# The half-life for radioactive decay (a first-order process) of plutonium-239 is 24,000 years. How many years would it take for one mole of this radioactive material to decay so that just one atom remains?

##### 1 Answer

#### Explanation:

Your strategy here will be to use **Avogadro's number** to calculate the *number of atoms* of plutonium-239 that you're starting with.

One you know that, use the equation that allows you to calculate the amount of a radioactive nuclide that **remains undecayed**, *initial amount* of the nuclide, **number of half-lives**,

#color(blue)(|bar(ul(color(white)(a/a)"A"_t = "A"_0 * 1/2^ncolor(white)(a/a)|)))#

Here you can say that

#color(purple)(|bar(ul(color(white)(a/a)color(black)(n = t/t_"1/2")color(white)(a/a)|)))#

where **half-life** of the nuclide.

So, you know that Avogadro's number acts as a conversion factor between the number of *moles* of a element and the number of *atoms* it contains

#color(blue)(|bar(ul(color(white)(a/a)"1 mole" = 6.022 * 10^(23)"atoms"color(white)(a/a)|))) -># Avogadro's number

Since you're dealing with **one mole** of plutonium-239, you can say that the initial amount of this isotope will be

#"A"_ 0 = 6.022 * 10^(23)"atoms"#

The amount that remains undecayed is

#"A"_t = "1 atom"#

Now, rearrange the above equation to solve for

#"A"_t/"A"_0 = 1/2^n#

#2^n = "A"_0/"A"_t#

This will be equivalent to

#ln(2^n) = ln("A"_0/"A"_t)#

#n * ln(2) = ln("A"_0/"A"_t) implies n = ln("A"_0/"A"_t)/ln(2)#

Plug in your values to get

#n = 1/ln(2) * ln( (6.022 * 10^(23)color(red)(cancel(color(black)("atoms"))))/(1color(red)(cancel(color(black)("atom"))))) = 78.99#

This means that it takes **half-lives** for your sample of plutonium-239 to decay from one mole to one atom.

Since the half-life of the nuclide is equal to

#n = t/t_"1/2" implies t = n * t_"1/2"#

#t = 78.99 * "24,000 years" = color(green)(|bar(ul(color(white)(a/a)color(black)(1.9 * 10^6"years")color(white)(a/a)|)))#

The answer is rounded to two **sig figs**, the number of sig figs you have for the half-life of the nuclide.

You can thus say that it will take **million years** for one mole of plutonium-239 to decay to one atom.