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I think one of the first things that has to be understood before we talk about radioactive decay is what can decay. Basically anything with more than 84 protons is unstable and will tend to decay. You can have isotopes with fewer protons decay, but its uncommon. The property that dictates how radioactive something is is the ratio neutron / proton ratio. If the isotope is found to be neutron rich, then its unstable. On the other hand if its neutron “light?” then it can also be unstable. 

We can have decay in a few manners that we may or may not be familiar with. We can have Alpha particles poppin off, or maybe some beta emission, or even gamma radiation. Further we could have a few types that are way less common like positron emission or electron capture. 

In the case of an alpha particle discharge, we shoot off 2 protons and 2 neutrons. That s a 4 mass swing! 

If we have a beta emission we shoot an electron from the nucleus. “WAIT A SECOND BOB, ELECTRONS AREN’T IN THE NEUCLUS!?” Hold your horses. 

We’ve done a deep dive on this topic, but in order to keep all of this in the same show, I’ll go over it quickly. Lets says we had a I-131 atom and it gives off a beta particle (an electron). So our I-131 remains with a mass of 131. The atomic number is 54 (which is 53 – (-1)) If we look at this purely at the atomic number of 54 it “identifies as Xenon. But the mass number doesn’t change going from I-131 to Xe-131, but the atomic number went down one. In the iodine neucleus a neutron was converted into an electron and shot out, what was left is a proton. Whew…ok, back to the regularly scheduled content.

If you could sit and watch a single atom of radioactive isotopes let's say uranium 238. The problem would be that you couldn't predict when that particular atom would decay. It might happen now later next week next Millennium there's really no way to understand.

So now let's say we have a big sample something big enough that we would be in where mathematicians call a statistically significant sample size, at this point we may be able to point out a pattern. Now we all understand that it takes time for atoms to decay and go away. So then you can also say that the same amount of time for half of the amount of atoms to decay which means that we also have half the amount still there can be calculated. So we call this time that it takes 1/2 of them to go away the half life of an isotope, or the t1/2if we had to calculate out the half life decay of a radioactive isotope the problem would be it's not a simple math equation. Meaning it's not linear. So for example you couldn't find the remaining amount of isotope if let's say it's 4.7 half lives by looking between 4 and 5 half lives. So if we wanted to find the amount or the times with a simple multiplication of a half life we would use this equation.

T1/2 = 0.6963/k

Then

Ln (Nt/No) = -kt

Now, maybe you're not a math nerd, then you’re gonna note that there's a little something different here this is what they would call an natural logarithm. There's also another one called AB's ilaug and that is not the base 10 log but it's on your calculator so don't worry about it. In here the end to the oh is the amount of radioactive isotope that we start with in terms of grams or maybe even a percentage. The end to the T is the amount of radioactive isotope that's left at so...