We saw how geology—specifically, the principle of uniformitarianism—helped to establish that the earth was old enough for evolution to occur. The early observations of Hutton and Lyell have more recently been corroborated by a series of techniques, collectively known as radiometric dating. Radiometric dating is a way of figuring out how old sediments, fossils, or even archaeological artifacts are. It is based on the fact that some radioactive isotopes occur naturally in the environment, and they change (or decay) at fairly predictable rates over time.
Let's back up a second. Remember that an atom of any given element has protons and neutrons in the nucleus, and electrons orbiting around the nucleus. The number of protons defines what kind of element it is…so carbon will ALWAYS have 6 protons, or it's not carbon. But, the number of neutrons can vary, resulting in different isotopes of elements. For example, carbon exists in three isotopes: 12C, 13C, and 14C. Most (>99%) of the carbon on earth is 12C, and most of the remaining 1% is 13C.
Both these isotopes are stable – that is, they are not radioactive and don't decay over time. 14C makes up a teeny tiny fraction of all carbon, and is radioactive. Over time, it decays into 14N, which is stable. Scientists know that, for any given amount of 14C, it takes about 5,730 years for half of it to turn into 14N. Thus, we say 14C has a half-life of 5,730 years. Radiometric dating requires that you know the half-life of the radioactive isotope you're using.
Since we've been talking about carbon so far, let's see how we could use radioactive carbon to figure out the age of something. We know how much 14C is currently in the atmosphere. Another important piece of information is that organisms can only gain 14C as long as they are alive. Once they die, they can't absorb any more. So, at the moment right before death, the amount of 14C would match the atmospheric levels, but it would steadily decrease thereafter—specifically, there would only be half the expected amount after 5,730 years.
By comparing the actual amount in a fossil sample (for example) to the amount in the atmosphere, you can calculate how many years have passed since that organism died. Pretty cool, right? This is called radiocarbon dating. Because the half-life of 14C is relatively short, after about 60,000 years there is so little 14C left that it's impossible to measure accurately. So if you want to know the age of something older than that, you've got to try another form of radiometric dating.
Potassium-argon (K-Ar) dating is another form of radiometric dating. 40K makes up about 1% of all K in the earth's crust, decays into 40Ar, and has a half-life of about 1.3 billion years. That's much longer than carbon's half-life – as a consequence, K-Ar dating doesn't work very well on things younger than 100,000 years old, because so little of the 4K will have decayed in that amount of time. K-Ar dating also isn't performed directly on fossils, as we saw with radiocarbon dating. Instead, usually volcanic rock that lies close to a fossil is dated, and that date is taken to approximate the age of the fossil or other surrounding sediments.
So why is volcanic rock so great for K-Ar dating? Well, at the time magma is hot, all the naturally occurring 40Ar lying around is able to escape from the molten rock, which basically means that, at the time the rock forms, there is absolutely no 40Ar in the rock. Once the rock cools, however, 40Ar can no longer escape, so any 40Ar being produced by decay of 40K gets trapped. By comparing the ratio of 40Ar to 40K in a rock sample and knowing the decay rate, scientists can figure out how much time has elapsed since the rock cooled.
There are many other kinds of radiometric dating, and all are a little different based on the half-life of the radioactive elements, the kinds of samples that can be dated, and the accuracy of the techniques. These techniques collectively confirm the observations of Hutton and Lyell and show with certainty that the earth is old enough for evolution to have occurred.