# The math behind radioactive carbon dating

Carbon has two stable, nonradioactive isotopes: carbon-12 (12C) and carbon-13 (13C).

There are also trace amounts of the unstable radioisotope carbon-14 (14C) on Earth.

Carbon-14 has a relatively short half-life of 5,730 years, meaning that the fraction of carbon-14 in a sample is halved over the course of 5,730 years due to radioactive decay to nitrogen-14.

The carbon-14 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) into organic compounds during photosynthesis, the resulting fraction of the isotope 14C in the plant tissue will match the fraction of the isotope in the atmosphere.

Radiocarbon dating (usually referred to simply as carbon-14 dating) is a radiometric dating method.

Thereafter, the concentration (fraction) of 14C declines at a fixed exponential rate due to the radioactive decay of 14C. ) Comparing the remaining 14C fraction of a sample to that expected from atmospheric 14C allows us to estimate the age of the sample.

Raw (i.e., uncalibrated) radiocarbon ages are usually reported in radiocarbon years "Before Present" (BP), with "present" defined as CE 1950.

Such raw ages can be calibrated to give calendar dates.

Libby and others (University of Chicago) devised a method of estimating the age of organic material based on the decay rate of carbon-14.

Carbon-14 is produced in the atmosphere when neutrons from cosmic radiation react with nitrogen atoms: C ratio of 0.795 times that found in plants living today. Solution The half-life of carbon-14 is known to be 5720 years. Radioactive decay is a first order rate process, which means the reaction proceeds according to the following equation: is the quantity of radioactive material at time zero, X is the amount remaining after time t, and k is the first order rate constant, which is a characteristic of the isotope undergoing decay.

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