Thorium-232 undergoes alpha decay and has a half-life of 1.4 * 10^10 years. How long will it take for a 1.00mg sample of thorium-232 to be reduced to 0.5mg?
The simple answer:
It will take 1.4E10 years.
If the half life is 1.4E10 and you want exactly 1/2 of the Th to decay, it will take just one half life. right?
You can work it out.
k = 0.693/t1/2
ln(No/N) = kt.
No = 1.0 mg
N = 0.5 mg
k from above.
Solve for t.
You should obtain 1.4E10 years.
To find the time it takes for a sample of thorium-232 to be reduced to a certain mass, we can use the formula for exponential decay:
N(t) = N0 * (1/2)^(t / T1/2)
Where:
N(t) is the final amount of the substance after time t,
N0 is the initial amount of the substance,
T1/2 is the half-life of the substance.
In this case, we need to solve for t, the time it takes for the thorium-232 sample to be reduced from 1.00mg to 0.5mg. Let's start by plugging in the values we know:
0.5mg = 1.00mg * (1/2)^(t / 1.4 * 10^10)
Next, let's simplify the equation:
(1/2)^(t / 1.4 * 10^10) = 0.5 / 1.00
Now, let's take the logarithm of both sides of the equation to solve for t:
log((1/2)^(t / 1.4 * 10^10)) = log(0.5 / 1.00)
Using the logarithm property log(a^b) = b * log(a), the equation becomes:
(t / 1.4 * 10^10) * log(1/2) = log(0.5 / 1.00)
Now, we can solve for t by rearranging the equation:
t = (1.4 * 10^10) * (log(0.5 / 1.00) / log(1/2))
Calculating this expression, we get:
t ≈ 9.86 * 10^9 years
Therefore, it will take approximately 9.86 billion years for a 1.00mg sample of thorium-232 to be reduced to 0.5mg.