 Welcome back. We have provided a proof of the Carnot theorem. That is not the only way to prove Carnot theorem. We have adjusted the working of the two engines that worked between the same T1 and T2 such that Q1 equals Q1r. You could have started by assuming Q2 equals Q2r or third option is assume W equals WR. You will take three different algebraic roots. However, whichever root you take while assuming eta is greater than eta r, you should be able to come to the conclusion that we can set up some entity, a one T heat engine for a refrigerator without work input which violates the second law of thermodynamics. I will leave this to you to try it for yourself. Now, what we have done is we have proven the Carnot theorem for engines, but there is also a Carnot theorem or what we call a Carnot theorem for refrigerators. Let us understand what it means. Again, let us assume that we have two reservoirs, one at temperature T1, another at temperature T2 such that thermodynamically T1 is greater than T2. And now instead of comparing engines, let us say we have a refrigerator r and we have a reversible refrigerator. Let me call it rr. This refrigerator absorbs it Q2 from the low temperature reservoir, rejects heat Q1 to the high temperature reservoir and absorbs work W. The reversible refrigerator absorbs Q2 r from the low temperature reservoir, rejects Q1 r to the high temperature reservoir and absorbs work equal to W r. For our refrigerator, the COP is defined as the heat absorb divided by the work required to run the refrigerator. That will be Q2 pi W. For the reversible refrigerator, we will have a COP for the reversible refrigerator equal to Q2 r by W r. The Carnot theorem for refrigerator claims that the COP of any refrigerator must be less than or at most equal to the COP of a reversible refrigerator working between the same two fixed temperatures T1 and T2. We can prove this exactly in a similar manner that we used for proving the Carnot theorem or the Carnot theorem for engines. You should be able to do it yourself. So, I will leave this to you to do as your homework. It is not difficult at all. I recommend that all of you try to do it. Thank you.