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.

**Carbon dating **(also called **carbon-14** or **radiocarbon dating) **is a method of determining the age of an organic substance by looking at the Concentration of carbon-14

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.

**Isotopes **are different forms of the same element that have a different number of neutrons.

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 (N_{2}). These cosmic rays contain high-energy neutrons, which react with the Nitrogen to form carbon-14, as shown below:

Carbon-14 then reacts with atmospheric oxygen (O_{2}) 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:

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, N_{0} 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**.

A species **half-life **is the time is takes for that species to decay to 50% of its original concentration. The formula for half life is:

$$k=\frac{ln(2)}{t_\frac{1}{2}}$$

or

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

Where k is the rate constant, and t_{1/2} is the time it takes for the species to reach 1/2 of it's concentration

The formula given is for a **first-order half-reaction. **The order of a reaction tells us how the concentration affects the rate of a reaction. The "order" refers to what power the concentration is raised to, so a First Order Reaction is k=[x]^{1}, while a second-order reaction is k=[x]^{2}, and so on. The first-order formula is given because the decay of carbon-14 is a first order-reaction

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.

**An archeologist digs up a human skull at a dig site. It was found that concentration of carbon-14 is 25% of its initial concentration. What is the age of the skull?**

Let's look at our formula:

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

Since there is 25% left of the carbon-14, that means the N_{t}/N_{0} ratio is 0.25. Plugging this into our equation, we can solve for the age of the skull (t):

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

$$t=\frac{(ln(0.25))}{(-0.693)}*5,730\,years$$$$t=2.000*5,730\,years$$

$$t=11,460\,years$$

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.

**A museum is testing the authenticity of a Leonardo da Vinci (1452-1519) manuscript. They send a paper sample to a lab and learn that it has 97.4% of its initial carbon-14 concentration. Is this manuscript authentic?**

To determine if this is authentic or not, we need to use our carbon dating formula to solve for the age in the paper. If the paper is ~500-550 years old, it is likely authentic.

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

$$t=\frac{(ln(0.974)}{(-0.693)}*5,730\,years$$

$$t=0.038*5,730\,years$$

$$t=218$$

The age of the paper suggests it was made in 1804, so this manuscript is *not *authentic (though it's still pretty old!)

## 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 CO_{2} emitted diluted the concentration of carbon-14 in the atmosphere. Thus making samples from the early 20th appear older.

## Carbon Dating - Key takeaways

**Carbon dating**(also called**carbon-14**or**radiocarbon dating)**is a method of determining the age of an organic substance by looking at the concentration of carbon-14**Isotopes**are different forms of the same element that have a different number of neutrons.- Carbon-14 is a radioactive isotope

- Carbon-14 is produced in Earth's atmosphere. It is formed by a reaction of cosmic rays with atmospheric nitrogen (N
_{2}). These cosmic rays contain high-energy neutrons, which react with the nitrogen to form carbon-14- When a life form is alive it has a stable supply of carbon-14, which will then decay once it dies

- 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
- The equation for carbon dating is $$t=\frac{(ln(\frac{N_t}{N_0}))}{(-0.693)}*5,730\,years$$
- Where t is the age of the sample, N
_{t}is the concentration of carbon-14 at time t, and N_{0}is the initial concentration

- Where t is the age of the sample, N

## References

- Fig.1 Nitrogen is converted into carbon-14 through radioactive decay (https://upload.wikimedia.org/wikipedia/commons/thumb/7/70/Carbon_14_formation_and_decay.svg/640px-Carbon_14_formation_and_decay.svg.png) by NikNaks (https://commons.wikimedia.org/wiki/User:NikNaks93) licensed by CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/)
- Fig.2-Carbon decays over time, which can be used to determine the age of organic materials (https://upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Radioactive_decay_of_Carbon-14.png/640px-Radioactive_decay_of_Carbon-14.png) by Kurt Rosenkrantz licensed by CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/)

What are isotopes?

**Isotopes **are different forms of the same element that have a different number of neutrons.

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##### Frequently Asked Questions about Carbon Dating

What is carbon dating?

**Carbon dating **(also called **carbon-14** or **radiocarbon dating) **is a method of determining the age of an organic substance by looking at the concentration of carbon-14

What is an example of carbon dating?

Carbon dating can be used to estimate the age of skeletal remains

How do you calculate age using carbon dating?

To calculate an organic sample's age, we use the following formula:

t=(ln(N_{t}/N_{0}))/-0.693 *5,730 years

Where N_{t} is the amount of cabron-14 at time t, N_{0} is the initial amount of carbon-14, and t is the age of the sample

How does carbon dating affect chemistry?

Carbon dating relies on different types of chemistry such as analytical and organic chemistry.

What is more accurate than carbon dating?

While carbon dating is the most common reliable method, methods such as Uranium-lead dating is more accurate.

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