How do you explain natural radioactivity when matter is so incredibly old?

Your question is a very comprehensive question. You should therefore split it into two parts.

If you look at the age of the universe, you can find that it is about 10¹⁰ year (estimated). Several radionuclides, such as Thorium-232 and Uranium-238, for example, have half-lives that are about the same. These elements were formed around the time of the Big Bang and are also added during supernovae. These nuclides move through the universe and can therefore also end up on Earth. Because of this and because of their long half-lives, we still have these nuclides on Earth. However, the activity that we now measure on these radionuclides is mainly due to their presence since the creation of the earth. The additional content is very small.

However, Thorium-232, as well as U-235 and U-238, decay via series decay. Among other things, the Ra-226 you quoted comes from such a series decay. U-238 will decay through several intermediate steps to this radium with a much shorter half-life. (see the 3 natural series at http://www.nrg.eu/public/abc/node735.html).

In addition to these primordial radionuclides, you also have radionuclides that are constantly being formed. For example, carbon-14. This is formed in the upper air layers under the influence of ionizing radiation from space from nitrogen 14, by absorption of a neutron and the ejection of a proton. You can of course also measure this by migration of carbon-14 in the environment.

Sincerely

Steven Lily