Question

The half-life of `""^131"I"` is 8.07 days. What fraction of a sample of `""^131"I"` remains after 24.21 days?

Answer 1

`1/8`

Explanation:

As you know, an isotope's nuclear half-life tells you how much time must pass in order for half of an initial sample of this isotope to undergo radioactive decay.

In other words, an isotope's half-life tells you how much must pass in order for a sample to be reduced to half of its initial value.

If you take `A_0` to be the initial sample of an isotope, and `A` to be the sample remaining after a period of time `t`, then you can say that

• `A = A_0 * 1/2 ->` after one half-life
• `A = A_0/2 * 1/2 = A_0/4 ->` after two half-lives
• `A = A_0/4 * 1/2 = A_0/8 ->` after three half-lives
  `vdots`

and so on. This means that you can express `A` in terms of `A_0` and the number of half-lives that pass using the equation

`color(blue)(A = A_0 * 1/2^n)" "`, where

`n` - the number of half-lives that pass in a given period of time

`color(blue)(n = "period of time"/"half-life")`

So, you want to know what fraction of an initial sample of `""^131"I"` remains after `"24.21 days"`.

How many half-lives do you get in that period of time, knowing that one half-life is equal to `"8.07 days"`?

`n = (24.21 color(red)(cancel(color(black)("days"))))/(8.07color(red)(cancel(color(black)("days")))) = 3`

This means that you have

`A = A_0 * 1/2^3`

`A = A_0 * 1/8`

Therefore, your initial sample of `""^131"I"` will be reduced to `1/8"th"` of its initial value after `"24.21 days"`.