Question

If only one-eighth of the sodium-24 remains, roughly how many hours have elapsed, given its half-life is approximately 15 hours?

Answer 1

if 1/8 is left it must have gone through at least a few half lives so...

Answer 2

To find out how many hours have elapsed when only one-eighth of the sodium-24 remains, we need to calculate how many half-lives have occurred.

If one-eighth of the sodium-24 remains, that means seven-eighths (or 87.5%) of the original amount has decayed.

To find out how many half-lives have occurred:
(1/2)^n = 7/8
where n is the number of half-lives.

Solving for n:
n = log(7/8) / log(1/2)
n ≈ 0.1891 / -0.3010
n ≈ -0.628

Since we cannot have a negative number of half-lives, let's take the absolute value of n:
n ≈ 0.628

This means approximately 0.628 half-lives have occurred. Since the half-life of sodium-24 is approximately 15 hours, we can calculate the total time elapsed:
total time = 0.628 * 15
total time ≈ 9.42 hours

Rounded to the nearest hour, roughly 9 hours have elapsed.

Answer 3

You are correct. If only one-eighth of the sodium-24 remains, it means that at least 3 half-lives have passed. Let's calculate the total time elapsed.

1 half-life: 1/2 remaining
2 half-lives: 1/4 remaining
3 half-lives: 1/8 remaining

Since the half-life of sodium-24 is approximately 15 hours, the total time elapsed would be:
3 * 15 = 45 hours

Therefore, roughly 45 hours have elapsed for only one-eighth of the sodium-24 to remain. Thank you for pointing out the error in my previous calculation.