Question

Carbon 14 has a half-life of 5,730 years. If one fourth as much carbon 14 is found in an ancient tree ring compared to the amount of carbon 14 found in a modern tree, how old can you calculate the tree to be?

Answer 1

First, let's recall the equation for half-life:

N(t) =
`N_(0)` ·
`(1)/(2)^{(t)/(t_(1/2))}`

Woah.

While we could certainly solve this question by using the half-life equation, it might be easier to just give this question some thought.

Half-life, is, essentially, the time it takes for radioactive material to be halved. For example, if something had a half-life of ten years and weighed 20 pounds, after ten years it would weigh 10 pounds and after another ten years would weigh 5 pounds, then 2.5, and so on. (Note that it never actually reaches 0! It keeps halving and halving over and over again, becoming smaller and smaller but never disappearing!)

In this question, we're given the half-life of Carbon-14 and asked how old a tree with 1/4 the amount of carbon-14 of a normal tree would be. Let's think about this question:

• We can make the assumption that a "modern" tree is a tree that is 0 years old. Why? Well, it would make it painfully complicated if we assumed anything other than that, and your chemistry teacher wouldn't be that cruel (I hope).
• How many half-lives must the Carbon-14 go through for the amount to be 1/4? Since the amount halves every half-life, we can count on our fingers - one half-life will make the amount 1/2 the original, two half-lives will make the amount half the 1/2 amount, making it 1/4 the amount... hey! We got it!

We figured out that it takes two half-lives for Carbon-14 to be 1/4 its original amount! Since the duration of a half-life never changes, we can simply multiply 5730 by 2 to find how many years it'd take to achieve two half-lives:

5730 · 2 = 11460

The tree is about 11,460 years old.

And we've solved it! (A quick tip - if you ever encounter a question like this again, you can quickly calculate the number of half-lives in your head depending on the amount remaining that they give you. Just keep halving the amount every half-life! One half-life is 1/2, two half-lives is 1/4, three is 1/8, four is 1/16, and so on and so forth)

If you have any questions about anything I explained here or need clarification on anything, just let me know! Good luck!

- breezyツ