 Welcome back, now it is time for us to proceed from cycles to processes. So let us make that transition and see what we discover. But before we do that we have to discover using the second law of thermodynamics a very important property, the entropy. Let us proceed from where we left off with the Clausius inequality. The Clausius inequality told us that integral d cube by T for any cycle is less than or equal to 0. But now let us look at the equality part of this and let us separate that out. If we consider the equality part separately, we will be able to write that the cyclic integral of d cube by T for a reversible cycle will be 0. And henceforth we will be using this subscript either something like this or something like this means this pertains to the reversible limit either an R as a subscript or ReV as a subscript. Now notice that this cyclic integral of this quantity d cube by T is true for any reversible cycle, this equality for the Clausius inequality. Now since it is true for any reversible cycle calculus tells us that this implies that d cube by T for a reversible process element must be an exact differential. This is the second time in this course that we are encountering this discovery of an exact differential. Another way of reaching the same conclusion which we will find in many textbooks is as follows. Consider a cyclic process. You can select either P and V or X1 and X2 as the coordinates to depict it. Consider a reversible cycle. Since it is reversible, it is definitely quasi-static. And let 1 and 2 be two distinct states on this cycle. And let us say that 1A2B1 is one way of traversing the cycle. And this equality part of the Clausius inequality tells us that expanding this, this will be cyclic integral of d cube by T reversible over the cycle 1A2B1. This is a reversible cycle. This will be 0. But now let us expand this as 1A2, one part of the cycle plus that over the second part of the cycle. Let me transpose a term, the second term to the right hand side and I will get. Now we notice going back to the sketch, the process 2B2B1 being part of a reversible cycle is a reversible process. So now if I execute it in reverse as 1B2, I am going to retrace the same path but all my interactions are going to be reversed or going to be inverted. Whatever was d cube will become minus d cube and so on. So the right hand side can now be written down as equal to integral of over 2B1 minus d cube by T which will be integral 1B2. Notice that the path has been reversed. It is being traversed in the other direction and hence this minus d cube will become d cube. Now compare the leftmost expression and the rightmost expression. Of course, everything here is reversible. I should never forget that. When you compare these two you will come to the conclusion that d cube by T reversible integral of this from 1 to 2 is independent of the path and since now our cycle was any general cycle and hence the process 1, 2 was any general process reversible. 1, 2 were any two states. So this implies that d cube by T for a reversible process element must be an exact differential. Now since d cube by T is an exact differential or integral 1 to 2 d cube by T being independent of the path, both these things independently mean the same thing. There must be a property which is represented by integral d cube by T or whose differential is represented by d cube by T provided this d cube by T is for a reversible process or reversible process element. That means this should represent the differential of some property or this would represent the change in some property from state 1 to state 2. This property or this property which is the same property we define this property as the entropy of the system and give it a simple s. Thank you.