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
60 hours
c
45 hours
d
30 hours
The correct answer is c) 45 hours.
In the first half-life, half of the sodium-24 decays, so only 1/2 remains.
In the second half-life, half of the remaining sodium-24 decays, so 1/2 * 1/2 = 1/4 remains.
In the third half-life, half of the remaining sodium-24 decays, so 1/2 * 1/4 = 1/8 remains.
Since the half-life of sodium-24 is 15 hours, after 3 half-lives (3 * 15 = 45 hours), only one-eighth of the sodium-24 remains.
To answer this question, we can use the concept of half-life. The half-life is the time it takes for half of a substance to decay or disintegrate.
Given that the half-life of sodium-24 is approximately 15 hours, we know that after 15 hours, half of the sodium-24 will decay, and only one-half will remain.
Since we are told that only one-eighth (1/8) of the sodium-24 remains, we can deduce that four half-lives have passed. Here's how we can calculate it:
1/2 -> 1/4 (1 half-life)
1/4 -> 1/8 (2 half-lives)
1/8 -> 1/16 (3 half-lives)
1/16 -> 1/32 (4 half-lives)
Each half-life takes 15 hours, so if four half-lives have passed, the total time that has passed would be 4 multiplied by 15 hours.
4 half-lives * 15 hours/half-life = 60 hours
Therefore, the correct answer is option b) 60 hours.
To determine the amount of time that has passed, we can use the equation for exponential decay:
N = N0 * (1/2)^(t/h)
Where:
N = remaining amount of sodium-24 (1/8 of N0)
N0 = initial amount of sodium-24
t = time passed
h = half-life of sodium-24 (15 hours)
From the question, we know that N = 1/8 * N0. Substituting these values into the equation, we get:
1/8 * N0 = N0 * (1/2)^(t/15)
To solve for t, we can take the natural logarithm of both sides:
ln(1/8) = ln((1/2)^(t/15))
Using the property of logarithms that ln(a^b) = b * ln(a), we can simplify further:
ln(1/8) = (t/15) * ln(1/2)
Dividing both sides by ln(1/2):
(t/15) = ln(1/8) / ln(1/2)
Now, let's calculate the right side of the equation:
ln(1/8) ≈ -2.0794
ln(1/2) ≈ -0.6931
(t/15) ≈ -2.0794 / -0.6931 ≈ 3
Multiplying both sides by 15:
t ≈ 3 * 15 ≈ 45 hours
Therefore, about 45 hours have passed if only one-eighth of the sodium-24 remains. So, the correct answer is c) 45 hours.