A rock contains  a radioactive element with a half-life of 100 million years. Tests show that the element in the rock has gone through three half-lives. How old is the rock?

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caledon | High School Teacher | (Level 3) Senior Educator

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Radioactivity involves the decay of an unstable isotope into another, more stable atomic configuration. This typically changes the number of particles in the atom by breaking off or changing the number of protons or neutrons. This means that the decayed element is different from the original one. 

Conceptually and broadly speaking, this is fairly easy to describe and observe, but on the scale of individual atoms, it turns out to be pretty unpredictable. The decay event depends upon events which, according to most quantum mechanical descriptions, are inherently random, with little in the way of further explanation or reason. This randomness is apparently built into the nature of the universe and the particle motions themselves. Therefore, it is impossible to know exactly when a decay will occur.

What can be determined is the average time it takes for half of a sample to decay. This is the half-life. A half-life of 100 million years means that half of the original sample will have decayed in 100 million years. It will take another 100 million for half of that amount to decay, and then another 100 million and so forth.

So:

Sample: 100% - no half life. Time = 0

Sample: 50% - 1 half life. Time = 100 million years

Sample: 25% - 2 half lives. Time = 200 million years

Sample: 12.5% - 3 half lives. Time = 300 million years (minimum).

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