 Welcome back. Now it is time for us to look at one of the most important derivations in thermodynamics, perhaps something which is at the heart of the second law of thermodynamics and that is the Carnot theorem. What is this theorem about? This theorem is about the efficiency of 2 T engines or 2 T heat engines. It provides a special status for reversible 2 T heat engines. Such reversible 2 T heat engines are sometimes known as the Carnot engines. The Carnot theorem compares the efficiency of any 2 T heat engine with a comparable reversible 2 T heat engine. What is this theorem about? Let us sketch a figure to appreciate what the theorem is about before proving it. Let us say that we have two systems, two thermal energy reservoirs at distinct temperatures T1 and T2. It is given that in a thermodynamic sense T1 is greater than T2 and we consider two engines. One, any engine E working between T1 and T2 and let it absorb heat Q1 from the reservoir at T1, reject Q2 to the reservoir at T2 and produce work W. The other one is a reversible engine working between the same two reservoirs. Let it absorb heat Q1R from the high temperature reservoir. Let it reject heat Q2R to the low temperature reservoir and let it produce work WR. Now the efficiency of this engine by the definition of efficiency is W by Q1. The efficiency of this reversible heat engine will be let us call it eta R that will be WR by Q1R. Carnot theorem compares this efficiency and this efficiency for fixed values of T1 and T2 and the Carnot theorem says that if you have a fixed value of T1 and another fixed value of T2 then the efficiency of any engine E must be less than or at most equal to that of the reversible heat engine working between the same two fixed temperatures. Now it is important to note that here we are just talking about an engine and a reversible heat engine. The given or the fixed things are the two temperatures, one is T1 and one is T2. There are no other qualifications on the engines except that the one on the right has to be reversible. Otherwise the engines may be made up of any materials, any structure, they may use any fluid, they may use any set of processes. That is the importance of the Carnot theorem. In a few moments we will be proving the Carnot theorem. Thank you.