Radioactive Decay Math Root
by Paul Doherty
Dating rocks by radioactive decay.
When a mineral crystallizes, it crystallizes with atoms in a specific ratio. For example a crystal of Uraninite has the chemical equation UO_{2}. When crystallization happens other elements are excluded. In particular, there is little lead in Uraninite. In addition, there are different isotopes of Uranium. All Uranium atoms contain 92 protons but the sum of the number of protons and neutrons in naturally occuring Uranium can have several different values including 238 and 235
Uranium238 decays to lead 206 by emitting eight alpha particles and six beta particles. The half life of this decay is 4.51 x 10^{9 }yrs.
Uranium 235 decays to lead 207 by emitting seven alpha particles and four betas. The half life is 7.13 x 10^{8} yrs.
Thus Uraninite ore contains two separate Uranium decay clocks that can be used to check each other.
In a Uraninite ore sample in which there were equal numbers of Uranium238 and lead 206 atoms, half of the Uranium238 would have turned into lead and it would be one half life old. 4.5 billion years. Which is older than any rock on Earth.
The equation for the number of radioactive atoms remaining, N, at a time, t, after a time when a sample starts with N_{0} atoms is.
N = N_{0} 2^{t/T}
where T is the halflife, the units of t and T don't matter as long as they both have the same units.
For Uranium, after one half life, T = 4.5 x 10^{9} yrs and t = 4.5 x 10^{9} so
N = N_{0 }2^{1} = 1/2 N_{0}
(A different version of this same equation uses the base e instead of 2, it is
N = N_{0} e^{}^{l}^{t}
where l is called the decay constant and is l = 0.693/T
for Uranium this is 1.5 x 10^{10} yr^{1}
Scientific Explorations with Paul Doherty 

10 July 2002 