How to calculate the age of the Earth (almost)!

Apparently nearly half of Americans believe that the Earth and all life on it were created in their present form less than 10,000 years ago. I don't think I've ever met any of them. But I know how these debates go. "Because scientists say so" doesn't appear to be a very convincing argument, since most of them seem to mistrust scientists anyway. It's much more satisfying to say "because I did the calculations myself." I don't know whether it's more convincing, but it's certainly more satisfying.

I'm not a biologist, so I'm not going to try to address evolution. But I can say that I have personally calculated the age of the Earth. And as long as you're not afraid of a little math, you can too! Yes, you!

Well, almost. Since the Earth's surface continued to be highly geologically active after its formation, we have no current rocks that actually go back that far. The currently accepted age of 4.5 billion years was actually calculated using meteorites, which were formed at the same time but cooled down quite a bit faster. Meteorite dating uses lead-lead dating methods, which are more complicated than I want to get into here. But using the simpler uranium-lead dating, we can get back almost that far!

I present to you the world's oldest known terrestrial material. It's a zircon crystal from western Australia. Zircon is special: every once in a while, a uranium atom can be substituted for a zirconium atom in its lattice structure. Lead is not incorporated at all. Yet old zircons have a good deal of lead in them. Why? Two different isotopes of uranium (²³⁵U and ²³⁸U) will radioactively decay, eventually, to two different isotopes of lead (²⁰⁷Pb and ²⁰⁶Pb, respectively). This takes a long time- the half-lives for these two isotopes are 704 million years and 4.47 billion years, respectively. This means that there is enough of both elements present in the crystal to determine the age of the rock.

Here comes the math. The radioactive decay law determines how much of a particular radioactive isotope will remain after a given amount of time:

Here, t is the time that has passed, U₀ is the amount of uranium you started with, and U_t is the amount of uranium that is left. λ is the decay constant for uranium, defined as ln(2) divided by the half-life. For ²³⁵U, λ = 9.85 × 10^-10 yr^-1. For ²³⁸U, λ = 1.55 × 10^-10 yr^-1. The rest of the uranium atoms (U₀ - U_t) are now lead:

Divide one equation by the other, multiply top and bottom by e^λt, and you get:

Now solve for t:

That's it! The paper I linked to earlier provides lead to uranium ratios for both decay paths (along with a ton of other information) measured at different sites on the crystal. Now we just plug 'n' chug. For example, in one sample, ²⁰⁶Pb/²³⁸U is measured to be 0.965. Using the appropriate value for λ (1.55 × 10^-10), we find that t = 4.36 × 10⁹ years, or 4.36 billion years. But that's not all! The ²⁰⁷Pb/²³⁵U ratio is given as 71.9, yielding t = 4.35 billion years. That's two independent methods of measurement yielding similar answers, which means we can put a lot of confidence in this result.

(If you look at the published results, you'll see that not all of the samples give similar answers, but enough of them are in the 4.3-4.4 billion year range that we can be sure this is correct. The anomalous results can be explained by lead escaping the crystal through cracks caused by radiation damage.)

Some important notes:

1. Thanks to the unique properties of zircon, we don't have to make any assumptions as to the relative concentration of lead at the time of the crystal's formation- we know that it's chemically impossible for it to be anything other than zero.
2. The decay equations are derived directly from the theory and backed up by mountains of empirical evidence. The relevant half-lives were well established before this form of dating was first used. This leaves no room for accusations of adjusting the theory to match the data.
3. The two decay chains are completely independent of each other. This means two independent measurements giving two independent results. Either method by itself would be valid; using both makes the findings (if you'll excuse the pun) rock solid.

Any questions? Was this clear enough?