Answer :
To determine which radioisotope Rachel has in her sample, we need to calculate the half-life of the unknown sample using the provided mass measurements and time interval. Here’s a step-by-step solution to solve the problem:
1. Understand the problem:
Rachel measured the mass of an unknown radioisotope at two different times:
- The initial mass was 104.8 kg at 12:02:00 P.M.
- The mass later decreased to 13.1 kg at 4:11:00 P.M.
2. Calculate the time elapsed:
- Initial time: 12:02:00 P.M.
- Final time: 4:11:00 P.M.
We convert these times to seconds past noon to find out how much time has passed.
- Convert initial time: [tex]\(12 \times 3600 + 2 \times 60 = 43200 + 120 = 43320\)[/tex] seconds
- Convert final time: [tex]\(16 \times 3600 + 11 \times 60 = 57600 + 660 = 58260\)[/tex] seconds
- Time elapsed = 58260 seconds - 43320 seconds = 14940 seconds
3. Use the formula for exponential decay:
The relationship for radioactive decay is described by:
[tex]\[
M_{\text{final}} = M_{\text{initial}} \times \left(\frac{1}{2}\right)^{\frac{\text{time elapsed}}{\text{half-life}}}
\][/tex]
Rearranging this formula allows us to solve for the half-life:
[tex]\[
\frac{\text{time elapsed}}{\text{half-life}} = \frac{\log\left(\frac{M_{\text{final}}}{M_{\text{initial}}}\right)}{\log\left(\frac{1}{2}\right)}
\][/tex]
[tex]\[
\text{half-life} = \frac{\text{time elapsed}}{\frac{\log\left(\frac{M_{\text{final}}}{M_{\text{initial}}}\right)}{\log\left(\frac{1}{2}\right)}}
\][/tex]
4. Calculate the half-life:
- With [tex]\( M_{\text{initial}} = 104.8 \)[/tex] kg and [tex]\( M_{\text{final}} = 13.1 \)[/tex] kg, calculate the ratio:
[tex]\(\frac{M_{\text{final}}}{M_{\text{initial}}} = \frac{13.1}{104.8}\)[/tex]
- Insert this into the formula to get the half-life and simplify to find that the half-life is approximately 4980 seconds.
5. Compare with the possible isotopes:
Given the half-lives of the isotopes:
- Potassium-42: 12.36 hours
- Nitrogen-13: 9.965 minutes
- Barium-139: 83 minutes
- Radon-220: 55.6 seconds
Comparing the calculated half-life of approximately 4980 seconds (or about 83 minutes), it matches closely with Barium-139, which has a half-life of 83 minutes.
Thus, Rachel's unknown sample is Barium-139.
1. Understand the problem:
Rachel measured the mass of an unknown radioisotope at two different times:
- The initial mass was 104.8 kg at 12:02:00 P.M.
- The mass later decreased to 13.1 kg at 4:11:00 P.M.
2. Calculate the time elapsed:
- Initial time: 12:02:00 P.M.
- Final time: 4:11:00 P.M.
We convert these times to seconds past noon to find out how much time has passed.
- Convert initial time: [tex]\(12 \times 3600 + 2 \times 60 = 43200 + 120 = 43320\)[/tex] seconds
- Convert final time: [tex]\(16 \times 3600 + 11 \times 60 = 57600 + 660 = 58260\)[/tex] seconds
- Time elapsed = 58260 seconds - 43320 seconds = 14940 seconds
3. Use the formula for exponential decay:
The relationship for radioactive decay is described by:
[tex]\[
M_{\text{final}} = M_{\text{initial}} \times \left(\frac{1}{2}\right)^{\frac{\text{time elapsed}}{\text{half-life}}}
\][/tex]
Rearranging this formula allows us to solve for the half-life:
[tex]\[
\frac{\text{time elapsed}}{\text{half-life}} = \frac{\log\left(\frac{M_{\text{final}}}{M_{\text{initial}}}\right)}{\log\left(\frac{1}{2}\right)}
\][/tex]
[tex]\[
\text{half-life} = \frac{\text{time elapsed}}{\frac{\log\left(\frac{M_{\text{final}}}{M_{\text{initial}}}\right)}{\log\left(\frac{1}{2}\right)}}
\][/tex]
4. Calculate the half-life:
- With [tex]\( M_{\text{initial}} = 104.8 \)[/tex] kg and [tex]\( M_{\text{final}} = 13.1 \)[/tex] kg, calculate the ratio:
[tex]\(\frac{M_{\text{final}}}{M_{\text{initial}}} = \frac{13.1}{104.8}\)[/tex]
- Insert this into the formula to get the half-life and simplify to find that the half-life is approximately 4980 seconds.
5. Compare with the possible isotopes:
Given the half-lives of the isotopes:
- Potassium-42: 12.36 hours
- Nitrogen-13: 9.965 minutes
- Barium-139: 83 minutes
- Radon-220: 55.6 seconds
Comparing the calculated half-life of approximately 4980 seconds (or about 83 minutes), it matches closely with Barium-139, which has a half-life of 83 minutes.
Thus, Rachel's unknown sample is Barium-139.
