Determine the total time that must elapse until only ¼ of an original sample of the radioisotope Rn-222 remains unchanged.
What's the half-life of Rn222? I think that's approx 4 days but you can look that information up in your text/notes.
It takes 1 half life to reduce the amount to 1/2 of what you started and another (to make 2 half lives) to reduce another 1/2 or 1/4 (that's 1/2 x 1/2 = 1/4) of the original.
To determine the total time until only ¼ of an original sample of the radioisotope Rn-222 remains unchanged, we can use the concept of half-life.
The half-life of Rn-222 is approximately 3.8 days.
1. Calculate the number of half-lives required for the amount to decrease to ¼ of the original sample:
- Since each half-life reduces the amount of Rn-222 by half, we need to find how many times we need to divide the original amount by 2 to get to ¼.
- ¼ can be written as 1/2^2, so we need to divide the original amount by 2 twice.
- This means we need 2 half-lives.
2. Multiply the half-life by the number of half-lives to determine the total time required:
- Multiply 3.8 days (the half-life) by 2 half-lives.
- The total time required is 7.6 days.
Therefore, the total time that must elapse until only ¼ of the original sample of Rn-222 remains unchanged is approximately 7.6 days.
To determine the total time that must elapse until only ¼ of an original sample of the radioisotope Rn-222 remains unchanged, we need to use the half-life of Rn-222.
1. Find the half-life of Rn-222: The half-life of Rn-222 is approximately 3.8235 days. This means that after 3.8235 days, half of the original sample will have decayed.
2. Calculate the number of half-lives required: Since we are looking for the time it takes for only ¼ of the original sample to remain, we need to calculate the number of half-lives required to reach this point. To do this, we can use the formula:
Number of half-lives = log(base 2)(Final fraction/Initial fraction)
In this case, the final fraction is 1/4 (since we want only ¼ of the original sample to remain), and the initial fraction is 1 (the original sample). So, plugging these values into the formula:
Number of half-lives = log(base 2)(1/4/1) = log(base 2)(1/4) = log(base 2)(1) - log(base 2)(4) = 0 - 2 = -2
Since the result is negative, we need to take the absolute value to get the number of half-lives required: |-2| = 2.
3. Calculate the total time: Now that we know that 2 half-lives are required, we can multiply the half-life of Rn-222 by the number of half-lives:
Total time = 2 * 3.8235 days ≈ 7.647 days
Therefore, the total time that must elapse until only ¼ of an original sample of Rn-222 remains unchanged is approximately 7.647 days.