Mathematics of carbon dating

Summary: Exponential and logarithmic functions are used in carbon dating—a method of determining the age of plant and animal fossils.

As is demonstrated throughout this encyclopedia, mathematics provides explanations for many interesting physical phenomena, and enables humankind to better understand its surrounding world. One of our ongoing intellectual projects is simply to make sense of the world we inhabit, based on the evidence that surrounds us. As anthropologists, archaeologists, and geologists have worked to determine the age of the earth and to track the evolution of species, radioactive isotopes have played a prominent role in efforts to create a timeline that charts a wide range of historical developments. In particular, carbon-14 dating has provided a fundamental test enabling scientists to accurately date certain plant and animal fossils that are approximately 60,000 years old or less. Willard Libby was one of the first to research radiocarbon dating, and he won a Nobel Prize in chemistry. Carbon dating is not an exact science, and statistical methods are used to enhance the reliability of the methods.

The Mathematics of Carbon Dating

Left alone, a radioactive quantity will decay at a rate proportionate to the amount of the quantity present at a given time. More specifically, a radioactive chemical element (such as uranium) is one that is unstable; as it decays, it emits energy and its fundamental makeup changes as the mass of the element is changed to an element of a different type. Because such an element is losing mass at a rate proportionate to the available mass at time t, an exponential function may be used to model the amount of the isotope that is present.

Letting M(t) represent the mass of the element at time t, it turns out that M(t)=M₀e^-kt, where M ₀ is the mass at initial measurement (at time t=0), and k is a constant that is connected to the rate at which the element decays. Furthermore, k is tied to the isotope’s half-life (the amount of time it takes for 50% of the mass present to decay). In the given model, if h represents the half-life, then when t=h, it follows that

That is, the equation

must hold. Dividing both sides by M₀, yields

and using the natural logarithm function, one may solve for k and thus rewrite the most recent equation as

This can be rewritten as

A property of the natural logarithm is that

so that in slightly simpler terms,

Therefore, the model for radioactive decay of an element having half-life h is

With this background in place, one is now ready to understand how carbon dating works.

All living things contain carbon, and the preponderance of the carbon present in plants and animals is its stable isotope, carbon-12. At the same time, every living being takes in radioactive carbon-14, and this carbon-14 becomes part of our organic makeup. While carbon-14 is constantly decaying simply by doing the normal things that come with being alive, each living organism continuously replenishes its supply of carbon-14 in such a way that the ratio of carbon-12 to carbon-14 in its body is constant.

When no longer living, a plant or animal lacks the ability to ingest carbon-14, and thus the ratio of carbon-12 to carbon-14 starts to change, and this ratio changes at the rate that carbon-14 decays. Chemists have long known that carbon-14 has a half-life of approximately h=5700 years, and this knowledge, together with the exponential model

enables people to determine the age of certain fossils. Consider, for example, the situation where a bone is found that contains 40% of the carbon-14 it would be expected to have in a living animal. With less than half the original amount present, but more than 25%, it can be determined that the bone is somewhere between one and two half-lives old; that is, the animal lived between 5700 and 11,400 years ago.

Through our understanding of exponentials and logarithms, this estimate can be made much more precise.

Specifically, let t=0 be the year the animal died. The present year t satisfies the equation M(t‗=0.4M₀, since 40% of the initial amount of carbon-14 remains. From the model, it is known that t must be the solution to the equation

First, divide both sides by M₀ to get0.4=e^{-(ln(2)/5700)t} and then, taking the natural logarithm of both sides of the equation, it follows that

Thus, solving for t yields

and the skeletal remains have been dated according to their carbon content.

Limitations of Carbon Dating

Carbon dating does have some reasonable limitations. One of these involves the complications of measuring only trace amounts of carbon-14, and emphasizes the behavior of functions that model exponential decay. For each half-life that passes, half of the most recent quantity of the element remains. That is, after one half-life,

The quantity rapidly diminishes from there. For instance, after 10 half-lives have elapsed, there is

0.0009766M₀ left. Because each living organism only contains trace amounts of carbon-14 to begin with (of all carbon atoms, only about one-trillionth are carbon-14), after 10 half-lives elapse, the remaining amount of carbon-14 is so small that it is not only difficult to measure accurately, but it is difficult to ensure that the measured carbon-14 actually remains from the organism of interest and was not somehow contributed from another source. Ten half-lives is approximately 60,000 years, so any organism deemed older than that needs to be dated in another manner, typically using other radioactive isotopes that have considerably longer half-lives.

Finally, because radiocarbon dating depends on naturally occurring radioactive decay, its accuracy depends on such decay not being accelerated by unnatural causes. In the 1940s, the Manhattan Project resulted in humankind’s development of synthetic nuclear energy and weapons; subsequent nuclear testing and accidents have released radiation into the atmosphere that makes the accuracy of carbon-14 dating more suspect for organisms that die after 1940.

New Developments

The exponential model M(t)=M₀e^-kt of radioactive isotope decay has enabled humans to better understand our surrounding world, and to know with confidence key information about the history of the existence of plant and animal life on Earth. Even today, there are new developments in the science of radiocarbon dating as experts work to understand how subtle changes in Earth’s magnetic field and solar activity affect the amounts of carbon-14 present in the atmosphere. In addition to continuing to help analyze fossil histories, carbon-14 dating may prove an important tool in ongoing research in climate change.

The Accelerator Mass Spectrometry method of dating directly measures the number of carbon atoms rather than their radioactivity, which allows for the dating of small samples. Other methods under development include nondestructive carbon dating, which eliminates the need for samples. A group of Russian mathematicians have proposed a new chronology of history based on other methods for dating; however, many have dismissed their work as pseudoscience. Physicist Claus Rolfs explores methods to accelerate radioactive decay in the hope of reducing the amount of radioactive material.

Bibliography

“Archaeological ‘Time Machine’ Greatly Improves Accuracy of Early Radiocarbon Dating.” Science News Daily (February 11, 2010) http://www.sciencedaily.com/releases/2010/02/100211111549.htm.

Brain, Marshall. “How Carbon Dating Works.” http://www.howstuffworks.com/carbon-14.htm.

Comap. For All Practical Purposes: Mathematical Literacy in Today’s World. 7th ed. New York: W. H. Freeman, 2006.

Connally, E. et al. Functions Modeling Change: A Preparation for Calculus. Hoboken, NJ: Wiley, 2007.