# 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?

##### 1 Answer

#### 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 * 1/2 = A_0/2 -># afterone half-life

#A_0/2 * 1/2 = A_0/4 -># aftertwo half-lives

#A_0/4 * 1/2 = A_0/8 -># afterthree 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

In your case, you know that the fossil is

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 **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")#