Hello all.
How did science arrive at a date for the formation of the earth?
Let's begin at the beginning. The first radiometric dating method to be used was the uranium-lead method. Various minerals contain uranium-238 (U-238), which decays into lead-206 (Pb-206) with a half-life of 4.468 billion years; they also contain U-235, which decays into Pb-207 with a half-life of 703.8 million years.
Every atom of U-238 that decays yields exactly one atom of Pb-206; every atom of U-235 that decays yields exactly one atom of Pb-207. Thus the sum of the number of atoms of U-238 and Pb-206 is the original number of U-238 atoms when the rock was first formed. (The same goes for U-235 and Pb-207.) Knowing both the present number and the original number of U-238 atoms yields the time when decay started, i.e. the time when the U-containing mineral crystallised from the magma.
Now comes an interesting bit. You must know from your geochemistry and mineralogy lessons that different elements have different chemical characteristics and are associated with different types of minerals. Uranium is chemically lithophile and occurs in silicate minerals. Lead is chalcophile and occurs in sulphide minerals. For example, as everybody knows, the main ore of lead is galena, or lead sulphide (PbS). Thus most of the lead in silicate minerals comes from the radioactive decay of uranium. Zircon (zirconium silicate) is particularly useful in this respect, since it practically excludes lead from its crystal structure. Notice also that lead ores contain essentially no uranium, so that the isotopic composition of the lead has not changed since the ore was deposited.
Moreover, the isotope lead-204 (Pb-204) is not produced by radioactive decay; all the Pb-204 in a mineral is primordial, it was in the mineral from the beginning. Knowing the amount of Pb-204 in a mineral and the ratios of Pb-206 and Pb-207 to Pb-204 makes it possible to subtract the primordial Pb, so that one is left with the Pb-206 and Pb-207 that is produced by decay.
Even the first U-Pb measurements of terrestrial rocks showed that the Earth was much older than Lord Kelvin had thought. By the early 1930s, the oldest known rocks had yielded ages of 1300 million years. However, these measurements could give only a minimum age for the Earth itself.
Now we come to the really interesting part, the determination of the age of the Earth as a planet. Because U-238 and U-235 have different half-lives, they produce Pb-206 and Pb-207 at different rates. Therefore the Pb-207/Pb-206 ratio in terrestrial lead ores has changed during the Earth's history. Let me make this quite clear. A Mesozoic lead ore will still have the lead isotopic composition that it had when it was deposited, since it contains no uranium, but its isotopic composition will be different from an early Proterozoic lead ore, since the terrestrial lead reservoir from which it was formed has had time to accumulate more Pb-206 and Pb-207. This difference would not exist if the Earth was young and lead ores were therefore all the same age.
Knowing the present Pb-207/Pb-206 ratio, and knowing the production rates of Pb-207 and Pb-206, we can calculate the time when the two lead isotopes began to be formed by the radioactive decay of uranium if we know the original Pb-207/Pb-206 ratio when the solar system was formed. This was what Clair Patterson did in the 1950s. He very carefully measured lead isotopes in the Canyon Diablo iron meteorite; iron meteorites do not contain uranium, so the lead isotopic composition in the meteorite has remained the same since it was formed. Knowing the original Pb-207/Pb-206 ratio and the present ratio, Patterson was able to calculate an age of 4550 million years for the Earth and meteorites.
This has been a long explanation, but I hope that you have been able to follow it. I have done my best to explain clearly and in detail how radiometric dating works, to show how possible errors can be avoided, and to distinguish between radiometric dating of rocks and measurement of the age of the Earth as a whole. However, if you are interested and want to learn more, you ought to read books on the subject; they are much better than the inevitably short and superficial explanations that you will find on Internet forums.