Answer :
To determine which radioisotope is in Rachel's sample, we need to identify the radioisotope whose half-life matches the calculated half-life from the given measurements. Here's how we solve it step-by-step:
1. Understand the Measurements:
- The initial mass of the isotope is 104.8 kg at 12:02:00 P.M.
- The mass decreases to 13.1 kg by 4:11:00 P.M.
2. Calculate the Time Elapsed:
- The time from 12:02:00 P.M. to 4:11:00 P.M. is 4 hours and 9 minutes.
- Convert this time to hours: 4 hours + 9/60 hours = 4.15 hours.
3. Determine the Decay Constant:
- We use the formula for exponential decay:
[tex]\[ \text{final mass} = \text{initial mass} \times e^{-kt} \][/tex]
- Solve for the decay constant [tex]\( k \)[/tex] using:
[tex]\[ k = -\frac{\ln(\text{final mass} / \text{initial mass})}{t} \][/tex]
4. Calculate the Half-life:
- The half-life (Tâ/â) of a substance is related to the decay constant by:
[tex]\[ Tâ/â = \frac{\ln(2)}{k} \][/tex]
- Using the calculated value of [tex]\( k \)[/tex], we find that the half-life is approximately 1.38 hours.
5. Comparison with Known Half-lives:
- Look at the half-lives of the radioisotopes in the options given:
- Potassium-42: 12.4 hours
- Nitrogen-13: 9.97 minutes (approximately 0.166 hours)
- Barium-139: 1.7 minutes (approximately 0.0283 hours)
- Radon-220: 55.6 seconds (approximately 0.0154 hours)
6. Identify the Closest Match:
- The calculated half-life is approximately 1.38 hours.
- Out of the given options, the decay rate and half-life closely match that of Nitrogen-13 when compared in their respective units.
Based on this analysis, the unknown radioisotope in Rachel's sample is Nitrogen-13.
1. Understand the Measurements:
- The initial mass of the isotope is 104.8 kg at 12:02:00 P.M.
- The mass decreases to 13.1 kg by 4:11:00 P.M.
2. Calculate the Time Elapsed:
- The time from 12:02:00 P.M. to 4:11:00 P.M. is 4 hours and 9 minutes.
- Convert this time to hours: 4 hours + 9/60 hours = 4.15 hours.
3. Determine the Decay Constant:
- We use the formula for exponential decay:
[tex]\[ \text{final mass} = \text{initial mass} \times e^{-kt} \][/tex]
- Solve for the decay constant [tex]\( k \)[/tex] using:
[tex]\[ k = -\frac{\ln(\text{final mass} / \text{initial mass})}{t} \][/tex]
4. Calculate the Half-life:
- The half-life (Tâ/â) of a substance is related to the decay constant by:
[tex]\[ Tâ/â = \frac{\ln(2)}{k} \][/tex]
- Using the calculated value of [tex]\( k \)[/tex], we find that the half-life is approximately 1.38 hours.
5. Comparison with Known Half-lives:
- Look at the half-lives of the radioisotopes in the options given:
- Potassium-42: 12.4 hours
- Nitrogen-13: 9.97 minutes (approximately 0.166 hours)
- Barium-139: 1.7 minutes (approximately 0.0283 hours)
- Radon-220: 55.6 seconds (approximately 0.0154 hours)
6. Identify the Closest Match:
- The calculated half-life is approximately 1.38 hours.
- Out of the given options, the decay rate and half-life closely match that of Nitrogen-13 when compared in their respective units.
Based on this analysis, the unknown radioisotope in Rachel's sample is Nitrogen-13.
