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Half-Life and Radioactive Decay

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Hi. It’s Mr. Andersen and this is AP Physics essentials 136. It is on half-life and radioactive decay. To model that my daughter and I started rolling some dice. We have 32 dice in the cup. And what she would do is she would roll the dice and then she would pull all of a certain number out. So in this first simulation she is going to pull all of the 1s out.

And so we put those to the side. That is generation one. And now generation two. And generation three. And generation four. Now when you get to the fourth generation, she has hit her first half life. What does that mean? Well if there were 32 to start, that is the amount of time it takes for half of those to decay. So that would be 16. So 16 have decayed right here. So she is going to keep doing that. And so if we watch what happens in the rest of the simulation you are not pulling as many dice because there are not as many left.

And eventually they all have decayed after 19 generations. But what you can see is that the half life is consistent. So this was to 16 and then to 8 and then to 4 and then to 2 and then to 1. And so radioactive nuclei will decay. What does that mean? They are going to quick off some kind of a particle, it could be alpha, beta or gamma.

And then they are going to form a new, usually more stable nuclei. And so in this radioactive decay, mass and energy are conserved. We have talked about that in previous videos. But what we are going to talk about here is what is the probability of that decay occurring? Well it is chance. So we never know when the next one is going to decay. But what we can use is the law of large numbers to calculate the half-life. The half-life is the amount of time it takes for half of the radioactive nuclei to decay. In other words if this represents all of the radioactive nuclei that would be the time it takes for half of them to decay. Now we have all of these ones left. And so we would have another half life like that. We could just keep going and going and going like that. And so there is an equation we can use to figure out how many are going to decay at each generation.

And so delta N, which is the change in the radioactive nuclei, so N is going to be our radioactive nuclei. So the change in N is minus, because we are losing those nuclei, so negative decay constant, I will get to that in a second, times N which is the number of radioactive nuclei times time. And so if we kind of work backwards for that, in this simulation here our time was advancing 1 generation after another. And so that is going to be 1 each time.

What was our decay constant? Well it is 1 in 6. So it is a one-sixth probability that you are going to roll a 1 and that that one is going to decay. And so we could model that. I will just use a quick little spreadsheet to do that. So at time 0 how many radioactive nuclei did we have? Well our N value was 32. So let me walk you through this formula. What is our time going to be? It is going to be 1. What is going to be our N value? It is 32.

So 1 times 32. And then what is our decay constant? It is one-sixth. And so it is going to be negative one-sixth times 32. So what is that value? It is going to be negative 5.33. Now if we look, how many actually decayed? It is 6, but it is close, -5.33. This is what it would predict to be and this is what we actually got with a really small number of dice. Now how do you do the next one? Well what we have to do is we have to take that 32, these ones decayed, so we are going to take 32 minus 5.33 and that is my N value for the next generation. And so now what do I do? I go back to this formula again.

So it is going to be T, which is 1 times my new N which is going to be 26.67 times my decay constant, which remains constant, and so what I am going to get is negative 4.45. Now how does that match up? It is pretty close to this. So then we subtract that value like that and we could just do this over and over again. So right here on the right is going to be what we would predict to occur. And this is what actually occurred in this little simulation. And the number of dice are so small compared to the number of nuclei in a sample. But if you look at my data, the green line represents the actual data that we find. The red line represents the predicted. And so you can see that it matches up pretty quickly. What would happen if we changed the decay constant. What if we changed it from one-sixth to one-half.

So how would you do that in the modeling? Instead of pulling 1s out, she is going to pull the 1s, 3s and 5s out. So what is going to happen? Well it is going to happen more quickly. So more of them are going to decay at each point. And so it is going to take less time for all of them to decay. In other words our half-life has gotten much much shorter. And so you should be able to calculate half-life. So how long would it be for half of them decay? You can see it is going to be somewhere around 4 generations. And so when you look at a curve, the first thing you want to do is figure out how long does it take to go from 100 radioactive nuclei to 50 percent of that.

And so if I look across here it took 1 year for 50 percent of them to decay. So let’s watch the next generation. So now we should go to 25. You can see it is consistent. 1 year. 1 year. 1 year. And so you could say for this perfect model it is going to be a half-life of 1 year. But on a test you are more likely to get something like this. Calculate the half-life decay of carbon 14. So if you are given this curve right here, we go from 100 to 50, so you could count across like that. So this is 50 percent of the carbon 14. And then you just read the time on the bottom.

So if this right here is 10,000 years, what is our radioactive half-life? It is around 6,000 years. But let’s keep going. So now let’s go down to 25 percent and you can see it is around 12,000. And so it is pretty consistent over time. Now each form of radioactive decay is going to kick off a different particle. Let’s start with alpha particles. An alpha particle is 2 protons and 2 neutrons. So if we look at the alpha decay of uranium 238, let’s make sure that the mass is conserved. And so mass on the left side is 238, mass on the right side is 238. So we are fine there. Let’s make sure charge is conserved. 92 positive charge on the left. 92 positive charge on the right. So that is conserved as well. Now what is the half-life of uranium 238 decay? It is 4.47 billion years. It takes a huge amount of time for just half of the uranium 238 to decay. But there are so many nuclei that we can actually measure this. And this is how we determine the age of the earth. Let’s look at beta decay. Beta decay we converted a neutron to a proton.

We also give off an electron and an electron antineutrino. So if we look across the top, mass is conserved, 14 and 14. Charge is conserved. On the left side we have 6 positive charges. On the right side we have 7 positive, 1 negative. And so we have 6 positive charges on the right as well. What is the half-life of carbon 14 decay? It is going to be 5730 years. So we had shown that just a few slides ago. It was around 6000 for half-life. And we can use this to date living material. And then we could look at gamma decay. Remember in gamma decay we are just giving off these gamma rays. We are going from barium 137 that is charged to barium 137 that is not charged. And so we are conserving charge and conserving mass. What is the half-life? 2.6 minutes. So it is really really short. And so half-life is going to change depending on that decay constant. But you should be able to take a graph like this. Figure out the half-life. And I hope that was helpful..

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