The Chemistry Behind Radiocarbon Dating and its Applications in Dating Organic Samples, Specifically in Art Authentication

 

ACCM Concepts

• I. F.3.a. - Radioactive half-lives are unique and may be used to identify what isotopes are present. 
• I. F.3.b. - Radioactivity may occur via alpha, beta, gamma or other decay events and the type of radioactivity observed is often an important component of identifying the decaying isotope. 
• I. F.3.c. - In a mixture, the amount of a radioactive isotope can be determined via quantitative measurement of radioactive events.

 

Sections:

• Introduction
• Formation of ¹⁴C
• What is Carbon Dating?
• How Does Carbon Dating Work?
• Applying the Theory 1
• Applying the Theory 2
• Modern Applications in Industry and Service

                                                           

Figure 1: Willard Libby, the inventor of Carbon Dating (source: UChicago on Twitter)

 

Carbon dating, also known as radiocarbon dating, was proposed by William Libby in 1946. He began his research soon after the discovery of radiocarbon (¹⁴C). Libby realized that radiocarbon from the atmosphere would be absorbed by living organisms. From there, he theorized that measuring the amount of ¹⁴C would yield the age of the object. For his theory to be reliable, the concentration of ¹⁴C in the atmosphere needed to have been constant for thousands of years, and ¹⁴C needed to have moved through the carbon cycle. In the absence of cosmic radiation data, Libby assumed that it was constant. For the next part, Libby calculated a 1:10^2 14C to ¹⁴C ratio and calculated the mixing of carbon across carbon reservoirs.¹

Formation of ¹⁴C

Earth’s atmosphere is composed of approximately 78% of N₂(g).² Cosmic rays are a form of high-energy radiation, originating outside of the solar system that move at a speed close to the speed of light.³ When these high energy particle clusters enter the Earth’s atmosphere, they induce a nuclear reaction that produces neutrons. These neutrons are therefore called cosmic ray induced neutrons.⁴ Once in the atmosphere these neutrons react with the nuclei of the nitrogen present in the atmosphere to form ¹⁴C . The reaction is shown in equation 1: Nitrogen reacts with 1 neutron to produce ¹⁴C  and 1 proton. 

 

\[^{14}_7 N + ^1_0 n \rightarrow ^{14}_6 C+ ^1_1 p\]

Equation 1: formation of Carbon-14

Most existing atoms are stable because they have a ratio of protons to neutrons close to 1:1.

The stability of elements based on the number of nucleons can be seen in figure 1. 

                                                                           

Figure 2: stability of atoms (data: http://www.sprawls.org)

Due to its ratio of protons to neutrons of 1:1.3, ¹⁴C is unstable. To become more stable, ¹⁴C undergoes negative beta decay (ß^-). This consists of the transformation of a neutron into a proton, and emission of a charged particle and an antineutrino. This process is shown in equation 2.⁵ 

\[{ }_6^{14} C \rightarrow_7^{14} N+e^{-}+\bar{V} e\]

Equation 2: negative beta decay of ¹⁴C

The mass of ¹⁴C present in a sample undergoing this process will decrease over time. To describe the time taken by ¹⁴C to decay we use its half-life. It is 5,730 +\- 40 years, meaning that it takes 7,730 +\- 40 years for the mass of ¹⁴C to be halved in a sample. 

Once in the atmosphere ¹⁴C reacts with Oxygen to form carbon dioxide (CO₂). This reaction is shown in equation 3.

\[{ }_6^{14} \mathrm{C}+\mathrm{O}_2 \rightarrow 14-\mathrm{CO}_2\]

Equation 3: formation of  ¹⁴C

Radioactive carbon dioxide enters the body of photosynthesizing organisms through photosynthesis. Most commonly this is plants, and the ¹⁴C will be deposited in their tissue. Other living organisms consume the plants and intake the ¹⁴C, and if in turn these are come by other living organisms, they will intake it as well. When a living organism dies, the ¹⁴C inside of it will start to undergo negative beta decay. As time passes, the rate of decay should slow down.⁶ 

 

What is Carbon Dating?

Carbon dating is the process through which organic materials can be dated by measuring the amount of ¹⁴C in a sample and calculating how long it would take to decay to that point. With this information, scientists can determine how old the tested object is and draw conclusions. However, as a method of dating, carbon dating has a few limitations as it is not the most precise method. It can date organic materials from a few hundred years ago to about 50,000 years ago, but it gets less precise the older the object is. Within 1,000 years, the calculated date may be +/- 20-40 years off. Over 3,000, the calculated date may have a +/- 100-300 year error. When working with art authentication, this lack of precision may render the results useless. Additionally, carbon dating requires a piece of the object to be removed, thus potentially damaging the art. The process initially required 10g of material, but with modern technology has been reduced to 1mg.⁷ 

 

How does Carbon Dating work? 

Carbon dating determines the approximate age of an organic sample, from a measure of the amount of ¹⁴C present in it and the rate at which it is decaying compared to the rate of a recent similar sample. This process only works for organic samples because they need to contain ¹⁴C.⁵ The only assumption in this model is the validity of the commonly accepted value for the half-life of ¹⁴C, this is shown in equation 4. 

\[t_{\frac{1}{2}}=5,730(\pm 40) \text { years for } 14 C\]

Equation 4: half-life of  ¹⁴C

The equations 5 – 9 are used for this process. Carbon decay is a first order reaction, the equation to find the rate of reaction is shown in question 6. 

Key Equations

Equation 5: 

\[N_t=N_0 e^{-k t}\]

Equation 6: 

\[k=\frac{-\ln (2)}{t_{\frac{1}{2}}}=\frac{-\ln (2)}{5730}=-1.209680943 \times 10^{-4}\]

Equation 7: 

\[N_t=N_0 e^{\frac{-\ln (2) \times t}{t_1}}=N_0 e^{k t}\]

Equation 8: 

\[\frac{N_t}{N_0}=e^{\frac{-\ln (2) \times t}{t_1}}=e^{k t}\]

Equation 9: 

\[\ln \left(\frac{N_t}{N_0}\right)=\frac{-\ln (2) \times t}{t_{\frac{1}{2}}}=k t\]

Where: 

t = time passed after the death of a living organism 

N_t = number of atoms of ¹⁴C or rate of decay of ¹⁴C after t time has passed 

N₀ = number of atoms of ¹⁴C or rate of decay of ¹⁴C at t=0

k = rate constant - calculated from the half-life 

To determine the amount of ¹⁴C  in an organic sample, the most common techniques are Accelerator mass spectrometry (AMS) and the use of Beta counters.  Accelerator mass spectrometry is the most efficient way because the sample needed for the test is around 1mg, while beta counters require a sample of at least 10g.⁸

 

Applying the Theory

                                                              

Figure 3: The Shroud of Turin is a famous example of the application of carbon dating

(source: https://aleteia.org/2022/04/22/new-technology-suggests-shroud-of-turin-is-2000-years-old/)

Example \(\PageIndex{1}\)

A Duke chemistry student determines that the wood panel of an old painting has a ¹⁴C  decay rate of 12.17 counts/minute/gram. If the decay rate of ¹⁴C  in fresh wood is 13.60 counts/minute/gram and the half-life of ¹⁴C  is 5,700 years, what is the age of the wood panel in years? When did it start decaying?

Solution

Step 1: Prepare the starting equation

\[\ln \left(\frac{N_t}{N_0}\right)=\frac{-\ln (2) \times t}{t_{\frac{1}{2}}}\]

Step 2: Substitute values from the question into the equation as follows

N_t = 12.17

N₀ = 13.60

t_1/2 = 5730 years

\[\ln \left(\frac{12.17}{13.60}\right)=\frac{-\ln (2) \times t}{5730}\]

Step 3: Calculate and simplify the calculations 

\[\ln \left(\frac{12.17}{13.60}\right)=-0.1110958857 \times t\]

\[\frac{-\ln (2)}{5730}=-1.209680943 \times 10^{-4} \times t\]

\[-0.1110958857=-1.209680943 \times 10^{-4} \times t\]

Step 4: Rearrange equation to isolate t (time) 

\[t=\frac{-0.1110958857}{-1.209680943 \times 10^{-4}}\]

Step 5: Calculate the final value (t = time passed from the beginning of the decay in years)

\[t=918.3899797 \approx 918 \text { years, } 5 \text { months }(\pm 20-40 \text { years })\]

Step 6: Subtract the age from the current date and adjust for months to find the original date of the start of the decay, then adjust for CE or BCE. 

\[2022-918.3899797=1103.61002 \approx \text { July of } 1103 \operatorname{CE}(\pm 20-40 \text { years })\]

 

Exercise \(\PageIndex{1}\)

A Duke biology student measures the amount in grams of ¹⁴C present in an organic fabric sample from an ancient ornamental artwork. They calculate that it contains 4.21/10 ¹⁴C with respect to a new sample made of the same organic fiber. What is the age of the sample? When did it start decaying?

Answer

Solution

\[\frac{4.21}{10}=0.421\left(=\frac{N_t}{N_0}\right)\]

\[\ln (0.421)=\frac{-\ln (2) \times t}{5730}\]

\[-0.865122=-1.209680943 \times 10^{-4} \times t\]

\[t=\frac{-0.865122}{-1.209680943 \times 10^{-4}}=7151.66\]

\[t=7151.66 \approx 7151 \text { years, } 8 \text { months }(\pm 100-300 \text { years })\]

\[2022-7151.66=-5129.66 \approx \text { March of } 5129.66 \operatorname{BCE}(\pm 100-300 \text { years })\]

 

Modern Applications in Industry and Services   

Carbon dating has a large range of applications. In the art world, it is used to approximate the age of art to establish a timeline or benchmark, or it can be used for authentication purposes. Many artistic materials such as paper, canvas, wood, paint pigment, and ivory can be tested. Even pottery can be tested, though it can be complicated to do so accurately.⁹ Carbon dating greatly benefited archaeology and geology as well. Archeology makes use of carbon dating through testing unearthed bones, pots, baskets, wood, and animal hides, while geology tests the organic components in soil. Forensic scientists can measure ¹⁴C levels in unidentified human remains to determine the year of death.  

 

Sources 

1. American Chemical Society National Historic Chemical Landmarks. Discovery of Radiocarbon Dating.  http://www.acs.org/content/acs/en/ed...s/radiocarbon- dating.html (accessed November 27, 2022). 

2. National Geographic Society, and Tyson Brown. “Atmosphere | National Geographic Society.” Education.nationalgeographic.org, Clint Parks, 20 Apr. 2022, education.nationalgeographic.org/resource/atmosphere. Accessed 11 Nov. 2022. 

3. CERN. “Cosmic Rays: Particles from Outer Space | CERN.” Home.cern, 11 Oct. 2019, home.cern/science/physics/cosmic-rays-particles-outer-space. Accessed 11 Nov. 2022.
4. Korff, S. A. “The Production of Neutrons by the Cosmic Radiation.” Proceedings of the American Philosophical Society, vol. 84, no. 5, 30 July 1941, pp. 589–603, www.jstor.org/stable/984841?seq=1#metadata_info_tab_contents. Accessed 11 Nov. 2022.
5. Jull, A., Pearson, C., Taylor, R., Southon, J., Santos, G., Kohl, C., . . . Major, I. (2018). Radiocarbon Dating and Intercomparison of Some Early Historical Radiocarbon Samples. Radiocarbon, 60(2), 535-548. doi:10.1017/RDC.2018.18 
6. Raney, Jordan R. “Capturing Carbon: The Carbon Cycle and Climate Change.” The New Atlantis, no. 22, 2008, pp. 99–103. JSTOR, http://www.jstor.org/stable/43152447. Accessed 11 Nov. 2022. 
7. Hendriks, Laura, et al. “Uncovering Modern Paint Forgeries by Radiocarbon Dating.” Proceedings of the National Academy of Sciences, vol. 116, no. 27, 2 July 2019, pp. 13210–13214, www.pnas.org/content/116/27/13210, 10.1073/pnas.1901540116. 
8. Scientific Art Tests. “Scientific Art Tests - Carbon Dating.” Www.scientificarttests.com, 2022, www.scientificarttests.com/carbon-dating.html. 
9. Kulkova, Marianna. "RADIOCARBON DATING OF ANCIENT POTTERY". Samara Journal of Science 3.3 (2014): 115-122. WEB. doi: 10.17816/snv20143212 

    

Further Reading 

For further reading, readers interested in other dating methods may refer to "Rubidium Strontium Dating" on Britannica:

 

Britannica, The Information Architects of Encyclopaedia. "rubidium-strontium-dating". Encyclopedia Britannica, 11 Nov. 2022,  

https://www.britannica.com/facts/rub...rontium-dating. Accessed 11 November 2022. 

 

Contributions

Claudia Carugati (Duke University), Anna R. Finkelstein (Duke University)