Question

Carbon-14 has a half-life of 5770 years. If a fossil is 23,080 years old and it has 3kg of Carbon-14, how much carbon-14 did it originally have?

Answer 1

`"48 kg"`

Explanation:

A radioactive isotope's nuclear half-life tells you how much time must pass in order for a sample of this isotope to the reduced to half of its initial size.

In essence, any sample you start with will be halved with every passing of a half-life. So if you start with a mass `A_0` of your isotope, you can say that you'll be left with

`A_0 * 1/2 = A_0/2 ->` after one half-life**

`A_0/2 * 1/2 = A_0/4 ->` after two half-lives**

`A_0/4 * 1/2 = A_0/8 ->` after three half-lives**

`vdots`

and so on. You can thus find a relationship between how many half-lives have passed in a given period of time and how much of your initial sample is left undecayed

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

`A` - the amount left undecayed
`n` - the number of half-lives that have passed

In your case, you know that the fossil is `"23,080"` years old and that carbon-14, the isotope of interest, has half-life of `"5,770"` years.

This means that you determine how many half-lives have passed in this time period

`n = (23080 color(red)(cancel(color(black)("years"))))/(5770color(red)(cancel(color(black)("years")))) = 4`

So, you know that `"3.0 kg"` of carbon-14 are left undecayed after the passing of four half-lives, which means that the sample originally contained

`A = A_0 * 1/2^n implies A_0 = A * 2^n`

`A_0 = "3.0 kg" * 2^4 = color(green)("48 kg")`