 All right, so let's explore this generalized version of the Van der Waals equation a little more now that it's written in terms of these reduced pressure, reduced molar volume, and reduced temperature variables. See if we can understand how that could possibly be true about real gases that we no longer need to know their Van der Waals coefficients, A and B, in order to make predictions about their properties. And also this statement that the compressibility factor is equal to three-eighths, according to the Van der Waals equation, for any gas at its own critical point where the pressure, molar volume and temperature are equal to the critical properties or the reduced pressure, reduced molar volume, reduced temperature are all equal to one. So to see how that works, first let's pull up a graph of the PV properties for a handful of gases. So here I'm not plotting pressure versus volume, I'm plotting the compressibility factor as a function of the pressure. So the compressibility factor, remember that will be equal to one. When the gas is behaving perfectly ideally, if intermolecular attractions dominate, Z will be less than one. If the finite molecular volume dominates as it does at high pressures, then Z will be greater than one. So as you'd expect, these graphs look different for all sorts of different gases. So gases H2, N2, O2, methane, and CO2 all behave differently on this graph. They have varying degrees of non-ideality, again dominated perhaps by intermolecular interactions for the more strongly interacting gases like CO2, or dominated more by finite molecular volume for the more weakly interacting gases, especially at high pressure. But notice that as we go to low pressure, all these gases approach ideal behavior. So regardless of whether it's a gas dominated up here by finite volumes or by intermolecular interactions, at low enough pressures, they approach ideal and then they'll behave ideally at low pressures. That's a little bit like Tolstoy's famous quote that every happy family is the same, but all unhappy families are unhappy in their own unique way. The ideal gas behaves the same. Once it becomes non-ideal, it behaves non-ideal in its own sort of way. But that doesn't seem like it's what this generalized Van der Waals equation is predicting. So first let's take a closer look at the critical point. So if I replace this graph with one showing the properties, again compressibility factor as a function of pressure, but specifically not at 300 Kelvin anymore, but plotting each gas at its own critical temperature. So now we see the curves are still different, but they have begun to look a little bit more similar. So h2 here, the open squares, the compressibility factor drops with increasing pressure until it goes through a minimum and then it starts increasing again after it gets to this point. Each one of the gases has the same sort of behavior. It drops for a while and then begins increasing after that. So the curves have a similar shape now. The reason they're not exactly identical to each other is because I've plotted them as a function of the actual pressure rather than the reduced pressure. So if I change the graph one more time and now plot for each gas, not compressibility factor as a function of absolute pressure in atmospheres, but it's reduced pressure, pressure relative to the critical point, now you can see that they're all behaving in fact remarkably similar. They all go through the same sort of drop, go through a minimum, very close to the critical pressure and begin increasing after that and we'll eventually cross above z equals 1. So this is again, actually this is a typo that shouldn't say 300 Kelvin, that one should say critical temperature. When we're at the critical temperature all these gases behave the same when plotted as a function of their reduced pressure. So let's do this not just at the critical temperature, in fact while we're on the critical temperature let me point out that the critical compressibility factor when I'm at the critical temperature somewhere on this isotherm and at the critical pressure if I read up the compressibility factor is indeed very close to 0.375 or 0.3 or something like that. So all the gases have the same critical compressibility factor or very close to it at the critical temperature and at the critical reduced pressure. So now if I see how these gases behave not just at the critical temperature but at other temperatures as well that'll be the next graph that we pull up. So now the same data we saw before at the critical temperature is here at a reduced temperature of 1.2 so 20% above the critical temperature that all of the gases are plotted on the points that fall near this line right here at 50% above the critical temperature they behave like this twice the critical temperature they behave like this. So you can see they're not all truly identical which one is that? CO2 is a little bit different than the rest of these so but they're not actually truly identical to one another but the general idea behind this generalized equation of state is seems to be holding pretty well that all gases do behave fairly similarly if not exactly the same when thought of in terms of their reduced pressure and their reduced temperature and their reduced molar volume. So that is a fairly remarkable statement. So now we can move on to this next graph which is what we would use to make use of this idea and say since we don't need to know the A and B quantities for a gas we can plot these curves here's the curve for compressibility as a function of reduced pressure at the critical temperature 10% above 20% above and so on so it doesn't matter what gas we're interested in this graph will describe the compressibility factor and therefore the PVT properties of that gas fairly well regardless of the identity of the gas. So just to get an example to see how that works let's do a numerical example let's say we're interested in CO2. CO2 is a gas that has a fairly low critical temperature in fact it's often used to do supercritical used as a supercritical solvent so let's see the critical temperature of CO2 is about 304 Kelvin its critical pressure is 72.8 atmospheres so let's say I'm interested so those are just properties of CO2 let's say I'm interested in doing something at a temperature let's use 350 Kelvin and a pressure of 100 atmospheres as with any gas or supercritical fluid calculation we can predict one thermodynamic variable in terms of two others so I could predict the molar volume as a function of P and T one way to go about that would be to use the Van der Waals equation we already know how to do that but this generalized version of the Van der Waals equation and this generalized compressibility factor diagram gives us a different approach for doing that so if I have the gas at 350 Kelvin I can say the reduced temperature is going to be temperature over critical temperature so temperature divided by critical temperature 350 over 304.1 that works out to about 1.15 so 15 percent above the critical temperature that tells me which of these isotherms I'm living on the gas is living on likewise the pressure the reduced pressure is actual pressure divided by the critical pressure 100 atmospheres over 72.8 atmospheres and if I imagine that I know that to a few sig figs we can calculate that ratio 100 divided by 72.8 that works out to be 37 percent above the critical pressure so that's enough information for me to find where I am on this generalized compressibility diagram so this is the critical isotherm the isotherm at a reduced temperature of one this is the isotherm at a reduced temperature of 1.1 here's 1.2 we're interested in 1.15 so that would be a curve that will be we'll have to interpolate it in our minds but that would be a curve that is somewhere in between roughly halfway between the 1.1 and the 1.2 curves likewise the reduced pressure of 1.37 so here's here's 1 here's 1.5 1.4 is going to be roughly here so I just need to read up from this point and find the location on this 1.15 isotherm roughly right about there and then read off on the volume axis what is the compressibility factor for a gas at this temperature and at this pressure and if I try to do that fairly carefully it looks like that's going to be fairly close to 0.65 or so at least my ability to read off of this diagram so we predict that the compressibility factor let me write that where you can read it is going to be z is equal to 0.65 under these conditions compressibility factor of course is pv bar over RT we know what the pressure in the temperature are we know what z is we just need to figure out what the molar volume is so the molar volume is going to be z RT over P so 0.65 for the compressibility factor R if we have pressures in atmospheres I'm going to use R in units of liter atmospheres per mole kelvin also in the numerator I've got a temperature of 350 kelvin numerator multiplied by 350 kelvin in the denominator divide by my pressure of 100 atmospheres and just to double check the units compressibility factor unitless atmospheres cancel atmospheres kelvin cancel kelvin and I'm left with the liters over moles which is good that should be in fact it reminds me that I have a molar volume that I'm calculating my molar volume should work out in units of liters per mole and if I plug these numbers into a calculator what I find is that the result is 0.19 liters per mole if I did have supercritical carbon dioxide at a temperature of 350 kelvin and 100 atmospheres in other words 15 percent above its critical temperature 37 percent above its critical pressure I can predict just graphically by reading off of this diagram that the molar volume is going to be roughly 0.19 liters per mole that is much easier not quite as precise as using the Van der Waals equation of state with as many sig figs as I know my Van der Waals constants to but most likely more accurate because these isotherms were obtained from empirical data from real gas molecules rather than this Van der Waals equation model using this compressibility diagram is essentially a way of doing PVT calculations you can calculate molar volume as a function of P and T or vice versa you can calculate P as a function of V and T using this this diagram and solve a problem graphically rather than algebraically this general approach as we've seen that's very useful when I have temperatures and pressures that are typically quite a bit higher than the critical temperature the critical these isotherms are for temperatures at the critical temperature and above pressures most of this diagram is covered by larger than the critical pressure these are conditions where we have either very hot gas or hot liquid or super critical fluid there's also interesting features of the phase behavior of gases down at colder temperatures well below the critical temperature in fact below the temperatures that cause the substance to solidify into a solid so that's what we'll do next is explore the portions of the phase diagram where we have solid phases