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
60 hours
b
30 hours
c
15 hours
d
45 hours
To solve this problem, we can use the half-life formula:
amount remaining = initial amount * (1/2)^(time elapsed / half-life)
We are given that only one-eighth of the sodium-24 remains. We can substitute this information into the formula:
1/8 = 1 * (1/2)^(time elapsed / 15)
To solve for the time elapsed, we can take the natural logarithm of both sides:
ln(1/8) = (time elapsed / 15) * ln(1/2)
Simplifying,
ln(1/8) = (time elapsed / 15) * ln(1/2)
ln(1/8) = (time elapsed * ln(1/2)) / 15
Now we can solve for time elapsed:
time elapsed = (ln(1/8) * 15) / ln(1/2) ≈ 45.102 hours
Therefore, about 45 hours have passed.
Answer: d) 45 hours
To find out how much time has passed, we need to determine how many half-lives have occurred for one-eighth of the sodium-24 to remain.
Since the half-life of sodium-24 is approximately 15 hours, we can calculate the number of half-lives that have occurred by dividing the total time by the half-life:
Total time / Half-life = Number of half-lives
Let's substitute the given values:
Total time / 15 hours = 1/8
To solve for the total time, we can rearrange the equation:
Total time = 15 hours * (1/8)
Calculating this expression, we get:
Total time = 1.875 hours
Therefore, about 1.875 hours, or approximately 1 hour and 53 minutes, have passed for only one-eighth of the sodium-24 to remain.
Since none of the given answer choices match this result exactly, we can round it to the nearest option. The closest option is 60 hours (answer choice A).
To determine how much time has passed when only one-eighth of sodium-24 remains, we can use the concept of half-life.
The half-life of sodium-24 is approximately 15 hours. This means that every 15 hours, the amount of sodium-24 is reduced by half.
Since we are given that only one-eighth of the sodium-24 remains, we need to find out how many half-lives it took for the initial amount to reduce to one-eighth.
To do this, we can use the following formula:
Remaining amount = Initial amount * (1/2)^(number of half-lives)
Let's substitute the given values into the formula:
1/8 = 1 * (1/2)^(number of half-lives)
To solve for the number of half-lives, we need to isolate it.
Taking the logarithm base 2 of both sides of the equation:
log base 2 (1/8) = log base 2 (1/2)^(number of half-lives)
-3 = (number of half-lives) * log base 2 (1/2)
Using the identity log base b(a^n) = n * log base b(a):
-3 = (number of half-lives) * (-1)
-3 = -1 * (number of half-lives)
Dividing both sides by -1:
3 = number of half-lives
Thus, it took 3 half-lives for the amount of sodium-24 to reduce to one-eighth.
Since each half-life is approximately 15 hours, we can multiply 3 by 15 hours to find the total time:
3 * 15 = 45 hours
Therefore, about 45 hours have passed. The correct answer is option d) 45 hours.