 Now, whenever we do statistical mechanics, we always go through thermodynamics. We don't need to go through thermodynamics actually. As I said, I can develop an alternative way approach to statistical mechanics as I discussed here. But standard thing of doing through thermodynamics has merit. The merit is that it allows you to go and discuss some of the things that statistical mechanics actually does. One of the main merit of statistical mechanics is that it calculates the thermodynamic properties. It calculates the entropy. It can give you the free energy. That is the main merit of statistical mechanics. Now, other than that, Einstein wrote many things about thermodynamics. He said the only thing that will remain, if everything gets, you want to send one thing, world is getting destroyed in a capsule to alien. Only thing that you should send is thermodynamics or one thing that will survive which will be, it will never be overthrown. Now, I will spend a little time on thermodynamics and the reason is that again I will tell it little bit differently. I will tell it the way I looked at it. So, it is very important to realize when does thermodynamics starts. Again, as I telling you again and again, if you take interest in the history of evolution, you will find it is much more interesting in whatever work that you are doing because you will be able to relate little bit. How am I going to start it with the industrial revolution? Why it is starting the industrial revolution? Because India played a very important role in this industrial revolution. Because of India, the 1757 that British, then the first thing they started doing, they started taking away the coal and England already had enough coal but they also had steel but those days huge thirst for the raw material. And India provided huge amount of raw material. Now, I write a blog which you might, there is a, you know, Raja Ramonrai who was considered the first Renaissance man in India, Raja Ramonrai made a fantastic calculation. About the money British was taking away. Anybody knows of that amount? He figured out those days by this raw material and everything, how much money British was taking away. Even those days, a few million pounds per week. And all in all, now the calculation is that British took away 45 trillion dollars. It is important that suddenly enormous amount of coal, steel went to England. Same happened in it also. France also equally efficiently robbed Africa and many other things. But coming back to thermodynamics, so steam engine came and so there was now the necessity to have an efficient engine and that is where thermodynamics can play a role. And I am telling you again and again one very important thing that everything is very practical. Thermodynamics was done completely because of this. Boyle was travelling in a steam ship and he is the one who formulated the first law. So quickly go through and again I will tell you as I said statistical mechanics provide the microscopic basis in terms of atoms and molecules. So when I say microscopic basis, I say atoms and molecules and they are interacting, they are interaction potential which when gave the meaning of entropy. There are other entropy and I told you Einstein gave the meaning to specific heat. All these are from comes from statistical mechanics. Thermodynamics limitation all of you know or some of you know that it does not provide numbers because it provide changes that all of us know delta G equal to delta H minus T delta S. But it does not tell you how to calculate except you have to go when they had to calculate free energy that is what Planck and Ernst introduced the third law. Because to calculate other than third law you cannot. Then third law combined with specific heat, temperature dependent specific heat gives you enthalpy and entropy. That is you have done that in A. So that was an imperfect law and why it was that one understands from Benjamin is equal to K, B, L and W. But those days people did not know of that. I will quickly go to thermodynamics and tell you something rather interesting about thermodynamics. Again I am not going to do a full course on thermodynamics that is beyond the scope. But many Statemac book has one very long chapter of I always thought that was not the right thing because I am going to here study statistical mechanics. Something students should learn something new. Statistical mechanics gives me the values of thermodynamic quantities. That is an important input that makes sense. Let us quickly go through the laws. So you know this is delta E, delta is missing. Q plus W, Q is the heat. So if you are going from one state to another state then internal energy change is delta E and this delta is a state function is a function of the state does not depend on the path but these two heat and work depends on the path. Now this is a very interesting law, very interesting change of internal energy is sum of two path functions. So it is a constraint. I can do Q. I am interested actually in a steam engine or think of you guys have not seen this old week travel on those railway where you have big coal and one poor guy was put the two guys in the engine section and one putting coal into the fire and boiler and all this thing what are going and that was the force was turning the, so all these things heat was generating the work. But one important thing is that is not easily accessible work is to be measured, there are certain problem but major problem is that this is one equation but we have two unknowns. You know I want to have a relation between Q and W but this equation other than that Q and W I can vary any way I want but I do not know anything more than that that they have to be together they have to be conserved. So one equation two unknown, this equation is not useful as you have one equation and three variables of the daily say I can measure but they are still I have two things. So this is very important limitation of the first law of thermodynamics that it does not let you do anything. So important thing is that I want to have the relation between Q and W that then took a long time that is the one that whole generation of scientists did that. Carnot engine and all this thing was just to get a relation between Q and W. So well statistical mechanics what is the exact equation that is not too important right now we have to do in terms of intermolecular, internal energy is the sum of kinetic and potential energy. So statistical mechanics gives you certain teeth into this quantity what is the energy? It does not tell you about Q and W. Now I will come back to that. One thing we separate out is the enthalpy because there is a work mechanical work which is thermodynamics. So we introduced the enthalpy with h equal to v plus pv. Now come back I will come back to intermolecular interaction that was a digression and I do not want to do the right now. So we continue with the second law of thermodynamics. Whole idea of Clausius Clapey, Clapeyron, Thomasson all these people were to find a relation between heat and work. And there are several statements Clausius statement it is see they are trying to find a relation between heat and work. It is impossible for any cyclic machine to transfer heat from low temperature to high temperature. Kelvin statement it is impossible any cyclic machine to convert heat completely into work. That was the dream to convert heat completely into work. So all are these statements of negation. You know all of these things you have done at least five times as I am telling you. So these statements are equivalent as all of you know but all of them are statements of what you cannot do. But then what we can do? One thing that everybody was interested how to find the maximum amount of work out of heat. I have delta e conserved. Now I wanted relation between q and w which satisfies that particular way of doing which was maximum. Those days in industrial revolution there is a tremendous it on the efficiency and you know efficiency. So Carnot engine and all these things was it was very practical. They wanted to increase the efficiency of a steam engine. And steam engine was there everywhere in train was pulled by it in ships everywhere. Then what the one thing they realized that well they cannot find a relation between heat and work. Because it is both a path functions but if they do it so slowly from one state to other state then they will be able to do something. That means if they move so slowly then I am what is the reversible path I can go from one state to other state. And along the path the Carnot engine Carnot showed that I can introduce the entropy. So this is the thing all of you know that it is t I do not know why. Okay the capital T and so this is the Clausius in equality. Important thing here is the equality that q is the reversible and t is the temperature. And this equality holds for reversible process. And the equality is the one that then allows us to derive all the free energy. Now why free energy is so important? Can anybody tell me why free energy is so important? Why so much work Carnot engine 2-3 chapters of thermodynamics at the end of the day we get a free energy. Why so much attempt was made to get the free energy? Absolutely it is the energy free to work. So the part that we cannot do anything is entropy. So now you understand the whole second law. So much work by so many smart people was done to derive an equation of this kind under different conditions. So the two new things somewhat new things that I have told you. The first law give you a relation between work and heat but it is an equation one equation two unknown. And so I need a between q and w and then there is huge amount of work went into all the smart people in Europe try to get the relation. They cannot do it if it is irreversible because it is arbitrary. But if you go in find and slowly by going from one equilibrium state to another equilibrium state to another equilibrium state piece together then you get what is Clausius equality part of the inequality. Inequality part is not too useful other than sometime universal thermodynamics but not in equilibrium thermodynamics. Equilibrium thermodynamics it is the equality part that is important. And that ultimately leads you to these things and this is the of the energy change. This is the one that is getting lost. So this is the one that is free to do work. Now I have an upper bound of the efficiency because I know how much entropy I can how much work I can extract from an internal energy. So this is just one energy surface that I will talk a little later. Let me go back. One important thing of statistical mechanics is that what started with van der Waals and Boltzmann not that much Maxwell but Boltzmann was that the two particles interact. So intermolecular interaction is place an essential role in our understanding because why? Both in physics and chemistry but particular mode in chemistry that we want to think of two molecules that are interacting. So the only property is whether it is specific heat, whether it is density or any dielectric constant or conductivity all are determined by the intermolecular interaction. That means two molecules are coming together and when there is one molecule I put it at the center of my coordinate system. Then I bring one more molecule and what is the interaction? When they overlap here then there is a large repulsion. So basic part then and then in a there is attraction because of electronic fluctuations of a function that we are not going to go into. So this is the separation between two particles and this is what is called intermolecular interaction. So then this whole idea of statistical mechanics is that if I give you this intermolecular interaction starting giving you this intermolecular interaction. Can you give us everything else? Can you give us the specific heat? Can you give us the phase transitions? Can you tell us why water freezes into ice at 0 degree centigrade and goes into vapor at 100 degree centigrade? So all these things the statistical mechanics claims that was the dream of Boltzmann that starting with a bare potential. There are three things given here. This is the one that I drew here is called Linnert Jones. This is the heart sphere of the billiard model that is no interaction till they touch and when they touch they go to infinity. Another one sometime used is called square well that means there is an attraction then 0 and then harsh repulsive potential. Interestingly such a simple potential is the simplest possible potential that describes liquid to crystal transition fairly accurately. And that captures the essence of freezing however surprising that may appear to you. This is the one and these two describe both gas to liquid, liquid to crystal many many other things. For example if I want to do water then water will be somewhat more complex potential but not to you know orientation dependent potential but it is like that. So now think of the following that we have done little bit of thermodynamics. First law that essentially would boil formulated second law by many people given to entropy and free energy. We have our agenda fixed and well there is not one agenda there are many agenda. One of them is that how can I calculate the properties of a matter and to start with the calculation of properties of matter. If I want to calculate entropy from first principle I want to calculate the specific heat. I want to calculate say conductivity or isothermal compressibility or phase transition. What is the latent heat of freezing? What is the latent heat of evaporation? I want to calculate what is the property of water and ethanol? Why there is a eutectic point at 95% of water and ethanol mixture? So why when I add little DMSO? Why does Lysosimes these cell oil breaking Lysosimes CO bond get accelerated by factor of 3? Why protein folds to its structure native state? So statistical mechanics has this agenda, the grand agenda that you give me this interaction potential and I give you answers to all the questions. That is the basic idea of statistical mechanics. And it can be broadly divided into two parts equilibrium and non-equilibrium. What you are going to say in the beginning is equilibrium because if you do not do equilibrium statement you won't be able to do non-equilibrium statement. You need to know how particles just like Gibbs did give away the path of Boltzmann because he realized equilibrium is much simpler or rather there is a simpler way to do the equilibrium statistical mechanics. So that vision and that brilliant understanding of Willard Gibbs but by the time of Willard Gibbs and by the time of Boltzmann and of course Einstein they already had this agenda. They knew that you give me an interaction potential. Then the idea was that how simple can be. It turned out as I told gas to liquid cannot be explained by that but liquid crystal can be explained by these very simple potential. They can explain both gas to liquid and liquid to crystal. This how do you get the interaction potential? I can give you a flow charge is that you start from quantum mechanics and quantum mechanics gives you the interaction potential. Once you get the interaction potential then you get from the interaction potential you get all the properties. I am not really lying to you and I am not bragging. I am not saying but that is really the agenda and promise of statistical mechanics which has been largely fulfilled this realization of the promises but it took more than 100 years particularly advances of computer simulation because many of the equations with Boltzmann could not do or Gibbs could not do we can do it by computer simulation. Now of course you always have to do analytical work because the analytical models tells you how to go then you do more complex things. So again to formulate the starting from Boltzmann who introduced intermolecular potential Boltzmann consider this potential. In his case it was a hard disk or hard spheres. Only two of them he considered he could not consider three as I said even now three body problem is unsolved in analytically. Then came of course much later in 1930s or these things these things were done. Now I think I will stop here today the interaction potential business and the subsequently we will start from the statistical mechanics just in quantum mechanics we start with Schrodinger equation in statistical mechanics we need certain postulates so these postulates of statistical mechanics as I understand repeatedly says the simplicity of a theory is based on how few assumptions you can make and then how far you can go. The statistical mechanics which goal is starting with an intermolecular potential to go to thermodynamic properties or realistic properties that builds only on two postulates and one hypothesis and so in the next class what I will do what I have done today give the history and did the thermodynamics and in the next class I will start with the postulates of statistical mechanics and then from postulates one of the postulate was done by Boltzmann which is the time average second postulate was done by Wheeler Gibbs which is the ensemble and these two combined with the help of hypothesis and that goes into the development of leads to the development so the way it happens is that you have the postulates and the hypothesis that allows you to go to the final state and that is how you can do that as I have done, we will go to this so we have no problem we have the postulates and we will go to the next class so the first form of the postulates as I have done is the postulates Then you do to rigid rotator, same you do quantum mechanics, we do in statistical mechanics rigid rotator, so you will find that the certain sense, the step by step thing of statistical mechanics, parallels quantum mechanics and the step by step thing was done, just Schrodinger did the quantum mechanics, it was the wheeler Gibbs who did all these things for statistical mechanics and in the process you will see again we derive ideal gas law will be deriving something like van der Waals, but now with an intermolecular potential, van der Waals had an imperfect and incomplete intermolecular potential, but we will now be able to do in a more complete way, so now I think I will stop here today, but I will take some questions, no this is one very you know, your equilibrium means you do not go anywhere, your equilibrium means you stay there, so here you are making infinitesimally small changes, you are making so small change that you can talk like the concept of virtual work, you do a very small amount of thing and you say that I can apply equilibrium principle, so you are not at equilibrium, you are inducing a change, but change is done such a small step that you can say okay I am going from A1 to A2, but I can now talk of, I can talk of thermodynamics, I can talk of equilibrium inducing the small change, so it is like that you are going from A1 to A2 and you do by infinitesimally change, now the next step you are doing from step 1 to 2, then step 2 to 3, all of them piece together give you a, it is not equilibrium, but it is reversible, okay, so if you do a reversible change then I think I do not have any control anymore, I go from in a big jump or finite jump from one state to another state, then all I have is that relation go back to your first law and you have an inequality, but inequality does not help you because it is ill-defined, so you strive to get the equality and you, it was derived in classical mechanics we have this virtual work, we do very small amount of work, but then we use thermodynamics principle to get that, so it is exactly that somewhat funny concept that you are doing infinitesimally slowly to that unique path which is the reversible path, it has, it is confusing all the way and except that it is what Karno showed by this reversible path that you can show that dq by T, dq reversible by T is a path, is independent of path, that was the, there is a beautiful proof and the only place such a good proof is given is in Castellan, where he showed that you can piece together this and get a state function, so whole idea is that dq reversible by T is a proving that is a state function for a reversible path, that is the Karno engine and then Karno engine, I then I have to do the thermodynamics which I do not want to do, but when Karno engine was that next proof came that dq reversible by T over a cyclic path.