Question

Question #d4494

Answer 1

Here's what I got.

Explanation:

The thing to remember about an isotope's nuclear half-life, `t_"1/2"`, is that it represents the time needed for half of an initial sample of that isotope to undergo radioactive decay.

In other words, the half-life tells you how much must pass in order for your sample to be halved.

If you take `A_t` to be the amount of a radioactive isotope that remains undecayed after a time `t` passes and `A_0` to be the initial amount of that isotope, you can say that you will have

• `A_t = A_0 * 1/2 = A_0/2 = A_0/2^color(red)(1) ->` after `color(red)(1)` half-life
• `A_t = A_0/2 * 1/2 = A_0/4 = A_0/2^color(red)(2) ->` after `color(red)(2)` half-lives
• `A_t = A_0/4 * 1/2 = A_0/8 = A_0/2^color(red)(3) ->` after `color(red)(3)` half-lives
  `vdots`

and so on. So with every passing half-life, you get to divide the initial amount by `2`. This means that if `color(red)(n)` half-lives pass in a given period of tiem `t`, you will divide the initial amount by `2` a total of `color(red)(n)` times.

`A_t =A_0/(underbrace(2 * 2 * ... * 2)_(color(black)(color(red)(n)color(white)(.)"times"))) = A_0/2^color(red)(n)`

This is equivalent to

`A_t = A_0 * (1/2)^color(red)(n)`

with

`color(red)(n) = t/t_"1/2"`

In your case, you start with `"200 g"` of this radioactive isotope and end up with `"25 g"`. This means that you have

`25 color(red)(cancel(color(black)("g"))) = 200color(red)(cancel(color(black)("g"))) * (1/2)^color(red)(n)`

Divide both sides by `200` to get

`25/200 = 1/2^color(red)(n)`

This is equivalent to

`1/4 = 1/2^color(red)(n)`

`1/2^2 = 1/2^color(red)(n) implies color(red)(n) = 2`

So you can say that in order for your sample to be reduced from `"200 g"` to `"25 g"`, two half-lives must pass.

This means that you have

`t = 2 * t_"1/2"`

`color(darkgreen)(ul(color(black)(t = 2 * "8.4 days" = "16.8 days")))`

I'll leave the answer rounded to three sig figs, but a more accurate answer would be

`t = "20 days"`

because you have only one significant figure for the initial mass of the sample.