(This post was linked earlier.) I'm now wondering what the process is to determine these radioactive half-lives for these isotopes. How do the scientists know that it takes 4.5 x 10^9 years for uranium to decay into lead, for example?
Sorry for the nit-picking but the value you quoted is for half life which is the amount of time 50% of the parent nuclei has decayed.
This leads to another important consideration; half life is a statistical median which is the midpoint of an exponential distribution.
Since statistics is involved, this can involve some very counterintuitive ideas.
I'll use the example of the half life of Xenon 124 to show the half life is calculated rather than measured using both theoretical and experimental physics.
The Xenon 124 nucleus can capture 2 electrons converting two protons into two neutrons which decays to ¹²⁴Te (Tellurium 124) and two neutrinos according to the reaction
¹²⁴Xe + 2e⁻ → ¹²⁴Te + 2νₑ
The half life for the decay is an astonishing 1.8 x 10²² years which is over a trillion times older than the age of the universe.
I imagine creationists would have a field day yet the decay of ¹²⁴Xe has been observed from which the half life can be calculated.
If ¹²⁴Xe decayed within the time fame of the age of the universe, the event would sit in the left hand side of the exponential distribution where the fraction of ¹²⁴Xe remaining dominates.
To observe a decay event in the time frame of say one year would require an enormous number of ¹²⁴Xe atoms which fortunately we have in the form of liquid Xenon dark matter detectors.
Here is the mathematics behind the calculation of the half life.
82 real decay events were recorded over a period of one year for a 1000 kg mass of liquid Xe from the XENON1T dark matter detector.
Assuming a knowledge of high school chemistry the molar mass of Xe is 131.3g/mol of which the natural abundance of ¹²⁴Xe is 0.095%.
1 mole of Xe contains 6.02 x 10²³ atoms of Xe and the number of ¹²⁴Xe atoms in 1000 kg of Xe is:
(1000/1.31) x 0.00095 x 6.02 x 10²³ = 4.35 x 10²⁴ atoms of ¹²⁴Xe.
The decay constant λ = [(number of decay events)/(number of ¹²⁴Xe atoms)] x t x ε where ε is the detection efficiency factor for the XENON1T and is ≈ 0.5
For t =1 year, λ = [82/(4.35 x 10²⁴ x 0.5)] = 3.77 x 10⁻²³/year
Now using the formula for half life t₀.₅ = ln(2)/λ = 0.693/3.77 x 10⁻²³ ≈ 1.8 x 10²² years.