 Welcome back. Before we go over to the exercises on property relations, let us derive one important relation, the Clausius-Clapeyron relation. What does the Clausius-Clapeyron relation do? Let us we sketch the phase diagram for water. Let us say this is the triple point and this is the critical point. So this is the liquid vapor saturation line. We have a solid vapor saturation line and then we have a solid liquid saturation line which is almost vertical but slightly leftward or westward leaning. The liquid vapor saturation line has a positive slope but the solid liquid saturation line has a negative slope. Let us see where thermodynamics comes into this. Let us consider a point on the liquid vapor saturation line. We know that it represents either the saturated liquid or dry saturated vapor or any state in between. What we are interested in is the slope of this line. Let me call this slope dp by dt of the saturation line between liquid and vapor. We want to derive an expression for this slope. Let us see how we can go about doing that. That was the pt diagram. Now let us project the same state space on the pv diagram and let us say that this point, this pressure let me extend here. Let us say now on the v axis the saturated liquid state and dry saturated vapor state can be shown distinctly. All the intervening wet vapor states can also be shown. So let this be the saturated liquid line. Let this be the dry saturated vapor line. So now this point is our saturated liquid state. This point is the dry saturated vapor state and these are the intermediate states which are at p constant and p is equal to the saturation pressure which we have selected and because we have a two phase situation here, this line also means that it is an isothermal line. So t is also constant and equals to corresponding saturation temperature. Now let us look at this process from f to g. Properties here are Sf, Hf, Vf, etc. Properties here are Hg, Vg, Hg, etc. Now let us consider the variation of entropy with specific volume on this line. Now since this line is both an isobaric line as well as an isothermal line, we can write the variation of entropy with respect to specific volume on this line at constant p to be equal to variation of entropy with respect to volume at constant t on this line. So this is from state f to state g and then we notice that both S and V between f and g are linear functions of the trinase fraction and hence this can be written down as simply Sg minus Sf divided by Vg minus Vf which in short is Sfg divided by Vfg. Now notice that both these partial derivatives are represented in the Maxwell's relation. So without spending much time on it, I will use the relations and we will be able to show I recommend that you check this out. Partial of S with respect to V at constant p equals partial of p with respect to t at constant S and partial of S with respect to V at constant t equals partial of p with respect to t at constant V. Now between f and g in the saturation domain or on the saturation line, both these partial derivatives are equal and hence both the right hand side should also be equal and that obviously means something which we already know that both these are equal and are equal to dp by dt along the saturation line for the liquid vapor combination of states. Now all that we have to do is use the right hand side of equation 1 and the right hand side of equation 2 and note that the left hand sides are equal hence the right hand side should also be equal and if you do that you will get this relation dp by dt along the saturation line, the liquid vapor saturation line is Sg minus Sf divided by Vg minus Vf which can also be written down as Sfg divided by Vfg. This is one form of the Clausius-Clapeyron equation. We can extend this form a bit by noting that the expression for dh is Tds plus Vdp. Now this expression we integrate from f to g along the line which is both isobaric and isothermal. Now notice that in this dp will be 0. Why? Because it is an isobaric process that we are looking at and what about T? T will be constant and will equal the corresponding saturation temperature. Hence integration of this will give us Hg minus Hf equal to T sat into Sg minus Sf. Now use the resulting expression for Sg minus Sf from this relation. Substitute back here and you will get dp by dt saturation along the liquid vapor line equals Hg minus Hf divided by T sat Vg minus Vf. It can also be written down as Hfg divided by T sat Vfg. Now this is true for the liquid vapor saturation line. We can apply the same principle to say the solid-liquid saturation line and if you follow the procedure you will get dp by dt for the saturation line between solid and liquid. It will turn out to be. Now here I will use S for the solid phase and L for the liquid phase. Instead of F for the liquid and G for the vapor in case of the liquid vapor case will equal Hl minus Hs divided by T sat in that particular case into Vl minus Vs. All these three relations and similar relations are known as the Clausius-Clapeyron relations. And now I will leave it to you to argue out that the slope of the liquid vapor saturation line is positive because both the terms in the numerator as well as the denominator on the right hand side are positive. The vapor specific volume is higher than the liquid specific volume. Vapor rises over the liquid. The latent heat anyway in the numerator is positive. Similarly, for the solid-liquid saturation line the latent heat is positive. But what about Vl and Vs? The solid floats over the liquid and hence Vl minus Vs is a negative number and that makes the line turn backwards with a negative slope. Thank you.