Okay, in this video I'm going to continue on my tutorial videos on thermodynamics and statistical mechanics. We're on video number four and this is going to be on the ideal gas law. It's going to be a bit back to front, I probably should have done this earlier, but I'm doing this in this order, so be it. The previous video, which the bonus has nothing to do with this, but it is the previous video nonetheless. I spoke about compression work and it went from work as a function of volume for compression of its PDV or an NKT, the log natural algorithm VI over VF. So the ideal gas law, well the ideal gas law came from a number of experiments that were formed. You had boiled law, you had challenges law and all these. I'm not really going to talk about those because the bonus, I don't think they're very, I don't think they, we really need to. What's important is the ideal gas law is empirical in that it wasn't derived from anything really, it's just an experiment, these are experimental facts and they're, they're, they're, they're, they're summarized quite well in what's called the ideal gas law. So let's write it down. The ideal gas law says that PV is equal to N or T. So let's start and look at the different parameters. We of course, we have pressure, we have volume, we have temperature. Now what are these two here? What is N and what is R? The answer is follows. If you're a chemist, you know that small N is the number of moles. Okay and a mole of molecules is average, average, average number of molecules. So if you've won all of the substance N sub A like this is average, average number. Just a constant, it's average, average number and it's six times 10 to 23 molecules. So if you have a few of the series of molecules that are packet of them and you have six times 10 to 23 of them, you have what's called one mole. So we like to talk about the number of moles, especially if you're a chemist. So small N is the number of moles. So let's say for example, you had three times 10 to the 23 molecules, you would have three moles. And R then is, I'm going to call it the molecular, so the, the molar gas constant, molar gas constant. Okay and it's, it has an empirical value. So R is equal to 8.31 Joules per mole per Kelvin. Now what's important here is temperature is measured in Kelvin. They're all in SI, pressure is measured in, and, and, and, and, and, and, and, and, and, of course, N, it's just a constant. All right. So what else can we say about this? So we have the molar, it's just, it's just a constant. So we don't need to worry about, about R. But there is something interesting here. We know that six times 10 to the 23 molecules is equal to one, one mole, R is N. N, N is the number of molecules. So it's, sorry, N is the number of moles. So capital N is the total number of molecules is, of course, going to be small N, the number of moles times Avogadro's number. All right. So let's just arrange this because we know that PV is equal to N RT. So let's see if we can somehow sub N for this N here. If we do that, we find that N, small N, the number of moles is the number of molecules divided by Avogadro's number, and that means that pressure times volume is equal to the number of molecules divided by Avogadro's number times R times T. Okay. So I'm going to write, write this in a new way. PV is equal to N times R over N A times T. Now, great. I'm going to tell you that we actually call it, although this is a constant, we call it something else. That we call it, we give it a placeholder K, and I'm sure you know what I'm talking about. We're talking about Boltzmann's constant. This is Boltzmann's constant. It has a value of 1.38, times 10 to the minus 23 joules per Kelvin. All right. So if you write it this way, we have a new ideal gas law. Well, I say new is just a new way of writing the ideal gas law. So we can have PV is equal to small N times the molar gas constant times temperature. Our PV is equal to capital N number of molecules times Boltzmann's constant times T. So now hopefully you can see why I call this the molar constant, and I call this the molecular constant. Okay. So if you're wondering which one do I use, do I use, you know, when do I use, with which of the ends do I use K and which of the ends do I use, or you know that with the molecular gas constant to use the number of moles, and with the, excuse me, with the molar gas constant to use the number of moles, and the molecular gas constant to use the number of molecules, pretty straightforward stuff. So we can summarize, or we can summarize the relationship between the molecular and molar gas constant by saying that we had R is equal to, have a gathers number times K. I like to think about it R is equal to N A times K. That's just the way I remember it from your own head. Now the ideal gas law is by good. I'm going to write it as N capital N KT. The molecular gas, or the idea gas law is great because for many years it predicted loads of things first. So units aside, the ideal gas law summarizes a number of important physical facts. So we see that for a given amount of gas, at a given temperature, doubling the pressure, squeezes the gas into exactly half as much space. So if we look, this side equals this side. So if we double the pressure, if we double the pressure, we must half the volume. That's just, that's one of the properties of the idea gas law, because we haven't changed the temperature, and these here are constants. So the only thing that can change is the volume. All right, or at a given volume, doubling the temperature causes the pressure to double. So if you hold the volume fixed, so P, V, N, KT. If you hold the volume fixed, that means these three here are fixed. So if we double the temperature, that means we must double the pressure. All right. So just actually, I think this is a better way of looking at it. If we have P, V is equal to N, KT. If we are holding the temperature fixed, that means this side here is all constant. So if we double the pressure, we must have the volume. If we double the volume, we must have the pressure, okay, in order for the equation to hold, to hold. So for nearly all the laws in physics, the ideal gas law is an approximation or like all the laws. It is an approximation, and it's never exactly true for real gas in the real world. However, it is valid in certain circumstances, and this is important. So when is the ideal gas law valid in the low density limit? So when your gas is of low density, then it is valid. So if you find, if somebody asks you a problem and there's a large density, well, then the ideal gas law will break down. And you'll find that, let's say at room temperature and at max-period pressure, the average distance between molecules in the air is roughly 10 times the size of a molecule. So the ideal gas law is accurate enough for most purposes. However, if you're talking pretty advanced physics, you're often, you are in chemistry, you're often going to be talking about high densities. And if you're talking about high densities like plasmids and so on, the ideal gas law absolutely will not hold and you must come up with something else. Okay. So that's all I've got to say about that. 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 a net. Thank you.