 But let me now spend time on second law and property relations. Now second law is perhaps the most complex part of thermodynamics and it is so because unlike the other laws of physics, second law has something different. All other laws of physics end up with giving you an equation or state something which will happen. For example, Newton's first law of mechanics says if you do not have any force acting, a body will continue to move in a uniform velocity in its original direction or if it is at rest it will remain at rest. Newton's law of second law of motion gives an equation, force is mass into acceleration. Second law gives vectorially action equals reaction with a negative sign equal and opposite and so on. So do our laws of momentum conservation and laws of energy conservation which is the first law of thermodynamics. 0th law is slightly different but it talks of existence and transitivity properties. The second law is a totally odd law and in fact many physicists are also not happy about it because this law is a negative, it says something cannot happen. Mathematically it gives rise to an inequality and physicists being comfortable in the mathematical domain are uncomfortable with the second law of thermodynamics. In fact there are some great physicists who are very uncomfortable with thermodynamics etc. But let us forget that and let us say that look. 0th law defines delta E defines Q provides conservation of energy, suppose to represent conservation of energy. When we says that look energy is always conserved that means the energy of two systems if they do not interact with any other systems will be 0 will not change. What does 0th law do? 0th law defines or helps us define temperature. Now if you look at these laws there are a few issues which are still unsolved. If you look at the first law we have Q equals delta E plus W. You specify a problem in which part of these three terms are specified the remaining term can be computed out. Then you specify the same problem with everything made negative. Negative of this also is a balancing equation. So it does not tell us whether the given process is possible or the reverse of that or the inverse of that is possible. But we know from our experience of real life situation that certain processes take place only in one direction. Certain processes cannot be made to occur at all. When it comes to 0th law we have a temperature but 0th law does not tell us which is a higher temperature which is a lower temperature. Using scales of temperature we have given them numerical values but we say that look 80 Celsius is higher than 40 Celsius but that is because the Celsius scale is defined like that. There is nothing great thermodynamically about 80 Celsius being higher or 40 Celsius being lower. And the history exists that Celsius wanted its temperature to be 100 C at ice point and 0 C at steam point thermodynamically nothing would have gone wrong except that conversion would have been more complicated rather than T Kelvin being T Celsius plus something it could have been some complicated term minus T Celsius and things like that. So we need something to tell us that some processes are possible, some processes are impossible. We need something to tell us why certain processes take place only in one direction. We need something to tell us which is a higher temperature and a lower temperature and why from a thermodynamic point of view. How do we define a higher and lower temperature? And we need, see 0th law gave us only the basis for temperature as a property it does not tell us how to quantify it from a thermodynamic basis. 0th law says you label isotherms and that is where my job ends. So we need a proper basis for thermometer and all this is done by the second law. So the second law we will do the following, it will put limits on processes, possibility or impossibility of processes, it will provide a thermodynamic basis for scales of temperature. Actually our Celsius scale depended on properties of glass and mercury, the ideal gas Kelvin scale depended on some imagined ideal gas approximation of a real gas. But why should a scale of temperature depend on the property of some material, you go to some place where maybe on some other planet where no mercury is available, no glass is available, no gas is available, everything is solid or everything is plasma. How do we explain to them what is meant by Celsius temperature with Kelvin temperature? Then it allows us to derive all sorts of property relations, funny esoteric property relations. All this is done by the second law of thermodynamics. So it is only of the say, after the second law of thermodynamics that the status of temperature in thermodynamics will be complete, although status the temperature is defined by the zeroth law, it just defines it, gives us an idea of what it is all about but does not do anything more than that. Historically if you see the way back in history, it was Karnu who first said something which finally we today call the second law of thermodynamics. He was followed by in some order Clausius and followed by Kelvin who developed it further. There are many others who contributed but finally it was Karateodori the mathematician who put the whole thing on a proper mathematical footing. Although we appreciated Karateodori's form of the first law of thermodynamics, Karateodori's second law of thermodynamics, although it is understandable the mathematics required to derive useful relations out of it is very complicated. So at the end we will see what is the form of Karateodori's for the second law of thermodynamics but we will base our ideas on the more classical statement of the second law. So we will consider our primary statement of the second law to be what is known as the Kelvin Planck statement in short Kp. What the Kelvin Planck statement says it is simply the following. You have a system, let that system execute a process such that there is no change of state, cyclic maybe and then let it execute, imagine that executes a positive amount of work and while doing this we have some other system from which it absorbs heat. Of course since there is no change of state that means delta E is 0 for our system and if you apply the first law to this system that would mean q equals w and hence it should be greater than 0 and the Kelvin Planck statement says that such a thing is not possible. It better to show this in this graphical manner rather than says that there cannot be a system which you know while cooling somebody has no other effect than providing a net positive amount of work. Although it is equivalent to this I always prefer to show something in terms of our thermodynamic scheme of things. So you show a system it says there should be no change of state that means either there is no change of state or it is executing cyclic processes, it is allowed to absorb heat from a system, it is allowed to do positive amount of work. If we say we want to set up such a scheme then the Kelvin Planck statement says that this is not possible. This is going to be our primary statement of the second law of thermodynamics and why do we call it a primary statement is for this reason is that we are talking of just one system and another system. System here is executing cyclic device, this is some other system with which it is interacting by means of a heat interaction. We are not talking of any temperatures here, temperatures will come in later and we are only talking of directions, we are not talking of magnitudes of work. The function of work here work done by the system heat absorbed by the system Kelvin Planck statement does not prohibit a situation where work is absorbed by the system where W is negative and this Q is also negative Kelvin Planck statement does not say anything about it and this in many textbooks is also the one of the primary states of the second law. There are other statements of the second law for example immediately after this many textbooks will talk about the Clausius statement of the second law of thermodynamics which says that you cannot have a cyclic device whose only effect will be to absorb heat from a low temperature system and reject it to a high temperature system, there will be no work in there. We do not look at it as a primary and a good statement of second law because it talks of a higher temperature and a lower temperature and so far using either the first law or the zeroth law we have not been able to define a higher temperature or a lower temperature. So if you ask someone what is higher temperature then you can say oh it is on the Kelvin scale then you will say why Kelvin scale what is so special about it, there is no answer to that Kelvin scale requires existence of an ideal gas and thermodynamics is basic principle. The principles of thermodynamics should not depend on the properties of any material or any system. Now based on this Kelvin Planck statement we will now be able to derive almost everything else and let us see how. Just to simplify matters, so one to simplify matters, two to replace calculus and integrals by algebraic relations and to be in consonance with a large amount of text books which present it in a very simplified algebraic passion, we will define some things by which our further progress will be simple. So although the progress will be simple remember that we need not have made these assumptions all that would happen is wherever we write Q we will have to write DQ and integral of DQ and whenever we write T1 we will have to maybe integral of DT and things like that we will have to do. So we will define what is known as a heat engine then we will define its efficiency and we will define what is known as a thermal reservoir or constant temperature or identified temperature reservoir and then we will define a 2T heat engine and all that. Heat engine you all know is a cyclic device by that means the moment we tap an engine try to start it it will execute exactly an integer number of cycles. In our analysis you cannot stop an engine in between once you start it it will execute a cycle and then come. If you want it to execute more than one cycle it will execute 2 cycles 3 cycles but it will never execute 1 and a half cycle or 3 fourth of a cycle or pi cycles or anything like that. Though one of the requirement of an engine is that it should produce a positive work output you can say per cycle per unit time that is left to you. For this naturally if it has to provide positive work output it has to absorb a positive amount of work. So this is the minimal realization of an engine. This is the net may not be from one system it could be from more than one system some may be absorption some may be rejection how much should be absorption rejection that we will see later. Now what we will do is we will define the efficiency of such an engine this is known as the thermal efficiency. For defining this what do we do is we take the same engine which is a cyclic device. At this stage we need not worry about splitting W into various components but what we will do is the cycle will consist of various processes and process elements. During every process element we check whether the engine is absorbing heat or rejecting heat. Some processes it may absorb it, some processes it may reject it. What we do is all processes in which it absorbs heat we will call it Q plus or Q absorbed. During some processes the heat interaction will be from the engine to some other systems it is possible. So this will be Q rejected from the system to some other system. So our first law here says W will be Q net and since here the Q net has been split into an absorbed part and a rejected part. The first law would say this would be Q absorbed minus Q rejected that is this is all because our delta E of the engine is 0. This E is for energy this E is a short form for engine and then we define the efficiency of this engine as W divided by Q absorbed. Sometimes this W because some W may be positive some W may be negative this is written as W net does not matter you can use W net if you are more comfortable. Notice that if Q rejected is 0 the efficiency of such an engine would be 1 or 100 percent but it automatically says that if Q rejected is 0 then this engine it is something which violates the Kelvin-Planck statement of the second law of thermodynamics. So having defined efficiency we can say that the Kelvin-Planck statement implies that the efficiency of any engine cannot be equal to 1 or 100 percent and that means efficiency of any engine has to be less than 1 or less than 100 percent. This is the first consequence. Now again from historical we define what is known as a reservoir which is a short form for thermal energy reservoir or identified temperature energy reservoir ITER. This is a device used as a prop. This is a system large enough and with such characteristic that any finite amount of heat absorbed or rejected does not change its temperature and we know that practically we can create such system. The requirement is a finite amount of heat transfer if you want a larger amount of heat transfer you will have to have a still larger system here. Such a system is known as a thermal energy reservoir and it helps us in our derivations because if it were not large enough then even a small amount of heat absorbed or rejected would change its temperature. So as the heat interaction takes place we will have to keep track of its temperature and integrate it over the process. This allows us to characterize each reservoir by its temperature T. So we can now talk about a reservoir at T1 or another reservoir at T2 and so on rather than a system with initial temperature T as the process takes place T will either rise or lower. We do not have to worry about it. In our arguments things become simple and things become algebraic. So we remain in the algebra domain rather than the calculus domain. Then we notice that we are already in the domain of what is possible and what is not possible. The second law starts by the Kelvin-Planck statement which says such a cyclic device is not possible. Then we defined an heat engine and its efficiency and we immediately concluded that no heat engine can have an efficiency of 1 so the efficiency should always be less than 1. And then for further ease of arguments we defined a thermal reservoir. Then we define what is known as a 2T heat engine. As all of us know is a heat engine working between two distinct thermal reservoirs. By distinct we mean we do not worry about which is higher which is lower. We have a heat interaction here, we have a heat interaction here and we have a work interaction. Such a heat engine is known as a 2T heat engine. Now the first thing which immediately comes to your mind is, is it possible for us to have a 2T heat engine working like this, I never said heat rejection is 0 is impossible. Whatever we have done so far, can you logically argue out that this is not possible? You have to show that it violates the Kelvin-Planck statement. How will you show that it violates the Kelvin-Planck statement? We have not yet talked about low temperature, high temperature because there are two ways of doing it. One way is to put it like this. Then you say that just look at this, this is an engine which produces work and it absorbs from some systems positive amount of heat, there is no nothing else. This happens to be not possible because it violates the second law of thermodynamics, I mean the Kelvin-Planck statement. Another more robust way of doing this is to show that look the two temperatures do not matter because we have said that T1 is not equal to T2. We then say that since T1 is not equal to T2, the zeroth law says that when allowed to interact across a diathermic wall, there will be a heat interaction, zeroth law does not tell us in which direction, but it tells us there will be a heat interaction. So all that we say is let us say this, this is one side. On the other side we will say that look, when we allow these two reservoirs to interact with each other because the temperatures are different, let us say the heat interaction is from here to here. Then for the engine we are given that this is Q1 and this is Q2. Now we invoke one of our earlier premises that thermodynamics is scale independent. So we can always adjust this heat interaction to be equal to Q2, thermodynamics does not prevent that. We can always adjust the scale of our interactions and scale of our existence. So this becomes equivalent to a single system T1 providing energy to the engine E. The two streams, one directly Q1 and one indirectly Q2 through some intermediate reservoir here. Now if you consider this together as an extended engine, you will say anyway this is a cyclic device. So every cycle it does not change its state and then you say this reservoir also does not come into picture because it is absorbing Q2 and it is rejecting Q2. So it is also not undergoing any change of state. Consequently this becomes an extended engine and this definitely violates Kelvin Planck statement. Anyway whichever argument you do, it is now clear that a 2T heat engine cannot be of this type absorbing from two reservoirs. So that means this implies that a 2T heat engine must be of the type T1 in this direction. So first law becomes and it automatically means that, sorry less than 1. Then you are back to this, less than 0 means you are back to this situation and if you say it is 0 then you are back to this situation, the original situation, this situation. So whichever way finally all those options considered finally you will say that a 2T heat engine must be of this type. Now let us look at this. At this stage we cannot say that look T1 is higher and T2 is lower. Our job now is to go on exploring this further to see whether we can arrange temperatures in some order so that one end of that arrangement we could say this is the hot end, the other end of the arrangement we could say this is the cold end. Let us see how to do this and for that I will show you just one hint but then I will leave it to you a number of exercises which you can do and these are very enjoyable exercises because simple logic works. You show that suppose there is an engine T1 not equal to T2 of this. Ask yourself the question, I have these two reservoirs and I allow them to interact directly with each other across a diatomic partition. What will happen? The two temperatures are different so there will be a heat interaction, will the heat interaction be this way or will the heat interaction be this way, which direction? The temperatures are different, no doubt about it because unless the temperature are different this engine cannot work. We have not defined higher lower yet. I will leave it, those who say both are possible are not right, I will leave it as an exercise to you, you should be able to quickly do it during the T time. We will show that if you assume this to be 2 then this with this violates Kp because it is simple, if this is true we will adjust this to be equal to Q2 and then this engine plus that Q2 will give you a 1T heat engine but if you assume this to be true then it does not violate because all that will increase, it will increase the magnitude of Q1, it will increase the magnitude of Q2 but both reservoirs will still be in action, whatever be the magnitude of this Q, consequently this violates Kp, if this is given to be true. This means that if this is so, that means if an engine works like this then if you allow direct heat transfer this is possible and not, now extending this I will ask you to do the following exercises, show that if this is possible that engine works by absorbing Q1 from T1 and rejecting Q2 to T2 and of course this work has to be greater than 0 otherwise it cannot be an engine then it is not possible for us to have an engine which produces work by absorbing some amount of heat from Q2 from T2 and rejecting some amount of heat to T1 that means we have T1 and T2 at two distinct temperatures and we say let an engine be working by absorbing heat from T1 and rejecting it to reservoir 2. Then arises that can I have another engine which absorbs heat from reservoir T2 and reject heat to reservoir T1 and produces some positive amount of work, by similar argument in about 5 to 10 minutes you should be able to show that the combination violates Kelvin Planck statement, so the combination is not possible since one of the combination is said to be true the other must be false, so that also means if this is true the inverse of this is not possible. Then we come to a very interesting thing this is very important and you should also be able to derive it, we will show that let there be three reservoirs such that T1 is not equal to T2, T2 is not equal to T3 and T3 is not equal to T1 that is all distinct then let T1 and T2 be such that between T1 and T2 an engine can work, let us say this is E1 and the heat interactions are in the direction shown and let it also be possible because T2 and T3 are different, let T2 and T3 be such that an engine let me say this is EA, this is EB, this is WA, this is WB, this is between T1 and T2, this is between T2 and T3 given that means if this and this. Now the question is T1 and T3 are also distinct, so there should be possible for me, it should be possible for me to run an engine between these two. The question is we know that given two reservoirs at two distinct temperatures engine will work only in one mode absorbing one heat reserve absorbing it from one of them and rejecting it to the other, it cannot be working the other way. Now I have made it complicated I have two engines one between T1 and T2 distinct another is between T2 and T3 also distinct temperatures and also I have given that T3 is not equal to T1, so there must be an engine possible to run between T1 and T3 which would be the direction of heat transfer, heat absorbed will be from T1 or heat absorbed will be from T3, why? What happens if we absorbed heat from T3? You can show that if you assume this combination of these three will lead to a violation of the Kelvin Planck statement, since then we say that since this is given to be true, this is given to be true the only falsity will be this, so that means this implies that if such an engine has to work, it has to work only in this mode and not in any other mode, let me call this Q1c, this is Q3c. Now we have seen earlier from here that if an engine works between T1 and T2 by absorbing at T1 and rejecting to T2 then a direct transfer of heat will only be from T1 to T2 and not from T2 to T1. So here this is equivalent to saying and if you want you can show that if T1 and T2 are such that when allowed to transfer heat, heat is transferred only from T1 to T2 and when allowed to transfer heat, heat is transferable only from T2 to T3 then you can show that when T1 is allowed to transfer it directly to T3 or exchange is allowed, it will only be between T1 and T3, T1 to T3 and not T2 to T1. If heat transfer ducts stays placed from T1 to T3 to T1, you can show that it violates Kelvin Planck statement. Extending this you can show that if you have a number of reservoirs say T1, T2, T3, T4 and let us say that we have arranged them in such a way that it is possible to transfer heat from T1 to T2 across a diathermic wall. These are distinct temperature. It is possible to transfer from T2 to T3, it is possible to transfer from T3 to T4. Now suppose I have I try to transfer from T1 to T3, what happens? Which way will it be possible? It will be possible only from T1 to T3 and not from T3 to T1. So if I try interacting T1 and T3, it will be like this and not the other way round. Similarly, if I try to interact between T1 and T4, it will be like this and not the other way round. And same thing you can extend, if I want to interact T2 to T4, it will only be like this and not the other way round. Now suppose I have another T0, I bring a reservoir. By doing appropriate experiment, I will be able to place it either here or here or here or here or even beyond this. Because when will I put it here? When I notice that it is able to transfer heat to T1 and not absorb from T1. And then I will be able to say that on itself it will transfer heat to T2, also to T3, also to T4. If I find that it is absorbing heat from T1 and rejecting heat to T2, I will put it here, locate it here in between these two. That will be the proper hierarchy. If I find that it absorbs from these two and rejects to these two, I will put it here. In fact, I need not do the whole experiment. If I notice that it absorbs from this and rejects to this, I will put it here. And then I will be able to show that it will necessarily absorb from here and reject here, okay. But do you understand that the idea is that given reservoirs distinct temperatures can always be arranged, say left to right, such that Q is greater than 0 from a reservoir at left to reservoir at right. Now I am talking of reservoirs, but instead of reservoirs and direct heat transfer, you could have engines in between them. Suppose after arranging this, you take a 2T heat engine and link it between a reservoir between two reservoirs. From which reservoir will it absorb heat? One at the left. To which reservoir it will reject heat? Always the one at the right. Whichever pair you choose, such arrangement we can always make. What this tells us after all these arguments in a student's class, I go through these arguments for over and over, but I understand that we are short in time, so I am leaving this to you as a homework. But this arrangement indicates that there is a hierarchy of temperatures and this tells us that using this we can define a higher value of temperature and a lower value of temperature. How do we do that? We now have going back to this, we have arranged our reservoirs from left to right, like real line, one after another. Arranging people in order of their heights for example. So that if you put a food rule between the heads of two people, it will always be slanted from the higher person to the taller person to the shorter person. Take any pair, it will always be tilted in that direction. So the tilt in that direction you can consider is as the heat flow. So once if you have arranged in such a way that heat flow is always from the reservoir at left to the reservoir at right. Traditionally which reservoir we say is at the higher temperature? The one at the left. Which reservoir we say is at a lower temperature? The one at the right. So because the right is always the recipient, the left is always the donor. So this is something like a left handed compliment to the right hand. So our definition based on all this argument is T1 is greater than T2 if and only if that means it is equivalent to saying that when allowed that is across diatomic wall, Q will always be from the reservoir at T1 to the reservoir at T2 or if there is an engine, it will always absorb heat from T1, reject heat to T2 and produce a positive amount of work. This is the definition of a higher temperature and a lower temperature. So look at our earlier school bookish argument. What was temperature? Degree of hotness. For us temperature is a label of isotope. First law Q is delta E plus W. So measure delta E during that process, measure W during that process, delta E plus W is Q. This is the principle. Implementing it may be difficult but this is the principle. And how do you determine delta E for that process? Actually one argument and one question I have not yet been asked. We said that look to determine delta E between point 1 and 2, execute an adiabatic process between 1 and 2 so that only work is done. Measure the work and that will be your delta E with a negative sign. Question arises is given two states of a system 1 and 2. Is it always possible to execute an adiabatic process from 1 to 2? That is a question which has not yet been answered. If answered you should say the answer is this. Well at this stage we cannot assert that it is always possible to have an adiabatic process from state 1 to state 2. But after studying second law and we say we are studying second law later because we have to go from lecture to lecture. I cannot tell you everything in one shot. After studying second law we will realize that given two states of a system 1 and 2, an adiabatic process is possible either from 1 or 2 or from 2 to 1 or both. So the second law asserts that if you find that it is not possible to execute an adiabatic process from 1 to 2, an adiabatic process will definitely be executable from 2 to 1. And in that case you calculate E21 and E12 is minus E21 because delta E just change in some property. So delta E21 is minus delta E12. End of this second law it will be clear that from 1 to 2 or 2 to 1 at least one way an adiabatic process is executable. For some special processes, special pairs of states it may be executable in either direction but any pair of state adiabatic process is executable in at least one direction. We will come to that. Only if you one adiabatic passing through a point. From a point, there are illustrations in your example. From a point you can have an adiabatic isowaric process. You can have an adiabatic isochoric constant volume process. So the adiabatic line is not unique line. Adiabatic means work transfer only. Which way you do that work is left to you. So the details of the process are left to you. But then if it is a constant interpressual process, don't you think some heat transfer What did you say? It is a constant entropy process. I did not say that. I mean adiabatic process you said can be an isobaric process. Yes, can be. But isobaric process demands some heat transfer to take me. No, no it does not. Solve the exercises in F1. There is an isobaric adiabatic process there. There is an isochoric adiabatic process there. If you want I can have an isothermal adiabatic process. But why I say one of the second law statement is there are points around a given point which cannot be accessed by adiabatic processes. That is one of Karathiya Dore's techniques. Yes, but that does not mean an adiabatic process is unique. Hello, but I am again saying you said if you are given any two points 1 and 2, we can have an adiabatic process connecting them in either or other direction. Yes, in one direction. May not be both. But if one direction is possible, see if one other direction how is it possible then if one point cannot be reached from the other point. So second point can be reached from the first point. Then this way I mean if it is connecting by an adiabatic process, direction need not be worried about. See what this is essentially Karathiya Dore's statement. Karathiya Dore's statement says that given two pairs of points using an adiabatic process either two is accessible from one or one is accessible from two. That is all he says. Possibly in one direction the other direction also should be possible. No, not necessarily. That is the fun of second law. Second law says processes are possible only in one direction. This is an implementation of that. I can stir a liquid but if I put in a stirrer I cannot ask the liquid to stir it. No adiabatic reversible isentropic are all different. Let me come to this that will soon be clear. Sir, I have one question. Second, both laws are negative statements. Second law is negative statement. Second that is two statements. If you take Clash's statement also. I am not considering Clash's statement. Kelvin-Planck's statement. Only Kelvin-Planck's statement is a negative statement and most of the deductions what we are doing is a logical deduction. Is there any, there is no mathematical proof for this? This is mathematics. Only thing is the mathematics is simplified because of our definition of the thermal reservoir. If we do not do thermal reservoir all these will be differential mathematics. That is all. Differentials and integrals will get bogged down by that. That is the only difference. If you want it can be done. And Karateodori's papers and works are all in all those details. Karateodori does not work about, worry about any reservoir or anything like that. Full mathematics. But that is why we end up into differential geometry and all that. Why get into those complications? The principles remain the same. Second law is system obeys second law. A system obeys first law. Second law is not a system. So second law is a statement of a law. So a system has to obey first law. A system has to obey second law. And a pair of system have to obey 0th law because which only says that they may be at states where they may not interact across a diacritic wall or they may not. So now this is our definition of a higher temperature and a lower temperature. So if a student after this asks you what do you mean by saying that the temperature of this bottle, water in this bottle is higher than the temperature of water in this bottle. The simplest way to say is allow them to interact across a diathermic wall that means put in thermal contact with each other. The heat interaction will be from this system to this system. Hence this system is at a higher temperature. Now after this, this is qualitative. Qualitative things we have come across is now there is a hierarchy of temperature and now given two temperatures we can always determine whether the two temperatures are equal for which we needed only 0th law or one of them is higher than the other. If so which is the higher one that is the second law.