Question

Question #5e9e6

Answer 1

`"2.25 years"`

Explanation:

Your tool of choice here will be the equation

`color(blue)(|bar(ul(color(white)(a/a)A_t = A_0 * 1/2^n color(white)(a/a)|)))`

Here

`A_t` - the amount undecayed after a period of time `t`
`A_0` - the initial mass of the radioactive isotope
`n` - the number of half-lives that pass in a period of time `t`

In your case, you know that it took `9` years for a sample of a given radioactive isotope to decay from `"448 g"`, the initial mass of the sample, to `"28 g"`, the mass that remains undecayed.

Your goal here will be to use the above equation to find the value of `n`, the number of half-lives that pass in `9` years. Plug in your values to find

`28 color(red)(cancel(color(black)("g"))) = 448 color(red)(cancel(color(black)("g"))) * 1/2^n`

Rearrange to find

`2^n = 448/28 = 16`

Since `16` can be written as a power of `2`

`16 = 2 * 2 * 2 * 2 = 2^4`

you will have

`2^n = 2^4`

This implies that

`n = 4`

So, you know that it takes `9` years for `4` half-lives to pass, which means that one half-life, `t_"1/2"`, is

`t_"1/2" = "9 years"/4 = color(green)(|bar(ul(color(white)(a/a)color(black)("2.25 years")color(white)(a/a)|)))`