 Yesterday morning session, the second part of the morning session, we began our discussion of the second law of thermodynamics and this session we will be continuing with that. What we did yesterday, if you look up our scheme of discussion from main topic 9, the second law, we have discussed I think the first 8 points. We have decided to define a higher temperature and a lower temperature and that we did by deriving from the Kelvin Planck statement that a 2T heat engine will work only in one direction given two reservoirs at two distinct temperatures and heat will flow on its own between two reservoirs at two distinct temperatures only in one direction and we look at that direction and the source we say as higher temperature, the think we say is the lower temperature. So that way our definition of higher and lower temperatures is established. One should note that the higher and lower temperatures decided here are thermodynamically higher and lower temperatures, they need not be higher and lower numerically on any given scale. However, to be consistent with the thermodynamic higher and lower scale, all our temperature scale to begin with historically and now formally of course, now we do not have a multiplicity of temperature scale, we only have one temperature scale and that is the Kelvin scale. We know that our Kelvin scale is based on some material, an idealized material called an ideal gas but now it is time to put that scale on a proper footing which we will do in item 11 of chapter 9. Yesterday's discussions pertain only to the direct derivations from Kelvin Planck statement and they only considered what is possible and what is not possible. We know that using that we can show that many simple processes as working of a 2T heat engine between two thermal reservoirs, working of or direct transfer of heat between two systems at two distinct temperatures, just possibility or impossibility. Now we come to a stage where we start quantifying this possibility and impossibility will soon be converted to greater than or equal to or less than or equal to. And for that we have to come to a most important theorem in the development of second law which perhaps is typically most important because that was proposed by Carnot and is today considered perhaps the first set of thoughts pertaining relating or leading finally to the second law of thermodynamics. Now we come to an important adjective, we started with adjectives like quasi-static then we said an adjective like adiabatic, the next adjective was diathermal then isothermal. Now the next adjective we are going to look at is reversible, reversible is an adjective which like quasi-static is applicable to processes. So we will define a reversible process and naturally since a cycle is also a process later on we will apply it to a reversible cycle and since many pieces of equipment particularly an engine is implement, engine implements a cycle so we will apply it to an engine. But the first thing to realize is what is a reversible process. Let me tell you that similar to a quasi-static process, a reversible process is defined a process which we can think about. You can say that it is a remember Einstein proposed experiments which can be about which we can think so called thought experiment, experiments can be thought about may be very difficult to implement or perhaps even impossible to implement them in practice. So similarly a quasi-static process is a process about which we can think about. So quasi-static process perhaps is a thought process in a similar manner a reversible process is a process which we can think about so it also is a thought process. A reversible process is a very special kind of process. Let us say that we have two systems system A and system B and let there be some interaction between them of any kind and because of these interactions state A execute a process from A1 to A2 with some detail may be a path B executes a process B1 to B2 with its appropriate path. Now if it is possible for this to occur change of state of A1 to A2 through an appropriate path perhaps interaction I with respect to B which simultaneously goes from B1 to B2. If this process is possible and if the inverted process that is a process in which A2 goes back to its state original state A1, B2 goes back to its original state B1 and all interactions are reversed. I may not show it with a reverse arrow but I will say reversed or inverted or you can even put minus i. This is also possible such that let us say this is process P and this is process Q just labels. When P is executed it has some detail when Q is executed all those details are inverted reversed take place in the other direction. In such a manner when P followed by Q is executed no trace remains that we first executed the process P followed by the process Q. List out all your favorite detectives Sherlock Holmes, Karamchand, some Jassous, all the heroes in CID, Hercule, Poirot bring all of them together. If a reversible process is executed in one direction and is reversed everything comes back to its original state all interactions get reversed. If any third system comes into the picture during the process and changes its state even that state is reversed absolutely no evidence is left no trace remains that the process was ever executed and then reversed. So that means if I have some system and it changes its state and comes back to its original state by executing a pair of reversible processes then there is no way I can determine whether the systems involved have ever left their original state or not. It is a very strict requirement and we know that all processes in real life take place in such a way that some trace or the other is left some evidence is left circumstantial otherwise whatever type. But if a reversible process is executed and is reversed so that the systems involved are brought back to their initial state then the interactions are also reversed to such detail that absolutely no trace ever remains that the processes were executed. So that is a definition of a reversible process and now remember we have a general process then we have a quasi-static process and now we have defined a reversible process. Anything happens in real life is a process quasi-static process is a model helps us with our analysis and we know that it takes great effort to process to make a process quasi-static but many real life processes can be approximated to be quasi-static. A reversible process is also a model but to execute a process in a reversible fashion perhaps is just impossible in real life because in real life absolutely nothing is reversible. So this is something which we will think about and we will use in our analysis but there is a big question mark about whether a process will ever be executed as a reversible process. The advantage of defining a reversible process is something which we will soon see. In fact we do not really have to worry about the reversibility of a process. The idea of a reversible process is used just to derive certain relations which finally lead us to the definition of entropy and many useful tools. So consider the reversible process as a limiting case of a real process and it is the limit or these limiting cases that we are going to study. Later on we will again have to think about when we derive property relation but how to execute at least in theory or in our mind a reversible process because that would lead us to certain relations between property, the basic relation in property. Till we do that it is enough for us to understand that a reversible process is such that you trigger it, it executes one way, trigger it, it gets executed in the reversed way by inverting, reversing or executing in the other direction all interactions that have taken place in the forward direction. So that if we execute it in the forward direction once and if you execute it in the reverse direction once, the sum effect is absolutely nothing, no trace of the forward execution followed by reverse execution ever remain. Now consider a 2T engine which is reversible and let us say that if we trigger it, tap it, it will execute one cycle. Now suppose I have two taps, one way if I tap it, it executes the cycle in the forward direction producing a positive amount of work. If it is a reversible engine I must be able to tap it maybe with my left hand so that it executes the cycle in the other direction absorbing the same amount of work exactly in the same fashion. It rejected Q2 to the low temperature reservoir, it would absorb exactly the amount Q2 from the low temperature reservoir and reject it Q1 as reject heat Q1 to the high temperature reservoir. So that if this happens one forward cycle and one reverse cycle when we are away from the engine and then if we look at the engine there is no way for us to say whether the engine was absolutely doing nothing or it executed one forward cycle and one reverse cycle because no trace at all is left. From reservoir T1, heat Q1 was absorbed in one part in one forward cycle, Q1 was rejected to it in the reverse cycle. Same thing Q2 was made available to the reservoir at T2, Q2 was absorbed from the reservoir at T2 and no trace is left in the reservoir T1 and T2. Similarly, the work produced would have been absorbed in some third system maybe in the rays of a weight or whatever so that also is absorbed back and the third system is also brought back to its original state without any trace being left. So if you understand this that is all that is needed for us to use the defined tool known as a reversible process. Now if you have understood what is meant by a reversible process the next step is to say that a reversible cycle is a cycle that consists of all reversible processes and if this cycle is implemented in an engine that becomes a reversible engine. Now we come to the most important derivation or one of the most important derivation known as the Carnot theorem. Now from this point onwards I am going to follow the procedure but some details I will leave for you to fill in because these details are well described and well derived in almost all books on thermodynamics. The Carnot theorem states that the situation which considered in Carnot theorem is the following. We have two reservoirs at temperatures T1 and T2 and it is given to us that T1 is thermodynamically higher than T2. Under yesterday's definition we can now confidently say that two temperatures will be such that one is higher than the other. That means if I try to run a 2T engine between T1 and T2 it will work in this fashion. It will absorb Q1 from the reservoir at T1 reject Q2 at the reservoir at T2 and produce a positive quantity of work W. The quantity shown in these derivations are positive in the direction which is shown. So Q1 is heat transferred from reservoir at T1 to the engine, Q2 is the heat transferred from the engine to the reservoir at T2 and W is the positive amount of work delivered by the engine to some other system which absorbs it. Carnot compared the efficiency of such an engine with that of a reversible 2T heat engine working between the same two reservoirs but in a reversible fashion. That means let us say that this is Q1 for the reversible engine, this is Q2 for the reversible engine and this is W for the reversible engine. The engine E is not claimed to be reversible, it could be any engine but this engine is reversible engine. So if this engine can work only in one direction, this engine can be made to work in that direction producing positive amount of work equal to WR but that engine can also be made to work in the reversed direction if there is any meaning in the direction. Then the same amount of WR will be absorbed by the engine, Q2R will be absorbed from the reservoir at T2, Q1R will be delivered at the reservoir at T1 and if we execute the reversible engine's processes once in the forward direction and once in the reverse direction absolutely no trace would be left that it ever occurred that is the meaning of the reversible engine. Now the Carnot's theorem states that under these conditions the efficiency of this engine let us say eta and the efficiency of this engine say eta R, eta must be less than or equal to eta R that is the Carnot's theorem. And how do you prove the Carnot's theorem, I will just give you the idea of the proof because the proof is well known. We say it is standard technique reductio ad absurdum, if this is what we have to prove then all that we say let the opposite of this be true that means we say let us assume that eta is greater than eta R, so again let me sketch it here. Now let us do something, we will use our premise that thermodynamics is scale independent, so we can always adjust the working of these engines in such a way that let us say some two parameters are equal. Now in the previous slide I said let the heat flows be Q1, Q2 for the engine and Q1R for the reversible engine and let us assume that W is the work output of the engine and WR is the work output of the reversible engine. What we can do is let us assume that we adjust our Q2 and Q2R to be the same, this is one way of doing it you can assume Q1 and Q1R to be the same. You can assume for example Q2 and Q2R to be the same, so Q2R and Q2R the same, let this be Q1R and let this be W. Now what you should do and this is the proof in text book, so I will not spend time on it, you reverse the reversible engine take this in the other direction, so this engine will absorb Q2R, this is WR I forgot to write here, it will absorb work WR and it will reject Q1R to the high temperature reservoir. Then combine the engine and the reversible engine and maybe extend that idea to this include the reservoir at T2. Now see what is happening, we have a cyclic device, we have an engine and a reversed engine together, we have a reservoir but the reservoir absorbs heat Q2 and the reservoir also rejects heat Q2 because we have said that let Q2 be equal to Q2R. Now what do you notice that what you have is a modified engine which is E plus R reversed plus the reservoir at T2. What is the net interaction? The net interaction is work produced would be W minus WR and what is the heat absorbed from the reservoir at T1, Q1 by the engine and supplied back Q1R by the reversed engine, reversed reversible engine as I have shown R in this. Now I am not going through the algebra because all of you know how to do that, now show that it can be shown that given this it turns out that W minus WR is greater than 0 from this implication turns out. Just use this and you will be able to show that W minus WR equals 0. What does this means? This means that there is a violation of the Kelvin Planck statement and violation of Kelvin Planck statement means a violation of the second law of thermodynamics and why does it get violated? We must have assumed something wrong and the only assumption which we have made is this assumption and that means this assumption is wrong. Hence, we prove that the original thing must be true. This is the proof of the Carnot theorem. Various textbooks have derived it in a different way. Some will assume Q2 equal to Q2R adjusted that way. You can also derive it by assuming Q1 equals Q1R. Then you will get a 1T heat engine with T2 as the reservoir. T1 will get consumed in this 1T purported 1T heat engine. Whichever way you choose you will finally come to a conclusion that assuming eta to be greater than eta R leads to a violation of the second law as stated by the Kelvin Planck statement. So, we come to the first conclusion which we have proved. Eta less than or equal to eta R for fixed T1 T2 and remember that for this we have considered only so called 2T heat engines. Now that is a very special case of engines and cycles also which are special as implemented in 2T heat engines. Now let us proceed by look at some corollaries of Carnot theorem. There is one major corollary of Carnot theorem although we can have a number of corollaries. And the most important corollary of the Carnot theorem is this. Let us again have our two reservoirs T1 greater than T2. And now let us have to begin with two reversible engines working between them. All that we say is that they are reversible and let the efficiency of one engine be eta R1, the efficiency of the other engine be eta R2. Then the principal corollary of Carnot theorem, the most important one says that eta R1 must equal eta R2. And proving this is straight forward all that we have to do is use the Carnot theorem twice. For example, if you consider the engine R1 to be reversible and neglect the fact that engine R2 is also reversible consider R2 to be any engine. Then we will get the requirement that from Carnot theorem that eta R2 must be less than or equal to eta R1 because we are considering R1 to be reversible and neglecting the reversibility possibilities of R2. Now consider R2 to be a reversible engine which is as given and neglect the fact that R1 is also reversible. Then the engine you will get efficiency of R1 to be less than or equal to efficiency of R2. So in the first case we had efficiency of R2 less than or equal to that of R1. Looking at it the other way efficiency of R1 is less than or equal to that of R2. And hence the efficiencies of both these conditions will be satisfied only when efficiency of R1 is less than efficiency of R2 will be equal to efficiency of R2. And why is this corollary important because that leads to the statement that the efficiency of any reversible engine working between fixed T1 T2 is the same and underlining reversible engine and fixed T1 T2. Now is the same means what it is or it does not depend on the working details working and other details of the engine. That means materials, fluids, processes, processes, processes, processes, details, etc. Except that the requirement is the any engine must be a reversible engine and any engine must be working between fixed temperatures T1 and T2. Only then we can compare their efficiency and all such efficiencies will have to be the same. Now we can turn this around and say that this means that efficiency of a reversible engine working between fixed temperature is a function only of T1 and T2 and of nothing else. So efficiency of reversible engine working reversible engine depends only on the T1, T2 between which and notice that this is important. It does not depend on the working and other details particularly does not depend on materials, fluids, processes, details, anything. The requirement is that the engine must be reversible and hence we can say that the efficiency of such a reversible engine would be a function only between T1 and T2. And since this relation does not depend on materials, we can say that this relation is a basic thermodynamic relation and not a relation which is based on any materials or any process, etc. The only condition is remember that this R is the most important thing here. The engine must be reversible and must be working between two distinct temperatures as represented by two reservoirs. Now this brings us to a situation where using this thermodynamic relation use this relation to set up thermodynamic scales of temperature. Why? The advantage is such scales depend on the properties of any material. This is important. Why is it important? Because the temperature scales which we have set up so far, Celsius scale, Pyramid scale, Kelvin scale, they depend on properties of some material. For example, the classical Celsius scale depended on the properties of mercury and of course, glass because it was expected that the glass has much smaller expansion coefficient than mercury. The ideal gas Kelvin scale of temperature depends on an idealized and approximated gas called an ideal gas. So, we take a real gas, use it in that zone of state space where it behaves like an ideal gas and we can measure the Kelvin temperature, Kelvin ideal gas temperature or ideal gas temperature on the Kelvin scale. The attraction on the thermodynamic scales of temperature based on the fact which we will soon set up is that efficiency of a reversible engine depends only on the temperatures between which it works and on nothing else. Hence, if we set up a temperature scale which is based on the efficiency of reversible engines, we would have set up a scale of temperature which is independent of any material. But what is the disadvantage? We have said that look number one, the simpler of the two disadvantages or less complex of the two disadvantages is that we need two constant temperature reservoirs. We can approximately create them to a very good approximation. But we also need a reversible engine and we have said that a reversible processes that reversible processes and a reversible engines are things about which we can only think. These are thought processes and thought engines. These are not real life processes and real life engines. So, if we set up a scale of temperature based on the characteristic of reversible engines, that scale is only something with which we can think about. We cannot really implement in the practical practice. We will live with this disadvantage for the time being, but soon we will be able to show that the Kelvin scale based on the ideal gas equation of state has a very proper thermodynamic basis. But before we do that, let us look at the characteristics of thermodynamic scales of temperature. Let us do the following. We are now going to use the fact that eta r is a function only of T1 and T2 to set up a scale of temperature. And the idea would be like this, T2 is fixed as a reference. We will always need that. There is no such thing as absolute temperature. Temperature will always with respect to some reference and on some scale. Then T1 can be defined in terms of, that means fixed temperature of some fixed state T2, that will be our fixed point. Then to determine some other temperature T1, all that we do is run a reversible engine between the two. And the efficiency of that reversible engine can then be used to define our scale of temperature. That is the basic idea. We do the implementation as follows. Rather than talk of efficiencies, we talk of something else which we will measure to determine the efficiency. What we do is we go back to our, now we know that the efficiency of this engine is a function only of T1 and T2. The efficiency is defined as W by Q1. Using first law, W will be Q1 by Q2 by Q1 and this means 1 minus Q2 by Q1. And that means if the efficiency is a function only of T1 and T2, then my Q2 by Q1 will also be a function only of T1 and T2. So, I turn it around and say that this implies that the ratio of heat absorbed from T1 to heat rejected to T2 will be some function, say f is a general function to be defined as of T1 and T2. So, this gives us a more detailed idea. To set up a scale, define T2 and reference state. Actually I should say reference state and T2, reference state 2 and T2. Then define f as a function and to determine T1 major Q1 and Q2. And how do you measure Q1 and Q2? For this, you will need to set up a reversible 2T heat engine between the system whose temperature is to be measured and the reference system whose temperature is defined at T2. And if the two systems are not reservoirs, we will have to create a reservoir at the reference temperature T2 and create another reservoir at a reference temperature T1. Now, if this is so, the question arises is what type of function should F be? For that we do the following derivation. Let us say that we have three reservoirs T1, T2 and T3 at successively higher temperatures. That means T1 is greater than T2, T2 is greater than T3. Now, let us work a reversible engine R12 between T1 and T2. Let us let it absorb Q1 by from the reservoir at T2 and let it reject Q2 to the reservoir at T2. Let it produce some work say W12. Now, let us set up another reversible engine R23 between the reservoir at T2 and the reservoir at T3. And let us adjust its working that it absorbs Q2. Notice this, it absorbs Q2 from the reservoir at T2 and rejects Q3 to the reservoir at T3. Let it produce some work W23. Now, the question is suppose I now set up a reversible engine R13 which works between the reservoirs at T1 and T3. And if I set it up in such a way that it absorbs heat Q1 from the reservoir at T1 and rejects say Q3 prime to the reservoir at Q3. And then let us ask ourselves the question what is the relationship between Q3 and Q3 prime. A simple thought would hint to you that Q3 must be equal to Q3 prime. And you can even prove that it must be equal to Q3 prime because if you do not assume it to be otherwise, you will notice that you are violating the Carnot theorem or even the corollary of Carnot theorem. So show that this is I am leaving it as a homework to you, show that Q3 must be equal to Q3 prime. Now the consequence of this is that because Q3 is Q3 prime, I can now write Q1 by Q2 multiplied by Q2 by Q3 equal to Q1 by Q3 prime which is equal to Q3. This prime is not really necessary. And now we have seen that Q1 by Q2 the ratio of the heat transfers by a 2T reversible engine is a function only of the two temperatures T1 and T2. We can write this as the requirement for the function F. F should be such that F of T1 T2 multiplied by F of T2 T3 must be equal to F of T1 T3. Any function F will do provided it satisfies this relation. And obviously by looking at it, we can see that one function or one type of function which would satisfy this is if we consider F to be of the type, if F is of the type some function of T1 divided by some function of T2. Maybe there are other possibilities, but this is definitely one possibility and the simplest of these possibilities would be T1 by T2. So remember there is nothing special this is one possibility if you use if you hunt out some other type of function we can use that maybe it will be more complicated than a ratio definitely. And this is the simplest of the lot. We could have considered T1 square divided by T2 square or logarithm of T1 divided by logarithm of T2 exponential of T1 and exponential of T2 or sine of T1 sine of T2 any complicated function you can put. But let us keep the matter simple because if simple things lead to simple results and useful results nothing like it. So let us consider this to be the simplest possibility. So the question is why not use Q1 by Q2 equal to T1 by T2 and Q1 by Q2 for a reversible 2 T heat engine to set up a temperature scale. And this will be a thermodynamic scale because it will depend only on characteristics of the temperature and will not depend on the details of that engine whether it works on air, whether it works on water, whether it works on mercury or whether it works on paper. If you can make it work as a reversible 2 T heat engine, Cardo theorem says that its efficiency between two temperatures will depend only on those two temperatures and hence we are able to use it as a basis of this temperature scale. Now this is a restricted definition of a temperature scale the restrictions have come about or a specialization has come about because we have decided that f will be modeled like this and then we have selected the simplest of the possibilities that f T1 T2 would simply be a ratio of T1 by T2. And it turns out that this works beautifully and not only it turns out that this gives us a scale which is perfectly aligned with our Kelvin ideal gas scale and we will see the advantages of that later. Now how do you implement this scale? Now let us define some thermodynamic temperature scale, but remember still we have possibility of more than one. All that we do is we set up a reference system at reference state. It could be the triple point of water, but if you are so esoteric you could have triple point of ammonia, triple point of mercury, maybe triple point of lead depending on what is the most abundant and easily available material in your surroundings. Water is one of the most abundant and easily available material on earth. If you go to some other situation where something else is more abundant maybe you could use the triple point or some other fixed point based on that. We have to define its temperature Tf then this is a system whose temperature is to be measured. What we do if the system cannot be assumed during our experiment to behave like a reservoir, we create a reservoir at the same temperature at the system T that we can do because that reservoir across a diothermic partition will not allow any or will not have any heat interaction with our system. And then we set up a reversible 2 T heat engine between the two systems, the reference system at the reference state and the system whose temperature is to be measured. Let us say without any the direction here are immaterial it could either be both this way or it could both be this way. Let this interaction be Q reference and let this interaction be Q, the heat interaction. There will be some work done by the engine but we need not measure that or worry about it. What we do is if this is thermodynamically higher than the reference temperature, reference temperature Tf then Q will be absorbed from our system and Qr will be rejected to the reference system. If the system temperature is lower than the reference temperature then Qr will be absorbed from the reference system and Q will be rejected to the system whose temperature is to be measured. But anyway let the interactions be Q and Qr and then because Tr is defined and Qr and Qr measured, we extract our temperature T from this relation T by T ref is Q by Qr. Notice the similarity of this with the definition of our ideal gas Kelvin scale. There the idea was T by T ref was P v by P v ref. For Kelvin scale, we said the reference system was water, a system containing water at its triple point and for convenience, we had defined 273.16 Kelvin as its temperature, reference temperature. In a similar fashion, we have now defined thermodynamic temperature scale and of course, depending on whatever reference states are and what we define as T ref, we will have different scales. So, we will have to be specific about this to define a particular type of scale. But even then, we appreciate the fact that we require a reversible 2 T heat engine which can only be thought about not really implemented in practice. So, at this stage let me take after 1 hour a small break for 5 minutes. I will soon be back with you. Thank you.