the radioisotope in Rachel's sample is most likely Barium-139.
1. Measure the Initial Mass and Final Mass:
- Initial mass of the radioisotope = 104.8 kg
- Final mass at a later time = 13.1 kg
2. Determine the Time Elapsed:
- Rachel takes her first measurement at 12:02 P.M. and the second one at 4:11 P.M.
- Convert these times to minutes since midnight:
- 12:02 P.M. is 12 hours and 2 minutes, which is [tex]\(12 \times 60 + 2 = 722\)[/tex] minutes.
- 4:11 P.M. is 16 hours and 11 minutes, which is [tex]\(16 \times 60 + 11 = 971\)[/tex] minutes.
- The time elapsed is therefore [tex]\(971 - 722 = 249\)[/tex] minutes.
- Convert minutes to hours: [tex]\(249 \div 60 \approx 4.15\)[/tex] hours.
3. Use the Decay Formula:
- The decay formula for radioactive substances is:
[tex]\[
N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{\text{half-life}}}
\][/tex]
- Rearrange to solve for the half-life ([tex]\(\text{half-life}\)[/tex]):
[tex]\[
\text{half-life} = \frac{t \times \log(2)}{\log\left(\frac{N_0}{N(t)}\right)}
\][/tex]
- Knowing [tex]\(N_0 = 104.8\)[/tex] kg and [tex]\(N(t) = 13.1\)[/tex] kg, and using the elapsed time [tex]\(t = 4.15\)[/tex] hours:
- Upon calculating, we find the half-life to be approximately 1.38 hours.
4. Identify the Isotope:
- Now, we compare the calculated half-life to the known half-lives of the isotopes listed:
- Potassium-42: 12.4 hours
- Nitrogen-13: 9.97 minutes
- Barium-139: 1.5 hours
- Radon-220: 55 seconds
- The calculated half-life of approximately 1.38 hours is closest to the half-life of Barium-139, which is 1.5 hours.
Therefore, the radioisotope in Rachel's sample is most likely Barium-139.
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