Half Life

Discussion

introduction

N = N02t

radiocarbon dating

Every time a living being dies a stopwatch starts ticking.
Death starts the stopwatch. Science can read it.

Radiocarbon dating is used to determine the age of previously living things based on the abundance of an unstable isotope of carbon. The isotopic distribution of carbon on the earth is roughly 99% carbon 12  (with 6 protons and 6 neutrons) and 1% carbon 13 (with 6 protons and 7 neutrons). These isotopes are stable, which is why they are with us today, but unstable isotopes are also present in minute amounts. About one carbon atom in a trillion (1012) contains a radioactive nucleus with 6 protons and 8 neutrons — carbon 14. This rare, unstable isotope is produced in the upper atmosphere from ordinary nitrogen 14.

In earth's upper atmosphere, on the edge of what is commonly called outer space, light atomic nuclei from unknown sources outside of our solar system traveling at speeds approaching the speed of light called cosmic rays rain down continuously. These highly energetic nuclear bullets wreak havoc on the atoms in the upper atmosphere: tearing electrons from their orbitals and setting them free, knocking neutrons and protons from the tight confines of the nucleus and setting them free, generating x‑rays and gamma rays as they decelerate, and creating exotic particles like muons and pions directly from their excessive kinetic energy. These secondary cosmic rays are also highly energetic and will ionize atoms, transmute nuclei, and generate x‑rays themselves. A secondary cosmic ray neutron of sufficient energy striking a common nitrogen 14 nucleus can force it to eject a proton.

147N + 10n → 146C + 11p

This is the process by which all of the carbon 14 on the earth is produced. (Produced naturally to be more precise. More on that later.)

All organic material contains carbon. (By definition, organic chemistry is the chemistry of carbon compounds.) Plants absorb 14C like they absorb other isotopes of carbon — through the respiration of carbon dioxide — and then use this carbon to produce sugars, fats, proteins, and vitamins. Bacteria, fungi, and animals eat these plants and each other. In this way, atmospheric carbon is distributed throughout the web of life until every living thing has the same ratio of 14C : 12C as the atmosphere. (Well, nearly the same ratio. Plants and animals tend to favor lighter nuclei just a bit. Serious technicians know how to compensate for this preference when dating samples.)

With a half life of 5730 years, 14C decays by beta emission back into the 14N from which it originated.

146C → 147N + 0−1e + 00ν̅

Since living creatures are constantly swapping atoms with their environment, the abundance of 14C within them remains fixed. After death, however, no new radioactive carbon comes along to replenish that which has decayed and the abundance of 14C decreases. The ratio of 14C : 12C in a piece of living organic matter will be the same as it is in the atmosphere but larger than in a piece of dead organic material. A timber found in a home built 5730 years ago (one half life) would have half the 14C : 12C ratio that a person living today would. A discarded oyster shell from someone's dinner eaten 11,460 years ago (two half lives) would have one quarter the 14C : 12C ratio that a cotton shirt worn today would. A tusk from a mammoth that died 17,190 years ago (three half lives) would have one eighth the 14C : 12C ratio that a cardboard box manufactured today would. And so on. Death starts the stopwatch. Science can read it.

age (half-lives) age (years) 14C (atoms) 12C (atoms) 14C : 12C (ppt)*
0 0 128 128 × 1012 1
1 5,730 64 128 × 1012 0.5
2 11,460 32 128 × 1012 0.25
3 17,190 16 128 × 1012 0.125
4 22,920 8 128 × 1012 0.0625
5 28,650 4 128 × 1012 0.03125
6 34,380 2 128 × 1012 0.015625
7 40,110 1 128 × 1012 0.0078125
Radiocarbon dating of a hypothetical organic sample
In the United States and a few other countries 1012 is called a trillion. The unit ppt means "parts per trillion".

Calculations of this sort are based on the assumption that the ratio of 14C : 12C in the earth's atmosphere has remained constant. As a first approximation one can assume this, but more accurate results must take into account fluctuations in the intensity of the cosmic rays entering the earth's atmosphere. These deviations were determined from the comparative dating of ancient tree rings (a field called dendrochronology) and the results were then compiled into a calibration curve. Current calibration curves go back as far as 26,000 years ago. The range of radiocarbon dating extends back to about 50,000 years. For items older than this, there isn't enough undecayed 14C left to measure the ratio reliably.

Radiocarbon dating in the future will have to include adjustments for human activities. Beginning in the late 1950s, considerable amounts of anthropogenic (human-produced) 14C have been added to the atmosphere, mostly as a result of nuclear weapons testing. This activity reached its peak in the early 1960s when an atmospheric blast occurred somewhere on earth every two to three days. Nuclear bombs generate large numbers of high energy neutrons, which can in turn transmute nitrogen 14 into carbon 14 in exactly the same way as naturally occurring secondary cosmic rays. By 1965, atmospheric 14C concentrations were double their pre "atomic age" values.

Coal and petroleum are the fuels that powered the Industrial Revolution. Coal is nearly pure carbon and petroleum is a mixture of hydrocarbons. By the end of the Twentieth Century, humans were adding carbon from fossil fuels to the atmosphere at a rate of about 10 billion tonnes per year (1013 kg/year). The total human contribution to the atmosphere passed one trillion tonnes of carbon (1015 kg) a few years into the Twenty-first Century. Fossil fuels are the remains of long dead plants that were buried in sediment tens to hundreds of millions of years ago (coal being made primarily from land plants and petroleum from plankton and algae). The fraction of the original 14C left in a plant that has been dead for 5,730,000 years (one thousand half lives) is …

1  ≈ 10−301
21000

For all intents and purposes, this number is zero. Coal and petroleum have been dead for so long they no longer contain any 14C. Burning these fossil fuels has the effect of diluting the 14C content of the atmosphere. This effect this has on radiocarbon dating was first measured by the Austrian chemist Hans Suess (1909–1993) in the 1950s and is now known as the Suess effect.

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