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I'm currently reading a book about Earth's geological history and the authors mentions radioactive dating as on of the methods used to estimate the age of given fossils. It obviously does makes sense to me and I fully accept it as a valid scientific method, but :

  • How do scientists measure the initial amount of the radioactive material present in a given object, for instance let it be C-14? Do they compare them to other objects of identical chemical structure in which the decay process is yet to start, and if so, how do they know if the process itself hasn't started already?
  • What method has been used to calculate the half-life for each isotope, because obviously it did not happen through scientific examination, since in many cases the radioactive decay takes thousands, if not millions of years.
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    $\begingroup$ Wikipedia has a good article on Carbon-14 dating. As for dating other isotopes, the concentration at the "starting time" depends on the element and the location of the element in the earth. For instance there was a natural nuclear fission reactor. In fact different isotopic compositions at different earth locations limits the precision which the atomic weights of the elements can be calculated. $\endgroup$
    – MaxW
    Mar 24, 2020 at 21:32
  • $\begingroup$ Similar: physics.stackexchange.com/questions/202537/…, physics.stackexchange.com/questions/7584/… $\endgroup$
    – BowlOfRed
    Mar 24, 2020 at 21:55
  • $\begingroup$ "It is based on the fact that radiocarbon (14 C) is constantly being created in the atmosphere by the interaction of cosmic rays with atmospheric nitrogen. The resulting 14 C combines with atmospheric oxygen to form radioactive carbon dioxide, which is incorporated into plants by photosynthesis; animals then acquire 14 C by eating the plants. " I'm not sure I follow, since plants absorb carbon dioxyde, not C-14. Is it created through the photosynthesis process? $\endgroup$
    – pq89
    Mar 25, 2020 at 6:34

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The C-14 dating method was calibrated by comparing its results with the results from another independent dating method (the counting tree-rings - dendrochronology).

Quoted from Radiocarbon dating - Calibration:

To produce a curve that can be used to relate calendar years to radiocarbon years, a sequence of securely dated samples is needed which can be tested to determine their radiocarbon age. The study of tree rings led to the first such sequence: individual pieces of wood show characteristic sequences of rings that vary in thickness because of environmental factors such as the amount of rainfall in a given year. These factors affect all trees in an area, so examining tree-ring sequences from old wood allows the identification of overlapping sequences. In this way, an uninterrupted sequence of tree rings can be extended far into the past. The first such published sequence, based on bristlecone pine tree rings, was created by Wesley Ferguson. Hans Suess used this data to publish the first calibration curve for radiocarbon dating in 1967.

The K-Ar dating method is based on the half-life of $^{40}K$ which is $1.248\cdot 10^9$ years. This half-life could be determined in the laboratory by measuring two things:

  • The isotope mixing ratios of natural potassium can be determined with a mass spectrometer. It contains $0.0117 \text{%}$ of $^{40}K$. The other isotopes are not radioactive.
  • The radioactive decay rate of a certain amount of natural potassium can be determined with a radioactivity counter (e.g. a Geiger-Müller counter). The decay rate is around 44 per second and per gram of potassium.

From these two measured numbers and Avogadro's constant the half-life of $^{40}K$ could be calculated in a straightforward way.

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