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# 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?

##"6.25 g"##

The 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## sample of a radioactive isotope, you can say that you'll be left with

- ##A_0 * 1/2 ->## after
**one**half-life - ##A_0/2 * 1/2 = A_0/4 ->## after
**two**half-lives - ##A_0/4 * 1/2 = A_0/8 ->## after
**three**half-lives - ##A_0/8 * 1/2 = A_0/16 ->## after
**four**half-lives ##vdots##

and so on.

Notice that you can express the amount of the sample that remains after ##n## half-lives like this

##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 ##"96,440"##, and so the remaining sample after this period of time will be

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

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