# The half-life of plutonium-239 is 24,100 years. Of an original mass of 100g, how much plutonium-239 remains after 96,440 years?

##### 2 Answers

#### Explanation:

*Note: I edited the question to use the half-life of plutonium-239 as the correct value of #24,100# years, not the value in the original question of #24,110# years, which I assume was a typo.*

Use the equation

Where

and

to find

So:

The mass remaining is

#### Explanation:

The nuclear half-life of a nuclide tells you how much time must pass before **half** of an initial sample of said nuclide undergoes radioactive decay.

Simply put, an initial sample of radioactive isotope is **halved** with *every passing* of a half-life.

This means that for a

#A_0 * 1/2 -># afteronehalf-life#A_0/2 * 1/2 = A_0/4 -># aftertwohalf-lives#A_0/4 * 1/2 = A_0/8 -># afterthreehalf-lives#A_0/8 * 1/2 = A_0/16 -># afterfourhalf-lives

#vdots#

and so on.

Notice that you can express the amount of the sample that remains after

#color(blue)(A_n = A_0 * 1/2^n)#

Now, notice that the time given to you is actually a *multiple* of the half-life

#n = 96440/24110 = 4#

This means that **four** half-lives of plutonium-239 will pass in

#m = "100 g" * 1/2^4 = "6.25 g"#

I'll leave the answer rounded to three sig figs, despite the fact that you only have one sig fig for the mass of the sample.