Question

The half-life of Iodine-131 is 8 days. What mass of I-131 remains from an 8.0g sample after 2 half-lives?

Answer 1

`"2 g"`

Explanation:

The key to this problem lies with how the nuclear half-life of a radioactive isotope was defined.

For a given sample of a radioactive isotope, the time needed for half of the sample to undergo decay will give you that isotope's nuclear half-life.

This means that every passing of a half-life will leave you with half of the sample you started with.

Let's say that you start with a sample `A_0`. Using the definition of a nuclear half-life, you can say that you will be left with

• `A_0 * 1/2 = color(purple)(A_0/2) ->` after one half-life

What about after the passing of another half-life?

• `color(purple)(A_0/2) * 1/2 = color(orange)(A_0/4) ->` after two half-lives

What about after the passing of another half-life?

• `color(orange)(A_0/4) * 1/2 = color(brown)(A_0/8) ->` after three half-lives

and so on. With every half-life that passes, your sample will be halved.

Mathematically, you can express this as

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

`A` - the mass of the sample that remains after a period of time
`n` - the number of half-lives that pass in that period of time

You know that your sample of iodine-131 has a half-life of `8` days. In your case, you are interested in figuring out how much iodine-131 will remain undecayed after the passing of `2` half-lives.

This means that here `n=2`. You will thus have

`A = A_0 * 1/2^2 = A_0/4`

Since you started with an `"8-g"` sample, you will be left with

`A = "8 g" * 1/4 = color(green)("2 g")`