 Hi, well I'm Stephen Nesheva, and I'm here to tell you a little bit about how to derive expressions for the compressibility z as a function of the gas density. So this is the operative equation here. z is defined by the pressure, kind of volume divided by rt. Now where the pressure is going to be given, not by the ideal gas law, but by some other equation of state, for example, the Berthelot or the Peng Robinson, and the other piece of this that we need to know is that the volume, and I'm using v instead of v sub m just for notational simplicity, but the volume is inversely proportional to the density, and so we'll be using that equation. So how does that work? The Berthelot equation of state for the pressure equals rt over v minus v, and that's just the same as the band of all species, so minus a over t v squared, that's a little bit different. So how do we construct z? Well, I'm just going to take that pressure, multiply by the volume, and divide by rt, and when you do that, of course the rt's go away, the volume's line up this way, and so on. After that, what we would like to do is we'd like to get this in terms of the density, and there all we do is I take everywhere that I see a volume, I'm going to replace it by 1 over the density, and if you roll out how that works algebraically, you'll find that this first term here turns into 1 over 1 minus b rho, and the second term, while we don't really have to do much with it, is just that same coefficient, and instead of the 1 over v, I write rho. So that's how we're done with that one. How about for Payne Robinson? Well, the first term is exactly the same as the Berthelon and the Vanderbals, and then the second term looks a little bit more complicated, but we already know what to do with this. I'm going to take that pressure, multiply it by the volume, divide by rt, and when we do the first term, of course, we'll end up, the first term of z, we'll end up looking just like it did for Berthelon. What are we going to do about this term? Well, it's pretty straightforward. Everywhere I see a volume, I'll just replace it with 1 over rho, which is what I've done right here, and I think I still need an 8t upstairs, and so now we have z in terms of the density for a Payne Robinson, and all the other gases work in the same way.