 We are now perhaps entering the most interesting part of thermodynamics and that is the second law. Of course, I can write off thermodynamics below this, but the second law of thermodynamics has a very special status in physics. Any person involved with physics, physicist, chemist will vouch to the fact that the second law of thermodynamics is the most important of the laws which they use. It is an odd law. Most of the other laws of thermodynamics, other laws of physics including laws of thermodynamics tell you that something is equal to something. The first law says there is something called energy and we know that delta E is Q minus W. The zeroth law says that there is something useful called temperature. And when you two systems are in states such that in spite of providing a diatomic partition between them, they refuse to have any heat interaction. That means they have the same temperature. We define them to have the same temperature. Second law unlike all these, finally ends up in an inequality. We define something for equality, but finally everything ends up in inequality. Delta S for an isolated system or delta S for an adiabatic system is greater than or equal to 0. We talk of possibilities and impossibilities. So second law unlike many, many other laws of physics is one of those laws which ends up being an inequality. And perhaps that is the reason why many people including some great physicists and mathematicians are uncomfortable with it. And again the second law has such a status that when you talk of a first law and we have a first law of thermodynamics, we have Newton's first law of motion, we have Kepler's first law of planetary motion. And I am sure there may be a few other first laws in physics. When you say the first law, people will say first law of what? But when you say the second law, a qualification is not necessary. The second law means the second law of thermodynamics. If you want to talk about the second law of Newton, you have to say Newton's second law. But when you say the second law, the default assumption is that it is the second law of thermodynamics. So that is the specialty of the second law. Now why do we need a second law? Why is it, what is it that we have not yet tackled? Let us review what we have done in the first law. First law tells us that we can define an interaction q and property of state e such that q is delta e plus w. It is an inequality and so long as that inequality is satisfied, first law has no objection to it. For example, let us take a system, we have a vessel containing water and we stir it and it is cooled. So the work done is by the system is negative and it is cooled. So the heat transfer takes place from the system to the surroundings and if we say that there is no change in the state of this, say the temperature does not change then delta e will be 0 and by default the things seem okay. We have if delta e is 0, we have q equals w s t both negative. This we see in practice. This is okay by first law. Then we propose that look, I propose a situation where this water absorbs heat from the surroundings and stirs the stirrer producing some power. That means if I say that q is greater than 0 and w stirrer is greater than 0, q is w s t and if q is w s t, first law says okay with this. But we know from our real life situation that this is not okay. This does not happen in practice. If I stir water, it will get heated up and if I leave it like that it will eventually get cooled. But then if I put a stirrer there, whatever I do, wait how long without I do something funny, explain more words, water will not absorb heat from the surroundings and move the stirrer. So first law is immune to direction of a process. If we have a process where a system goes from state 1 to state 2 with some interactions q w and delta e, then first law says it is okay if it comes back from 2 to 1, then the change in energy will be delta e with interaction minus q and interaction minus w. This is okay by first law. But we know that in practice almost always it is only one of these two processes which is likely to occur. In fact, one of the questions which is asked and which for some reason what was not asked is when we define delta e is minus w adiabatic using first law. We say if you want to determine the change in energy between, difference in energy between two states of a system 1, 2, 1 to 2, calculate, execute an adiabatic process from state 1 to state 2, measure the work done and e2 minus e1 is minus w adiabatic 1 to 2. The question which you will be asked once in a while by a smart student is that sir is it always possible to go from state 1 to state 2 by an adiabatic process. Before that he will ask sir which adiabatic process, our explanation should be any adiabatic process. So select one which is most convenient to you. Then he or she will ask is it always possible to go from a given state 1 to another state 2 by an adiabatic process. The answer should be well we know that it is not always possible but if it is not possible to go from 1 to 2 by an adiabatic process. Later on when we study the second law of thermodynamics it will assert to us that if it is not possible to go from 1 to 2 it is definitely possible to go from 2 to 1. So all that you do is execute an adiabatic process from 2 to 1. So minus w adiabatic 2 to 1 will be then be equal to e1 minus e2. So just take the negative of that and you will get e2 minus e1. But we always know from our daily experience that processes take place only in one direction. In colloquial language we say that in real life everything occurs. Kalaya tasma in the maha, time moves only in one direction. You cannot go back into the past. You always travel from the past to the present to the future. When things come into being, things get destroyed, they cannot be recreated, things are irreversible. Then you have all philosophies of life, death and all that. But the bottom thing is, bottom line is things occur only in one way in nature and whatever we do we cannot prevent nature from doing things which are one way. That is about the first law. What about the zeroth law? Zeroth law brings in the idea of a useful property called temperature. We define isotherms in the state space of systems. It allows us to determine given states of two systems separated by a diathermic partition, whether q will be equal to 0 or q will be non-zero. It does not tell us about the direction of q. It does not tell us anything about higher T or lower T. Remember we have solved problems. We have defined scales of temperature and using those scales of temperature like Celsius or the ideal gas Kelvin scale, we have defined some temperatures to be higher, some temperatures to be lower. For example, which of the two systems have a higher temperature? What shall we do? We will take an ideal gas in a cylinder piston arrangement, bring it in thermal contact with say system A, let it reach thermal equilibrium, find out the PV product, let me call it PV A. Then bring it to in thermal contact with system B, let it come in thermal equilibrium, measure its pressure and volume, call it PV B. Our definition of the Kelvin scale and higher and lower temperature based on Kelvin scale is that if PV B is higher than PV A, then the temperature of system B is higher than the temperature of system A. But this depends on our definition of what is higher and lower. There is nothing thermodynamic about it. We are using an ideal gas, which is an approximation of a real gas. Maybe tomorrow you go to a situation where there are no gases at all. So, there is no question of a Boyle's law being discovered. In which case, how do we define a higher and lower temperature? See the first law does not require the property of any particular substance or a particular type of system, not a zeroth law. But the higher and lower temperature which we have defined requires a system called an ideal gas. That is not a good idea. We need a thermodynamic definition higher and lower temperatures. And we want a thermodynamic basis for temperature scales, which does not or which do not depend on the properties of any specific material like mercury in glass, alcohol in glass, a thermocouple of copper and constanton or chromium and alluminium or some real gas or an idealization of a real gas called an ideal gas. So, all this thing is done by second law. So, what will second law allow us to do? The second law will allow us to do the following It will allow us to determine whether a specified process is possible or not. Then it will provide limits on processes. Either it will provide limits on end states saying you cannot go to pressure higher than this or temperature higher than this or it will put limits on interaction saying you cannot transfer more heat than this or you cannot do more work than this. Then it will dictate to us various relationships between properties. It will say that look if pressure varies with respect to volume like this, entropy we should vary with temperature like this. It will not dictate what value a particular property should have, but variation of one property with respect to a second property will be related to the variation of a third property with respect to fourth property. In the morning we notice that for a fluid like water, the solid liquid vapor line on the P T diagram is inclined towards the left and we all know this that if you put pressure on ice, it is melting point reduces. That means at higher pressure the melting point reduces, but the boiling point increases. As you go to higher and higher pressure, boiling point increases. But as you go to higher and higher pressure, the melting point reduces. That is the reason why you can do ice skating and skiing and ice sleds can work on in the north pole and south pole. Thermodynamics will relate this to the fact that the liquid of water is denser than the solid of water, ice floats on water, but steam is not denser than liquid. So the density variation, density difference which is higher which is lower dictates how saturation temperature behaves with pressure. All this is done by defining a very useful property called entropy. Entropy is a very misused and very improperly understood and hence misused property in thermal analysis. Perhaps in all of physics because people talk of entropy being a measure of disorder and all that. Without defining what a disorder is, we are going to do absolutely nothing of that sort. In fact, if you look up the formulations of classical or classical formulations of thermodynamics, whether by Kelvin Planck, whether by Gibbs, whether by Caratheo-Dory or the latest absolutely esoteric topological formulation by Giles, entropy is not linked to disorder at all. In fact, disorder is something which is not mentioned at all. We will go through a process by which we will put all these things in proper order. So let us begin by considering a statement of the second law. There are a number of statements of the second law and the number of candidates. Historically many of these are important. Perhaps the oldest statement is that of Carnot. He never proposed it as a statement of the second law of thermodynamics. When Carnot proposed his theory of the efficiency of engines, he never knew what thermodynamics was. In fact, the caloric theory was very popular. Caloric was a fluid which flowed from material at a higher temperature to material at a lower temperature. So steam had higher caloric than water. Ice had still lower caloric than water and such things. So when ice comes in contact with steam, what happens is caloric flows from steam to ice. So the caloric content of ice goes up, it becomes water and caloric content of steam goes down. So steam also becomes water, such stuff. Carnot was an engineer working and looking after equipment in mines. So one of his jobs was to maintain and run machines, engines which would pump water out of the mine and move material in the mine. And naturally the engines have to be fed fuel, coal, wood or whatever. And naturally that cost money and if you could save coal, you could save money for the mine. And he had this idea of something called efficiency. He said, what shall I do to improve the efficiency of my machines, efficiency of my engines. And in his flights of fancy and that is where the greatness of Carnot lies, at that time we had essentially reciprocating steam engines of some funny kind, very crude ones, very leaky, very inefficient, nowhere near the so called ideal efficiency. But he came up with the idea that look, if I do something by which these machines will work in a funny way, which he called reversible way, then I will have the highest efficiency. That was his thought. Now his machines and his processes were nowhere near reversible. Even the so called relative efficiency is compared to an ideal machine where nowhere near one, there will be a small fraction of a percent for all you. But he could think through all that in his flights of fancy and said that look, if I have something called a reversible machine and he explained his idea of reversibility, then I will have the most efficient machine. And he never said that the most efficient machine will have a 100 percent efficiency. He will have that machine which is of this kind, will have the highest efficiency. Any other machine will have a low efficiency. So, he said that the efficiency of an engine will be less than or equal to that of a machine which will have a maximum efficiency. That was the idea of Carnot and that was the germination of thought processes leading to the second law of thermodynamics. His thinking and his working was not mathematically very robust. And hence although we give credit to him that as the proponent of the thermodynamic thought processes which led to the second law of thermodynamics, his statement is not today considered the basic statement of the second law of thermodynamics. We considered it as a derived statement and that statement we will today use as Carnot theorem. Next came Clausius and Kelvin and Planck. The Clausius and Kelvin Planck statements are the standard statements presented in every text books as the basic statements of the second law of thermodynamics. The Clausius statement says that you cannot have a cyclic device which will not be absorbing any work, but whose only task will be to absorb heat from a system at a lower temperature and reject it to a system at a higher temperature. So, this is the Clausius statement. All of us know that I will not spend time on this. This is proposed to be a statement of the second law of thermodynamics. This is an okay statement, but the problem here is we have not yet defined what is a lower temperature and what is a higher temperature. We have lower and higher temperature on the Celsius scale and lower and higher temperature on the Kelvin scale, but there is nothing thermodynamic about it. We could have defined all higher temperatures to have lower numbers as Celsius originally proposed to do and lived with it. In which case, maybe ice would have at a melting point of 100 degrees C and steam would have boiled at 0 degrees C and maybe boilers would have worked at minus 360 degrees C and the pressure of 40 bar and 50 bar. Today we find this to be laughable because that is not according to convention, but there is thermodynamic, there is nothing wrong in setting up a scale of temperature which goes to higher values at lower temperature because these are arbitrary scales of temperature based on properties of some material and the way we define that temperature scale. So, the Clausius statement is not considered a useful statement of the second law of thermodynamics. It is only a part of the history of thermodynamics. The statement by Kelvin and Planck is a useful statement. The Kelvin and Planck statement says that if you have a system which either does not undergo a change of state or executes a cyclic process, then that system will not be capable of executing one process. In that process is absorb heat from some system and convert it completely into work. So, if you have a system and if you have this device which is either no change in state or only cyclic processes, that means effectively no change in state when it executes its process, then absorbing heat from a system and convert it purely into work because cyclic means delta E is 0. This is what we really want, whether it is cyclic or otherwise what we want is delta E is 0. Now, Kelvin-Planck statements that says that this is impossible and this statement we will consider today as the basic statement and a very useful statement of the second law of thermodynamics. Of course, there are other statements which we will not talk about. For example, perhaps the neatest mathematical statement of the second law of thermodynamics is that of Karatheodori. But, to study that and derive the idea of entropy from that requires calculus of very high order, calculus of exact difference. Differentials, differential geometry and so on. The statement says that in the state space of any system, in the neighborhood of any state, there exist states, the typical mathematician statement of existing existence and all that. Of course, there is no uniqueness involved. But, he says that given a state in the state space of any system, in its neighborhood there are some states which are not accessible from this state through quasi-static adiabatic paths. This is the statement, just listen to it and leave it at that. We will come to it as a consequence, because our primary statement of second law is going to be the Kelvin-Planck statement. We will use K p to represent the Kelvin-Planck statement. Graphically, I have some system and this is a cyclic device. Actually, I should not say cyclic device. It is device which when you ask it to work, may be execute some set of processes, but finally finds itself in the original state, leading to data e equal to 0. If you propose a scheme by which it will absorb heat from some system and convert it completely into work, then the Kelvin-Planck statement says that this is impossible. Now, that is the basic characteristic. Second law statement says that something is impossible. Now, what is the consequence of this? In fact, all the further consequences will now be shown to be consequences of this. All further derivations will be shown to be consequences of this and for that, we will be using the first law of thermodynamics, we will be using zeroth law of thermodynamics and we will be using tricks of mathematics, logic and all the normal mathematical and logical tools which we use. We will now define certain terms which make our derivation simpler and it will bring us in line with typical textbook terms, which are used in textbooks on thermodynamics. We will define what is known as a heat engine. We will define what is known as the efficiency of a heat engine. We will define what is known as a thermal reservoir and then we will define certain secondary terms like a 2 T heat engine and all that. All this only simplifies our algebra. In fact, definition of a heat engine, a 2 T heat engine and a reservoir helps us do our derivation in an algebraic fashion rather than worry about details of calculation and integration of states or in the state space. When necessary finally, we will go back into the differential form. So, the first definition which we will use is that of a heat engine. First a heat engine is a cyclic device. By cyclic device, we mean a device which when we start or tap it will execute one cycle or an integer number of cycles. You cannot stop it in between. Suppose it is a 7 stroke cycle or 7 detailed processes cycle. If you say start, it will immediately execute 7 processes and then stop or it will execute those 7 processes twice and then stop or twice and then stop. But it will not stop after half a cycle or two thirds of a cycle or three of the 7 processes, nothing like that. The moment you say go ahead, it will execute one cycle or two cycles or three cycles as needed. You may even specify the number of cycles, but always one complete cycle or a number of complete cycles. Consequently, every time it executes a process, delta E will be 0. We do not have to worry anything about it. So, an engine must be a cyclic device that the first requirement. The second requirement is every time it works, it should produce work which is greater than 0. That means it should deliver power or deliver energy as work. Something which leads to net absorption of work is not okay. This does not mean that every time there is a work interaction it has to be positive. Part of the cyclic process may have positive work interaction. Part of the cyclic process may have negative work interaction. That does not matter. It should have a net positive work interaction. During a cycle, it may produce 50 kilo joules of positive work, but absorb 20 kilo joules work during say processes of compression. Well, the net work output is plus 30 kilo joules from the engine to some surrounding system. That is okay by us. So, W greater than 0 is the second requirement. First requirement is a cyclic process. Second requirement is that work should be greater than 0. Obviously, if delta E is 0, W is greater than 0. There has to be a set of systems from which it will be absorbing Q and net heat absorption will naturally have to be equal to the net work done. So, this is the scheme of an engine. But immediately, we will now define what is known and to define the efficiency of an engine, we will do the following. I said that look, this work may be made up of some positive components of work and may be made up of some negative components. At this stage, we are not going to worry about positive and negative components of work. We are worried about the net work output W net. You may call it W net as many text books call, but we are going to look at Q. Similarly, Q need not all be absorption. It is possible that some heat interactions are from the surrounding to the system. Some heat interactions are from the system to the surroundings. The net would be Q, which will have to be equal to W by applying first law to this device. The efficiency of an engine and now since we have understood an engine, we will simply call that engine E. So, E in a circle or E in a bubble would mean an engine, cyclic device as required. It will have to have W greater than 0. But then what we are going to do is whenever it absorbs heat from surrounding systems, we are going to combine all those things, heat interactions together and call it Q absorbed. Similarly, during some other part, it may be rejecting heat to some surrounding systems, the same ones or other ones. We will call that interaction Q rejected and just to be in conformity with conventions, the Q absorbed will be shown to be from the surrounding systems to the engine. Whereas, the rejected will be shown to be from the engine to the surrounding systems, which absorb that heat from the engine. This is just to be conformant with the textbooks of engineering all over the place, including the ones which are listed. Similarly, W by first law which needs to be equal to Q net absorption will turn out to be in this nomenclature Q absorbed minus Q rejected, W or if you want to write W net. So, this is the first law as applied to engine and then we define the efficiency of an engine as efficiency equal to W divided by Q absorbed. And using first law, you can show this to be Q absorbed minus Q rejected divided by. So, this is definition, this is a derivation which will be 1 minus Q absorbed sorry Q rejected by Q absorbed. And the visual thing of this is if this is an engine from some systems, it will be absorbing heat that is summed up as Q absorbed to some systems, it will be rejecting heat that is Q rejected, this is W. Now to simplify matters, we now define what is known as a reservoir or thermal reservoir or as many books say constant temperature energy reservoir. That is to take care of the fact that this Q absorption will be from different systems. Both systems may undergo a change of state, a change of temperature, a change of other variables of their property as the heat absorption proceeds during that cycle. And we will have to take care of how the temperature changes etcetera.