 Welcome back. Now let us try to prove the Carnot theorem. We will prove it by the method of reductio ad absurdum. We have to prove that eta is less than or equal to eta r for fixed values of T1 and T2. We will assume that that is not true. So let us assume that eta is greater than eta r and then we should be able to demonstrate that this violates the second law of thermodynamics. Let us try. Let us again sketch our figure. Two systems, two reservoirs at T1 and T2 such that T1 is greater than T2 and T1 is fixed at some convenient value, T2 is also fixed at some convenient value. We have an engine E and we have a reversible engine R E. Let us say that the engine absorbs heat Q1, rejects heat Q2, produces work W. Let the reversible engine at shown earlier absorb Q1 R, reject Q2 R and produce work W R. What we are going to do is we are going to note that this is a reversible engine. So I am able to run it as a refrigerator as well as as an engine. If I run it as an engine, the interactions are as shown whereas if I run it in reverse that is as a refrigerator what will happen is Q2 R will be absorbed from the low temperature reservoir, Q1 R will be rejected to the high temperature reservoir and W R the work will be absorbed. The three arrows will get reversed or inverted in their direction that is something which we can do and we will do to demonstrate the proof of Carnot theorem. So we will say that to prove Carnot theorem let us assume that eta less than or equal to eta R is false. This implies that if it is false then the efficiency of our engine E must be higher than the reversible engine. Now let us see the consequences of this. To see the consequence of this we will adjust the two engines first in such a way that Q1 R is Q1. This is an adjustment we will do as we have done earlier. Then what we will do is we will notice that the efficiency of our heat engine is W by Q1. Efficiency of our reversible heat engine is W R by Q1. Notice that Q1 R is assumed equal to Q1. The working of RE has been adjusted. Now since eta is greater than eta R, your denominators Q1 are the same. Since eta is greater than eta R we have this work interaction must be higher than this work interaction. W greater than W R. And consequently since the same amount of heat is absorbed a larger amount of work is produced by this standard engine. Q2 must be less than Q2 R. Also we must have Q2 less than Q2 R. Now what do we do is we reverse this engine. What we do is we reverse the reversible engine. What is the consequence of this? The first consequence of this is that the three interactions have their directions interchanged. So this becomes in this direction. This also in this direction and this work interaction goes into the reversed engine. So this is RE reversed as shown now. Let us now notice that the reservoir at T1 absorbs heat Q1 from the reversible engine. It rejects it Q1 to the other engine E. Consequently the reservoir at T1 does not really interact. It only anchors the temperature at T1. And now let us consider a complete system which is made up of our engine E and the reversible engine in its reversed working mode. Now what is this? Notice that this is engine plus reversible engine reversed plus the reservoir at T1. What are its interactions? First it produces work which is a combination of this work done by the engine and the work absorbed WR by the reservoir. So it produces work which is equal to W minus WR. It absorbs heat from the low temperature reservoir which is at T2. And the amount of heat absorbed is Q2R from the reverse reversible engine minus the one rejected by engine E Q2. So the heat absorbed will be Q2R minus Q2. And now you will notice that not only is the work done greater than 0, heat absorbed greater than 0 but the resulting entity happens to be a 1T heat engine. Now 1T heat engine obviously violates the second law. That means we must have done something which has led to the violation of second law and that something must not be true. And what did we do? We assume that the Carnot theorem statement eta less than or equal to eta R is false. Consequence of assuming this false is that we have violated the second law of thermodynamics and that means this particular assumption is untrue. We cannot assume eta to be less than or eta R to be a false statement and this implies that eta must be less than or equal to eta R and that proves the Carnot theorem. Thank you.