## Question

A piece of furniture was recovered from an archaeological site and carbon dating was used to estimate its age. It was assumed that the proportion of Carbon-14 to natural carbon (Carbon-12) in living wood was 1.25 x 10^{-12}. If the number of particles emitted from 5 g of carbon prepared from the specimen was 21 per minute, how old is the specimen?

[The half-life of Carbon-14 is 5600 years and the mass number of natural carbon is 12.]

## Answer

Since the proportion of Carbon-14 is very small compared to Carbon-12, we can assume for the sake of finding the number of carbon atoms that all of the carbon in the specimen is Carbon-12.

Given that the proportion of Carbon-14 to Carbon-12, we can find the original number of Carbon-14 atoms when the piece of furniture was living wood using the following approximation. (Refer to bottom for further discussion of this approximation.)

Using the half-life, we can calculate the decay constant of Carbon-14.

Hence, we can find the initial activity when the specimen was still living wood.

The final activity of the specimen is given by

Now, we can find the age of the specimen.

## Discussion on Approximation

Based on the phrasing of the question that

“the proportion of Carbon-14 to natural carbon is *k*”,

However, since the *k* is small compared to 1,

Hence, we arrive at