In the following derivation, N is the number of radioactive parent atoms at time t, N0 is the original number of radioactive parent atoms, D is the number of stable daughter atoms, λ is the decay constant, and t½ is the half life of a radioactive atom. The situation I use as an example is one parent and one stable daughter. Slight modifications are needed for multiple parents (the (U-Th)/He system), multiple daughters (40K decay), or the use of unstable daughters in a decay chain (U-series dating).
The activity, or rate of decay of a radioactive isotope is proportional to the number of radioactive atoms present
By introducing a proportionality constant λ (which we will call the decay constant), we can turn the above equation into
Because the λ is in units time-1, and the number of atoms decreases with time, we also throw in a negative sign. Rearrange this equation
and integrate from N0 (original number of radioactive parent atoms) and t=0 (time zero)
and we get
since t0 = 0, this leads to
which can be rearranged to
The above equation describes the number of radioactive atoms left after time t. The decay constants, along with their uncertainties, are readily available (I'll list some of the geologically important ones at the end of this post). Decay constants are inherent properties of the nucleus that describe the probability that an atom will decay. I blogged about demonstrating this phenomenon with M&M's here
OK, so the above equation is handy, but when we work with geologic samples, we don't know N0, the original number of parent atoms in the crystal, so can we put this equation into terms of things we can measure today?
In a system with one parent and one daughter, N0 is the sum of N, the number of parents left, and D, the number of daughters produced.
So if we put the last two equations together, we get this
Which is entirely in terms of things we can measure in the lab. The next steps are simple mathematical rearrangements
Until we get an equation that solves for time and whose variables are all things that we can measure.
I've been using the decay constant up until now, which as I said earlier describes the probability that an atom will decay. The half-life of an isotope is a more intuitive measure, the half-life is a unit of time over which half of the radioactive atoms in a given population will decay. You can easily see the relationship between the decay constant (λ) and half-life (t½) with this simple exercise. At t½, half of the parents have decayed and turned into daughters, so N=D, and D/N = 1. So let's plug that into our age equations
and we get the relationship between λ and t½
So the equations are pretty straightforward. As I said, they get a bit more complicated for more complicated decay schemes, but the basics are all the same.
Here are some of the more important radioactive isotopes, decay constants, and half lives. I got this information from table 1.1 of Alan Dickin's Radiogenic Isotope Geology (2ed), he has the references for the appropriate scientific papers listed as well.
|Isotope||Decay Constant (yr-1)||Half-Life|
|40K (40Ar)||5.81 X 10-11||11.93 Ga|
|40K (40Ca)||4.962 X 10-10||1.397 Ga|
|40K (total)||5.543 X 10-10||1.25 Ga|
|235U||9.8485 X 10-10||703.8 Ma|
|238U||1.55125 X 10-10||4.468 Ga|
|232Th||4.9475 X 10-11||14.01 Ga|
|87Rb||1.402 X 10-11||49.44 Ga|
|147Sm||6.54 X 10-12||106 Ga|
|176Lu||1.86 X 10-11||37.3 Ga|
And, as usual Ga means billion years, and Ma is million years. Now you too can enjoy hittin' the geochronic.