This equation allows us to figure out how many radioactive atoms are left after any amount of time.
There are many general forms of the equation that deal with chains of nuclides, but here we only worry about the basics. Number of atoms at any given time We start by noting that the speed of radioactive decays occurring in a sample is proportional to the number of radioactive atoms in the sample.
Atoms that decay faster have larger decay constants, and so on. Now that we know the rate at which our sample is decaying, we can integrate this rate to figure out the total number of atoms that have decayed at any given time.
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Relating the half-life to the decay constant Excellent. The final equation there is the well-known radioactive decay equation.
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The blue curve represents a nuclide that has a year half-life. Compare this with the green curve, which represents a nuclide with a year half-life.
Decays of several example nuclides with different half-lives from 50 to years. The Python code to generate this image is available here. These may be useful for radiogenic dating of objects like the Earthunderstanding nuclear waste behavior, etc.
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Notice that the sum of these two curves is always 1. The decay of Uranium eventually leads to Lead There are many steps along the way, but they all have very short half-lives compared to that of U so they can be ignored on these long geologic time scales. Find details of the decay chains here.
See Also The Radioactive Decay [Wikipedia] page has a much more detailed and thorough description of radioactive decay, including situations with more complex decay chains.