 Welcome back. Now it is time for us to do some basic exercises and basic derivations in property relations and we will always remember that by default unless otherwise said our system is a fluid system that is simple compressible system and it is at rest and that means there is only one two-way work mode expansion compression represented by PDV and all other components of energy do not change except the thermal energy. So we can replace DE by DU. We start with our basic energy relation which is TDS equals DU plus PDV. First we are going to give a proper treatment to the thermal energy differential and for that what we do is transpose DU to the left hand side and write DU equals TDS minus PDV. Now remember here the left hand side being the differential of a property is an exact differential and hence the right hand side must also be an exact differential. On the right hand side we notice that we have differentials of S and differentials of V specific entropy and specific volume respectively and hence looking at our exact differential formula that if Z is a function of X and Y then DZ is something into DX plus something into DY and here that something are DS and DV. So let us consider U to be a function of S and V. Expanding DU in terms of DS and DV we will get DU equal to partial of U with respect to S at constant V into DS plus partial of U with respect to V at constant S into DV. Now compare these two relations. Left hand side is the same and hence comparing the coefficients of the differentials of the independent variables that is coefficients of DS which would be the same and coefficients of DV which would also be the same we get T equal to partial of U with respect to S at constant V and P equal to minus partial of U with respect to V at constant S. Let me complete the full derivation and then we will discuss something further about this. Now this is the first derivative relation. Now because the right hand side of this equation is an exact differential we must have the cross derivative relation also applicable and hence we get using the second derivative relation partial of T with respect to V at constant S is minus partial of P with respect to S at constant V. Let us call this relation which we will refer to again later as equation M1. Now before we proceed any further let us note the following. Notice that here we have considered U as a function of S and V and this triad of properties U, V, S is very important because we have noticed that any thermodynamic system should have a volume so V is a very natural variable. The first law provides a property E or U in this particular case and the second law provides a property S. So you can say that this triad of property is a basic thermodynamic property triad and then if you consider this to be the property triad then naturally you can consider U to be a function of S and V and in that case these two relations T equals partial of U with respect to S at constant V is the defining relation for temperature in this case in terms of U, S and V and the second relation P which is minus partial of U with respect to V at constant S now becomes the defining relationship for pressure. Let us generalize this a bit further. We considered U, V, S as the basic property set because we were looking at a simple compressible system at rest. If you consider a complex system in which case we cannot replace D E by D U straight away. You will have changes in thermal energy component plus changes in other components of energy. So let us remain with E. Basic property V will remain, second law will provide S but because apart from PDV work or expansion compression work we will have other modes of work and let they be of the type Y DX or Y1 DX1, Y2 DX2 whatever you have any number. Let me write just two of them here, some X1, X2 but you could write any number. Now in that case our definition of temperature would turn out to be T equal to, now here we will have to write partial of E with respect to S at constant V and also at constant X1, X2 and if you extend the list of dependent variables here, one pertaining to each two way work mode that list will also have to be extended here. And also the definition of pressure would be minus partial of E with respect to V not only with constant S but also at constant X1, X2 etc. Thank you.