For radioactive materials with short half-lives, you use a very sensitive calibrated detector to measure how many counts per second it is producing. Then using the exact same set up you do the same at a latter time. You use the two readings and the time between them to determine the half-life. You don’t have to wait exactly a half-life, you can do the math with any significant time difference. Also, you don’t need to know the absolute radioactivity, as long as the set up is the same you only need to know fraction by which it changed.
For radioactive materials with long half-lives that won’t work. Instead you approach the problem differently. You precisely measure the mass of a very pure sample of the radioactive material. You can use that to calculate the number of atoms in the sample. Then you put the sample in a counter that is calibrated to determine the absolute number of disintegrations happening in a given time. Now you know how many of them are disintegrating every second. You use the following equations:
Decays per Second = (Number of Atoms) x (Decay Constant)
Half-life = (Natural Log of 2) / (Decay Constant)
And you can calculate the half-life
Hope it helps :)
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Look at these images, the oldest is at the bottom, the newest is at the top, so the drawings are next to time zones are properly set up, so next to the time scale is when the drawing next to it was made.
