 Welcome back. It is now time for us to do some exercises. So we have these exercises set E. There are a number of exercises in that and many of these are you know sort of standard exercises. Actually most of these exercises ask you to essentially derive partial of x with respect to y at constant z where x, y, z is a triplet of some thermodynamic properties. You select some thermodynamic property for x, some other for y and the third one for z and you can derive an expression. However there are certain basic exercises, some standard derivations which all of you should know. These are like standard moves in chess which all reasonably good chess player should know or standard strokes in tennis which all good tennis players should be very comfortable with. Let us look at a few of these. Some of these are formally part of this exercises set and some of these are hidden somewhere and are required. We know the definitions of CP and CV. Now we will derive some alternative forms of expression for CP and CV. One form is obtained by considering the expansion of H. We know that DH has a relation TDS plus VDP. Now consider H as a function of T and P because we are looking for an expression for CP and CP is defined as partial of H with respect to T at constant P. So for the definition of CP, H has already been considered as a function of temperature and pressure. So let us continue with that. Now on the right hand side we have a DS. So let us consider S also to be a function of T and P and of course we have a DP but P is already an independent variable. Now expanding this we will get the expansion of DH on the left hand side is partial of H with respect to T at constant P DT plus partial of H with respect to P at constant T DP and on the right hand side we will get T into DS is partial of S with respect to T at constant P DT plus partial of S with respect to P at constant T DP. The second term plus VDP remains as it is. Now since P and T are independent variables, the coefficient of DT on the left hand side and right hand side must be equal and the coefficient of DP on the left hand side and the right hand side must also be equal and that gives us comparing the coefficients of DT partial of H with respect to T at constant P equal to T into partial of S with respect to T at constant P but DH by DT is already defined as CP the so called specific heat at constant pressure and hence we now have an alternative expression for CP in terms of S, T and P. I will leave it to you as an exercise to show that CV which is defined as partial of U with respect to T at constant V turns out to be equal to T into partial of S with respect to T at constant V. Now the second term here is also useful. We have a coefficient of DP on the right hand side and a coefficient of DP on the left hand side. Comparing them we will get partial of H with respect to P at constant T equal to T into partial of S with respect to P at constant T plus V. Now notice that we have a partial derivative here and let us see whether we can replace it using the Maxwell's relation. So we have partial of S with respect to P at constant T expressed in the Jacobian form as Jacobian of S, T with respect to P, T. Now here we have T, S in the reversed order in the numerator so we multiply this with the Jacobian of VP with respect to S, T which is 1 and now since the numerator here and the denominator here are similar we can essentially cancel that. So we get the Jacobian of VP with respect to P, T. So this equals negative of partial of V with respect to T at constant P and hence partial of H with respect to P at constant T turns out to be equal to minus T partial of V with respect to T at constant P plus V. I will leave it to you to obtain a similar form for this partial of U with respect to V at constant T. This is to be derived as homework and for both this expression and this expression you will have to start from the relation for U, T dS minus P dV. Start with this and you will get the extended expression for CV as well as the expression for partial of U with respect to V at constant T. Thank you.