Application: Radiocarbon Dating
Skills to Develop

Identify the age of materials that can be approximately determined using radiocarbon dating.
When we speak of the element Carbon, we most often refer to the most naturally abundant stable isotope ^{12}C. Although ^{12}C is definitely essential to life, its unstable sister isotope ^{14}C has become of extreme importance to the science world. Radiocarbon Dating is the process of determining the age of a sample by examining the amount of ^{14}C remaining against the known halflife, 5,730 years. The reason this process works is because when organisms are alive they are constantly replenishing their ^{14}C supply through respiration, providing them with a constant amount of the isotope. However, when an organism ceases to exist, it no longer takes in carbon from its environment and the unstable ^{14}C isotope begins to decay. From this science, we are able to approximate the date at which the organism were living on Earth. Radiocarbon dating is used in many fields to learn information about the past conditions of organisms and the environments present on Earth.
The Carbon14 cycle
Radiocarbon dating (usually referred to simply as carbon14 dating) is a radiometric dating method. It uses the naturally occurring radioisotope carbon14 (14C) to estimate the age of carbonbearing materials up to about 58,000 to 62,000 years old. Carbon has two stable, nonradioactive isotopes: carbon12 (^{12}C) and carbon13 (^{13}C). There are also trace amounts of the unstable radioisotope carbon14 (^{14}C) on Earth. Carbon14 has a relatively short halflife of 5,730 years, meaning that the fraction of carbon14 in a sample is halved over the course of 5,730 years due to radioactive decay to nitrogen14. The carbon14 isotope would vanish from Earth's atmosphere in less than a million years were it not for the constant influx of cosmic rays interacting with molecules of nitrogen (N_{2}) and single nitrogen atoms (N) in the stratosphere. Both processes of formation and decay of carbon14 are shown in Figure 1.
Figure 1: Diagram of the formation of carbon14 (forward), the decay of carbon14 (reverse). Carbon14 is constantly be generated in the atmosphere and cycled through the carbon and nitrogen cycles. Once an organism is decoupled from these cycles (i.e., death), then the carbon14 decays until essentially gone.
When plants fix atmospheric carbon dioxide (CO_{2}) into organic compounds during photosynthesis, the resulting fraction of the isotope ^{14}C in the plant tissue will match the fraction of the isotope in the atmosphere (and biosphere since they are coupled). After a plants die, the incorporation of all carbon isotopes, including ^{14}C, stops and the concentration of ^{14}C declines due to the radioactive decay of ^{14}C following.
\[ \ce{ ^{14}C > ^{14}N + e^} + \mu_e \label{E2}\]
This follows firstorder kinetics.
\[N_t= N_o e^{kt} \label{E3}\]
where
 \(N_0\) is the number of atoms of the isotope in the original sample (at time t = 0, when the organism from which the sample was decoupled from the biosphere), and
 \(N_t\) is the number of atoms left after time \(t\).
 and \(k\) is the rate constant for the radioactive decay.
The halflife of a radioactive isotope (usually denoted by \(t_{1/2}\)) is a more familiar concept than \(k\) for radioactivity, so although Equation \(\ref{E3}\) is expressed in terms of \(k\), it is more usual to quote the value of \(t_{1/2}\). The currently accepted value for the halflife of 14C is 5,730 years. This means that after 5,730 years, only half of the initial 14C will remain; a quarter will remain after 11,460 years; an eighth after 17,190 years; and so on.
The equation relating rate constant to halflife for first order kinetics is
\[ k = \dfrac{\ln 2}{ t_{1/2} } \label{E4}\]
so the rate constant is then
\[ k = \dfrac{\ln 2}{5.73 \times 10^3} = 1.21 \times 10^{4} \text{year}^{1} \label{E5}\]
and Equation \(\ref{E2}\) can be rewritten as
\[N_t= N_o e^{\ln 2 \;t/t_{1/2}} \label{E6}\]
or
\[t = \left(\dfrac{\ln \dfrac{N_o}{N_t}}{\ln 2} \right) t_{1/2} = 8267 \ln \dfrac{N_o}{N_t} = 19035 \log_{10} \dfrac{N_o}{N_t} \;\;\; (\text{in years}) \label{E7}\]
The sample is assumed to have originally had the same ^{14}C/^{12}C ratio as the ratio in the atmosphere, and since the size of the sample is known, the total number of atoms in the sample can be calculated, yielding \(N_0\), the number of ^{14}C atoms in the original sample. Measurement of N, the number of ^{14}C atoms currently in the sample, allows the calculation of \(t\), the age of the sample, using the Equation \(\ref{E7}\).
Note
Deriving Equation \(\ref{E7}\) assumes that the level of 14C in the atmosphere has remained constant over time. However, the level of 14C in the atmosphere has varied significantly so time estimated by Equation \(\ref{E7}\) must be corrected by using data from other sources.
Example 1: Dead Sea Scrolls
In 1947 samples of the Dead Sea Scrolls were analyzed by carbon dating. It was found that the carbon14 present had an activity (rate of decay) of d/min.g (where d = disintegration). In contrast, living material exhibit an activity of 14 d/min.g. Thus, using Equation \(\ref{E3}\),
\[\ln \dfrac{14}{11} = (1.21 \times 10^{4}) t \nonumber\]
Thus,
\[t= \dfrac{\ln 1.272}{1.21 \times 10^{4}} = 2 \times 10^3 \text{years} \nonumber\]
From the measurement performed in 1947 the Dead Sea Scrolls were determined to be 2000 years old giving them a date of 53 BC, and confirming their authenticity. This discovery is in contrast to the carbon dating results for the Turin Shroud that was supposed to have wrapped Jesus’ body. Carbon dating has shown that the cloth was made between 1260 and 1390 AD. Thus, the Turin Shroud was made over a thousand years after the death of Jesus.
Describes radioactive half life and how to do some simple calculations using half life.
History
The technique of radiocarbon dating was developed by Willard Libby and his colleagues at the University of Chicago in 1949. Emilio Segrè asserted in his autobiography that Enrico Fermi suggested the concept to Libby at a seminar in Chicago that year. Libby estimated that the steadystate radioactivity concentration of exchangeable carbon14 would be about 14 disintegrations per minute (dpm) per gram. In 1960, Libby was awarded the Nobel Prize in chemistry for this work. He demonstrated the accuracy of radiocarbon dating by accurately estimating the age of wood from a series of samples for which the age was known, including an ancient Egyptian royal barge dating from 1850 BCE. Before Radiocarbon dating was able to be discovered, someone had to find the existence of the ^{14}C isotope. In 1940 Martin Kamen and Sam Ruben at the University of California, Berkeley Radiation Laboratory did just that. They found a form, isotope, of Carbon that contained 8 neutrons and 6 protons. Using this finding Willard Libby and his team at the University of Chicago proposed that Carbon14 was unstable and underwent a total of 14 disintegrations per minute per gram. Using this hypothesis, the initial halflife he determined was 5568 give or take 30 years. The accuracy of this proposal was proven by dating a piece of wood from an Ancient Egyptian barge, of whose age was already known. From that point on, scientist have used these techniques to examine fossils, rocks, and ocean currents and determine age and event timing. Throughout the years measurement tools have become more technologically advanced allowing researchers to be more precise and we now use what is known as the Cambridge halflife of 5730+/ 40 years for Carbon14. Although it may be seen as outdated, many labs still use Libby's halflife in order to stay consistent in publications and calculations within the laboratory. From the discovery of Carbon14 to radiocarbon dating of fossils, we can see what an essential role Carbon has played and continues to play in our lives today.
Summary
The entire process of Radiocarbon dating depends on the decay of carbon14. This process begins when an organism is no longer able to exchange Carbon with their environment. Carbon14 is first formed when cosmic rays in the atmosphere allow for excess neutrons to be produced, which then react with Nitrogen to produce a constantly replenishing supply of carbon14 to exchange with organisms.
 Carbon14 dating can be used to estimate the age of carbonbearing materials up to about 58,000 to 62,000 years old.
 The carbon14 isotope would vanish from Earth's atmosphere in less than a million years were it not for the constant influx of cosmic rays interacting with atmospheric nitrogen.
 One of the most frequent uses of radiocarbon dating is to estimate the age of organic remains from archeological sites.
References
 Hua, Quan. "Radiocarbon: A Chronological Tool for the Recent Past." Quaternary Geochronology4.5(2009):378390. Science Direct. Web. 22 Nov. 2009.
 Petrucci, Raplh H.General Chemistry: Principles and Modern Applications 9th Ed. New Jersey: Pearson Education Inc. 2007.
 "Radio Carbon Dating." BBC Homepage. 25 Oct. 2001. Web. 22 Nov. 2009. http://www.bbc.co.uk.
 Willis, E.H., H. Tauber, and K. O. Munnich. "Variations in the Atmospheric Radiocarbon Concentration Over the Past 1300 Years." American Journal of Science Radiocarbon Supplement 2(1960) 14. Print.
Problems
 If when a hippopotamus was breathing there was a total of 25 grams of Carbon14, how many grams will remain 5730 years after he is laid to rest? 12.5 grams, because one half life has occurred.
 How many grams of Carbon14 will be present in the hippos remains after 3 halflives have passed? 3.125 grams of Carbon14 will remain after 3 half lives.
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