Given a 800 mg sample of Thorium-231 decomposes via beta emission with a half life of 25.5 hours, how long would it take, in days, for the sample to decay down to 25 mg?
231 0 231
Th -> Beta + Pa
90 -1 91
800 mg/ 25 mg
25/800 = 1/32 = 1/2^5
so, it will take 5 half-lives
To solve this problem, we can use the radioactive decay formula:
N(t) = N₀ * (1/2)^(t / T₁/₂)
Where:
N(t) represents the remaining amount of the sample at time t,
N₀ is the initial amount of the sample,
t is the time that we want to find,
and T₁/₂ is the half-life of the substance.
In this case, we want to find the time it takes for the sample to decay down to 25 mg when the initial amount is 800 mg, and the half-life is 25.5 hours.
Plugging in the values, we have:
25 mg = 800 mg * (1/2)^(t / 25.5)
To solve for t, we need to isolate t on one side of the equation. Let's divide both sides of the equation by 800 mg:
(25 mg) / (800 mg) = (1/2)^(t / 25.5)
0.03125 = (1/2)^(t / 25.5)
Next, we can take the logarithms of both sides to eliminate the power:
log(0.03125) = log[(1/2)^(t / 25.5)]
Using a logarithm property, we can bring down the power as a coefficient:
log(0.03125) = (t / 25.5) * log(1/2)
Now, let's solve for t by multiplying both sides of the equation by 25.5 and dividing by log(1/2):
t = (25.5 * log(0.03125)) / log(1/2)
Using a scientific calculator, we find:
t ≈ 706.65 hours
Finally, to convert hours to days, we divide by 24:
t ≈ 29.44 days
Therefore, it would take approximately 29.44 days for the sample to decay down to 25 mg.