*Dear Dr. Zoomie – every now and again I read about isotopic dating being used to tell the age of the Earth or the age of some meteorites, when an ancient home was built, or something like that. But I also read about “problems” with isotopic dating that sound pretty convincing. What gives? And can you tell me how isotopes are used to tell us how old something is?*

Yeah – this one took me a little while to get a feel for when I first came across the idea in a college Geology class; then I ended up with a Master’s advisor who had helped to develop a lot of isotopic dating methods and took a couple of classes in undergrad and grad school. So let’s see if I can help out with some of the concepts – the second part of your question. For the first part, there are some other blog posts here that do a great job of explaining why the arguments made against isotopic dating don’t hold water – those are well worth a read.

So…we’re going to start by taking a walk. Say you’re strolling the length of Central Park in NYC – the park runs from 59^{th} St to 110^{th} St – call it 50 blocks (we’ll round 59^{th} off to 60^{th} St to make the counting easier). You start off at a brisk pace – walking a block in just over a minute. But as you walk, you’re getting tired and slowing down. So you might be able to walk the first 25 blocks in half an hour, but by the time you get to 85^{th} St you’re beat; in the next half hour you’re only able to make it to 98^{th} or so – about half the remaining distance. By now you’ve walked nearly 2 miles and, if you’re like most of us, you’re even more tired and maybe your feet and knees are starting to hurt – in the next half hour you manage to stagger on to 104^{th} St. In other words, in the first half hour you can make it half the length of the park, in the next half hour you can make it about half of the remaining distance; in the next half hour you can make it half the remaining distance, and so forth.

So think about it – if I know that every half hour you can walk half of the distance from where you’re at to the north end of the park then I can look at where you’re standing (or walking or limping) and know how long you’ve been on your trek. If you’re at 85^{th} St (25 blocks from where you started) then I know you’ve been walking for half an hour; if you’re at 98^{th} (half of the remaining half of the park) then I know you started walking an hour ago; if you’ve made it to 104^{th} St (half the remaining distance) then you must have been walking for an hour and a half.

This is how isotopic dating works. But let me make it a little more formal.

When you’ve covered half the length of the park you have an equal amount of park ahead of you as you do behind – the ratio of ground covered to ground remaining is 1 (25 blocks walked:25 blocks to go). At the end of two half-hours you’re ¾ of the way to 110^{th} St and that ratio is 3:1 (37 ½ blocks walked:12 ½ blocks remaining); if you’re at 104^{th} you’ve walked 44 blocks and have 6 remaining; a ratio of 44:6 (about 7:1) means that you’ve been walking for an hour and a half.

We can do the same thing with radionuclides – and there are a few ways to tackle the problem.

Take potassium-40 (K-40), for example. K-40 decays to form argon-40 (Ar-40) with a half-life of 1.28 billion years (there are a few other factors that make it a little less straightforward than this, but let’s keep it simple). So say you’re trying to find out how old a rock is; you might start by trying to find a mineral grain in the rock – something with a lot of potassium in it like orthoclase feldspar (also called potassium feldspar). You can analyze the grain of orthoclase, using an instrument called a mass spectrometer to measure the amount of potassium and the amount of argon it contains. If you find as much Ar-40 as you do K-40 then it’s the same as if I find you at 85^{th} St. in your trek to the northern end of Central Park – we know that half the K-40 has decayed away to form Ar-40, so the rock must be one half-life (1.28 billion years) old. If there are three times as many Ar-40 atoms as there are K-40 then the rock is 2 half-lives (2.56 billion years) old…and so forth. By counting the number of “parent” atoms and comparing to the number of “progeny” atoms to which they decay we can calculate the age of the rock (or whatever it is that we’re dating), as long as we know the half-life of the parent radionuclide.

Under some conditions you can do isotopic dating another way – by looking at the amount of radioactivity in a sample and comparing it, not to the number of progeny atoms, but to the concentration you *expect* to see were the sample brand-new. Say, for example, you expect that one out of every 1000 atoms of a particular element will be a radioactive isotope that you know has a half-life of one million years – but when you analyze your sample you find that only about one atom in a million is radioactive.

If we calculate the fraction of radioactive atoms we expect to see for every half-life that passes it looks something like this:

Number of half-lives | Fraction of original radioactive atoms remaining |

1 | ½ |

2 | ¼ |

3 | 1/8 |

4 | 1/16 |

5 | 1/32 |

6 | 1/64 |

7 | 1/128 |

8 | 1/256 |

9 | 1/512 |

10 | 1/1024 |

Using this with an eye towards our problem, we see that it takes just under 10 half-lives for the number of radioactive atoms to drop by a factor of 1000; thus, our sample is a little less than 10 half-lives – about 10 million years – old. This is the basis for C-14 dating, with the caveat that the expected concentration of C-14 changes with time because its formation rate depends on solar activity, which is always changing (more on that at https://www.ntanet.net/radiocarbon.htm).

So this is the basic idea behind isotopic dating and how it works, but it gets a lot more complicated fairly quickly – I have the better part of a shelf filled with books and scientific papers that go into the minutest of details of various isotopic systems, how they’re affected by geochemistry, and all the disparate types of information that can be gleaned by simply counting the atoms of parent and progeny. But once you master the basics, all the rest is elaborations on the general theme, so I’ll stop here.

If you want to explore the subject more thoroughly, there’s a great book called *The Age of the Earth*, written by G. Brent Dalrymple, one of the scientists who helped to found this branch of science. This book gives a great explanation of not only the basic principles of isotope geology, but also gives a wonderful summary of previous attempts to learn how old our planet is and some of the fascinating details of various isotopic systems. It’s so well-done, yet approachable that I considered using it as a textbook when I was considering teaching an undergraduate class on this subject.