Temperature, kinetic theory, and the ideal gas law
Thermodynamics part 1: Molecular theory of gases
After all the work we've been doing with fluids, you probably have a pretty good sense of what pressure is. Now let's think a little bit about what it really means, especially when we think about it in terms of a gas in a volume. Remember, what was the difference between a gas and a liquid? They're both fluids, they both take the shape of their containers, but a gas is compressible, while a liquid is incompressible. Let's start focusing on gases. Let's say I have a container, and I have a bunch of gas in it. What is a gas made of? It's just made up of a whole bunch of the molecules of the gas itself, and I'll draw each of the molecules with a little dot-- it's just going to have a bunch of molecules in it. There's many, many, many more than what I've drawn, but that's indicative, and they'll all be going in random directions-- this one might be going really fast in that direction, and that one might be going a little bit slower in that direction. They all have their own little velocity vectors, and they're always constantly bumping into each other, and bumping into the sides of the container, and ricocheting here and there and changing velocity. In general, especially at this level of physics, we assume that this is an ideal gas, that all of the bumps that occur, there's no loss of energy. Or essentially that they're all elastic bumps between the different molecules. There's no loss of momentum. Let's keep that in mind, and everything you're going to see in high school and on the AP test is going to deal with ideal gases. Let's think about what pressure means in this context. A lot of what we think about pressure is something pushing on an area. If we think about pressure here-- let's pick an arbitrary area. Let's take this side. Let's take this surface of its container. Where's the pressure going to be generated onto this surface? It's going to be generated by just the millions and billions and trillions of little bumps every time-- let me draw a side view. If this is the side view of the container, that same side, every second there's always these little molecules of gas moving around. If we pick an arbitrary period of time, they're always ricocheting off of the side. We're looking at time over a super-small fraction of time. And over that period of time, this one might end up here, this one maybe bumped into it right after it ricocheted and came here, this one changes momentum and goes like that. This one might have already been going in that direction, and that one might ricochet. But what's happening is, at any given moment, since there's so many molecules, there's always going to be some molecules that are bumping into the side of the wall. When they bump, they have a change in momentum. All force is change in momentum over time. What I'm saying is that in any interval of time, over any period or any change in time, there's just going to be a bunch of particles that are changing their momentum on the side of this wall. That is going to generate force, and so if we think about how many on average-- because it's hard to keep track of each particle individually, and when we did kinematics and stuff, we'd keep track of the individual object at play. But when we're dealing with gases and things on a macro level, you can't keep track of any individual one, unless you have some kind of unbelievable supercomputer. We can say, on average, this many particles are changing momentum on this wall in this amount of time. And so the force exerted on this wall or this surface is going to be x. If we know what that force is, and we you know the area of the wall, we can figure out pressure, because pressure is equal to force divided by area. What does this help us with? I wanted to give you that intuition first, and now I'm just going to give you the one formula that you really just need to know in thermodynamics. And then as we go into the next few videos, I'll prove to you why it works, and hopefully give you more of an intuition. Now you understand, hopefully, what pressure means in the context of a gas in a container. With that out of the way, let me give you a formula. I hope by the end of this video you have the intuition for why this formula works. In general, if I have an ideal gas in a container, the pressure exerted on the gas-- on the side of the container, or actually even at any point within the gas, because it will all become homogeneous at some point-- and we'll talk about entropy in future videos-- but the pressure in the container and on its surface, times the volume of the container, is equal to some constant. We'll see in future videos that that constant is actually proportional to the average kinetic energy of the molecules bouncing around. That should make sense to you. If the molecules were moving around a lot faster, then you would have more kinetic energy, and then they would be changing momentum on the sides of the surface a lot more, so you would have more pressure. Let's see if we can get a little bit more intuition onto why pressure times volume is a constant. Let's say I have a container now, and it's got a bunch of molecules of gas in it. Just like I showed you in that last bit right before I erased, these are bouncing off of the sides at a certain rate. Each of the molecules might have a different kinetic energy-- it's always changing, because they're always transferring momentum to each other. But on average, they all have a given kinetic energy, they keep bumping at a certain rate into the wall, and that determines the pressure. What happens if I were able to squeeze the box, and if I were able to decrease the volume of the box? I just take that same box with the same number of molecules in it, but I squeeze. I make the volume of the box smaller-- what's going to happen? I have the same number of molecules in there, with the same kinetic energy, and on average, they're moving with the same velocities. So now what's going to happen? They're going to be hitting the sides more often-- at the same time here that this particle went bam, bam, now it could go bam, bam, bam. They're going to be hitting the sides more often, so you're going to have more changes in momentum, and so you're actually going to have each particle exert more force on each surface. Because it's going to be hitting them more often in a given amount of time. The surfaces themselves are smaller. You have more force on a surface, and on a smaller surface, you're going to have higher pressure. Hopefully, that gives you an intuition that if I had some amount of pressure in this situation-- if I squeeze the volume, the pressure increases. Another intuition-- if I have a balloon, what blows up a balloon? It's the internal air pressure of the helium, or your own exhales that you put into the balloon. The more and more you try to squeeze a balloon-- if you squeeze it from all directions, it gets harder and harder to do it, and that's because the pressure within the balloon increases as you decrease the volume. If volume goes down, pressure goes up, and that makes sense. That follows that when they multiply each other, you have to have a constant. Let's take the same example again, and what happens if you make the volume bigger? Let's say I have-- it's huge like that, and I should have done it more proportionally, but I think you get the idea. You have the same number of particles, and if I had a particle here, in some period of time it could have gone bam, bam, bam-- it could have hit the walls twice. Now, in this situation, with larger walls, it might just go bam, and in that same amount of time, it will maybe get here and won't even hit the other wall. The particles, on average, are going to be colliding with the wall less often, and the walls are going to have a larger area, as well. So in this case, when our volume goes up, the average pressure or the pressure in the container goes down. Hopefully, that gives you a little intuition, and so you'll never forget that pressure times volume is constant. And then we can use that to do some pretty common problems, which I'll do in the next video. I'm about to run out of time. See you soon.
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