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
45 hours
c
30 hours
d
60 hourss
The correct answer is b) 45 hours.
To solve this problem, we can use the formula for exponential decay:
Amount remaining = (initial amount) * (1/2)^(time elapsed / half-life)
We are given that only one-eighth of the sodium-24 remains, so the amount remaining is 1/8 of the initial amount. Therefore, we can set up the following equation:
1/8 = (1) * (1/2)^(time elapsed / 15)
To solve for the time elapsed, we can take the logarithm of both sides of the equation:
log(1/8) = log((1/2)^(time elapsed / 15))
Simplifying the equation, we get:
-3 = (time elapsed / 15) * log(1/2)
Now we can solve for the time elapsed:
time elapsed / 15 = -3 / log(1/2)
time elapsed = -3 * 15 / log(1/2)
Using a calculator, we find that log(1/2) is approximately -0.30103. Therefore:
time elapsed ≈ -3 * 15 / (-0.30103) ≈ 45 hours
So about 45 hours has passed.
To answer this question, we need to use the concept of a half-life. The half-life of a radioactive substance is the time it takes for half of the substance to decay or break down.
In this case, the half-life of sodium-24 is approximately 15 hours. This means that after 15 hours, half of the original amount of sodium-24 will have decayed, and only half will remain.
Now, the question states that only one-eighth of the sodium-24 remains. One-eighth is the same as half of half of half. In other words:
(1/2) * (1/2) * (1/2) = 1/8
Since the half-life of sodium-24 is 15 hours, we need to determine how many half-lives it takes for the amount to decrease from 1 to 1/8.
To do this, we can set up an equation:
(1/2)^n = 1/8
Simplifying the equation, we get:
(1/2)^n = (1/2)^3
So, n must be equal to 3.
Since each half-life is 15 hours, and we need three half-lives to reach one-eighth, the total time that has passed can be calculated as:
3 * 15 = 45 hours.
Therefore, the answer is option b) 45 hours.
To answer this question, we need to calculate the number of half-lives that have passed.
Since the half-life of sodium-24 is 15 hours, in one half-life only half of the original amount remains. Therefore, after one half-life, the remaining amount of sodium-24 would be one-half (1/2).
If one-eighth (1/8) of the original amount remains, we need to figure out how many half-lives it would take to reduce the amount to one-eighth.
1/2 --> 1/4 --> 1/8
We can see that it takes 2 half-lives to reduce the amount to one-eighth.
Since each half-life is 15 hours, the total time passed would be 15 hours × 2 = 30 hours.
Therefore, the answer is c) 30 hours.