Question

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?

Answer 1

`1.9 * 10^6"years"`

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, `"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 sig figs, 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.