Sodium-24 has a half-life of approximately 15 hours. If only one-eighth of the sodium-24 remains, about how much time has passed?
A. 15 hours
B. 30 hours
C. 45 hours
D. 60 hours
The correct answer is C. 45 hours.
Since sodium-24 has a half-life of 15 hours, this means that after 15 hours, half of the original amount will remain.
If only one-eighth (1/8) of the sodium-24 remains, then 7/8 of the original amount has decayed.
Since each half-life is 15 hours, multiplying 15 by the number of half-lives (x) we get:
(1/2)^x = (7/8)
To solve for x, we take the logarithm of both sides:
log((1/2)^x) = log(7/8)
x*log(1/2) = log(7/8)
x = log(7/8) / log(1/2)
x ≈ 2.80735
Since x represents the number of half-lives, and each half-life is 15 hours, the total time that has passed is approximately:
2.80735 * 15 hours = 42.11025 hours.
Rounding to the nearest hour, this is equal to 42 hours.
Thus, the closest answer is C. 45 hours.
To solve this problem, we can use the concept of half-life.
The half-life of sodium-24 is 15 hours, which means that every 15 hours, the amount of sodium-24 is reduced by half.
If only one-eighth of the sodium-24 remains, it means that it has undergone 3 half-lives.
To find the time passed, we need to multiply the half-life by the number of half-lives.
3 half-lives * 15 hours per half-life = 45 hours
Therefore, the correct answer is C. 45 hours.
To determine the amount of time that has passed, we can use the concept of half-life.
The half-life of a substance is the time it takes for half of the original quantity to decay. In this case, the half-life of sodium-24 is approximately 15 hours.
If only one-eighth of the original sodium-24 remains, it means that three half-lives have occurred. Here's how we can calculate that:
1. Start with 100% (the original amount of sodium-24).
2. After the first half-life (15 hours), we have 50% remaining.
3. After the second half-life (30 hours), we have 25% remaining.
4. After the third half-life (45 hours), we have 12.5% remaining, which is one-eighth of the original amount.
Therefore, if one-eighth of sodium-24 remains, approximately 45 hours have passed (option C).