Introduction to Radio-Carbon Dating

Introduction to Radio-Carbon Dating This article is a brief outline of the basic mathematics that serve as the foundation of our ability to approximate any particular artifact’s age with spectacular accuracy. In order to properly demonstrate radio-carbon dating, we first must understand the basic principles of logarithms, exponential decay, and half-life. Logarithms You should be familiar with logarithms from high-school algebra. As addition is to subtraction, and multiplication to division, the logarithm is the inverse operation to exponentiation. Written as logbN, the logarithm of a positive number N to a given base b is the exponent of the power to which b must be raised to produce N. For example, 2 is the logarithm of 9 to base 3, written log39 = 2. The most important logarithmic function is the natural logarithm, in which the base e is an irrational number (non-terminating) approximately equal to 2.71828. e is known as Euler’s number, named for the 18th century mathematician Leonhard Euler, and is the most famous irrational number next to π. The natural logarithm is written ln N, and is equal to 2.3026 logeN. Euler’s number and natural logarithms take a central role in the calculations of radioactive decay and carbon dating, and it is exceedingly important that you understand these concepts prior to moving forward. _____________________________________________________________________________ Exponential Decay Exponential functions arise naturally in situations where the rate of change of a quantity is proportional to itself. The change in a physical quantity may be in either a positive or negative direction, here we are concerned specifically with decay (change in a negative direction). Exponential decay displays a pattern in which a quantity decreases rapidly at first, and more slowly as time has passed. Radioactive decay is the exponential decay of unstable (radioactive) elements. Atoms which have the same number of protons but differ in the amount of neutrons are called isotopes which may decay into isotopes of the same atom, or different atoms entirely. The processes through which radioactive decay takes place can vary. Alpha decay is the emission of a helium nucleus, beta decay is the emission of an electron or positron, and gamma decay is the emission of a very energetic photon. The law of radioactive decay states that the number of atoms to survive these processes is given by N = N0e-λt, where N0 is the initial number of atoms, λ is the decay constant, and t equals time elapsed. The decay constant is in s-1 (per second) and is given by ln(2)/t1/2. _____________________________________________________________________________ Half-Life and Decay Rate A sample of atoms reduces overtime as they transform in such a manner that after a characteristic time called the half-life, only one half of the original sample remains. For example, after two half-lives only one-quarter (½ of ½) of the original sample will survive, and after three half-lives only one-eighth. The decay rate (also called activity) is the number of decays per second, which is a product of the decay constant λ, and the number of radioactive nuclei N present at a particular time: R = N λ where R equals the decay rate. _____________________________________________________________________________ Carbon-14 Dating Carbon occurs in two different isotopes, its typical form being Carbon-12, which has a total of 12 protons and neutrons. The radioactive form is Carbon-14, being made of 14 protons and neutrons 14C has a half-life of 5,730 years and is produced in a nuclear reaction when solar neutrinos collide with our atmosphere. Radioactive carbon has the same chemistry as stable carbon, allowing it to be easily consumed by our ecosphere where it becomes part of every living organism. Living things maintain the same ratio of 14C/12C during their lives as they incorporate atmospheric carbon into their tissues, upon death the ratio begins to steadily decline. By measuring the relative amount of 14C to 12C in an animal or plant remain, we can date the specimen. According to the law of radioactive decay, N = N0e-λt, Since we know that in the case of 14C, t1/2 = 5730 yr, λ = ln(2)/t1/2 =.693/5730 = 1.21 x 10-4/yr It should be noted that isotopes of other atoms can also be used for radioactive dating. Sometimes very old rocks are dated based on the decay of Uranium-238. However, since 238U has a half-life of 4.5×109 (4.5 billion) years, it is only useful for dating proportionally old materials, showing for example, that the oldest rocks on Earth solidified approximately 3.5×109 years ago. Using radioactive-dating techniques on meteorites, most researchers believe the earth itself formed roughly 4.56 billion years ago. _____________________________________________________________________________ Example: Carbon-Dating Fossilized Wood We are now prepared to determine the age of a fossilized piece of wood which has a 14C/12C ratio that is 10% of that found in a living tree. In this case we know the remaining fraction N/N0 and need to find the elapsed time. From the equation N = N0e-λt, we know that e-λt = N/N0 = 10% so that t = [ln(N/N0 )] / λ Substituting N/N0 = 10% and the computed value of λ in the above expression yields t = 19045 yr _____________________________________________________________________________ Have suggestions about future content or need help solving a physics/math problem? Tweet @ThePhysicsBlog