 Welcome back. Let us look at a few more exercises are from E04 to E07. In E04 we are asked to derive an expression for partial derivative of Cv with respect to v at constant t in terms of pv and t. Let us proceed. We have to start with a definition of Cv and we start with the expression which represents Cv which we have shown to be t into partial of s with respect to t at constant v. Now we have to differentiate Cv partially at constant t with respect to v. So let us do that. What we want is partial of Cv with respect to v at constant t. Let us differentiate the right hand side with respect to v at constant t. Because the differentiation is at constant t, this t which is a factor will come out of the process of differentiation. So we have t and then we have, let me write in big brackets, partial of s with respect to v at constant t of the second term which is partial of s with respect to t at constant v. Now we will change the order of integration because this is a second partial derivative. So the order of integration should not matter. So when you interchange the order of integration we have to be careful because the variable which is maintained constant also gets flipped. So we end up with t. Now the inner differentiation was with respect to t at constant v. So we have that as the outer. The outer one was with respect to v at constant t. So that becomes the inner. Now we notice that this particular partial derivative appears in our Maxwell's relations. If we do the exercise, partial of s with respect to v at constant t, the Jacobian form, it will be Jacobian of s t with respect to v t. We want to convert this t s to p v. So we write s t same order as this but instead of t s in the numerator we write p v in the same order s t s. Now this s t here and this s t here sort of cancel each other and you will end up with Jacobian of v and t with respect to Jacobian of v and t which means partial derivative of t with respect to t at constant v. Substituting this back into our previous expression we will get partial of c v with respect to v at constant t is t outer bracket partial differentiation with respect to t at constant v and the inner differentiation now replaced by partial of p with respect to t at constant v. Now you notice that the inner differentiation and the outer differentiation are both with respect to t. The inner differentiation is at constant v, outer differentiation is also at constant v. So we can write this as simply the second partial differentiation of p with respect to t both at constant v. This is the expression which our exercise E04 expects us to derive. Now in the next exercise we have to show that for a van der Waals gas c v is a function of t alone. Before going to the van der Waals gas do the following exercise. In the right hand side of this expression here which is a function only of p v and t. We substitute say the ideal gas equation. If you substitute p v equals r t or p equals r t by v. You will notice that the second differential of p with respect to t at constant v is 0 and you will get partial of c v with respect to v at constant t it is 0. This is the ideal gas and that means because there is no variation of c v with t when volume is maintained constant that means c v is a function only of t and not of v. Now use the van der Waals equation of state. Remember that the van der Waals equation of state is linear in pressure. You can straight away solve it for pressure p equals some simple function of t and v. And you will notice that variation of c v with respect to v at constant t is 0 even for the van der Waals gas and that means even for a van der Waals gas c v is a function of t alone. It is not a function of v. The next exercise asks you to determine c p minus c v for a van der Waals gas. That is easy to do. We have an expression for c p minus c v in terms of p v t. So substitute the van der Waals equation of state and you will get an expression for c p minus c v and check whether that expression is a function of t alone or depends on v or p also apart from t. If you find that c p minus c v for a van der Waals gas is an equation only of t then well we can say that since c v for a van der Waals gas is a function only of t, c p for a van der Waals gas is also a function only of t. Otherwise we will have to come to a conclusion that for a van der Waals gas although c v is a function only of t, c p is a function of t as well as either v or p. Thank you.