To determine which radioactive isotope takes the least amount of time to become stable, we compare the half-lives of various isotopes listed in the provided table:
1. Rubidium-91: 58.4 seconds
2. Iodine-131: 5 days
3. Cobalt-60: 5 years
4. Carbon-14: 5730 years
5. Cesium-135: [tex]\(2.3 \times 10^6\)[/tex] years
6. Uranium-238: [tex]\(4.5 \times 10^9\)[/tex] years
Let's begin by converting all the half-lives into the same units for accurate comparison. For simplicity, we'll convert days and years into seconds because the smallest given time unit is seconds.
### Conversion to Seconds:
1. Iodine-131 (5 days):
- 1 day = 24 hours = 24 60 minutes = 24 60 60 seconds
- Hence, 5 days = 5 24 60 60 seconds = 432,000 seconds
2. Cobalt-60 (5 years):
- 1 year = 365 days = 365 24 hours = 365 24 60 minutes = 365 24 60 60 seconds
- Hence, 5 years = 5 365 24 60 60 seconds = 157,680,000 seconds
3. Carbon-14 (5730 years):
- 5730 years = 5730 365 24 60 60 seconds = 180,835,200,000 seconds
4. Cesium-135 ([tex]\(2.3 \times 10^6\)[/tex] years):
- [tex]\(2.3 \times 10^6\)[/tex] years = [tex]\(2.3 \times 10^6\)[/tex] 365 24 60 60 seconds = 72,529,800,000,000 seconds
5. Uranium-238 ([tex]\(4.5 \times 10^9\)[/tex] years):
- [tex]\(4.5 \times 10^9\)[/tex] years = [tex]\(4.5 \times 10^9\)[/tex] 365 24 60 60 seconds = 141,912,000,000,000,000 seconds
### Comparison of Half-lives in Seconds:
- Rubidium-91: 58.4 seconds
- Iodine-131: 432,000 seconds
- Cobalt-60: 157,680,000 seconds
- Carbon-14: 180,835,200,000 seconds
- Cesium-135: 72,529,800,000,000 seconds
- Uranium-238: 141,912,000,000,000,000 seconds
From the converted half-lives in seconds, it is clear that Rubidium-91, with a half-life of 58.4 seconds, has the shortest half-life compared to the other isotopes.
Thus, the radioactive isotope that would take the least amount of time to become stable is:
Rubidium-91
