 Hi, I'm professor Nesheva and I am here to tell you a little bit of about This question, which is why is utv that quantity? Why do we get to call it CV? Let me explain so CV is supposed to be What we call a heat capacity the little subscript V means we're thinking about a heat capacity that happens when we when we try to heat up a system but Maintain its volume constant and a constant volume we use this word isochoric So the isochoric heat capacity is CV. Let me give you an example I'm just imagining I'm going to add a little bit of heat to this system Which starts off at some temperature, but because heat goes in the temperature goes up and And Well, you can imagine that is long that there should be some proportionality there that the amount of heat that goes in should result in a proportional temperature increase and that's what this equation expresses Okay, we say the amount that he goes in is proportional to the amount that the temperature went up so for example if I put 210 joules into my system over there and I saw that the temperature went up the change in temperature was 10 Kelvin Then I would use that equation and say well I can solve that that means that CV must be equal to 21 and the units are are going to be Joules per Kelvin. Okay, so that would be the heat capacity of that system that we determine from from that experiment at some particular Temperature. Okay, so that's the heat capacity What does this have to do with these geometric pictures of the U as a state function? well What we have argued before is that the that the slope of the U in a state space of temperature and volume the slope there in the in the temperature direction we we gave it A name called it CV But formally we would say that that slope in that temperature direction is partial of you with respect to temperature holding the volume constant Okay, so that's that's CV that we're calling it CV. The big question is you know How do I know that that's the same CV that we've got over here? Okay, well, how are we going to go about that? We have two really great equations one of them is the differential equation of state and This you know just describes the slope and in in this graph remember that slope is pi sub T And the idea was that any a change in temperature or in volume can be used to calculate the change in the internal energy Just by using those coefficients CV and pi sub T. Okay, that's a differential equation of state and now in this particular case since I'm imagining an Isochoric heating I'm not changing the volume at all. So I get to throw away that term So we just have that left put it to be differential equation of state another good Key equation here is the first law in differential form that would be the amount of heat that goes in is sorry the amount of energy that that that the system Changes and must be able to dequeue the heat that goes in and work that's done again since this is an isochoric Process, there's no change in the volume. So the work must be zero and I have du is equal to dequeue Now I'm looking at these two equations. This is equal to that. This is equal to that the same thing So I think I can pretty readily solve this to say that dequeue must be equal to CVdT Okay, that's what those two equations put together say and that's of course exactly the same thing that we started off by when we have defined what the heat capacity is and So that's how we know that uvT uTV is the same as is legitimately called the isochoric heat capacity