 So, how we describe the pressure of a gas as it depends on its temperature and its volume differs depending on whether we believe the gas might behave ideally or maybe whether it might obey the Van der Waals model or in fact since those are both only models and there's the actual real world gas has its own pressure as a function of V bar and T that isn't perfectly described by any one of these models. Regardless of whether you were using a Van der Waals model or a more complicated equation of state we're often interested in the fact how ideal in fact is either the model or the gas itself. So we can define a quantity that we can think of, I'll give it a name in just a second, but we can think of it as a description of how ideal the molecule is. If I say what's the actual volume of the gas divided by the volume it would have if it were an ideal gas. So the name for that quantity is something called the compressibility factor which is a somewhat unfortunate name because we're usually not explicitly talking about the compressibility of the gas when we describe the compressibility factor but it's most useful to think of it as this measure of the ideality. If in fact the volume is exactly the same as the ideal gas law would have predicted then if we have Z equals one, V equals V ideal then the gas is behaving perfectly ideally and that also means let's go ahead and say if it's behaving perfectly ideally then PV equals NRT works just fine. In fact if I rewrite the ideal gas volume as pressure sorry not pressure as RT divided by pressure so I've just used the ideal gas law to rewrite V ideal as RT over P if I clean that up a little bit moving the pressure to the numerator. Another way to think about the compressibility factor is it's the ratio of PV bar to RT and again if the gas is behaving ideally PV is equal to RT the volume is equal to the ideal gas volume and the compressibility is one those are all identical ways of saying the same thing. On the other hand if the compressibility factor is smaller than one because this ratio is less than one because V bar is less than V ideal the gas takes up less volume than it would have if it were behaving like an ideal gas or in this case PV bar is less than RT and of course the opposite is true if the compressibility factor is greater than one that means the gas is taking up more volume than an ideal gas would and PV bar is bigger than RT. The first of these cases the volume is smaller than the ideal gas volume that happens when the most important deviation from ideal gas law is the attraction between the molecules. The molecules attract themselves occupy a smaller volume than they would have that PV product is smaller than we expect it should be. On the other hand when the volume is bigger than what the ideal gas law says is because the molecules the deviations are driven by the the finite volume of the molecules so it's making them take up more space than the ideal gas law predicts they would. So one thing that the compressibility factor is useful for is a prediction of whether or a measurement of whether it's the intermolecular attractions or the finite volumes that are dominating the deviation away from ideality and we can see that a little more explicitly if we bring up a graph of what the compressibility factor looks like so let's go ahead and add a graph here so this is a graph for a real gas this is real data for nitrogen at various different pressures measuring the compressibility factor z at a particular temperature of 200 Kelvin so the flat line here z equals one shows what the compressibility factor would be if the gas behaved ideally and we see in fact that the compressibility factor is not one it starts out at one and it's no surprise that at very low pressures at pressures near zero the the gas doesn't behave does in fact behave like an ideal gas at low pressures we expect gases to behave ideally but as we increase to pressures of one or two or ten or all the way up to 50 or 100 bars 100 atmospheres worth of pressure we can see that the compressibility factor has gotten much smaller only about 80 percent the volume is only about 80 percent of what the ideal gas volume would be and for this type of behavior what's going on as i change from let's say the ideal gas where i've got molecules in a box if i've compressed the the gas down to a much smaller volume the main differences the molecules are now much closer to each other than they were over here so there's closer to each other means they can interact more favorably and more strongly with one another so the effect of the intermolecular attractions has become more important so that's what decreases the compressibility factor but if i continue increasing the pressure up to 200 or 300 then what i see is now i've got the molecules packed so closely together that they're almost overlapping and on top of one another and now the size of the molecules has come into play and they've started to to bump into one another so that's generally what happens both at 200 kelvin as we see here also at at higher temperatures like 300 kelvin is at first the attractions between the molecules dominate and then later on the finite volume of the molecules begin to dominate and notice that at the higher temperature of 300 kelvin the effect of the interactions is is weaker than it was at 200 kelvin and that's for boltsman type reasons the interaction between the molecules is relatively weak and at temperatures of 300 kelvin thermal energy is more than enough to break those molecules apart fairly often so the effect of the intermolecular interactions is not as strong and in fact if we go to even higher temperatures of a thousand kelvin we see that the effect from the intermolecular interactions has disappeared completely and the only deviations from ideality that we see are in the positive direction so it's only the finite volume of the molecules that is affecting deviations from ideality and at these high temperatures the weak interactions between the molecules are not terribly important so that's an introduction to this idea of the compressibility factor and both this graph and this way of understanding that the compressibility factor might be sometimes lower than one because of intermolecular interactions or higher than one from finite volumes and we do in fact see both of those behaviors in real gases that brings up the interesting questions of what happens when coincidentally or otherwise those deviations from ideality happen to exactly cancel one another and that's the topic of the next video lecture.