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
45 hours
b
30 hours
c
15 hours
d
60 hours
To solve this problem, we need to determine how many half-lives have passed.
Since the half-life of sodium-24 is approximately 15 hours, we can divide the total time passed by 15 to find the number of half-lives.
Let's call the total time passed "t" hours. Since only one-eighth of the sodium-24 remains, this means that seven-eighths (or 7/8) of the sodium-24 has decayed.
So, after one half-life, only half of the original amount remains. This can be represented as 1/2.
After two half-lives, only one-fourth (1/2 * 1/2 = 1/4) of the original amount remains.
After three half-lives, only one-eighth (1/2 * 1/2 * 1/2 = 1/8) of the original amount remains.
This means that three half-lives have passed.
Therefore, the total time passed, "t" hours, is equal to 3 half-lives * 15 hours per half-life.
t = 3 * 15 = 45 hours.
Therefore, the answer is a) 45 hours.
To determine how much time has passed, we can use the formula for half-life:
t = t₀ * (1/2)^(n)
Where:
t is the time that has passed
t₀ is the initial time (in this case, 15 hours)
n is the number of half-lives that have occurred
Since only one-eighth of the sodium-24 remains, this means that there have been 3 half-lives (2⁻³ = 1/8).
Plugging these values into the formula, we get:
t = 15 hours * (1/2)^(3)
t = 15 hours * (1/8)
t = 1.875 hours
Therefore, about 1.875 hours have passed. This is closest to option: b) 30 hours.
To determine the amount of time that has passed, you can use the formula for calculating half-life:
N(t) = N₀ * (1/2)^(t / half-life)
Where:
- N(t) is the remaining amount of the substance at time t,
- N₀ is the initial amount of the substance,
- t is the amount of time that has passed, and
- half-life is the time it takes for the substance to reduce to half its original amount.
In this case, we know that only one-eighth of the sodium-24 remains. Let's assume that the initial amount is represented by N₀.
So, we have:
N(t) = N₀ * (1/2)^(t / 15)
Since only one-eighth of the sodium-24 remains, N(t) = N₀ / 8.
Therefore:
N₀ / 8 = N₀ * (1/2)^(t / 15)
To solve for t, we can eliminate N₀ from both sides of the equation:
1/8 = (1/2)^(t / 15)
Now, we can take the logarithm of both sides with the base of 1/2:
log₁/₂(1/8) = t / 15
To evaluate log₁/₂(1/8), note that 1/8 is equal to 2^(-3), so:
log₁/₂(1/8) = log₁/₂(2^(-3)) = -3
Therefore:
-3 = t / 15
To find t, we can multiply both sides by 15:
-3 * 15 = t
t = -45
Since time cannot be negative in this context, we can discard the negative value.
So, the correct answer is: c) 15 hours