About twenty radioisotopes with ultra-long half-lives, from 82 million years (244- Plutonium) to 6 million billion years (50-Vanadium) are found in the earth’s crust, yet no radioisotopes with half-lives less than 70 million years (146- Samarium) are found. Why not? Use this fact to estimate a lower limit for the age of the earth. Hint: How many half-lives does it take for a radioisotope to decay into background levels?
I can't find it anywhere on the web but I think the number of half-lives at which a radioactive isotope is no longer a threat (from the radioactivity) is 10 half-lives.
To understand why no radioisotopes with half-lives less than 70 million years are found in the Earth's crust, we need to consider the process of radioactive decay. Radioactive isotopes decay over time, transforming into stable isotopes or other elements. The half-life of a radioactive isotope is the time it takes for half of the isotope to decay.
Let's take the example of an isotope with a half-life of 70 million years. After one half-life, half of the original isotope remains. After two half-lives, only one-fourth remains, and so on. As more and more half-lives pass, the amount of the original isotope remaining becomes significantly smaller.
Now, let's consider the radioisotopes found in the Earth's crust. If a radioisotope with a half-life less than 70 million years were present, it would have decayed away to almost undetectable levels by now. This is because multiple half-lives would have passed, reducing the initial amount of radioactive isotope to an extremely small fraction.
However, we do find radioisotopes with ultra-long half-lives in the Earth's crust. These isotopes have decayed relatively slowly over billions of years, allowing a significant fraction of the original isotope to be present even after such a long time.
Using this information, we can estimate a lower limit for the age of the Earth. Since no radioisotopes with half-lives less than 70 million years are found, it indicates that the Earth must be older than the time it takes for a radioisotope with a half-life of 70 million years to decay to background levels.
Each half-life reduces the amount of the radioactive isotope by half. To reach background levels, the amount would need to decrease to a negligible amount. As a rough estimate, we can assume it takes around 10 half-lives for a radioisotope to decay to background levels, as this reduces the original amount to approximately 1/1000th.
Using 70 million years as the half-life, we can calculate the time it takes for 10 half-lives: 70 million years x 10 = 700 million years.
Therefore, based on the fact that no radioisotopes with half-lives less than 70 million years are found, we can estimate a lower limit for the age of the Earth to be approximately 700 million years.