To determine which isotope is approaching its fourth half-life, let's follow these steps:
1. Understand the Problem:
You need to find which isotope among the given options is approaching its fourth half-life given that the Earth is approximately 4.5 billion years old.
2. Calculate Number of Half-Lives Elapsed:
For each isotope, calculate the number of half-lives that would have elapsed over the 4.5 billion years.
3. Check If it is Approaching its Fourth Half-Life:
An isotope is considered to be approaching its fourth half-life if the number of half-lives elapsed is between 3.5 and 4.5.
4. Perform the Calculation:
Let's calculate the number of half-lives for the given isotopes:
- Potassium-40:
- Half-life = 1.25 billion years
- Number of half-lives = 4.5 billion years / 1.25 billion years = 3.6
- Thorium-232:
- Half-life = 14.0 billion years
- Number of half-lives = 4.5 billion years / 14.0 billion years ≈ 0.32
- Uranium-235:
- Half-life = 704 million years (0.704 billion years)
- Number of half-lives = 4.5 billion years / 0.704 billion years ≈ 6.39
- Uranium-238:
- Half-life = 4.5 billion years
- Number of half-lives = 4.5 billion years / 4.5 billion years = 1.0
5. Determine the Correct Isotope:
- Potassium-40 has approximately 3.6 half-lives elapsed, which falls into the range of 3.5 to 4.5.
- Thorium-232 has significantly fewer half-lives elapsed (0.32), far from approaching the fourth half-life.
- Uranium-235 has completed more than six half-lives (6.39), which is beyond the range of 3.5 to 4.5.
- Uranium-238 has only completed one half-life (1.0), which is also far from the fourth half-life.
Therefore, the isotope that is approaching its fourth half-life is Potassium-40.
