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?

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, $"A"_t$, in terms of the initial amount of the nuclide, $"A"_0$, and the number of half-lives, $n$, that pass in a given period of time $t$.

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

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 $t_"1/2"$ is the 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 $n$

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

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

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 $78.99$ 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 $"24,000 years"$, it follows that you have

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

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

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

You can thus say that it will take $1.9$ million years for one mole of plutonium-239 to decay to one atom.