Question

How many days does it take for 16 g of palladium-103 to decay to 1.0 g?

The half-life of palladium-103 is 17 days.

Answer 1

It will take 68 days for the palladium to decay.

Explanation:

The formula for the half-life of a first-order process is

`color(blue)(bar(ul(|color(white)(a/a) t_½ = (ln2)/kcolor(white)(a/a)|)))" "`

∴ `k = (ln2)/t_½ = (ln2)/"17 days" = "0.0408 day"^"-1"`

The integrated formula for a first-order process is

`color(blue)(bar(ul(|color(white)(a/a) ln("A"_0/"A") = ktcolor(white)(a/a)|)))" "`

`ln("A"_0/"A") = ln((16 color(red)(cancel(color(black)("g"))))/(1.0 color(red)(cancel(color(black)("g")))))= 2.77 = "0.0408 day"^"-1"t`

`t = 2.77/ "0.0408 day"^"-1" = "68 days"`

If we had recognized that the `"Pd"` was decaying to `1/16`of its original value, we could have said that it takes 4 half-lives (4 × 17 days) for the decay.

This is easier, but it works only for an integral number of half-lives.