Radio-Isotope Dating

Nuclear reactions are all first order reactions. That means that all nuclear reactions have a half-life. An analogy may help you to understand half-lives.

Half-lives have entered the common vocabulary (although most people may not really understand it). For instance, you may have heard people discussing the half-life of nuclear compounds when they discuss nuclear power.

Because half lives are so consistent, they can help us determine the age of many things. For instance, imagine that you are digging in your neighbor's garden and you find a femur (the big bone in your upper leg). Should you call the police or the museum? The answer depends on how old the bone is.

Fortunately for us, all living things (including chemistry students) contain a radioactive isotope of C called cabon-14 (carbon with 8 neutrons and therefore a mass number of 14). Although this radioactive isotope is slowly decaying in your body, you are constantly eating more (from the mildly radioactive plants and animals that you consume) so that the amount of ¹⁴C in your body doesn’t change. Until you stop eating, that is.

When you stop eating, you stop replenishing the ¹⁴C in your body and the amount begins to go down. Assuming that the previous owner of the femur stopped eating when he or she died (a pretty safe assumption) we can use the amount of ¹⁴C in the bone to figure out the age.

Put into a nice word problem the question would look something like this:
A femur is discovered with only 1/4 th the amount of ¹⁴C found in the bone of a living animal. If the half life of ¹⁴C is 5260 years, how old is the bone?

We can solve this problem simply

We now know that the bone is 10,500 years old (approximately). We should call the museum.

This process is called carbon-dating and has been used to date many artifacts from early human settlements.

Two things to realize:

All radioactive isotopes decay in first order, so ANY isotope can be used to date something. All you need is enough in the sample to measure it, to know how much there was at the beginning, and to know how long the half-life is. With those facts we have been able to date bones of early humans, and rocks that formed billions of years ago on the earth.

Here are some problems for you to try.

Of course, not all things have been through an exact number of half lives. There is a more complex way to do the math (with partial half-lives) called the integrated rate law, but that is beyond the scope of this text.