How this instrument works
Radiocarbon dating works because living organisms constantly exchange carbon with their environment, keeping the radioactive isotope carbon-14 at a roughly constant, known fraction of their total carbon. Once an organism dies, that exchange stops, and the carbon-14 already present decays away at a fixed, predictable rate — while the far more common, stable carbon-12 stays put. Measuring how much carbon-14 remains relative to what a living sample would have, Nt/N0, tells you how many half-lives have elapsed since death, and therefore how old the sample is.
Every radioactive isotope decays exponentially, and carbon-14 is no exception: after one half-life, exactly half the original amount remains; after two half-lives, a quarter remains; and so on, following Nt = N0 × (1/2)^(t/half-life). Solving that relationship for elapsed time t gives the formula this instrument uses — age = −(half-life / ln 2) × ln(Nt/N0) — which works for any remaining fraction, not just the neat halves and quarters that fall exactly on a half-life boundary.
Carbon-14's half-life is conventionally taken as 5,730 years (the 'Cambridge half-life'), refined from the original 5,568-year 'Libby half-life' that early radiocarbon labs used and that some older published dates still implicitly assume. Because decay is exponential, this method's usable range is limited: past roughly eight to ten half-lives, so little carbon-14 remains that it becomes very hard to measure precisely, which is why radiocarbon dating tops out at somewhere around 50,000 years.
- Enter the measured carbon-14 fraction into Remaining fraction, Nt / N0 (0-1) — 0.5 means half remains, 0.25 means a quarter remains, and so on.
- Enter the isotope's half-life in years into Half-life (years) — it opens at 5730, the standard Cambridge half-life for carbon-14.
- Read the result off Estimated age (years) — how long ago the sample stopped exchanging carbon with its environment.
- The remaining fraction must be greater than 0 and no larger than 1; a fraction of exactly 1 means no decay has occurred yet, giving an age of 0.
- This same age = −(half-life / ln 2) × ln(Nt/N0) formula works for any radioactive isotope, not just carbon-14 — swap in a different half-life for uranium-lead, potassium-argon or any other decay system.
Worked example — a sample at exactly one half-life
Enter 0.5 into Remaining fraction, Nt / N0 (0-1) and leave Half-life (years) at its default, 5730 — a sample where exactly half of its original carbon-14 has decayed away. Estimated age (years) reads 5730.0.
This result follows directly from what a half-life means: by definition, the time for half a radioactive sample to decay is one half-life, so a remaining fraction of exactly 0.5 must give an age exactly equal to the half-life entered. Checking the formula confirms it: ln(0.5) = −0.693147, which is −ln(2), so age = −(5730/ln 2) × (−ln 2) = 5730 years exactly, with the ln 2 terms cancelling cleanly.
Questions
Why does the remaining fraction have to be between 0 and 1?
Because Nt/N0 is a fraction of the original amount still present — it can't exceed 1 (you can't have more of the isotope left than you started with) and it must be positive, since ln(0) is undefined and a fraction of exactly 0 would imply an infinite age. The instrument enforces both bounds and asks for a fraction in range if you enter something outside it.
What's the difference between the Libby half-life and the Cambridge half-life?
Willard Libby's original 1949 measurement put carbon-14's half-life at 5,568 years; later, more precise measurements revised it to 5,730 years, the value known as the Cambridge half-life and now the accepted standard. Some older published radiocarbon dates were calculated using the Libby value by convention, even after the more accurate figure was known, so it's worth checking which half-life a cited date actually assumes.
How far back can carbon-14 dating reliably reach?
Roughly 50,000 years, which corresponds to about eight to nine half-lives of decay. Past that point, so little carbon-14 remains in a sample that it becomes extremely difficult to measure precisely against background radiation, so older material is typically dated with other methods, such as potassium-argon or uranium-lead dating, that use isotopes with much longer half-lives.
Does this formula only work for carbon-14?
No — the underlying exponential decay law applies to every radioactive isotope, so the same age = −(half-life / ln 2) × ln(Nt/N0) formula works for any isotope once you supply its own half-life. Uranium-lead dating, potassium-argon dating and other radiometric methods all rest on this identical mathematics; only the isotope, its half-life, and the measured remaining fraction change.
Why is a remaining fraction of exactly 1 a trivial case?
A remaining fraction of 1 means all of the original isotope is still present, so no decay has happened yet — and since ln(1) = 0, the formula returns an age of exactly 0 regardless of the half-life entered. It's a useful sanity check confirming the formula behaves correctly at its boundary, rather than a scenario you'd expect from a real archaeological or geological sample.