Article — Carbon-14 Dating Calculator
Carbon-14 Dating Calculator: Age from Radiocarbon Fraction Remaining
Carbon-14 dating uses the formula t = −(t½ / ln 2) · ln(N/N₀), where t½ is the C-14 half-life of 5730 years and N/N₀ is the ratio of radiocarbon remaining compared to a living sample. A bone with half the modern C-14 level is one half-life old (5730 years). A quarter level is two half-lives (11,460 years). Modern accelerator mass spectrometry (AMS) pushes the practical limit to about 57,000 years — 10 half-lives — where only 0.1% of the original C-14 remains. Willard Libby developed the method between 1946 and 1949 and won the 1960 Nobel Prize in Chemistry for it; modern radiocarbon labs date Egyptian mummies, charcoal from cave paintings, and forensic remains with precision of ±30 to 50 years for samples younger than 5,000 years old.
This calculator runs in two directions. Enter the fraction of C-14 remaining (between 0 and 1) and get the age. Or enter the age and get the expected fraction. Default half-life is the Cambridge value of 5730 years; the calculator lets you change it for special applications (Libby's original value was 5568 years).
What is carbon-14 dating
Carbon-14 dating, also called radiocarbon dating, is a way to estimate the age of organic materials by measuring how much carbon-14 remains in a sample. Living organisms continuously exchange carbon with the atmosphere through photosynthesis and the food chain, so their tissues hold the atmospheric ratio of C-14 to C-12. When the organism dies, exchange stops and the C-14 begins to decay at a known rate. Measuring the remaining C-14 and comparing it to the modern atmospheric ratio gives the elapsed time since death.
The method works because C-14 is unstable and decays to N-14 by emitting a beta particle (an electron and antineutrino). C-12 and C-13 are stable, so the ratio C-14/C-12 decreases with time at a predictable rate.
Atmospheric nuclear tests between 1945 and 1963 nearly doubled the C-14 content of the atmosphere — the so-called "bomb spike." Modern wines, oils, and forensic tissues can be dated within a few years by where they fall on the bomb curve as atmospheric C-14 declines back toward natural levels. The bomb spike is now Earth's most precise short-term radiometric clock.
Carbon dating formula and half-life
The carbon dating formula is t = −(t½ / ln 2) · ln(N/N₀), derived from the exponential decay law N = N₀·e^(−λt). λ = ln 2 / t½ is the decay constant, equal to 1.21 × 10⁻⁴ per year for C-14. The half-life is 5730 years (the Cambridge value), measured precisely in the 1960s. Libby used 5568 years originally; the difference is small but matters for very old samples and is automatically handled by calibration curves.
An equivalent form uses base-2: N(t) = N₀ · (1/2)^(t/t½). This makes it easy to count half-lives mentally: 50% remaining is one half-life (5730 yr), 25% is two (11,460 yr), 12.5% is three (17,190 yr), and so on.
1 half-life 5730 yr / 50%2 half-lives 11,460 yr / 25%5 half-lives 28,650 yr / 3.1%10 half-lives 57,300 yr / 0.1%λ for C-14 1.21×10⁻⁴ /yrHow carbon-14 is produced in the atmosphere
Carbon-14 is produced when cosmic rays strike nitrogen in the upper atmosphere. A neutron knocks out a proton from N-14, creating C-14: n + ¹⁴N → ¹⁴C + p. The resulting C-14 atom quickly bonds with oxygen to form ¹⁴CO₂, which mixes through the atmosphere. About one C-14 atom exists for every 10¹² C-12 atoms — vanishingly small but measurable.
The production rate has varied through history with solar activity (which modulates cosmic-ray flux) and the Earth's magnetic field strength. Calibration curves correct for these long-term variations.
Carbon dating age range and limits
The carbon dating age range runs from about 500 years to 50,000 years for routine measurements, with AMS pushing to ~57,000. Below 500 years the signal is too close to modern atmospheric C-14 to be precise; recent atmospheric variability (Suess effect from fossil fuels, bomb-test spike) adds noise that calibration handles only partially. Above 50,000 years, only 0.2% of the original C-14 remains, and instrument background becomes a serious source of error.
Calibrating radiocarbon years to calendar years
Calibrating radiocarbon years to calendar years uses the IntCal20 (Northern Hemisphere) and SHCal20 (Southern Hemisphere) calibration curves. These are built from tree rings (back ~13,000 years), corals, foraminifera, and speleothems with independently known ages. Radiocarbon dates published as "cal BP" (calibrated years before 1950) account for these variations and are the standard for archaeological publication.
The raw radiocarbon age and the calibrated age can differ by hundreds of years, especially for samples between 1700 and 1900 CE (the Suess effect from fossil fuels) and across the late Pleistocene "Hallstatt plateau" (~2500 BP).
Materials that can be carbon-dated
Materials that can be carbon-dated include anything that contains organic carbon: wood, charcoal, bone collagen, antler, tooth dentin, textiles, papyrus, leather, hair, peat, lake-bottom sediments, and shell carbonate from molluscs and corals. The carbon should be from a single biogenic source — laminated or composite materials can give average ages weighted by carbon content.
Bone collagen is usually the best substrate for archaeological dating because collagen is chemically stable and the cleaning procedures effectively remove contaminants. Tooth dentin works similarly. Shell carbonate and freshwater fish bone need correction for the marine or freshwater reservoir effect, which can offset apparent age by hundreds of years.
Carbon dating precision and AMS
Carbon dating precision depends on the technique. Classic beta counting (Libby's method) needs grams of material and counts decay events over hours or days. Modern accelerator mass spectrometry (AMS) counts atoms directly using a particle accelerator, needs only milligrams, and finishes in minutes. AMS gives precision of ±30 to 50 years for Holocene samples and ±100 to 200 years for samples near the limit. Sample preparation, contamination control, and laboratory standards drive the final uncertainty more than the count statistics.
Common carbon dating mistakes and myths
The most persistent myth is that carbon dating "always" gives ages of millions of years. It cannot — the method tops out at ~57,000 years. Anything older needs K-Ar (potassium-argon, good for volcanic rocks), U-Pb (uranium-lead, geological time), or other isotope systems. A second common error is assuming the calibration is one-to-one with calendar years; in reality, radiocarbon years and calendar years differ systematically, especially across the bomb spike, the Suess effect, and the Hallstatt plateau.
Modern carbon contamination of an old sample makes it look much younger than it is. Just 1% modern carbon in a 50,000-year-old sample halves the apparent age. Sample handlers wear gloves, work in clean labs, and use multiple extraction and purification steps. For very old samples, results below 0.5 percent modern carbon are reported as "infinite age" rather than risk over-stating precision.