To determine which radioactive isotope would take the least amount of time to become stable, we need to compare the half-lives of the given isotopes. Here is the detailed breakdown of each isotope's half-life:
1. Rubidium-91: 58.4 seconds
2. Iodine-131: 8 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
To compare these half-lives accurately, it's ideal to convert them to a common unit, such as seconds, but for simplicity, we will recognize that shorter half-lives indicate quicker stabilization.
- Rubidium-91 with a half-life of 58.4 seconds is clearly the shortest duration compared to others.
- Iodine-131 has a half-life of 8 days, which is definitively longer than 58.4 seconds.
- Cobalt-60 with a half-life of 5 years is even longer.
- Carbon-14 with a half-life of 5730 years.
- Cesium-135 with a half-life of [tex]\(2.3 \times 10^6\)[/tex] years.
- Uranium-238 with a half-life of [tex]\(4.5 \times 10^9\)[/tex] years is the longest among the list.
So, comparing all the given half-lives, the isotope Rubidium-91, having the half-life of 58.4 seconds, would take the least amount of time to become stable.
