Half-life | Physics | Khan Academy

This is a Neanderthal skull. Neanderthals are an extinct species of humans, and we believe they went extinct about 35 to 40,000 years ago. This is Earth, and we believe Earth to be about 4.5 billion years old. But my question was always, how do we know these things? How do we figure these things out?

Turns out one common method is radiometric dating, where we make use of radioisotopes to figure this out. But how does that work? How do you use radioactivity to figure out how old something is? Let's find out.

Carbon-14 is a radioisotope of carbon, so it decays into a more stable isotope, nitrogen-14. Now, if you take any amount of carbon-14 you want, take any amount you want. Let's just take some number—a nice number, 100 grams, let's say. Now, as time passes by, the atoms will start decaying, and so the amount of carbon-14 that you have will start reducing, isn't it?

Now, it turns out that after about 5,730 years, 50 grams of carbon-14 would have decayed to nitrogen-14, and only 50 grams is left with you. Now, my question to you is, what do you think will happen if we wait for another 5,730 years? My intuition says it should now go to zero—all of the carbon-14 should decay. I mean, it makes sense, right? Like, in the first phase, in the first 5,730 years, 50 grams got decayed. Now we wait for the same amount of time, another 50 grams would decay, so obviously it should go to zero.

But turns out that that's not what happens. Instead, we find that now half of this amount gets decayed. So from 50 grams, 25 grams get decayed, and we are left with another 25 grams. The same thing continues. If you wait for another 5,730 years, half of this value gets decayed, and so on and so forth. That's how radioactivity proceeds.

So this means that regardless of whatever amount of carbon-14 you have with you, it doesn't matter what amount you have. But if you wait for 5,730 years, it will reduce to half its value—half of whatever you have right now. And therefore, that number is called the half-life. It's a number that tells you how much time you have to wait for 50%—half of the amount of stuff that you have—to decay. It could be half of the mass that you have, it could be half of the number of atoms that you currently have to decay, the number of moles, whatever—it's half of the amount of stuff.

How much time you have to wait for half the amount of stuff to decay? Let's take another example. If you take uranium-238, it turns out that it is a radioisotope, and it turns out that whatever isotope you get after the decay also undergoes another radioactive decay; there's a chain like that, but eventually it ends in lead.

Now, that's besides the point. What's important is the half-life of uranium-238 happens to be about 4.5 billion years. Now what does that mean? Well, what it means is that if you take some amount of uranium-238 with you, again it doesn't matter what amount you take with you. Right now, let's take some random 68 grams—you have 68 grams of U-238 with you. Now, if you wait for 4.5 billion years, that amount will reduce to half.

What happens if you wait another 4.5 billion years? Well, again that amount will not vanish. It will reduce to half, and that'll keep on happening. Half-life. So each radioisotope will have its own half-life. But the big question is, why does radioactivity proceed like this?

To gain some insights into this, let's play a game. Let's put 100 million people in a room and decide to toss a coin every minute. Now, if you get a tail, you're done; the game is over. You leave the room. But if you get a head, you stay in the room, wait for another minute, and toss the coin again. You keep on doing this.

So my question to you is, what's going to happen after a minute? We start the game, we wait for a minute, everybody tosses a coin—how many people will be left in the room after one minute? Well, you would say that tossing a coin is a random event. So there's a 50% chance of getting heads or tails. If I focus on any one individual, I have no clue whether that person is going to get heads or tails.

But since I have 100 million people, statistics come into play. So 50% of this group is going to get heads, and about 50% will get tails, which means half of them will stay in the room, and the other half would have left. So we would have 50 million in the room, isn't it? Now we continue; we wait for one more minute.

What do you think is going to happen? Do you think now all 50 million would leave the room? No. You would say, "Hey, the same thing's going to happen again," because again, there's a random chance. 50% of them will get heads, 50% of them will get tails.

Again, only 50% of this will stay—that is, 25 million will stay—and the game will continue like that. Do you see a pattern between these two? I mean, tossing a coin is almost a random event. If I were to focus on one specific individual, that person might survive the game for another 100 tosses or that person might leave the room the very next minute. I have no clue; I cannot comment on what happens individually.

But because we have 100 million of them, statistics become more significant. I can tell that 50% of this must keep decaying; 50% of them must leave. Something very similar is happening over here. If you take a single atom, there's no way to predict what's going to happen. It might decay the very next second, or it may not decay for another billion years. There's absolutely no way to talk about that.

But if you take billions and trillions and trillions of atoms together—if you take a lot of them together—then statistics become significant. So in 4.5 billion years, if half of them decay, well then, in another 4.5 billion years, again only half of them must decay, and the same thing must continue. Doesn't that make sense?

Think about this for a while. The key over here is that radioactivity is a random process, and that's why the chances of a radioactive atom decaying at any given moment is 50/50. There's a 50% chance of decaying, and a 50% chance it will not decay. This is the reason why the statistics work out.

And by the way, there is absolutely no way you can influence those chances. There's nothing you can do that will say you force it to decay or will stop it from decaying. All you can do is just sit back and watch it. This also means that the half-life of a radioactive sample is fixed. For example, carbon-14 always has a half-life of 5,730 years—period. It doesn't matter how much carbon-14 you take, under what conditions you take it—none of that matters.

Similarly, the half-life of U-238 is always going to be 4.5 billion years—done. And that's awesome because now, just by looking at half-lives, you can comment about which isotopes are more radioactive. I mean, which of these two do you think is more radioactive? Well, carbon-14 only takes about 5,730 years to decay to half the value, but uranium takes about 4.5 billion years.

So you can immediately see that because it has a shorter half-life, carbon-14 must be more radioactive. The shorter the half-life, the quicker it decays. More radioactive something is, there are isotopes which have half-lives in mere seconds.

Anyways, now there's one big difference between the game that we played and how radioactivity unfolds in reality. What's that difference? Well, let me just move this thing to the side. So if we go back to our game and we try to draw a graph, on the y-axis we'll plot the number of people who are remaining. On the x-axis we will draw the number of half-lives. Initially, we had 100 million people to begin with, so 100% of them were remaining.

They stayed in the room for about a minute, then they tossed a coin, and immediately that number reduced to half—half of it, 50%—and then again that stayed in the room for a minute, and then again it immediately reduced to half. And that kept on going, right?

Well, the difference is, in actual radioactivity, it doesn't happen like this. It's not like the 68 grams will stay 68 grams for 4.5 billion years and then instantly reduce to 34 grams. Instead, that radioactivity is a continuous process. The amount of the radioisotope that you have will continuously decay. This number will continuously reduce, and so the actual graph that you will get will be more of a curve that looks like this.

But the point is, after if you wait for one half-life—which could be 5,730 years for carbon-14 or 4.5 billion years for uranium-238—if you wait for one half-life, look, the number reduces to 50%. And then if you wait for the second half-life, another half-life, the number reduces to 50% of that, which is 25%, and so on and so forth.

This now brings us to our original question: how do you figure out the age of things? For example, how do you figure out how old Earth is? Geologists use what we call zircon crystals. I'll tell you what's so special about these crystals: they absolutely hate lead. We need not worry about why that is the case, but that turns out to be true.

But let's say you find a zircon crystal, and inside you'll find some traces of uranium and some lead as well. And just to keep things simple, I'm just taking 10 milligrams of uranium and 10 milligrams of lead over here, okay? But then you question, like, where did this lead come from? You realize, hey, a lot of this uranium is radioactive, and so this lead must have come from the decay of uranium.

There's no other way this lead could have come over here. And so this means you conclude that when this crystal was formed a long time ago, all of those atoms of lead must have been uranium to begin with. And then you know, because of the decay, they turned into lead. That's how you conclude that probably when the crystal was formed, you must have had about 10 plus 10, which is 20 milligrams of uranium.

But wait a second—look at what this means. This means that about half the uranium atoms have decayed, and since we know that the half-life of uranium is always 4.5 billion years, you say, "Aha! About 4.5 billion years must have passed since the time that crystal was formed."

Now, I know this is an oversimplification, but you get the idea, right? Now, to age the entire Earth, well, we looked at a lot of such rocks, which we believe to be very old, and we found that almost a lot of the old ones date back to about 4.5 billion years. That's how we concluded that maybe Earth was formed about 4.5 billion years ago.

Okay, what about the Neanderthal bones? We used a very similar technique, but instead of using uranium—because uranium happens, you know the decay happens over a much longer time span—we use carbon dating. All living organisms, including you and me, have radioactive carbon-14 inside of us. And so by looking at how much carbon-14 is left, again this is an oversimplification, but by looking at that, we can estimate how long ago that person lived.

This is why I feel radioactivity is incredibly cool.