Microstate (statistical mechanics)

Transcription

SAL: I've done a bunch of videos where I use words like pressure and-- let me write these down-- pressure and temperature and volume. And I've done them in the chemistry and physics playlist. Especially the physics playlist, but even in the chemistry playlist, I also use words like kinetic energy. I'll just write e for energy. Or I use force and velocity. And you know, a whole bunch of other types of, I guess, properties of things, for better or for worse. And in this video what I want to do is I want to make a distinction. Because it becomes important when we start getting a little bit more precise, especially when we get more precise in thermodynamics, or, I guess, you know, the study of how heat moves around. So these properties right here, these are properties of a system. Or we could call them macrostates of a system. And these could be macrostates. So for example, let me make it clear, when I call a system, if I have some balloon like this, and it has a little tie there and, you know, maybe it has a string. This has these macrostates associated with it. There is some pressure in that balloon. Remember that's force per area. There is some temperature for that balloon. And there's some volume to the balloon, obviously. But all of these, these help us relate what's going on inside that balloon, or what that balloon is doing in kind of an every day reality. Before people even knew about what an atom was, or maybe they thought that there might be such an atom but they had never proved it, they were dealing with these macrostates. They could measure pressure, they could measure temperature, they could measure volume. Now we know that that pressure is due to things like, you have a bunch of atoms bumping around. And let's say that this is a gas-- it's a balloon- it's going to be a gas. And we know that the pressure is actually caused-- and I've done several, I think I did the same video in both the chemistry and the physics playlist. I did them a year apart, so you can see if my thinking has evolved at all. But we know that the pressure's really due by the bumps of these particles as they bump into the walls and the side of the balloon. And we have so many particles at any given point of time, some of them are bumping into the wall the balloon, and that's what's essentially keeping the balloon pushed outward, giving it its pressure and its volume. We've talked about temperature, as essentially the average kinetic energy of these-- which is a function of these particles, which could be either the molecules of gas, or if it's an ideal gas, it could be just the atoms of the gas. Maybe it's atoms of helium or neon, or something like that. And all of these things, these describe the microstates. So for example, I could describe what's going on with the balloon. I could say, hey, you know, there are-- I could just make up some numbers. The pressure is five newtons per meters squared, or some number of pascals. The units aren't what's important. In this video I really just want to make the differentiation between these two ways of describing what's going on. I could say the temperature is 300 kelvin. I could say that the volume is, I don't know, maybe it's one liter. And I've described a system, but I've described in on a macro level. Now I could get a lot more precise, especially now that we know that things like atoms and molecules exist. What I could do, is I could essentially label every one of these molecules, or let's say atoms, in the gas that's contained in the balloon. And I could say, at exactly this moment in time, I could say at time equals 0, atom 1 has-- its momentum is equal to x, and its position, in three-dimensional coordinates, is x, y, and z. And then I could say, atom number 2-- its momentum-- I'm just using rho for momentum-- it's equal to y. And its position is a, b, c. And I could list every atom in this molecule. Obviously we're dealing with a huge number of atoms, on the order of 10 to the 20 something. So it's a massive list I would have to give you, but I could literally give you the state of every atom in this balloon. And then if I did that, I would be giving you the microstates. Or I would give you a specific microstate of the balloon at this time. Now when a system-- and I'm going to introduce a word here, because this word is important, especially as we go-- is in thermodynamic equilibrium. So let me write that down. Equilibrium. We learned about equilibrium from the chemistry point of view. And that tells you, that the amount of something going into forward reaction is equivalent to the amount going in the reverse reaction. And when we talk about macrostates, thermodynamic equilibrium essentially says that the macrostate is defined. That they're not changing. If this balloon is in equilibrium, at time 1 its pressure, temperature, and volume will be these things. And if we look at it a second later, its pressure, temperature, and volume will also be these things. It's in equilibrium. None of the macrostates have changed. And actually, I'll talk about in a second, in order for these macrostates to even be defined, to be well defined, you have to be in equilibrium. I'll talk about that in a second. Now, at second number, at time equals 0, you might have this whole set of-- I went and I listed 10 to the 20th-something microstates of all the different atoms in this molecule. But then if I look at these gases a second later, I'm going to have a completely different microstate right? Because all of these guys are going to have bumped into each other, and given each other their momentum. And all sorts of crazy things could have happened in a second here, so I would have a completely different microstate. So even though we're at thermodynamic equilibrium, and our macrostate stayed the same, our microstates are changing every gazillionth of a second. They're constantly changing. And that's why, for the most part, in thermodynamic, we tend to deal with these macrostates. And actually most of thermodynamics, or at least most of what you'll learn in a first-year chemistry or physics course, it was devised or it was thought about well before people even had a sense of what was going on at the macro level. That's often a very important thing to think about. And we'll go into concepts like entropy and internal energy, and things like that. And you can rack your brain, how does it relate to atoms? And we will relate them to atoms and molecules. But it's useful to think that the people who first came up with these concepts came up with them not really being sure of what was going on at the micro level. They were just measuring everything at the macro level. Now I want to go back to this idea here, of equilibrium. Because in order for these macrostates to be defined, the system has to be in equilibrium. And let me explain what that means. If I were to take a cylinder. And we will be using this cylinder a lot, so it's good to get used to this cylinder. And it's got a piston in it. And that's just, it's kind of the roof of the cylinder can move up and down. This is the roof of the cylinder. The cylinder's bigger, but let's say this is a, kind of a roof of the cylinder. And we can move this up and down. And essentially we'll just be changing the volume of the cylinder, right? I could have drawn it this way. I could have drawn it like a cylinder. I could have drawn it like this, and then I could have drawn the piston like this. So there's some depth here that I'm not showing. We're just looking at the cylinder front on. And so, at any point in time, let's say the gas is between the cylinder and the floor of our container. You know, we have a bunch of molecules of gas here, a huge number of molecules. And let's say that we have a rock on the cylinder. We're doing this in space so everything above the piston is a vacuum. Actually just let me erase everything above. Let me just erase this stuff, just so you see. We're doing this in space and we're doing it in a vacuum. Just let me write that down. So all of this stuff up here is a vacuum, which essentially says there's nothing there. There's no pressure from here, there's no particles here, just empty space. And in order to keep this-- we know already, we've studied it multiple times, that this gas is generating, you know things are bumping into the wall, the floor of this piston all the time. They're bumping into everything, right? We know that's continuously happening. So we would apply some pressure to offset the pressure being generated by the gas. Otherwise the piston would just expand. It would just move up and the whole gas would expand. So let's just say we stick a big rock or a big weight on top of-- let me do it in a different color-- We put a big weight on top of this piston, where the force-- completely offsets the force being applied by the gas. And obviously this is some force over some area-- right, the area of the piston-- over some areas so that we could figure out its pressure. And that pressure will completely offset the pressure of the gas. But the pressure of the gas, just as a reminder, is going in every direction. The pressure on this plate is the same as the pressure on that side, or on that side, or on the bottom of the container that we're dealing with. Now let's say that we were to just evaporate this-- well let's not say that we evaporate the rock. Let's say that we just evaporate half of the rock immediately. So all of a sudden our weight that's being pushed down, or the force that's being pushed down just goes to half immediately. Let me draw that. So I have-- maybe I would be better off just cut and pasting this right here. So if I copy and paste it. So now I'm going to evaporate half of that rock magically. So let me take my eraser tool. And I just evaporate half of it. And now what's going to happen? Well, this piston is now applying half the force. It can't offset the pressure due to this gas. So this whole thing is going to be pushed upwards. But I did it so fast. I did it so fast. And you could try it. I mean, this would be truth of a lot of things. If you had a weight hanging from a spring, and you would just remove half the weight, it wouldn't just go very, you know, nice and smoothly to another state. What's going to happen is-- and let me see if I can do this using their cut and paste tool-- it'll essentially, right when I evaporate half of it, the gas is going to expand a bunch, and then this weight is going to come back down, it's going to spring and go down. So let me do it again. It's going to expand, because that gas is going to push up, and then it's going to come back down. And then, it's just going to oscillate a little bit. And then eventually it'll come back to some stable and maybe it'll go back. It'll look, like right about there. And let me fill this in. This shouldn't be white, it should be black. Let me put some walls on it, on the container. So if we wait long enough, eventually we'll get to another equilibrium state, where this thing, the piston on top isn't, or the ceiling isn't moving anymore. And now the gas has filled this container. Now, at this point in time we were in equilibrium. The pressure throughout the gas was the same. The temperature throughout the gas was the same. The volume was in a stable situation. It wasn't changing from second to second. So because of that, our macrostates were well defined. Now, when we wait long enough, this thing will get to some stability where this thing stops moving. When this thing stops moving our volume stops changing. And hopefully our pressure will start to become uniform throughout the container. And our temperature will become uniform. And we'll now be a higher volume or lower pressure, probably a lower temperature if we assume that there's no other heat being added to the system. And then we'll be well defined again. So we could say what the pressure, and the volume, and the temperature's going to be. But what about right when I removed this rock? And this thing flew up and it oscillated, and for a while the pressure at the top was lower than the pressure down here. Maybe the temperature at the top was lower than the temperature down here. The whole thing was in a state of flux. It was not an equilibrium. And at that point, when we're-- let me let me draw that really-- so you know, when we were in that state, where everything was just crazy, right when we evaporated the rock. You know, we have a little rock up here. Everything is going up and down. Maybe the pressure up here was lower than the pressure down here. Everything did not have a chance to reach an equilibrium. At this state-- and this is important, especially as we go into talking about things like reversible reactions, and reversible processes, and quasi-static processes. At this point in the reaction, when we just did this, none of these macrostates were well defined. You couldn't tell me what the volume of this system is, because it's changing for every second to second, or microsecond to microsecond, it's fluctuating. You couldn't tell me what the pressure of the system is, because it's changing every second. You couldn't tell me what the temperature is. Maybe the temperature could be something there. It could be something there. All sorts of crazy things are happening. So when the system is in a state of flux, your macrostates are not well defined. And I really want to hit that point home. So me just draw that in a diagram. Let me draw that in a PV diagram. And we're going to use these fairly heavily. So on my y-axis I'm going to put pressure. In my x-axis I'm going to put volume. So our initial state here, when we had the rock sitting on top of the ceiling, this movable ceiling or this piston, maybe we had some well-defined pressure and volume. So my y, this is pressure and this is volume. So this is where we started off. So it was well defined. This is state 1. Let me label it right there. Now when we evaporated half the rock, we eventually waited long enough, and this got to an equilibrium. We got to state 2, and our pressure volume and out temperature was well defined. And I'll just put it on this pressure volume. So maybe this is state 2. We got down here. And just as an aside, I could maybe put temperature as an extra dimension, but temperature is completely determined by pressure and volume, especially if we're dealing with an ideal gas. Remember, and we did this in multiple videos, you have PV is equal to nRT. These are constants. The number of moles isn't changing. This is the universal gas constant, not changing. So if you know P and V you know T. So that's the only two things we have to plot. But I'll talk a lot more about that in future videos. But the important thing to realize is, I started off at this state, where pressure and volume were well defined. I finished in this state, where pressure and volume were well defined. But how did I get there? And because this reaction I did, all of a sudden it happened super fast, and it was essentially thrown out of equilibrium. I don't know how I got here. The pressure and volume were not well defined from going from that state to this state. Pressure, volume, and temperature are only well defined if every intermediate step is still almost in equilibrium. And we'll talk a lot more about that in the next video. But I want to really make this point home. It would be nice if we could draw some path. We could say, we moved from some pressure and volume to some other pressure and volume, and we moved along a well-defined path. But we cannot say that. Because when we went from there there, our definitions just disappeared for pressure and volume. We cannot define those macrostates in these intermediate non-equilibrium states. Now, just as a little aside, we could have defined the microstates. The microstates never change. At any given snapshot in time, I could have listed every particle that's in this thing. And I could have given you its kinetic energy. I could have given you its position. I could have given you its momentum. And there's no reason why I couldn't have done that. So I could have actually made a plot of one particular particle. And I could have said what its kinetic energy, and over a course of time, is at any given moment in time. And this is really important. So microstates are always well defined. The microstate is what's exactly happening to the atom in terms of its force and its velocity and its momentum. While macrostates are only defined, I should say well defined, when the system-- in this case it's the balloon, in this case it's this piston on top of this cylinder, this movable ceiling-- the macrostates are only well defined when the system is in equilibrium, or when you can essentially say, the pressure is x, the pressure is the same throughout. Or the volume isn't changing from moment to moment. Or the temperature is the same thing throughout. Anyway, I'll leave you there and we'll talk more about why I went through all this pain in the next video.