Carbon Dating: How It Works, Formula and Applications
Carbon dating estimates the age of once-living things — wood, bone, cloth, seeds — by measuring how much radioactive carbon-14 remains in them. Living organisms constantly exchange carbon with the air, so their carbon-14 fraction stays constant while alive. At death, exchange stops and carbon-14 decays away like a ticking clock.
It matters because it dates archaeological finds up to about 50,000 years old: ancient tools, Egyptian mummies, cave paintings, and old manuscripts.
Key Concepts
- Carbon-14 (¹⁴C): produced in the upper atmosphere when cosmic-ray neutrons hit nitrogen-14.
- Half-life: 5,730 years for carbon-14.
- Ratio method: compare ¹⁴C/¹²C in the sample with that in living tissue; the fraction remaining gives the age.
Formulas
N = N₀ (1/2)^(t/T½), so t = T½ × ln(N₀/N) / ln 2, with T½ = 5,730 years. Equivalently, activity falls: A = A₀ e^(−λt) with λ = 0.693/5730 per year. Assumes atmospheric ¹⁴C levels were constant (calibration curves correct for small variations).
Worked Example
A wooden tool shows 25 percent of the carbon-14 activity of living wood. How old is it?
- 25 percent = 1/4 = (1/2)², so two half-lives have passed.
- t = 2 × 5,730 = 11,460 years.
- Check with formula: t = 5730 × ln(4)/ln(2) = 5730 × 2 = 11,460 years.
Common Misconceptions
- Students think carbon dating works on rocks and fossils of dinosaurs. It works only on once-living material up to about 50,000 years; dinosaur fossils are far older and contain no measurable ¹⁴C.
- Many think all carbon in a living body is carbon-14. Only about one in a trillion carbon atoms is ¹⁴C; most is stable carbon-12.
- Some believe the method gives exact years. It gives a range, refined by calibration with tree-ring records.
Real-World Applications
Carbon dating verified the Dead Sea Scrolls' age, dated the Iceman mummy Ötzi to about 5,300 years old, authenticated paintings and manuscripts, and tracks climate history in ancient plant remains.
A practical detail: scientists measure either the ratio of carbon-14 to carbon-12 with a mass spectrometer (accelerator mass spectrometry) or count the beta decays from a larger sample. Both approaches rely on knowing the starting ratio in living tissue. Contamination matters — even a small amount of modern carbon from handling or groundwater can make an old sample look younger, so laboratories clean samples chemically before measuring.
In short, carbon-14 decays with a 5,730-year half-life after an organism dies, and measuring what fraction remains — through t = T½ × ln(N₀/N)/ln 2 — reveals its age.