# 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.

Oct 18, 2016

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

$t = \frac{2.77}{\text{0.0408 day"^"-1" = "68 days}}$

If we had recognized that the $\text{Pd}$ was decaying to $\frac{1}{16}$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.