 Hi, well, I'm Professor Steven Nesheva, and I'm here to help you out with this process of deriving critical constants from a model equation of state. So just to bring you up to date on this, here's an indicator diagram for a substance, and we have pressure on this axis, volume on that axis. Here we have an isotherm at a temperature greater than the critical temperature, so it just follows that kind of curve, it's got a little bit of a bump, but at the critical temperature, that's at a lower temperature at the critical isotherm, it develops a bump there or a point at which the first derivative is zero, and also the second derivative is zero, that's an inflection point. So I've kind of indicated that mathematically here, at that point star, the critical volume, the critical pressure and the critical temperature, we have these two conditions that are true, the slope of pressure with respect to volume along an isotherm, that's at constant t, is zero, and also the curvature, the second derivative of the pressure with respect to the volume along that isotherm is zero. So those are two equations, two conditions that are going to help us find what are called the critical constants, because what we want from this is we want to figure out what is the critical pressure, the critical volume, and the critical temperature. So how do we do that? Well, we have to start with a model equation of state, so here's a possible model equation of state, we say pressure is equal to RT over the volume minus those terms right there, so if we could imagine that that expression is what reproduces these curves here, alright? So how do we go about deriving what the critical volumes, etc., are? Well, analytically, you can take the first derivative of that pressure, and I've done it for you here, first derivative of RT over V is obviously minus RT over V squared, and so forth. So that's the first derivative, how do we take the second derivative? Well, you just take the derivative of that thing that we just did, so for example the derivative of minus RT over V squared is plus 2 RT over V cubed, how does that help us? Well, what we're going to do is we're going to say I am, that's equal to zero, and I'm also going to set that one equal to zero, so now we have two equations, and what appears in here are two unknowns, the two unknowns are the critical temperature and the volume, okay? So you're going to have to figure out a way, you would have to figure out a way to solve those two equations for those two unknowns. Here's one possible way to do it, I can notice that the difference between here and here is just a factor of 2 over V, so I can multiply both sides of that equation by 2 over V. Now they match, but they have opposite signs, so I could add those two equations to each other, and you can kind of tell that what would be left would just be something that depends on the volume, okay? And after that, so you would get the volume out of it that way. Once I know the volume, I could plug that volume back into this equation and solve for the temperature, so that would give me the critical temperature. Once I know the critical temperature and the critical volume, I could plug those into that equation to get the critical pressure, and then you would have expressions for all those three parameters, those critical constants in terms of parameters B and C. Okay, so that's how that works.