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# Exponential FunctionsCarbon Dating

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A group of archaeologists has discovered a new tomb in the Egyptian desert. They carefully open the hidden entrance, climb through several rooms filled with ancient treasures, until they arrive in the burial chamber. The sarcophagus is still sealed and contains the mummy of a Pharaoh.

After cataloguing every item in the tomb and carefully transporting them to a nearby museum, they try to answer their most pressing question: who is this Pharaoh, and when did he die?

Unfortunately, none of the drawings and inscriptions on the walls of the tomb seem to contain any names or dates. However, there is an ingenious method to accurately determine the age of ancient artefacts like mummies or fossils, which solely relies on physics and mathematics: Carbon dating.

All living organisms on Earth – plants, animals and humans – contain carbon atoms. Usually, the core of a carbon atom consists of six protons and six neutrons, but a small proportion of carbon atoms contain additional neutrons. These different “variants” of carbon are called isotopes:

Carbon-12
6 protons, 6 neutrons
98.8%

Carbon-13
6 protons, 7 neutrons
1.1%

Carbon-14
6 protons, 8 neutrons
0.1%

The proportion at which these isotopes occur is almost exactly the same, everywhere on Earth – and this will be very important later. In our example, we are particularly interested in carbon-14, which is abbreviated as 14C. It contains 6 protons and 8 neutrons, and it is created when cosmic rays coming from outer space hit particles high in our atmosphere.

Any sample of carbon atoms consists of % of these special 14C atoms. You might think that this is an insignificant amount, but your body contains around 8×1026 carbon atoms, which means that you also contain around 8×1023 carbon-14 atoms. That’s almost a million-million-million-million atoms!

Carbon-14
6 protons
8 neutrons

Carbon-14 is useful because it is radioactive. The atom is unstable, and it might decay into other, more stable elements. We are actually surrounded by many radioactive materials, but their concentration is not high enough to be dangerous.

During our life, as we eat and breathe, our body absorbs 14C atoms. When we die, we stop absorbing new 14C atoms, and the ones that are already in our body slowly start to .

All radioactive elements decay at a very predictable rate – this is determined by their half-life. Carbon-14, for example, has a half-life of approximately 6,000 years. This means that if you have a block of 14C atoms, it will take 6,000 years for half of them to decay. After another 6,000 years, half of the remaining atoms will have also decayed, so you’re left with just of the original amount.

Let’s assume we start with a block of 1,200 carbon-14 atoms. Using the half-life, we can calculate the remaining amount of 14C atoms over time:

 Years 0 6000 12,000 18,000 24,000 Amount 1200

As you can see, we’re multiplying by 12 at every step, just like a sequence.

Using exponents, we can write down an equation for the amount left after t years:

amount=1200×12t6000

Of course, 1200 and 6000 were just arbitrary numbers. A more general equation is:

amount=initial×12thalf-life

• Using the exponents laws, we can flip the fraction 12 into a 2, if we multiply the exponent by 1.
• This equation describes how many atoms are left after t number of years.

Since the equation contains an exponent and the number of atoms decreases, we call this process exponential decay.

We can plot the amount of 14C atoms over time in a coordinate system. If we start with an initial amount of \${format(x0)} atoms, and the half-life is \${format(hl)} years, then the decay looks like this:

The points on the graph show when the number of atoms has halved. Notice that we can calculate the remaining number of atoms at any point in time, not just these specific intervals. This is the main difference compared to geometric sequences.

The decay of radioactive atoms is random, and it is impossible to predict when exactly an individual 14C is going to decay. The graph shows the average number of atoms we expect to be left at a specific time. That’s also why the remaining number of atoms might not always be an integer – even though you can’t have “half an atom”. You will learn more about this in our course on probability.

Now we have all the information needed to determine the age of the Pharaoh. The archaeologists decided to cut a tiny sample out of the mummy’s skin. Using a complex machine called a mass spectrometer, they were able to “count” the number of 12C and 14C atoms in the sample.

In our example, they found 800 carbon-14 atoms. Given the ratios of 12C and 14C atoms, they also estimate that the same sample should have contained 1200 14C atoms when the Pharaoh was still alive.

Accelerator Mass Spectrometer in the Oxford Radiocarbon Accelerator Unit

All we have to do now is calculate how long it takes for the 400 missing 14C atoms to decay. That number is precisely the .

We can use the equation we found above and fill in the required parameters:

×2t=

• Fill in the three parameters from above!
• Let’s start by dividing both sides of the equation by 1200.
• We can find the decimal value of the fraction on the right-hand side of the equation.
• Now, we have to deal with the exponent on the left-hand side. To do that, we can use a special function called the Logarithm, which you’ll learn more about later.
• Using a calculator, we can find the value of log20.667.
• The rest should be simple: let’s multiply both sides of the equation by 6000.
• We can simplify the right-hand side of the equation.
• We can also remove the – sign on both sides of the equation.
• Thus, we see that it takes 3510 years for the required number of 14C atoms to decay.

This means that the Pharaoh died approximately 3510 years ago, or in BC. This time period was the beginning of the New Kingdom in Egyptian history: a “golden age” which marked the peak of Egypt’s power. And all we needed was a tiny piece of skin tissue, together with clever mathematics!

Geologists and biologists can use the same method to determine the age of fossils. This helps them understand when certain layers of rock in our Earth’s crust formed, or the evolutionary ancestry between extinct animals.

Carbon dating was developed in the late 1940s at the University of Chicago, by Willard Libby, who received the Nobel Prize in Chemistry for his work in 1960. It has become an indispensable method in many areas of science.

Note that we have greatly simplified the process of carbon dating in this chapter. There are many other things to consider, such as sample contamination, or how the concentration of carbon-14 in our atmosphere has changed over time.

Archie