The half-life of the plutonium isotope is 24,360 years. If 10 grams of plotonium is released into the atmosphere by a nuclear accident, how many years will it take for 80% of the
isotope to decay?
To find out how many years it will take for 80% of the isotope to decay, we can use the concept of half-life. The half-life is the amount of time it takes for half of a radioactive substance to decay.
In this case, the half-life of the plutonium isotope is given as 24,360 years. This means that after 24,360 years, only half of the original quantity will remain. After another 24,360 years, half of that remaining quantity will decay, and so on.
To find the time it takes for 80% of the isotope to decay, we need to calculate how many half-lives it will take for the amount to decrease to 20% (100% - 80%) of the original.
Since each half-life reduces the quantity by half, we can use the formula:
n = t / hl
Where:
n is the number of half-lives
t is the total time
hl is the half-life
Rearranging the formula to solve for n:
n = log(remaining / initial) / log(0.5)
Given that 80% (or 0.80) of the isotope will remain after decay, we can substitute the values into the formula:
n = log(0.20) / log(0.5)
Using a calculator, we can evaluate:
n ≈ 2.3219
This means it will take approximately 2.3219 half-lives for 80% of the isotope to decay.
Now, to find the time it takes for this to happen, we multiply the number of half-lives by the length of one half-life:
time = n * hl
time ≈ 2.3219 * 24,360 years
time ≈ 56,522 years
Therefore, it will take approximately 56,522 years for 80% of the plutonium isotope to decay.