We appreciate your visit to Rachel has an unknown sample of a radioisotope listed in the table Using a special technique she is able to measure the mass of just. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
To determine which radioisotope Rachel has, we'll need to calculate the half-life of the unknown isotope using the mass measurements and the time elapsed between them. Here’s how you can understand the step-by-step process:
1. Understand the Problem: We have two masses measured at different times:
- At 12:02 P.M., the mass of the radioisotope is 104.8 kg.
- At 4:11 P.M., the mass is 13.1 kg.
2. Calculate Time Elapsed:
- The time elapsed from 12:02 P.M. to 4:11 P.M. is 4 hours and 9 minutes.
- Converting the time to hours, 9 minutes is 9/60 = 0.15 hours.
- Therefore, the total time elapsed is 4.15 hours.
3. Equation for Radioactive Decay:
- Radioactive decay can be described by the formula:
[tex]\[
\text{final mass} = \text{initial mass} \times \left(\frac{1}{2}\right)^{\frac{\text{time elapsed}}{\text{half-life}}}
\][/tex]
- Here, we rearrange this to solve for the half-life by noting that:
[tex]\[
\left(\frac{\text{final mass}}{\text{initial mass}}\right) = \left(\frac{1}{2}\right)^{\frac{\text{time elapsed}}{\text{half-life}}}
\][/tex]
- Taking natural logarithms, we get:
[tex]\[
\frac{\text{time elapsed}}{\text{half-life}} = \log_2 \left(\frac{\text{initial mass}}{\text{final mass}}\right)
\][/tex]
4. Calculate the Half-life:
- Given that the initial mass is 104.8 kg and the final mass is 13.1 kg, calculate the fraction of the mass remaining:
[tex]\[
\frac{13.1}{104.8} \approx 0.125
\][/tex]
- Solving for half-life:
[tex]\[
\text{half-life} = \frac{\text{time elapsed}}{\log_2(104.8/13.1)}
\][/tex]
- Logarithm results indicate that [tex]\( \log_2(104.8/13.1) \)[/tex] approximately equals to 3 (since [tex]\( (0.5)^3 = 0.125 \)[/tex]).
- Therefore, half-life [tex]\( \approx \frac{4.15}{3} = 1.0375 \)[/tex] hours
5. Identify Radioisotope:
- Compare the calculated half-life of approximately 1.0375 hours with the known half-lives of the isotopes provided:
- Potassium-42: about 12.4 hours
- Nitrogen-13: about 9.97 minutes
- Barium-139: about 83 minutes (1.38 hours)
- Radon-220: about 55.6 seconds
- Barium-139 has a half-life that closely matches our calculated value of approximately 1.0375 hours.
Therefore, the radioisotope in question is likely Barium-139.
1. Understand the Problem: We have two masses measured at different times:
- At 12:02 P.M., the mass of the radioisotope is 104.8 kg.
- At 4:11 P.M., the mass is 13.1 kg.
2. Calculate Time Elapsed:
- The time elapsed from 12:02 P.M. to 4:11 P.M. is 4 hours and 9 minutes.
- Converting the time to hours, 9 minutes is 9/60 = 0.15 hours.
- Therefore, the total time elapsed is 4.15 hours.
3. Equation for Radioactive Decay:
- Radioactive decay can be described by the formula:
[tex]\[
\text{final mass} = \text{initial mass} \times \left(\frac{1}{2}\right)^{\frac{\text{time elapsed}}{\text{half-life}}}
\][/tex]
- Here, we rearrange this to solve for the half-life by noting that:
[tex]\[
\left(\frac{\text{final mass}}{\text{initial mass}}\right) = \left(\frac{1}{2}\right)^{\frac{\text{time elapsed}}{\text{half-life}}}
\][/tex]
- Taking natural logarithms, we get:
[tex]\[
\frac{\text{time elapsed}}{\text{half-life}} = \log_2 \left(\frac{\text{initial mass}}{\text{final mass}}\right)
\][/tex]
4. Calculate the Half-life:
- Given that the initial mass is 104.8 kg and the final mass is 13.1 kg, calculate the fraction of the mass remaining:
[tex]\[
\frac{13.1}{104.8} \approx 0.125
\][/tex]
- Solving for half-life:
[tex]\[
\text{half-life} = \frac{\text{time elapsed}}{\log_2(104.8/13.1)}
\][/tex]
- Logarithm results indicate that [tex]\( \log_2(104.8/13.1) \)[/tex] approximately equals to 3 (since [tex]\( (0.5)^3 = 0.125 \)[/tex]).
- Therefore, half-life [tex]\( \approx \frac{4.15}{3} = 1.0375 \)[/tex] hours
5. Identify Radioisotope:
- Compare the calculated half-life of approximately 1.0375 hours with the known half-lives of the isotopes provided:
- Potassium-42: about 12.4 hours
- Nitrogen-13: about 9.97 minutes
- Barium-139: about 83 minutes (1.38 hours)
- Radon-220: about 55.6 seconds
- Barium-139 has a half-life that closely matches our calculated value of approximately 1.0375 hours.
Therefore, the radioisotope in question is likely Barium-139.
Thanks for taking the time to read Rachel has an unknown sample of a radioisotope listed in the table Using a special technique she is able to measure the mass of just. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
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