Fick's law of diffusion
So I have a picture for you of Adolf Fick. And this is probably the second most well-known Adolf in history, but this Adolf was well known for his science. He actually came up with some fantastic laws that we use in all sorts of different branches in science today. And we're going to talk about one of his laws right now. So I drew for you a little box, and I thought one of the most fun ways to think about some of these laws that Mr. Fick came up with was to do a little game. So I'm going to give you a challenge. And the challenge is that, let's say that you're a person standing right here, maybe standing behind this box. And the part of the box that's facing you, that's nearest you, is that blue wall that I kind of shaded in. This blue wall is the back wall of the box. And on the front wall, I'm actually going to put in some little molecules. Let's say there are some molecules I'll put in, I don't know, let's say three or four molecules here. And the challenge is this, if a molecule gets from the front of the box-- I'm going to call it one, this first side is side one-- if it gets from side one, which is here, over to side two, which is the back wall, if it gets from side one to side two of the box, then you get $5 for each particle that makes it over. So to put that into words, we're interested in the amount of particles that move over some period of time. And that period of time can be one hour or 10 hours, whatever period of time we want. But I'm going to-- just for argument's sake, let's just say that we're going to do this for one hour. So let's say I do it for an hour and these molecules started moving around, they're migrating around, because, of course, they're bouncing, these molecules don't stay stationary. And I come back, and it turns out that only one molecule ever eventually made it over to this side. So I say, hey, good job, you get $5 because I promised that to you. And so of course you get your $5, and you're happy and smiling, right? But I'm feeling in a generous mood, and I say, you know what, let's start this experiment over. Let's just start over, and this time I'm actually going to give you a chance to tweak the experiment. You're going to actually get a chance to modify the experiment. And you can do whatever you want to try to maximize your profits. So think about this, you want to try to maximize this right here. And how do you do that? How do you maximize the amount of particles that make it to that blue wall over some period of time? And I'm going to write your good ideas down here, so start brainstorming some good ideas for how you might want to tweak the game, or play the game, to maximize your profits. Well, if you're thinking about this, you might think, well, maybe the first obvious thing is why do you make this so darn far? Why not make it a little bit closer? So let's get rid of this back wall and scooch it nearer, so that the molecules don't have to go so far. And that's a pretty good idea it seems to me, right? So let's just make this a little bit smaller. That's your first idea, and I would say, that's a brilliant strategy. Now let's just make it half the size, so it's less thick, and these molecules don't have to go nearly as far. And actually, let me maintain the blue wall so we can keep seeing properly what this should look like. And this is going to be dashed back here, and like that, and of course the blue wall is going to look like that. So now you just basically make it come closer and so the molecules don't have to go nearly as far. So idea one is less thick wall. What's another idea? Well you remember from Graham's law, we learned that these molecules, the big ones, actually don't move. Their diffusion rate is not as quick, and that it's the smaller molecules that actually have a faster diffusion rate. So if I'm waiting at that back wall to see how many molecules can get over, I want tiny little molecules to make their way over because they're going to have a faster diffusion rate. When I say tiny, I really mean smaller molecular weight. So small molecular weight molecules, that's the second idea. Change up the molecules, make the molecular weight smaller, and as Graham's law tells us, they'll move faster. So what's a third idea? Well, maybe you could just have more of them. Maybe in this number one plane, which is the leftmost plane of this box, why don't you just jam it full of more little molecules? If you have more molecules moving around-- that's another way of saying just increasing the pressure, that's increasing the pressure at that position one-- then you're going to have a better chance of having molecules move across. So increased pressure at one. And what's a fourth idea? I'm just going to make a little bit of space. What's a fourth idea that we can maybe put on here? Well, if you're thinking really outside the box, and this is actually literally outside the box, then you might think, well, why not just expand this entire thing? Make it a larger area. What about that? Why not just make a larger area. So that's the last idea. Maybe you can actually just make it a bigger surface area. So maybe something like this, you could expand it in all directions. And maybe you can do something like that. I'm going to have to make sure I draw it carefully so don't confuse you, but basically something like this, where you now have the same thickness, I'm not changing the thickness, but I'm basically-- you're going to make this wall bigger. Actually, I screwed that up a little bit. Let me just fix that. So this is what my new back wall is going to look like and maybe I should do it in blue just to make sure we stay consistent with the colors. But of course, this is just going to extend out like that. And this is my new back wall. Right, this whole thing is my back wall. And so if I expand the area, now I have, of course, much more chance of getting some of these molecules back there. And the partial pressure is going to stay the same, so if I expand the area, I still have more molecules on this initial leftmost face. And so the pressure is going to stay the same. This is the P1 that we just got through talking about. But because I have more area there's more chance that somewhere along this entire area a molecule will make its way across the thickness and hit that back wall. So something like that. And let me fill this in. So this is a fourth idea. Let me write that down as the fourth idea, is increase the area. So these are four good ideas, four good ideas for how you can maximize the amount of particles over time and hopefully make as much money as you can. Probably much more than $5. So this is exactly what Fick's law talks about. It talks about the idea of amounts of particles moving over time. So let me write out Fick's law, and this is actually how you most commonly will come across it, although there are some variations on it. It's going to look something like this. And I'm actually going to try to color code it to go along with the ideas that we already presented. So we said there's some things you can do with pressure, some things you might do with the surface area, and also, remember we had that diffusion constant. And you divide all this stuff by the thickness of the wall. So it's very colorful, but this is Fick's law as you usually see it. There are some other variations I'll talk about. So to go through this piece by piece, this is V with a dot over it. This is the rate of particles moving. And when I say rate, you know that that means that there's some time component. So this gets to what the challenge was. We said, how many particles can you get to go to that blue back wall over some period of time. And sometimes when we talk about particles we can think of them as giving that in terms of an amount. You might think of that as like a moles or some numerical value, or the volume of a gas that's moving across. So that's why sometimes you'll see it as V, to refer to volume. On the other side, this bit makes perfect sense. If you have more molecules, that's going to cause more pressure, we said earlier. And if there's a bigger pressure difference between what's on the first side versus what's on the second side-- remember the second side is the back wall-- then of course that's going to mean that more of the molecules are going to move over. So this is a bigger difference. So sometimes you'll see this as delta P, and delta just means difference. This A, we said, refers to just surface area. Of course if you have a greater surface area, that's going to allow for more of the molecules to get across. This D, that's an interesting one. This is diffusion constant. And remember when we think about diffusion constant, there are two laws that might jump into your head. In fact, the first one was Henry's law, and you remember we talked about solubility, in terms of the amount of molecules that go from, for example, air to liquid. This is Henry's law that told us about that. And of course if something is very soluble, then maybe that would be an increased P1, going back to the idea of pressure at the first wall. And then you have to divide by the molecular weight. The square root, in fact, of the molecular weight. So that's the idea we had, and this actually comes from Graham's law. So whenever I talk about the diffusion constant, remember there are two laws that are in play here, Henry's law and Graham's law, that are coming together to offer us some information about the diffusion constant. And that's actually why we said, well, if you have a small molecular weight molecule, maybe that'll help you out, because it's in the denominator it's going to cause the rate of particles moving across to go up. And finally, this T, this is thickness. This is the thickness of the wall. And this is totally intuitive. If you have a thick wall, it's going to be harder for molecules to make it across very quickly. So without even knowing it, you kind of derived Fick's law all by yourself just kind of using intuition, and that's kind of the best way to learn this stuff. And sometimes, as I mentioned, you might see this formula written differently. In fact, let me actually just rearrange this formula in a different way. I'm just going to sketch out how you might also see it, which is that sometimes you see area on this side of the equation. Of course, that's just rearranging it, right, dividing both sides by area. And then you might see the P actually up here, like P1 and then minus P2, something like this. And in the denominator over here, you'll see the T. So you'll get this, and then very separately you'll see times D. So this might not look so different, but what happens is that then people lump things, and that's when things get kind of tricky. They'll say well, let's lump this together, and let's lump this together. And they'll call this flux. And this second part they'll call gradient. So you may see Fick's law written this way, where it says flux equals the gradient times diffusion constant. Because of course the diffusion constant hasn't changed, it's the same thing, it just carries on down. And if you see it that way, let me just give you an example of what these things mean. Let's start with flux, which is basically the net rate of particles moving through an area, moving through some area. And you can kind of follow through the equation and that makes sense. And here the important part is the idea of a net rate. It's not total rate, but you're actually looking for what is the net gain or net rate that you're seeing. And this gradient over here, this is just change in pressure over-- divided by or over-- some distance, or over a distance. And occasionally you'll see pressure written out as particles in a volume, which is kind of the same thing. Conceptually it's the same thing. Particles in a volume, of course, are going to exert some pressure, so sometimes you'll even see it written this way. So I guess the point is that you'll see these different terms, and I just want you to be familiar with them. But at least now you know that the most common way you'll see Fick's law is written out this way, and that it's completely intuitive. In fact if you had to come up with it, given some time you'd probably come up with Fick's law yourself.