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Q

How do scientists know the half-life of carbon-14 is exactly 5730 years when no one has lived that long?
https://www.quora.com/How-do-scientists-know-the-half-life-of-carbon-14-is-exactly-5730-years-when-no-one-has-lived-that-long/answer/Hans-Georg-Lundahl

Hans-Georg Lundahl

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Answered just now

First of all, we assume it is a half life. This means, it is not a decrease by arithmetically even decrease, but by a geometrically even decrease.

Second, this means we can count on roots of 1/2 to figure out how much would be left after so and so many fewer years.

For instance, for 1/3 of a half life, you would have the third root of 1/2.

1910 years ago left organic material which now has 79.37 % of original carbon content.

So, we find artefacts which were left behind 1910 years ago, in 108 AD. That could be a papyrus from Egypt or a piece of wood from Egypt, or a mummy portrait from Fayyum:

Description English: The Mummy-portrait is wax-tempera-made on wood. A young man in roman clothing. Hair and beard are made in the modern way this time. He belongs to the greco-roman upper-class. Early 3d Century, 37,5 cm high, Inv.-Nr. 15013. Deutsch: Mumienportrait eines jungen Mannes in römischer Kleidung. Frühes 3.Jhdt. 37,5 cm hoch, Staatliche Antikensammlungen, Inventarnummer 15013 . Date 15 November 2008 Source Own work Author MatthiasKabel Licensing This work is in the public domain in its country of origin and other countries and areas where the copyright term is the author's life plus 100 years or less.

This one is actually a bit more recent, so we would have a bit more than 79.37 % of present carbon 14. It is in medium 1880 years old**, meaning a sample of it should have 79.66 % of modern carbon.

Or we take an object from 1/4 of the half life and it should have fourth root of 1/2 as carbon content, or square root of square root. 1432.5 years ago, 585.5 AD. Let’s make that 585.

King Childebert II, age 15, takes up his sole rule of Austrasia. A Frankish army under King Guntram marches to Comminges (Pyrenees), and besieges the citadel of Saint-Bertrand.

July – Gundoald, Merovingian usurper king, and his followers are defeated during the siege of Saint-Bertrand. He is executed and Guntram stages a triumphal entry in Orléans.

585 - Wikipedia
https://en.wikipedia.org/wiki/585

So, Gundoald died in 585. A bone of his or wood from his coffin would have sth like 84.09 % of modern carbon.

This means, for carbon 14, we can check the physicists’ assumptions on how long the half life would be against objects which can be dated historically too. This is the reason why the original half life by Libby has been corrected since he published it, from his estimate 5568 years to 5730 years.

Note well, this type of checkup cannot be done for halflives like Uranium or Potassium 40.

* 0.7937005259840997 ** Wiki article said AD 125–150, that leaves a medium of 137.5 which deducted from 2018 leave a medium of 1880.5 years.