How are we able to find fresh radioactive materials

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How can we be mining for radio active material if the earth was formed so long ago, would these materials not have run the course of depletion already? or is man kind creating them ? like Uranium or Plutonium

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The way radioactive half life works is it is the time period over which statistically half of the radioactive material will have decayed. This does not mean that two half lives will result in *all* the material having decayed, it would be half of the remainder decaying. So for any particular radioactive atom there is no telling how long it will take to decay, it might be in the next minute or not for dozens of half lives.

For radioactive uranium its half life is around 4.5 billion years, about the same as the age of Earth. This means we can expect that about half of the radioactive uranium which existed on Earth has decayed throughout its lifespan. That is the straightforward answer to why that material hasn’t “run the course of depletion already”.

Plutonium on the other hand is a byproduct of uranium decay in reactors so we are in fact creating that material. It has a half life of between 87 and 24,065 years depending on the isotope, and while probably some exists in natural deposits it is generally produced by humanity.

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