 Hi, well I'm Stephen Nesheva and I'm here to help you out with this idea of differential equations of state for the internal energy in a state space of temperature and volume. So here's what the geometric view is. State space of temperature across there, I'm going into the board, I'm writing U as a function of that. And you can see that the internal energy has some dependence on the temperature and on the volume. And the way we like to think about this is incremental tiny little changes in either the temperature or the volume would obviously create a little change in the internal energy. And that's given by this equation right here. This is the differential equation of state for U of Tv and it just says that a small change in U could result from a change in temperature multiplied by a coefficient. Or it could result from a change in the volume multiplied by a different coefficient. Formally these coefficients are the slopes. It's how U changes when the temperature changes holding the volume constant, i.e. an isochore. So that would be that slope right there which I'm labeling as Cv. This other number here, pi sub T, that's that slope right there. So then it would be written formally as the partial view with respect to volume while we hold the temperature constant. Now if we're thinking about a process in which a system is being heated up isochorically, that means it's being heated up while the volume is constrained to say the same, then we have an easier job of it right because I know that there's no change in the volume so I can strike that term out here and I know that a temperature change multiplied by that coefficient will produce a small change in the internal energy. Now what if we were going to not just go from some small temperature increase like that but we wanted to increase the temperature quite a lot. Well you can see that the slope might be changing along there. So if I want to get at the full change of the internal energy which could be indicated that way, I probably have to solve an integral. So that's what's written down here. The total change in the internal energy and going from one temperature to another, T1 to T2 would be the integral of Cv dt that comes straight from that. So now then we want to think about a couple of examples. The simplest case is the case where Cv is just a constant. If Cv doesn't change if it's temperature independent and you can imagine that Cv comes out of that integral and then what we're left with is the integral of dt and the integral of dt is just the change in temperature. So in the case of Cv constant we just get that delta u equals Cv times the change in the temperature. If you want to think about that graphically the idea is something like this. I'm integrating Cv over some range of temperatures but if Cv is constant then I have this dashed line right here and this area right here which is the height times the width. The height of Cv the width is delta T. So that would be what we would be calculating that area would be delta u in the case of Cv this constant. Now there might be a little bit harder case where you can imagine that Cv is temperature dependent and I've drawn a linear case here where Cv looks like A plus B times T. That would be that curve. Geometrically obviously that's just that area now. That's what I want. How do you get that area under the straight line? Well you just put it into there and use the rules of integral calculus to get the change in u that results from those temperature changes. Okay.