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Answer :
To identify the radioisotope Rachel has, we need to figure out the half-life of the sample by using the initial and final masses along with the elapsed time.
Here's how we approach this:
1. Initial and Final Mass: Rachel starts with 104.8 kg of the radioisotope at 12:02 P.M. and ends with 13.1 kg at 4:11 P.M.
2. Time Elapsed: Calculate the time difference between 12:02 P.M. and 4:11 P.M.:
- 12:02 P.M. is equivalent to [tex]\(12 \times 60 + 2 = 722\)[/tex] minutes after midnight.
- 4:11 P.M. is equivalent to [tex]\(16 \times 60 + 11 = 971\)[/tex] minutes after midnight.
- The time elapsed is [tex]\(971 - 722 = 249\)[/tex] minutes.
3. Half-Life Calculation: Use the decay formula to express the relationship between initial mass, final mass, and half-life. The decay formula is:
[tex]\[
\text{final mass} = \text{initial mass} \times (0.5)^{(\text{time elapsed} / \text{half-life})}
\][/tex]
Rearrange this formula to solve for the half-life:
[tex]\[
\text{half-life} = \frac{\text{time elapsed}}{\log_2(\text{initial mass} / \text{final mass})}
\][/tex]
Plug in the values:
- Initial mass = 104.8 kg
- Final mass = 13.1 kg
- Time elapsed = 249 minutes
When these are calculated, it results in a half-life of approximately 83 minutes.
4. Identifying the Isotope:
- Now that we have calculated the half-life as 83 minutes, we can compare it to commonly known half-lives of radioisotopes:
- Potassium-42 has a half-life of about 12.4 hours (not a match).
- Nitrogen-13 has a half-life of about 10 minutes (not a match).
- Barium-139 has a half-life of about 85.5 minutes (close match).
- Radon-220 has a half-life of about 55.6 seconds (not a match).
Considering the calculated half-life is closest to 83 minutes and matches most closely with Barium-139, it is reasonable to conclude that Rachel's unknown radioisotope sample is Barium-139.
Here's how we approach this:
1. Initial and Final Mass: Rachel starts with 104.8 kg of the radioisotope at 12:02 P.M. and ends with 13.1 kg at 4:11 P.M.
2. Time Elapsed: Calculate the time difference between 12:02 P.M. and 4:11 P.M.:
- 12:02 P.M. is equivalent to [tex]\(12 \times 60 + 2 = 722\)[/tex] minutes after midnight.
- 4:11 P.M. is equivalent to [tex]\(16 \times 60 + 11 = 971\)[/tex] minutes after midnight.
- The time elapsed is [tex]\(971 - 722 = 249\)[/tex] minutes.
3. Half-Life Calculation: Use the decay formula to express the relationship between initial mass, final mass, and half-life. The decay formula is:
[tex]\[
\text{final mass} = \text{initial mass} \times (0.5)^{(\text{time elapsed} / \text{half-life})}
\][/tex]
Rearrange this formula to solve for the half-life:
[tex]\[
\text{half-life} = \frac{\text{time elapsed}}{\log_2(\text{initial mass} / \text{final mass})}
\][/tex]
Plug in the values:
- Initial mass = 104.8 kg
- Final mass = 13.1 kg
- Time elapsed = 249 minutes
When these are calculated, it results in a half-life of approximately 83 minutes.
4. Identifying the Isotope:
- Now that we have calculated the half-life as 83 minutes, we can compare it to commonly known half-lives of radioisotopes:
- Potassium-42 has a half-life of about 12.4 hours (not a match).
- Nitrogen-13 has a half-life of about 10 minutes (not a match).
- Barium-139 has a half-life of about 85.5 minutes (close match).
- Radon-220 has a half-life of about 55.6 seconds (not a match).
Considering the calculated half-life is closest to 83 minutes and matches most closely with Barium-139, it is reasonable to conclude that Rachel's unknown radioisotope sample is Barium-139.
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