 Good morning for day 5 and today is the big day, we are going to study the second law of thermodynamics. Now today's topic is second law. This is perhaps one of the most celebrated to some extent controversial law of thermodynamics and perhaps the most celebrated law among any law of physics. There are many reasons for this. One of the reasons is that there has always been confusion as to whether this is an exact law or an approximate law. Particularly, people who study the classical thermodynamics and dabble somewhat in microscopic or statistical thermodynamics where everything is about averages. They say any law including the second law will be valid on an average, but not exactly in each and every instant. Some people say that the molecular thermodynamics or statistical thermodynamics can derive the second law of thermodynamics. That also is not true. There are also premises involved which are equivalent to the second law of thermodynamics. You can derive second law of thermodynamics, but then you have to replace what we consider as a premise, the second law by some other premise. Many physicists and mathematically oriented physicists do not like the second law of thermodynamics because it is an inequality. The inequality in the second law as we derive it makes many people uncomfortable. They say this is not a complete law of physics yet and I fail to understand why they should insist that each and every law must be an equality. We will look at the second law of thermodynamics from the classical perspective. What we are going to do is we will study the background. For that, we will look at quickly revisit the first law and the zeroth law. We will see what they can do and what are their restrictions. What is it that they tell us and what is it that they do not tell us? In particular, when it comes to zeroth law, we will be looking at the status of temperature and then we will look at the need for an additional premise or an additional law. Then we will develop the second law of thermodynamics and just the way there were various ways of looking at the first law of thermodynamics, there are various ways at looking at the second law of thermodynamics. Many people have proposed a statement of the second law of thermodynamics. Each one in a slightly different way. At a gross level, they are equivalent but there are out of these. There are two methods which are considered reasonably robust. One method is that of Karate Odori whose formulation of the first law we have used. Unfortunately, his formulation of the second law is too mathematically rigorous and complex for us to appreciate, for us engineers to appreciate. But perhaps that happens to be, except for the formulation of Giles, the neatest formulation of the second law of thermodynamics. We will be using a formulation due to Kelvin, Kelvin's or Kelvin Planck's formulation. The reason for that it is robust enough for our purpose and it talks about engines and heat transfer and conversion of heat into work, cyclic processes with which we engineers, particularly mechanical engineers, we find ourselves on home ground. So, we will be using Kelvin's formulation. Then we will have a large number of definitions and we will derive many, many things. But we will do, we will note that the second law will help us do the following. First, it will provide us a basis for temperature, a thermodynamic basis for temperature. So, instead of an ideal gas scale of temperature or a Celsius scale of temperature, we will see how we can define a thermodynamic scale of temperature. Remember, Celsius temperature scale depends on a system, the mercury in glass thermometer. So, to some extent it depends on the properties of mercury and may be to a very small extent on the properties of glass. The ideal gas Kelvin scale of temperature or any ideal gas scale of temperature depends on the existence of a substance called an ideal gas. Why should that be so? If temperature and a scale of temperature is something basic in thermodynamics, then thermodynamics itself should come up and say this is the way one should define temperature and that definition should be independent of the properties of any substance, not of mercury, not of glass, not of two different metals which expand at different rates as temperature changes nor of any gas real or ideal. The second thing is will help us decide what is possible and what is not possible. It will help us put limits on processes, then it will help us derive relations between properties. For example, yesterday when we drew the T s diagram of sorry P v diagram of P t diagram, the phase diagram of water, we drew the line I will just show it here. Let us start from the triple point. Liquid vapor line goes to the north east, the solid vapor line goes to the south west. Then I said that the solid vapor solid liquid lines bends slightly towards the west, it is not vertical, it is not tilted towards the right, it is slightly tilted towards the left. So, for water it is something like this if I show it as a exaggeration. For many other fluids, it tilts to the right. It is not very obvious to us, but the property relations which we derive will tell us that for water it tilts to the left because ice floats on water. You try to tell this to a schoolboy, schoolboy realizes the two things, but it is unable to link the two up. That link is something which we will derive. If the solid phase has a lower density, then this line will tilt to the left. If the solid phase has a higher density, then this line will tilt to the right. Then for example, for an ideal gas the equation of state is p v equals r t that is Boyle's law. Then Joule hunted out another characteristic that the internal energy of an ideal gas is a function only of temperature. The property relations derived using the second law will tell us that if the equation of state is of the type p v equals r t, then its thermodynamically consistent and proper that u be depend only on temperature. So, thermodynamics by itself will give us such sometimes funny and unexpected relations between properties. Thermodynamics will not dictate what the properties are. It will not say that look your saturation pressure at this temperature should be like this, but it will say that look if your saturation pressure varies with temperature in this form. Some other property should vary with some third property in this form. So, such relations or restrictions it will put on various property relations. Now, let us look at what we may call limitations of first law and zeroth law. The first law talks about interactions. It also defines delta e and the final relation which we say is delta e is q minus w. So, long as this equation is satisfied, first law is satisfied. We multiply a process in which first law is satisfied, not multiply. Take a process in which first law is satisfied and change the signs of everything. Change the sign of q, change the sign of w and change the sign of delta e. The first law still remains satisfied, but quite often we know that one of these two processes only is possible. For example, you take a system containing a liquid, put a stirrer and stir it. If it is insulated adiabatic, then naturally the energy of the liquid will go up, its temperature will rise. We know this, but suppose I cool it. If I take stirrer work and let me show this with proper direction and q and I am maintaining delta e is 0. I am maintaining the temperature of water. We know that stirrer doing work on the system w less than 0, heat being lost to the surroundings q less than 0 and delta e equal to 0 is a possible process. We have seen this happen, but now we say that look we propose that in such a situation, the system is absorbing heat from the surroundings and it is stirring the stirrer and doing work on whatever is the system which was earlier stirring the or moving the stirrer. We know that this never happens or both satisfy the first law. First law does not say that this is possible, this is impossible, but we know from our experience that this never happens, this is not possible. Similarly, the fact that there are limits on processes, take say one metal block at say 80 degrees C, put it in contact with another metal block at say 20 degrees C. If they are of the same metal and have the same mass and if you keep them insulated from the rest of the world, then we know that the final temperature will be 50 degrees C for either block if you wait long enough and we will say there is a transfer of heat from one block to the other. So, temperature of one reduces, temperature of the other increases, but we know that if we keep the two blocks at 50 degrees C next to each other in an adiabatic system and wait long enough, the reverse is never going to happen. So, there is a direction of processes and not only that we also know that if we start with 80, 20 the process is going to stop when the temperature resets 50 degrees C for either. This 80 degrees C going to 40 degrees C and this 20 degrees C going to say 60 degrees C, this is not going to happen, but this is from the first law point of view. We need another law to tell us and to generalize the fact that this is not okay. We can give enough examples like this of the limitations of first law. Now, come to zeroth law. Zeroth law helps us define a useful thermodynamic property called temperature. So, the zeroth law, the first part of zeroth law, the existence of isothermal states finally, leads us to the idea that there is a useful property called temperature, but zeroth law stops there. It does not even tell us how to measure temperature properly. So, we created the science of thermometer, but there is no thermodynamic basis measurement of T. That means, no thermodynamic basis for any scale of temperature that we have come across so far or we have defined so far. That is one thing. Second thing is it does not tell us, does not help us define which is higher temperature and which is a lower temperature. It only tells us that T 1 is not equal to T 2. If there is a thermal interaction between two systems, zeroth law helps us say that the two temperatures are different because the heat interaction is taking place across a diathemic boundary. If heat interaction does not take place and zeroth law through the existence part asserts that if you take two systems A and B, you can always find out states of the two, at least one state of A and at least one state of B in such a way that say A 1 and B 1. So, that even if you allow them to interact across a diathemic partition, they will not interact simply enough, but that means the way we have defined temperature, the two temperatures are different, but which is higher and which is lower zeroth law does not tell us. Not only that, it does not say anything about the direction of heat transfer. Zeroth law only says that if you have a system A and if you have a system B and they are not isotherm, they are normal that means they interact with each other across a diathemic wall. Then the way we have defined temperature all that we can say is T A is not equal to T B, but we cannot say whether the heat transfer will be from A to B or from B to A. So, direction of Q not decided by zeroth. So, that is the status of temperature which has come out of zeroth law that it is a useful property, it is the same for it should have the same value for all states of a system or all states of all systems which do not interact with each other across a diathemic wall, but you ask zeroth law A which is higher temperature and which is lower temperature zeroth law says no, I do not know do not ask me which is higher and lower. And then you ask the guardian of zeroth law to tell us there are two systems which are a non isothermal the guardian will say yes. So, if you allow them to interact across a diathemic wall there will be a heat interaction, but if you ask in which direction will the heat interaction take place zeroth law guardian says no I do not know do not ask me zeroth law knows nothing about it. And that is the reason why we need a second law and that is why people started looking at it from the physics point of view. If you look at the history as I have already said the germination of the idea that there is some limit some restriction in nature on the behavior of things came from Karnaugh at least that is what we think may be there were the earlier philosophers whose work is not published or not recorded, but the known earliest ideas came from Karnaugh. He was a mine engineer and looked at and studied the behavior of steam engines that was his job manage steam engines which pump out steam from mines and move material in the mine power plants burning some wood coal any junk thing which came across. And naturally he got an idea of what is the efficiency of such an engine he related efficiency to the amount of work done which he could measure so much ton of ore to be brought up the mine shaft. So, raise of a weight that was the work done and to run the steam engine the amount of fuel burned you would have noticed that some engines burn less fuel for the same amount of work some engines burn more fuel for the amount of work done. And naturally for the fuel burn may not be Karnaugh, but the owner of the mine will have to shell out some money. So, naturally everybody is interested in doing the same amount of work, but burning the lesser and lesser amount of fuel so improving efficiency. So, Karnaugh started thinking about the ways of improving efficiency I do not know what the tricks he used to improve efficiency, but he started thinking as to why is it that efficiency cannot be improved. And if I improve the efficiency will I be able to I do not think it thought about 100 percent efficiency he said what would be the absolute limit of efficiency. If you look at that time the steam engines were even now reciprocating steam engines are not super efficient they have a poor efficiency at that time they had even poorer efficiency, because engineering and technology had not developed to the extent it has developed today. But the greatness of Karnaugh was that he had the ability of flights of fancy which came up with an idea which today we say we celebrate it as the Karnaugh theorem. We will prove it later today or I will tell you how to prove it you are senior students or teachers of thermodynamics. So, small small parts I will leave it to you as homework. It is statement today we know as the Karnaugh theorem at that time it was not a formalized theorem we will formalize it today by defining all the terms which will come up in this theorems. The Karnaugh theorem said that what Karnaugh said was that if you use the symbol efficiency of an engine he said that the efficiency of an engine can be maximized provided you do everything reversibly. The great contribution of Karnaugh is the idea of reversibility. We imagine that if you could have no pressure drop he could have no friction and he could have no temperature difference not unnecessarily heat the boiler vessel to produce steam. Then things would be reversible and he says then I will get the maximum efficiency of out of my engine. And he says so long as I cannot make it reversible the efficiency of my engine will be less than that of my reversible engine later on we added the equate. So, his idea that efficiency of an any engine can be can never exceed that of a reversible engine working under similar condition is today Karnaugh's theorem, but historically that is the first time anything pertaining to the second law of thermodynamics was thought about and quantified in a mathematical sense. Today we realize that Karnaugh's theorem we cannot accept Karnaugh's theorem easily as the basic statement of the second law of thermodynamics because efficiency of an engine reversibility many things have to be defined. So, after Karnaugh there were many people who looked at the second law and proposed statements formal statements of second law one was Clausius. The Clausius statement of second law goes something like this Clausius looked at refrigerators he said look let us have two systems one at T 1 another at T 2 and he said let on the ideal gas scale T 1 be greater than T 2. See all our scales of temperature Celsius Fahrenheit or ideal gas Kelvin they are set up in such a way that what we feel hot to our hand has a higher numerical value of temperature and what we feel cool like ice is given numerically a lower value of temperature. So, right from our childhood a higher temperature means higher number and hot stuff a lower temperature mean lower number and cold stuff is ingrained into our mind. So, this is higher and lower from that point of view not from any thermodynamic point of view and then he said that if I want a machine here which we today call the refrigerator which should do the job of extracting some heat from the low temperature system say amount q 2 and we want that heat to be if you extract heat from some system you must reject it or make it available to some other system. We want to make it available to the system at a higher temperature and let us say this is an engine. So, like steam engine it will work in cycles you know same thing it will do again and again then he said that I can create such an engine provided I provides work to it. So, it will consume work without consuming work it is not possible to move heat from a low temperature system to a high temperature system. Here work must be made available or the refrigerator will absorb work and that also means a corollary immediate corollary of this is heat by itself cannot flow from a lower temperature to a higher temperature. Now, this is a candidate statement for the second law of thermodynamics and all of us will appreciate that because we know it does not flow from a lower temperature to a higher temperature it flows only from a higher temperature to a lower temperature. But the weakness in the Clausius statement is this assumption that T 1 is higher than T 2 and there is no thermodynamic basis for this. So, the Clausius statement is considered a weak statement of the second law of thermodynamics and hence is not considered to be the candidate for the primary statement. Let me come to Kelvin statement later let us first look at the Karateodori statement. The Karateodori statement of the second law of thermodynamics is today considered with some minor restriction that he is considering almost continuous state space. Perhaps the neatest and mathematically robust statement of the second law of thermodynamics and but that statement is something like this. We understood what is meant by a state space we have understood what is meant by a quasi static process we have understood what is meant by an adiabatic process what Karateodori statement says is the following let me write x 1 x 2 or you can even write p v if you feel like I would prefer to write x 1 x 2. He says that take consider a state of any system and then he says that in the neighborhood of this state there are a few states which are not accessible from this given state by quasi static adiabatic processes what it means is if you start from a given state and execute as you wish processes which are quasi static and adiabatic you will reach only a certain part of the state space quasi static adiabatic process can take you here. But there will be other for example you cannot go here you cannot go here you cannot go here you cannot go here these are states which are not accessible and that is the basic idea for the essential idea of the Karateodori's form of the second law of thermodynamics it talks of non accessibility. So, remember the second law it is some sort of a negative if you go back even Karno theorem was negative it is not possible to exceed the efficiency of a reversible engine in Clausius it is not possible to transfer heat without doing any work from a low temperature system to a high temperature system. And Karateodori's form also says that in the neighborhood of any state there are states which are not accessible or you cannot reach those states by executing quasi static adiabatic processes. We will now come to the Kelvin statement of it was properly developed by Kelvin and Planck. So, we will say the Kelvin Planck statement of the second law K p in a simple sketch it can be said like this you take a system at some temperature T 1 and remember here we are not talking of higher and lower temperature because we are talking of a system at some state let the temperature be T 1. Then if I take another system which executes only cyclic processes I try to extract heat from this system and aim is to convert it completely into work the cyclic process during any cycle I want only two types of interaction heat interaction of heat absorption from the system which we are looking at and producing a positive amount of work that means raising a weight. And the Kelvin Planck statement says that this is not possible and we should understand this for what it is we may write it in different types of big wordy statements that you will find being done in any thermodynamics book that the simple idea here is it is not possible for us to set up a cyclic device for a device which executes only cyclic processes which will do only the following extract heat from a single system and produce a positive amount of work. This is the basic idea of the Kelvin Planck statement and now all other derivations everything else from the second law pertaining to the second law definition of higher and lower temperature thermodynamic scale of temperature maximum efficiency of an engine maximum coefficient of performance of a refrigerator the idea of entropy all the things will follow from this single statement the Kelvin Planck statement of the second law of thermodynamics. Now before we proceed further some definitions so that instead of saying something in 20 words we can say it in 2 or 3 words some definitions are needed from the understanding point of you some definitions are needed from the proper for proper quantification some definitions are needed only for simplification of arguments. The first definition is that of a heat engine and then we will also define its efficiency and then heat engine is a cyclic device which will produce a proper positive amount of work that is our heat engine we define it properly. We will also define its efficiency because that is needed to prove Carnot's theorem and to do many other things. So, this is for short form understanding this is for quantification and for simplicity of arguments we will define what is known as a thermal reservoir which we will quite often shorten only to a reservoir actually the full name is constant temperature or fixed temperature thermal energy reservoir but we will shorten it to thermal reservoir or simply reservoir. First the idea of a heat engine our definition of a heat engine symbol will be E and it will be a cyclic device that means anytime I try to work an engine say tap it or press a start button it will execute exactly one cycle or an integral number of cycles it will never stop in between. So, all processes of an engine which we will consider will be cyclic processes unless we are after developing the second law of thermodynamics completely if we want to look at the internal workings of an engine then we will split the cycle into its individual processes but so long as we are doing the arguments pertaining to the second law of thermodynamics an engine is something which is a cyclic device and the moment we try to do or make it work it will execute one cycle or two cycles or three cycles or no cycle. Because of this remember that it every time we try to do something for this engine delta E will always be 0 for any process and no change of state moment you try to execute a cycle and there is no change of state that is first requirement of an engine. The second requirement of engine is that it must produce a positive amount of work that means it must have the ability to help us raise the weight in a gravitational field or do something useful which is similar to that which we all agree is a work interaction and of course because delta E is 0 and we may say see some part of the cycle it may absorb work some part of the cycle it will do work but the net work has to be greater than 0. So, sometimes we will put the subscript net here and of course first law has to be satisfied so if delta E is 0 the net work done in 0 the net heat absorbed also has to be positive if this is positive this also has to be positive q net has to be greater than 0 w net has to be greater than 0 this is our idea of a heat engine. Now let us look at the idea of efficiency of heat engine now we look at the working of an engine for engine we will typically use a circle with a symbol E in it let sum up all work interactions and we know that work has to be positive net work. Now let us look at the details of the heat interaction and you will agree that during part of the cycle during some part of the cycle heat will be absorbed some other part of the cycle heat will be rejected. Let us separate out all those interactions in which heat is being absorbed from those interactions in which heat is being rejected I will show the heat transfer in the other direction. This thing we will say this is the amount of heat absorbed during the cycle and what comes out is the amount of heat rejected in the cycle. We are modifying our sign convention a bit this arrow should have been inside and the q rejected should have a negative number. But we will say that just for of algebraic simplicity and just to keep ourselves aligned with all the text books on thermodynamics we will now modify our sign convention for heat to say heat absorbed by the system here is positive and heat rejected will be noted as q rejected with a arrow outside. So, that the first law for an engine delta e any way it is 0 this net work will be or if you want you can even say w net is q absorbed minus q rejected and both are positive numbers. But this heat interaction is from the engine to some surrounding systems heat absorbed are all sum up of all interactions of the heat kind from some surrounding systems to the engine. Now with this thing the definition of the efficiency of a heat engine this is defined to be w net divided by q and if you apply first law this will turn out to be 1 minus q rejected divided by q this is the definition this is a consequence of using the definition as well as the first law. So, we have defined what is a heat engine for convenience we have defined what is the efficiency of a heat engine and while defining that we have defined w or w net q absorbed and q rejected at least for the purpose of thermodynamics we do not have to split w net into its components all we should be satisfied is that w net should be greater than 0 unless it is greater than 0 it is not an engine. So, net q has to be split into q absorbed and q rejected. Now we come to the third definition and that of a thermal reservoir this is required only for convenience we can derive everything without a thermal reservoir, but then with thermal reservoir we can get away with algebra without thermal reservoir we will have to work with the differential and integration and the calculus will become mathematics will become more complicated. The thermal reservoir is a thermodynamic system such that the finite amount of or from it does not change it is. So, the idea is you take away some amount of energy in the form of heat or you supply some energy in the form of heat the temperature does not change in actual practice you can create such things you know large bodies a large tank containing water heat it up slightly extract heat slightly the temperature is not going to change significantly. So, you can say it is approximately a thermal reservoir, but on the other hand you can make an exact thermal reservoir by a trick something like this you can have a system a cloth system containing say water comes steam and at a pressure of say 1 atmosphere what will be its temperature temperature will be 100 degree c and let us say it is half water half steam trinaceous fraction 0.5 and now I supply some amount of heat or I extract some amount of heat and allow it to expand and contract while the pressure is maintained at 1 atmosphere. What will happen? So, long as I do not extract such a large amount of heat or supply such a large amount of heat that all the steam condenses or all the water evaporates the temperature is going to remain at 100 degree c because it is going to move along that isobar in the wet zone pressure is maintained constant. So, temperature will also be maintained constant. So, this is a situation where I can take out a finite amount of heat or supply a finite amount of heat and still maintain the temperature of that system to be constant. Another way of doing this is what is done in engines you have a chamber in which you burn fuel by supplying enough amount of oxygen or air and you take out the exhaust extract the heat as required and you can maintain the temperature by burning just the right amount of fuel similarly, you could have a system containing a coil of cooling water and you can supply some amount of heat and supply enough flow of water at a low enough temperature. So, that the temperature on this surface is maintained constant. So, there are enough systems in practice in which you can create a thermal reservoir, but the issue is we do not have to create a thermal reservoir the thermal reservoir here will be used only for ease of argument and ease of mathematics. Now, after having done that after having defined these three entities heat engine its efficiency and a thermal reservoir. Now, let us get into the real scheme of things we will now define modify or qualify the definition of a heat engine by an engine a 1 T heat engine a 2 T heat engine etcetera and an N T heat engine 3 T 4 T 5 T this 1 T 2 T represents the number of heat reservoirs that an engine may interact with. For example, a 1 T heat engine will look like this it absorbs heat only from one reservoir at temperature say T 1 I think I forgot to mention when we talk of a reservoir here I can change the pressure and I can change the temperature at which the reservoir is maintained. Here I can change the amount of fuel and air and maintain the temperature at which the furnace is maintained I can change the temperature of water and its flow rate and change the temperature at which this surface is maintained. So, the moment you have a thermal reservoir it will be characterized by its temperature T. So, I can talk of a reservoir at temperature T 1 a reservoir at temperature T 2 and so on. So, a 1 T heat engine will look like this it interacts thermally only with a single thermal energy reservoir whereas, a 2 T heat engine will look like this T 1 T 2 work must be greater than 0 I will not write it every time, but that will be assume this is Q 1 and Q 2 could either be like this or like this anyone of them could be in and out here it has to be in here it could be in and out here it could be in and out this is a 2 T heat engine and N T heat engine will be 1 which produces work and interacts with T 1 T 2 a number of them T n. Now, immediately you will notice something the first thing we notice is that the Kelvin Planck statement says that it is not possible for us to have a 1 T heat engine. Not possible why because it violates Kelvin Planck statement and quite often we will come to this not possible because this violates the Kelvin Planck statements of statement of the second law and we will now simply shorten it to this is not possible because it violates the second law Kelvin Planck statement if it violates it will violate the second law. Now, we are at a stage where we can get into the nitty gritty of the development of the second law of the model we have come to the conclusion that 1 T heat engine not possible what about 2 T heat engines let us see now the fun begins. Now, the moment we say a 2 T heat engine we can show it like this T 1 T 2 and naturally if you want it to have 2 T that means we must have T 1 not equal to T 2 otherwise it is equivalent to a single heat engine because the energy can come from the same temperature and because we are saying T 1 not equal to T 2 we are not getting into any trap we are not saying which is higher end which is lower we are just saying that they are different and 0 th law will allow us to check whether they are different or not just allow them to interact across a diathermic wall if there is a heat interaction some way or the other we say that T 1 is not equal to T 2. Now, let us assume that because W is positive the net absorption of heat from T 1 and T 2 has to be positive, but it is possible that that means at least 1 of them must supply heat to this engine the other one can supply or the engine can reject some heat to T 2. Let us assume that the heat absorbed from Q 1 is positive and let us assume that the heat absorbed from Q 2 is also positive the question we ask now is this possible and now we will demonstrate something funny we will go like this we say that if it is not possible then we must be able to show that it violates the second law and our trick will always be this by simple means can we argue out that this is equivalent to a 1 T heat engine if we can show that it is equivalent to a 1 T heat engine then naturally it violates the second law of thermodynamics. We start by saying that T 1 is not equal to T 2 that means T 1 and T 2 are not isothermal reservoirs and hence if I allow them to interact with each other not through the engine directly there will be heat transfer either from T 1 to T 2 or from T 2 to T 1. Let us assume and we are not losing any generality here let us assume that the when we allow direct interaction between T 1 and T 2 the interaction takes place like this. From T 1 to T 2 and this would be through some means of conduction or radiation and that area or that conductance we can always adjust. So, that what comes from here to here is Q 2 and if we do that what we would have set up is an engine like this an engine which does a net amount of work and which has the following from 1 to T 2. From a reservoir at T 1 it absorbs heat Q 1 and from T 1 through an intermediary system T 2 it absorbs heat Q 2. Now, I can notice that this reservoir absorbs Q 2 and rejects Q 2. So, essentially it does nothing I might as well take it out of our contention or I can say that this whole thing is now my extended engine and it is working only by absorbing energy from this reservoir at T 1. So, that means this is equivalent to a 1 T heat engine that means this violates the Kelvin Planck statements of the second law and that means our assumption that a 2 T heat engine can work by absorbing heat from 1 reservoir and absorbing heat also absorbing heat from another reservoir at a different temperature is not possible. Look at our argument all that we have done is used 0th law and the fact that T 1 is not equal to T 2 thus making it a 2 T heat engine to assert that heat will either flow from T 1 to T 2 or from T 2 to T 1. I have assumed that let it flow from T 1 to T 2 and said that it violates the second law, but you can also assume it to go from T 2 to T 1 and still show that it violates the second law that is the first item of homework for you. But for us to realize now is that we cannot now assume that for a 2 T heat engine the engine will absorb heat from each one of the two reservoirs that is not possible. So, now let us check the other possibility let us say that we have the two reservoirs T 1 and T 2 we have our engine E which produces work and let us say that I am not committing the same mistake which we did earlier and assuming that it absorbs from both reservoirs let us say it absorbs from T 1 and rejects to T 2 and let us say that as earlier because it is a 2 T heat engine T 1 is not equal to T 2. Now, remember here it is absorbing here it is rejecting. So, now does it violate second law you will notice that if you assume that when you connect T 1 and T 2 the heat flow is like this nothing goes wrong because anyway this is absorbing and this also is going from T 1 to T 2. So, we are not in a position to violate the second law what about the other way round we are on to something here. So, if you say this is Q 3 greater than 0 is no violation of second law. Now, what about the other way round let us say that maybe I should go on the next page I have T 1 I have T 2 I have my engine E which produces W T 1 is not equal to T 2 and the engine is working in such a way that it is absorbing Q 1 it is absorbing Q rejecting Q 2 to T 2. Now, let me say that since T 1 is not equal to T 2 the 0th law said that look one way at least is possible heat will go from T 1 to T 2 if you allow it to go directly that means allow them to interact across a diatomic partition or it will go from T 2 to T 1. We said let it go from T 1 to T 2 and any amount of Q 3 goes from T 1 to T 2 we say that look we are not getting into any violation of second law. I can extend this engine all that will happen is its efficiency will be lower and lower. But now let us say and let me assume that when the engine is working like this and if I allow this system 2 and system 1 to interact across a diatomic partition let there be a heat flow Q 3. Now, the question is this possible and you will immediately now realize that by adjusting the heat flow Q 3 if I make this Q 3 equal to Q 2 I am getting into trouble why am I getting into trouble because now I notice that this system absorbs it Q 2 and rejects that Q 2 to T 1 that means if I consider this as my cyclic device the state here is not changing anyway this is a cyclic device. So, the state of this is not changing state of this will not change because it is absorbing Q 2 and rejecting Q 2 and this totally device it is such that it absorbs Q 1 minus Q 2 from the reservoir at T 1 and produces a positive work W. So, this is equivalent to a 1 T heat engine which violates the Kelvin Planck statement of the second law by the way we should mention here that the second law of thermodynamics the word adjective second law is not unique to thermodynamics we have Newton's second law of motion Kepler's second law of planetary motion of course, that is not a fundamental law that is derivable from Newton's law of motion and the law of gravitation. But the second law of thermodynamics has such a impressive status such a heavy weight associated with it that anywhere whether it is an argument in engineering argument in physics chemistry or even in philosophy when you mention second law nobody will ask you what second law you are talking about by default it is assumed to be the second law of thermodynamics. If you want to say that you are talking about the second law of motion of Newton you will have to say so, but if you simply say second law everybody by default will assume the second law of thermodynamics anyway we are studying thermodynamics. So, our second law is definitely the second law of thermodynamics. Now what does this mean this violates that means what we have derived using the Kelvin Planck statement and simple arguments and the zeroth law of thermodynamics is the following is that if an engine works like this absorbing heat from a reservoir at T 1 rejecting heat to another reservoir at T 2 such that T 1 is not equal to T 2 then a direct transfer of heat from T 1 to T 2 is whereas, a direct transfer of heat let me write this as q 1 to 2 write this as q 2 to 1 is not possible and now you will say many of you will jump and many of my students jump. So, but sirs that is that is because now T 2 is lower temperature than T 1, but remember we have not defined T 2 to be a lower temperature than T 1 all that we have said is let T 1 be not equal to T 2 and we have already shown that it is not possible to have a 2 T heat engine absorbing heat from both system 1 and system 2, but this seems to be possible all that we have shown is that if this is possible there is no violation with this and this, but this and this is impossible of course this gives us an idea of what a lower temperature could be defined as and what a higher temperature could be defined as.