Radiocarbon dating calculation: the C-14 age formula

Why carbon-14 works as a clock

Cosmic rays keep producing radioactive carbon-14 in the upper atmosphere. Plants take it up with CO₂, animals eat the plants, so every living organism carries roughly the same share of C-14 as the air. When the organism dies, the intake stops and the C-14 decays with a half-life of about 5,700 years (the ENSDF evaluation used by the IAEA nuclide chart is 5,700 ± 30 years, and many textbooks still quote 5,730). Measuring how much is left tells you how long ago the exchange stopped.

The age formula step by step

Radioactive decay follows F = e^(−λt), with the decay constant λ = ln 2 ÷ T½. Solve for t:

t = ln(1 ÷ F) ÷ λ = ln(1 ÷ F) × T½ ÷ ln 2

In Excel: =LN(1/B3)*$D$8/LN(2). For the oak beam, ln(1 ÷ 0.705) = 0.3496, and 0.3496 × 5,700 ÷ 0.6931 = 2,875 years, about half a half-life (0.50). The same formula with a different half-life gives the drug clearance times in half-life and elimination rate constant, because both follow first-order decay.

Worked example: five samples, two half-lives

The fraction of modern carbon for each sample is an input (yellow). Column C calculates the age with the physical half-life, column D the conventional radiocarbon age with the Libby half-life, and rows 10 to 14 add the mean life, the share left after 10,000 years and the oldest measurable age.

Worked example: five samples, two half-lives

	A	B	C	D
1	Sample	Fraction modern F	Age (T½ 5,700 y)	Conventional age BP
2	Charcoal, hearth	0.9120	757	740
3	Oak beam	0.7050	2,875	2,808
4	Bone collagen	0.4620	6,350	6,203
5	Peat, 3 m	0.2180	12,526	12,236
6	Mammoth tusk	0.0405	26,368	25,757
7
8	Half-life of C-14, years (ENSDF evaluation)			5,700
9	Libby half-life, years (reporting convention)			5,568
10	Libby mean life = 5,568 ÷ ln 2			8,033
11	Half-lives elapsed, oak beam			0.50
12	Fraction left after 10,000 years			29.6%
13	Detection limit (fraction modern)			0.003
14	Oldest age measurable at that limit			47,771

Fictional samples with illustrative values. Real results also need a fractionation correction and calibration. Educational calculations only, not radiation-safety or medical guidance.

Where F comes from

F is the sample's C-14 level divided by a modern reference level. With decay counting, it is the measured activity per gram of carbon divided by the reference activity, roughly 0.226 Bq per gram of carbon for modern carbon. With accelerator mass spectrometry (AMS), the standard method today, it is the measured ¹⁴C/¹²C ratio divided by the reference ratio. Laboratories also correct for isotope fractionation, because some organisms take up the heavier carbon isotopes less readily, using the measured ¹³C value. Results are often given as percent modern carbon (pMC): 70.5 pMC is F = 0.705.

Conventional radiocarbon age and the Libby half-life

By an international convention dating from the 1950s, laboratories still report ages with the Libby half-life of 5,568 years, which gives a mean life of 5,568 ÷ ln 2 = 8,033 years: t = −8,033 × ln F. The ages are given in years BP, "before present", where present means AD 1950. Because 5,568 is about 2% lower than 5,700, conventional ages are about 2% younger: 2,808 instead of 2,875 years for the oak beam, and 25,757 instead of 26,368 years for the mammoth tusk. Keeping the old value means published results stay comparable, and the difference disappears in the next step anyway.

From radiocarbon years to calendar years

The C-14 level of the atmosphere has not been constant over time. It has changed with solar activity, the Earth's magnetic field and the carbon cycle, and was disturbed again by fossil fuels and nuclear bomb tests in the 20th century. A conventional age is therefore converted into a calendar age range with a calibration curve built from tree rings and other dated archives, currently IntCal20 for the Northern Hemisphere. The result is a probability range such as "1040–910 BC at 95%", not a single year, and the formula alone cannot replace that step.

How far back does it work?

After 10,000 years, =EXP(-LN(2)*10000/D8) = 29.6% of the C-14 is left. After about 50,000 years, less than 0.3% remains and the signal sinks into the background and into contamination with modern carbon. If a laboratory can measure down to F = 0.003, the oldest age it can report is ln(1 ÷ 0.003) × 5,700 ÷ ln 2 = 47,771 years. Older samples are reported as "beyond the limit". The method also needs material that once exchanged carbon with the atmosphere: it dates wood, charcoal, bone, shell, textiles and peat, but not stone or metal.

Common mistakes

• Using LOG instead of LN. LOG is base 10 in Excel. The formula needs natural logarithms, or the same base in both places.
• Mixing up F and 1 − F. F is the fraction of C-14 that is left. If 30% has decayed, F is 0.70, not 0.30.
• Comparing ages with different half-lives. Conventional ages use 5,568 years by convention. Do not mix them with ages computed with 5,700 or 5,730.
• Reading a conventional age as a calendar date. Years BP count from 1950 in radiocarbon years. Calibrate before quoting a calendar age.
• Dating beyond the limit. Below about 0.3% of modern carbon, tiny contamination changes the age by thousands of years.
• Ignoring reservoir effects. Marine shells and fish-eating animals took up older carbon from the ocean and can look hundreds of years too old.