Okay, in this video I'm going to continue on with my tutorial videos on thermodynamics and statistical mechanics. This is video 18 and I'm going to discuss the multiplicity of an ideal monotonic gas, for a monotomic means a single particle. And we'll see that the multiplicity for a wooden dimension is the volume and position space, multiplied by the volume and momentum space, divided by actually H barric cubed or H cubed. The fact of 2 pi is not very important because the multiplicity is so large. Now in the previous few videos I did a bit of a, I went down to the side road and I discussed the Taylor and McLaren series. I discussed the binomial theorem and I also discussed sterling approximation. So we want to come up with the multiplicity of an ideal gas with one particle. So what does multiplicity mean? It means the number of different states the particle can be in. So what characterizes a state for a single, let's say an oxygen molecule? Well, its position is important. If its position changes so does, you know, it becomes a new state. Its energy is also important. But if we talk about energy, energy is not the most fundamental quantity because we usually like to analyze energy in terms of velocity. Or in a similar fashion we could use momentum. Where we know that the energy of a particle, non-renovistic particle, is p squared divided by twice the mass of the particle, for p is the momentum of the particle. So it should make sense that in general, what else can we think about? Well, there is no potential energy. We'll assume that there's no potential energy. So that doesn't affect it. There are not all the particles which could interact with it. So it's an uninterracted particle. What else could there possibly be to change the state of the particle? So even if this isn't fully correct, let's suggest that the only factors which will change the state of the particle are its momentum and its position. So how do we find out the number of possible states? Well, we need to talk about the number of possible positions it has and the number of possible ways its momentum can be. So if that's the case, very generally, we can say that the multiplicity of an ideal gas with one molecule in one dimension, most of which is proportional to the volume and position space. Because, think about it. If you have one molecule and it puts it in a box and you close the box, well, it has a certain number of states. In other words, it would be the volume of your box divided by the volume of your particle. And no, if you have to see, this is approximately the number of position states that are available. But if you double the size of your box, you're going to double the number of possible states. Or even if that isn't fully correct, let's just suggest that it's correct because it sounds quite reasonable. So we're going to say that the multiplicities, these proportions, the volume which you're particles inside in position space. And next, we need to talk about momentum. So instead of characterizing, we know that it's a E is p squared over 2m. So that means the energy of your particle is equal to p of x squared plus p of y squared plus p of z squared over 2m. Because we're talking about a single particle. All right. So let's rearrange this. We're going to get 2mE. Let's say, actually, by the way, this is, let's talk about an average energy of the particle. To be specific, and we know later on, we can say that the average energy of the particle, multiplied by the number of particles, is equal to the total energy U, or the total average energy. But we aren't an actual fact, say that because the fluctuations are so small, we say that it's actually the total energy. So let's just deal with the average energy of a single molecule at the moment. So that's going to be equal to all of these to be square, piece of x squared plus piece of y squared plus piece of z squared. But look, that really is just another place older, or squared. And if we look at that, that's just the equation of a sphere. Where the radius of the sphere is the square root of piece of x squared plus piece of y squared plus piece of z squared. So if we want to look at momentum space, if we look at momentum space, the energy of our particle, the energy of our particle, and momentum can be analyzed, by looking at the number of momentum states in a volume or surface area in momentum space, whose radius is the square root of 2mE. So think about this. There's in momentum space, there's piece of x, there's piece of y, there's piece of z. So we're saying that the momentum and energy are related. So if we plot a sphere whose radius is square root of 2mE, or square root of 2mE bar, we get some space. Some sort of a sphere. And the number of momentum states inside that sphere gives the number of possible momentum states, or the number of possible momentum values inside that sphere gives the momentum. Now, I understand that we're not talking about a volume, really. We're talking about, we're here after saying, it should be the volume in momentum space. But here I'm really talking about a surface area. So just bear with me for a second, it's not going to be very eager. So I'm going to leave the V sub P, where a public should be surface area sub P. So, right, there we have so far. So what we're saying is that the multiplicity of an ideal gas, with however many molecules in 3D, should be proportional to the volume in which the molecules are in, and the volume in momentum space, even though it is really a surface area. So how do we calculate these volumes? Well, particles are very small. Molecules are very small. So it's best to talk about these, talk about lengths and all that sort of thing using the, using the broy relationship. Alright, so if that's the case, we say that each particle has a wavelength. Okay, so here, let's say in one dimension, this is the length of my box. Sorry, there's the length of my box in one dimension. Well, my particle can occupy a lot of different states in here, where delta x, the position that they would say the length in the box, which has to be traveled before you become into a new state, is delta x. So the total number of states, of course, is the length divided by the, you know, the distance you have to move in order to change state. So this gives the total number of states in one dimension in position space. So let's do something very similar and we talk about momentum space. We can talk about a length in momentum space. Okay, let's say if it's just in one dimension, in one dimension, say it's piece of x, okay, let's say it's piece of x like that. And, you know, the, that means the radius is square root 2mU. The length of this would be 2mU. But once again, we talk about having some form of a wavelength. So this time, what we're going to talk about, the length in position space, divided by the change in momentum or delta p, or the, you know, the amount by which the momentum must change before you become a new state. So let's put both of those together and go back up to our initial suggested guess for the multiplicity and we're going to get the length in position space divided by delta x times the length in momentum space, length in momentum space, divided by delta p. But delta x times delta p, according to the uncertainty principle, is approximately its hH bar, or its h over 2pi. So it's something like that. So I'm just going to say, but look, this 2pi, you know, it's quite small in terms of the numbers we're using. So in this further moment, we're going to call it h. So that means we've L times L sub p divided by h. Okay? That's in one dimension, of course. So if you want to actually use this, if you want to scale this up into three dimensions, as I've actually said here, then we need to get rid of all these L's and make it, we'll say v, we have vp divided by h cubed, like that. So that means that the number of states for a monotonic ideal gas in three dimensions is proportional to the volume of the, the, in position space, times the volume of momentum space divided by h cubed. Okay? And in actual fact, I'm just going to say it is in fact, it's equal to it. All right? So that's all I've got to say about that. In a future video, I'm going to discuss the multiplicity of an ideal gas with multiple particles. So thanks for watching. Please pass down to your friends, subscribe to my channel. And if you're in a good mood, you might also click on an ad. Thank you.