Imagine you are an archeologist at a dig site in Rome. After hours upon hours of digging, you strike gold (not literally). You find a worn clay pot that you believe is from the ancient Rome period.

So how do you check to see its true age? Well, one method is called carbon dating, which is used to date organic samples. This method can be used to date artifacts like our theoretical archeologist found, but it can also date plants and animals as well.

In this article, we will be learning about carbon dating and see how this process is used to date deceased organic materials and life forms.

• This article covers the topic of carbon dating.
• First, we will define what carbon dating is.
• Then, we will walk through the science that carbon dating is based on.
• Next, we will learn the formula for carbon dating.
• After that, we will use the formula in a few example problems.
• Lastly, we will discuss the accuracy of carbon dating.

Carbon Dating Definition

Let's start by stating the definition of carbon dating.

This process was developed by chemist Willard Libby (1908-1980) and revolutionized the fields of archeology and paleontology. Using this method, organic matter that lived as far back as 50,000 years can be dated.

Carbon Dating Method

Carbon dating focuses on the concentration of carbon-14, a carbon isotope, that is found in a deceased life-form, such as a plant or animal.

Carbon's most abundant form is carbon-12, which has 6 neutrons. However, carbon-14 has 8 neutrons.

So why carbon-14?

Well, let's talk about where carbon-14 comes from.

Carbon-14 is produced in Earth's atmosphere. It is formed by a reaction of cosmic rays with atmospheric nitrogen (N₂). These cosmic rays contain high-energy neutrons, which react with the nitrogen to form carbon-14, as shown below:

Fig.1-Nitrogen is converted into carbon-14 through radioactive decay

Carbon-14 then reacts with atmospheric oxygen (O₂) to form carbon dioxide. This carbon dioxide is taken in by plants through photosynthesis and by animals through eating those plants.

Carbon-14 is radioactive, so it decays over time. Essentially, radioactive decay occurs when a species is unstable, so it emits energy and/or particles to stabilize itself.

When a species is alive, it will take in more carbon-14 as mentioned above, so it will have a constant supply. However, when that species dies, it is no longer replenishing its carbon-14 supply, so the concentration of carbon-14 will decrease over time as it decays.

Below is a chart showing how carbon-14 decays over time:

Fig.2-Carbon decays over time, which can be used to determine the age of organic materials

The way carbon dating works is that it compares the initial concentration of carbon-14 to the current concentration in the deceased sample to determine how old the sample is. The less carbon-14 is present, the older the sample is

Carbon dating formula

So, how do we actually calculate the age of the sample? Well, we use the radioactive decay formula:

$$N_t=N_0e^{-kt}$$Where N_t is the concentration at time t, N₀ is the initial concentration (i.e. at t=0), k is the rate constant for the decay reaction, and t is the time.

But, what is k exactly? The constant k is a rate constant, which basically tells us how fast or slow the decay occurs. The value of k for carbon-14 is 1.21 x 10^-4 year^-1, which comes from the half-life formula.

The currently accepted value for, t_{1/2 (carbon-14)}, is 5,730 years, meaning it takes 5,730 for the concentration of carbon-14 to decay to half its original value.

Going back to our decay equation, we are solving for "t", where, t, is the age of the sample. First, we set up our equation and solve for, t:

$$N_t=N_0e^{-kt}$$

$$\frac{N_t}{N_0}=e^{-kt}$$

$$ln(\frac{N_t}{N_0})=-kt$$

$$\frac{ln(\frac{N_t}{N_0})}{-k}=t$$

Next, we want to plug the expression for k into our equation:

$$k=\frac{-0.693}{5,730\,years}$$

$$\frac{ln(\frac{N_t}{N_0})}{\frac{-0.693}{5,730\,years}}=t$$

To clean up our equation a bit, we multiply the entire equation by 5,730/5,730 (i.e 1)

$$\frac{ln(\frac{N_t}{N_0})}{\frac{-0.693}{\cancel{5,730\,years}}}*\frac{5,730\,years}{\cancel{5,730\,years}}=t$$

This simplifies to our final equation:

$$t=\frac{(ln(\frac{N_t}{N_0}))}{(-0.693)}*5,730\,years$$

Carbon Dating Examples

Now that we have our formula and a basic understanding of carbon dating, let's work on some examples.

That's one old skull! While carbon-dating can be used to simply learn the age of something. It can also be used to check the authenticity of items.

Carbon Dating Accuracy

Carbon dating is reliable and can give a relatively accurate date when compared with other dating systems. The main accuracy problem is related to how much carbon-14 was believed to be in the sample before it died. The key word here is "believed".

Scientists have many ways to calibrate and calculate the initial concentration of carbon-14. However, there can be some issues with this. Since this is an estimation, it can never be perfect.

Essentially, carbon-14 concentrations in the atmosphere fluctuate because of things like time, geographic location, and the burning of fossil fuels. For example, fossil fuels started being burned significantly during the industrial revolution of the 19th century. Because of this, the CO₂ emitted diluted the concentration of carbon-14 in the atmosphere. Thus making samples from the early 20th appear older.