Let's, return to the law of corresponding states now and look at it in a little bit more detail. So, when last we, paused to have some fun with the different phases of dry ice, carbon dioxide. I showed you this, table of critical parameters, temperature, pressure, molar volume for a variety of gases. And noted that although the individual parameters, the critical temperature, or the critical pressure or the critical molar volume, was really very substance dependent. an enormous range from 5 Kelvin all the way up to 304 Kelvin for the gases on this slide. Nevertheless, when you instead take a property like the compressibility measured at the critical point, you get near constant behavior. And I rationalized why the value might be near where it is based on the Van der Waals expressions for these critical properties. So, there's this correspondence between different real gases independent to the equation of state. So, let me actually define then, a reduced equation of state, a so-called reduced equation of state. So, this is just the Van der Waals equation of state, again showing what we derived the relationship to be. Between Van der Waals parameters and critical parameters. So, if now I rewrite the Van der Waals equation of state, changing its variables, so instead of using pressure, I'm going to use pressure divided by critical pressure. And I'll use molar volume divided by critical molar volume. These are, of course, measurable quantities, they're just numbers, the critical pressure and the critical molar volume. Once I have measured them and I've written them down in table somewhere, I'll just be dividing by some constant value. But I'm going to give those variables a name, I'm going to call it the reduced pressure, the reduced molar volume, and there's also a reduced temperature. So, you can think of this as being the quantity has a value of 1, these reduced quantities have values of 1 at the critical point. And they'll either be above 1 or below 1, if they exceed or are less than their respective critical values. So, when I do that, when I rewrite the equation of state. Not using a and b because by inserting, by doing these divisions by the critical parameters, which involve a and b. All of the a and b terms drop out. Entirely equivalently to the derivation in the last slide. So, you end up with this universal equation for all gases, that the reduced pressure, plus 3 divided by the reduced molar volume squared, times the quantity reduced molar volume, minus the 3rd. Is equal to 8, 3rd, times the reduced temperature. So, this is an example of the Law of Corresponding States. It says that all gases will have the same properties if compared at corresponding conditions, where corresponding conditions really means relative to their critical conditions. They'll all have different critical conditions, but when you look their behavior relative to their critical conditions, they all behave the same. So, let's take a loo at another property which shows corresponding state behavior. And in particular, let's look at the compressability factor Z, and in this instance, you'll recall the compressability factor is defined as pressure times molar volume, divided by, the universal gas constant times temperature. And I want to take a moment for an aside here. Compressibility factor is really the best thing to call this quantity Z. Occasionally, I've been a bit colloquial and I've referred to it as just compressibility, which is, in some sense a little counter-intuitive. If you think about the values the compressibility factor takes on. If it's a number greater than 1, which is to say that the product of pressure in molar volume is larger than it would be for an ideal gas. the way to think about that is given the pressure, perhaps the molar volume is larger than it would be for an ideal gas, which is to say It's a bit hard to compress the real gas compared to the ideal gas. So, there's this unusual feature in a way that the compressibility factor is larger when the actual compressibility, if you want to think of that as ability to be compressed, is smaller. But, okay, we'll ignore that paradox for a moment, and I'll try to be a little more clear and say compressibility factor here. But if we want to think of a universal compressibility factor within the context of the Van der Waal's equation of state. If I replace pressure using the Van der Waal's equation of state with its expression in terms of v bar and t. And I work at the critical temperatures, for instance, and I substitute in all the a's and b's with their corresponding relationships that are defined for the Van der Waal's parameters in relation to the critical parameters, pressure temperature and molar volume. Then, with a lot of algebra, and I won't go through all the algebra, it's a little tedious. But if, if you wanted to work backwards. You could actually plug in for these reduced values. What they are by definition, and then replace the critical values with the dependents on the Van der Waals constants. And find that you would ultimately walk your way back to the Van der Waals equation of state value for pressure plugged into compressibility factor. But I'll, I'll leave that for the people who really want to do that algebra. But, the take home message is, that the compressibility factor becomes expressed exclusively in terms of two reduced variables. So, here are the reduced molar volume and the reduced temperature. And, of course, I could, continue to, manipulate this to have it in terms of any two variables. But what it means is, I ought to be able to plot the compressibility factor against any two reduced variables. So, in this particular instance, I'm going to have the reduced pressure on the x axis. And then a series of reduced isotherms, that is constant values of reduced temperature ranging from 2 to 1. So, 1 of course would be actually at the critical temperature. And although the symbols are probably a little bit small that, to be completely clear. These different symbols filled circles, open circles, triangles, half filled, and so on, all correspond to different gases. But when we plot them using their reduced temperatures, and reduced pressures, they all fall on equivalent reduced isotherms. And so if I want to know, say, the compressibility of any gas when it is at its critical temperature, that would be tr equal 1, and at its critical pressure, that would be Pr equal 1, which would be right around here, I can just read it off. I don't need to know the nature of the gas. They all ought to behave about the same. They all ought to have a compressibility factor a little in excess of 0.2. So, again, that's a, a, an example of corresponding conditions leading to equivalent behavior for all gases. So, I'm going to let you take a moment to take a closer look at that reduced compressability graph. And maybe gain some appreciation for certain key points on it. Assess yourself, and then we'll return. Alright well I just want to drive home the key point one more time, associated with this particular piece of lecture video, and that is the corresponding states. They correspond when they are considered at conditions relative to their critical condition. So, let's just look at a, a couple specifics instead of trying to plot 25 different gases all on one plot. So, shown here is the behavior of Ethane gas at 500 Kelvin which is a reduced temperature of 1.6375. All right? So, it is 1.6375 times higher in temperature, than the critical temperature. So, could work out in a little calculator if you like, what the critical temperature must be by dividing 500 by that. But we won't worry about that for the moment. I'll just show you that the experimental data, those are the open circles. And also illustrate how two equations of state are doing here. So, the Van der Waals equation, has the right shape, but it dips down a little too much in the compressibility. That's what's being plotted on the left. And we're plotting that against molar volume. The Redlich–Kwong equation of state is also shown. And it seems to do extremely well actually for ethene over this temperature range. Now, let me show you a different gas, this is argon. And this is not argon at 500 Kelvin as ethane was. This is argon at 247 Kelvin, so only half the temperature. It's quite a lot cooler, but you would burn yourself rather badly if you stuck your hand into a 500 Kelvin gas whereas 247 Kelvin, that might feel good on a hot summer day. And if we again look at the experimental data, and the behavior of the equation's estate, they show sort of similar behavior. Again the Redlich-Kwong is a somewhat better equation of state than the Van der Waals. But the key point I want to make is Argon's critical temperature is very different than Ethane's critical temperature. When I divide 500 by ethane's critical temperature, I get 1.6375. When I divide 247 by argon's critical temperature, I get 1.6363, very close to 1.6375. That is, each of these two plots is for the same reduced temperature, even though it's for a completely different actual temperature. So, if I do indeed compare them at the same critical temperature, and in this case I will do 1.64, if I round this to only two digits they're right at the exact same temperature. That's the beauty of rounding, I guess. I'm going to plot, now I'll actually plot reduced pressure against compressibility just to have something to illustrate. And you see that ethane and argon behave exactly equivalently, this is a good example of the Law of Corresponding States. But, the take home message was, they're not at the same temperature, they're at the same reduce temperature. Alright. Well we have, explored several equations of state now. We've mostly focused on the Van der Waals. Although we've looked at Redlich-Kwong and Pang Robinson. In the next lecture, I want to take a look at yet another equation of state. And one that has properties that make it especially useful particularly appropriate within the room a statistical mechanics and that is the the Virial equation of state that will come next