Graham's law of diffusion

Welcome to a new planet. This is actually a planet I'm going to call Planet Graham. And you can already see the blue alien on this planet. And the reason I want to call it Planet Graham, I'll be honest, is because we're going to talk about Graham's law. And I thought it would be kind of a fun way to remember it and think about it. So, before I came to this planet, I actually brought along with me a giant pot. And this pot is like a cooking pot. But inside of it-- and I put a lid on it of course-- but inside I've got some molecules of oxygen, actually. From our home planet Earth, I brought some molecules of oxygen. And I'm drawing them as O22 molecules, or two atoms, rather, of oxygen in one molecule of O2. And I've also got some carbon dioxide here. And of course, carbon dioxide-- in the name you can already hear it-- it's got dioxide. So, it's got two oxygens as well. It almost looks like a little bow tie. So, I've got a few of these molecules. And I was very careful actually, made sure that in my pot I had 50% carbon dioxides and 50% oxygens. So, I've got equal proportions of both. So, I have a friend over here, an alien friend, and I asked my friend to stand some distance away from my pot. And you can see I'm actually cooking my pot. I've got some fire with firewood underneath it. And I say, please stand 10 feet away from my pot, sir. And this little alien friend is a good friend of mine, so he says, no problem. And the reason I'm asking him to help me with it is because he has a very special nose, a very, very special nose. He's never in his a life smelled oxygen or carbon dioxide. He's lived on Planet Graham his whole life. And Planet Graham has these little green molecules. But he has such a special nose that he can actually detect whether he's smelling carbon dioxide or oxygen. So, I'm going to actually take this lid off. And I'm going to say, hey, if you detect with your special nose either one of these-- let's say that these molecules, one of them goes over and kind of goes into his nose-- if you can detect it, please let me know which one you're smelling. And that's my test. And I want to know which of these molecules, oxygen or carbon dioxide, is going to reach his nose, which is 10 feet away, first. So it's basically a race. And you can make a prediction right now as to which molecule you think is going to get to his nose first, the oxygen or the carbon dioxide. Now you might think, oh, it's very easy. There's a direct path. But actually remember, these molecules, these green molecules in the planet atmosphere, are whizzing around. They're going in all sorts of different directions. And as a result, they're going to smack into our carbon dioxide or oxygen molecules as they try to make their way over there. And kind of a random fact, but an interesting one to think about, is that in our atmosphere we have a lot of nitrogen, a lot of nitrogen gas. Now, if you took one nitrogen gas molecule, which is N2, and let it go, and measured its speed and kind of clocked it, it would be going at about 1,000 miles an hour. But the only reason it doesn't actually go that speed in reality is because the molecules of nitrogen, they actually will smack into each other and bounce off of each other millions and millions of times every second. And so because they're smacking and colliding constantly, they never really reach those real potential speeds. They go much slower. So really what we're talking about is when molecules are bouncing and clanging into each other and slowly making progress towards our little alien's nose, that is the idea of diffusion. They're going to kind of rattle around and slowly make their way over to his nose. And maybe if I came back, let's say, 10 minutes later, maybe this little oxygen would be right here. Maybe you might have a little carbon dioxide right here. Slowly making progress towards the nose. That's what we're trying to figure out-- which one will get over there first. So, you've had time to think about it. And I'm actually going to tell you how I think we should approach the problem, which is thinking back to kinetic energy. We're heating this thing up, so we're putting thermal, or heat energy, into the molecules. Both types of molecules are getting the same amount. I've got the oxygen getting some kinetic energy. I'm going to put a little o for oxygen. And remember, the formula is 1/2 mass times velocity squared. And it's going to equal, or should equal, the amount of energy that my carbon dioxide is getting. And I'm going to do that as a little c for carbon dioxide. So these two molecule types should be getting the same amount of energy. Now remember, it's not like it's one molecule we're thinking of. We're thinking of many, many molecules. So first, I'm going to have to change these units a little bit. m, or mass, is going to change to molecular weight. Because again, I'm thinking about the individual molecule. So I've got to figure out what these molecules weigh. And v is going to change over to rate, or diffusion rate. And the reason I'm doing that is because, again, I'm thinking about the overall diffusion of the gas. It's not like I'm betting on any one molecule. I'm betting on the entire population of carbon dioxide molecules beating out the population of oxygen molecules, or vice versa, the oxygen molecules beating out the carbon dioxide molecules. But not an individual molecule. So I have to think of the average rate that those molecules are moving. So, let me rewrite this equation. It's going to now be 1/2 times molecular weight, I put it in parentheses, times diffusion rate. I'm just going to call it rate. And we'll call it rate 1. And 1 will be for the oxygen. In fact, molecular weight 1 can be for the oxygen as well. And over here, I'm going to say it equals 1/2 times the molecular weight 2. And 2 refers to carbon oxide, and rate 2 refers to carbon dioxide as well. And I really don't need to keep carrying on with these halves. I can just multiply both sides of the equation by 2 and get rid of them. So that makes it a little bit easier. And I almost forgot, I have to square both sides. That would have been a mistake. I forgot to square them earlier. So now I've squared them. And let me actually rearrange it to make it a little bit neater in a new color. So let's do this. So let me write it out nice and neat. And this is actually going to be Graham's law. So all I'm doing is rearranging the formula. I've got rate 1. This is the diffusion rate of one molecule divided by the diffusion rate of a second molecule, and then the molecular weight on the other side of the second molecule divided by the molecular weight of the first molecule. And you do a square root of this side. So, that's just a rearrangement of the formula. But what I've written out for you is now Graham's law. It's basically taking the kinetic energy rule and rearranging it to make sense for molecules. And let me make a little bit of space here. And so that an extension of this would be if you're just thinking about one molecule, then the rate, the diffusion rate-- when I say rate I mean diffusion rate-- is going to be proportional to the square root of the molecular weight. So, let's figure out how to apply this to our little riddle. We wanted to know whether oxygen or carbon monoxide is going to diffuse faster. And I can now go back to our the periodic table and look up oxygen. And I know that the molecular weight of 16 here and carbon is 12. And that means that O2 is just 16 times 2. So the molecular weight is 32. And carbon dioxide is going to be 2 oxygens plus 12 more. So it's going to be 44. So these are the molecular weights of carbon dioxide and oxygen. So basically what I do is I just plug them in. And I say, OK, let's plug them into the formula. Let's use this one right here. And I'm going to call rate 1 my oxygen rate. So what's happening with rate 1? We'll say, well, rate 1 is rate of oxygen-- I'm going to write a big o here-- equals the square root of-- let's make sure I stay consistent-- I said 1 was oxygen, so it's going to be 32 down here and 44 up here. And then that's going to be multiplied by rate of carbon dioxide. And I'll put a c for carbon dioxide. So what does this work out to be? That's 1.17. I just punched it into the calculator. So really, the diffusion rate of oxygen is 1.17 times faster-- this is our answer-- than the rate of carbon dioxide. So that's our answer. The oxygen is going to be the winner. So it's going to move faster. This is going to move a little bit faster. And it's going to get to our alien friend's nose first. So this is the power of Graham's law. It's basically telling us that, hey, if you have a small molecular weight, you're going to be able to diffuse pretty fast.