 Welcome back. Let us consider a 3-T machine, a cyclic device which could be an engine could be a refrigerator. Let us say that we can show the machine like this. This is our 3-T and let it have 3 interactions with 3 reservoirs, one at T1, another at T2, third at T3 and let the heat absorbed from the 3 reservoirs be Q1, Q2 and Q3 respectively and let the machine do a work W. Let us say all this thing is per cycle and again remember here that I call this a machine because it is possible that W is positive in which case it will be an engine. It is possible that W is not positive, it is negative in which case it will be a refrigerator. It is also possible that W is 0, such machines also exist. For example, we have what is known as the free fluid refrigerator, way back almost 18, 90 years ago or maybe even 100 years ago. Such a machine was patented by two famous scientists, Albert Einstein and Leo Zillard. So that is a machine where heat is extracted from a low temperature reservoir and heat is rejected to another reservoir, the ambient and heat is also supplied from a high temperature reservoir. There is no work interaction. So that is a refrigerator, a 3-T refrigerator which works without any work interaction, does not require a pump or a compressor. So here Q1, Q2, Q3, W, these four interactions, any one of them could be positive, any one of them could be negative and W could be 0. Of course, I am not considering the case where one of Q1, Q2, Q3 would be 0 because then it could be a 2-T machine, not a 3-T machine. Now first thing we note that because the machine executes a cycle, we must have W equal to Q1 plus Q2 plus Q3. This is the first requirement. Now what does the Clausius inequality say? The Clausius inequality, if it pertains to this machine would be, if I extend the boundaries of my machine to the three reservoirs like this, then for this extended system, the Clausius inequality would be Q1 by T1 plus Q2 by T2 plus Q3 by T3 is less than or equal to 0. This is what we want to prove. Again, let us attack this problem by the technique of reductio and epsilon. We say, let us assume that what we attempt to prove is not true. So we assume that Q1 by T1 plus Q2 by T2 plus Q3 by T3, we say, let us assume it to be greater than 0. And let us see what the consequence of this is. And of course, in second law, we know the trick. We see whether with this assumption, we can set up a 1T heat engine. If we can set it up, that means we have violated the second law of thermodynamics, the Kelvin-Plank statement of the second law of thermodynamics. And if we are able to violate that, that means our assumption is wrong and then the Clausius inequality is proved. Let us do the following. Let us have a reservoir at some temperature T0. And let me have three machines. The first machine, a reversible one, let me call it R1, takes the required amount of heat from the reservoir at T0 and provides to the reservoir at T1 exactly the amount that the reservoir T1 provides to the system. Let that be Q1 and let this be Q01. And let this machine produce a work, a reversible machine produced a work, W01. Similarly, this machine R2, let it absorb Q02 from the reservoir at T02 and provide exactly Q2 to the reservoir at T2. Let it produce work equal to W02. And let the third reservoir absorb heat equal to Q03 and provide heat equal to exactly Q3 to the reservoir at T3. This is also a reversible machine and let it produce a work, W03. Now, remember R1, R2, R3 are not necessarily engines. If for example, T1 happens to be lower than T0 and if Q1 happens to be a positive number, then R1 will be an engine. In some other cases, it could be a refrigerator. The signs of W, Q1 and Q01 will appropriately change. But our symbolism is as indicated. Now, let us look at what W01 will be. We will notice that W01 is Q01 minus Q1, first law. And then let us write this as Q1 into Q01 by Q1 minus 1. So, this is the first law. This is algebra. And then we notice that because the machine involved R1 is a reversible to T machine, Q01 by Q1 can be written down as equal to T0 divided by T1. So, this is because R1 is a reversible to T machine. In a similar fashion, we will be able to show that W02 is Q2 into T0 by T2 minus 1 and W03 equals Q3 into T0 by T3 minus 1. Now, let us go back. Consider a machine which is this. It contains everything except the reservoir at T0, this machine. What does it do? Let me go back and sketch a separate. The combined machine interacts with T0. The interactions are three heat interactions, Q01, Q02, Q03. And the work output is W from our original machine plus the three engines or three machines which produce W01, W02, W03. And this system is a cyclic device because it contains our original system which is a cyclic device. It contains the three reversible machines which are also cyclic devices. And it contains the three reservoirs which now do not come into picture because they receive the same amount of heat that they reject. So, they also one can say go through a cycle. So, this also is a cyclic device. And now, let us see what is the amount of work done. The total work done by this machine is the sum of these four work interactions. And that will be W plus W01 plus W02 plus W03. Now, let us substitute for each one of these. W is Q1 plus Q2 plus Q3. W01 is Q1 T0 by T1 minus 1 plus Q2 T0 by T2 minus 1 plus Q3 T0 by T3 minus 1. And now, you will notice that Q1 cancels with this Q1 here, Q2 cancels with this Q2 here, Q3 cancels with this Q3 here. And you end up with Q1 by T1 plus Q2 by T2 plus Q3 by T3. And this we have assumed to be greater than 0. Now, if our assumption is true and is valid, what we have done is set up a 1 T heat engine which obviously we cannot do because it violates the second law. And that means our assumption which we have made and which we tried to prove this one is incorrect. It is a false assumption. And hence, this implies by reductio ad absurdum that Q1 by T1 plus Q2 by T2 plus Q3 by T3 must be less than or equal to 0, which is the Clausius inequality for a 3 T machine. The next step is to prove the Clausius inequality in general. Thank you.