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
30 hours
b
45 hours
c
60 hours
d
15 hours
To solve this problem, we can use the formula for exponential decay:
N(t) = N0 * (1/2)^(t/h)
Where:
N(t) is the amount of the substance remaining at time t
N0 is the initial amount of the substance
t is the time that has passed
h is the half-life of the substance
In this case, we know that only one-eighth of the sodium-24 remains, which means that N(t) = N0/8. We are trying to find out how much time has passed, so we can let t be the unknown variable. The half-life of sodium-24 is 15 hours, so we can substitute these values into the formula:
N(t) = N0 * (1/2)^(t/h)
N0/8 = N0 * (1/2)^(t/15)
Now we can simplify the equation:
(1/8) = (1/2)^(t/15)
To solve for t, we can take the logarithm of both sides of the equation:
log((1/8)) = log((1/2)^(t/15))
Using logarithm properties, we can bring down the exponent:
log((1/8)) = (t/15) * log((1/2))
Now we can solve for t:
8 = (2^(-t/15)) * log((1/2))
We can rewrite 8 as 2^3:
2^3 = 2^(-t/15) * log((1/2))
Since the bases are the same, we can set the exponents equal to each other:
3 = -t/15 * log((1/2))
We can rearrange this equation to solve for t:
t/15 = -3 / log((1/2))
Now we can solve for t:
t = -3 * 15 / log((1/2))
Using a calculator, we can evaluate this expression:
t ≈ -3 * 15 / (-0.6931)
t ≈ 45.46637502
Therefore, about 45 hours has passed. Answer choice b is the correct option.