 Today what we will start on is an important area of statistical mechanics important both for physics and chemistry and as I keep telling you that these are fairly regularly taught in most places in the world at least in serious places. Now why these things are important is the following till now we have done ideal gases and ideal gas the particles do not interact though they have many interesting properties and particularly useful expressions like entropy specific heat that we use later. However, a system under is interesting because they have interact with each other they undergo phase transition and ideal gas does not go under go phase transition you plot PV or P against P 1 over V you know it just never shows any bend like if I plot P against 1 over V then remember it is a straight line and this is the that is how it goes ok that is where universal gas constant what found out. In the real world however when you high temperature it is something like that but when you go to low low temperature then it undergoes a phase transition and this is gas this is liquid and then it goes undergoes one more phase transition is the crystalline state. So these breaks and this is the coexistence are because the phase of the system goes discontinuously from gas to liquid to crystal and ideal gas which does not interact does not show the phase transitions it just goes on linearly. People started looking into that then what the first thing that you find that was noted by van der Waals that if this is the ideal gas which was known long time then a dense gas when density of gas I will little bit quantify what is the density because it is very important then it starts deviating from the A then it goes like that but it starts deviating from the ideal gas behavior. So this is the ideal gas which is PV equal to RT this is the real gas it goes at lower temperature it undergoes phase transition but you do not need to talk about it right now we will talk about it at length but right now let us see that we are just not that low temperature we are intermediate temperature but we start seeing the departure from the ideal gas behavior as I said this was observed by van der Waals and many other people. So the people then try to explain this it was clear very early that this is because of interactions between atoms and molecules now in order to talk of interactions I need to talk of a separation length scale between two things and the way we talk is introduce a dimensionless quantity rho star which is number of particles in a volume V then cube by the molecular diameter and this so sigma is the molecular diameter so n sigma cube is kind of a fraction of the volume occupied by molecules so this gives you a measure that if I have this kind of molecules and if this is sigma then pi by 6 sigma cube would be the volume each one occupies then you get a measure of that so in the roasts and roasters are very significant by making a dimensionless we get some very important what is called the convergence for example the roaster typically in the liquid is between 0.8 or 0.85 to 0.95 gas is between about 0.1 or so so in between these gas and liquid there is a huge coexistence which really takes you from 0.1 to 0.6 or so these are very important numbers why it is important because now I tell you some numbers like water the roaster is 0.76 then acetyl I think as far as you remember something like 0.85 methanol little bit more dense it is about 0.9 so in a very small so when you in number density if you look at them number density this part varies quite dramatically from one liquid to another liquid even of the order of one order of magnitude but however when you do the roaster there is an universality that appears the universality is that this is that we have a suddenly a dimensionless quantity to characterize the liquid okay now I have a number I can put okay this is the density roaster where I have a liquid this is where it is I do not have to talk of those numbers like 10 to the power 22 or a density of water 10 to the power 23 per cc those kind of big numbers are really doesn't make many sense to a scientist a scientist must make them understandable something we can grasp we can kind of think and these roaster also at the same time is a kind of fraction is proportional to the fraction of the total volume occupied by the molecules or atoms if I consider them as hard spheres they are not hard spheres but there is a big repulsive potential and because the rough potential you have size that is why Pauli was fond of saying that that he created the solid state or condensed matter physics because because of Pauli exclusion principle and all the you have the in a hard repulsive part of the potential and solid exist because of that hard repulsive part of the potential Pauli was known for making many many interesting jokes okay so this is an important thing that the number 10 said now as I just said in the gas ideal gas which we derived last two three lectures that we have these expressions they are very nice expressions but they are not we cannot explain phase transitions phase transitions exist as Pauli also said called condensed matter because particles interact even me we exist because we interact for some people it does not bother that whether you exist or not or you know or why you exist but for some crazy people it is important to know why we exist so these teaching in India which is a very hot based I remember our school questions were always what is this famous for what is the capital of Hawaii or whatever what is the capital of Madagascar so what are the important goods made in Nigeria it is all about what and you go very far by just doing what we do not say why and why these why these things happen why you get this particular fruit in that Mediterranean then come even deeper level is how how it happens so you see no fine man's you must be joking Mr. fine man there is a very interesting chapter you should read Richard fine man's father was a salesperson he wanted to make his son a scientist so whenever he had time he was at home he would make scientific toys and then he would take his son into the park their little track mountainous region he would go and he would teach them you teach them these the bird that is the this is but that is bad and then I think fine man was always a show off right so he showed up little bit and then whole locality every father and mother started taking their children out for tracking and then they all knew names of every bird or everything and they now in turn asked fine man hey do you know the name of that bird and by the time fine man forgot it then fine man's father told him a very important thing which fine man said was important all his life that the name is not important it is important that what it looks like it is important this kind of bird does this kind of things and you find in this area so we know many of our education system is this what best we don't even go to why and then of course how is people far so coming back we are now going to ask a few questions why a phase transition takes place and how do we explain that so these are long and very just like chemical reaction dynamics which has got probably one of the most celebrated theories and most celebrated Nobel laureates coming from Rudy Marcos, Herzberg Palliani or Ahmitz Well and many others who recently essentially the the paper of 1976 which gave Levitt and Warshall Nobel Prize was also essentially doing enzyme kinetics now in physics and also part of physical chemistry is equally respected and celebrated is the phase transition and the condensed matter so because that's how materials form and now that's how we understand how something forms then of course thermodynamic properties are important like compressibility as we discussed and other resistivity then come of course the electronic properties which somehow people are more focused on but you have to make the material first before you study electronic properties so basically why and how the phase transition occurs that gas liquid liquid so this is the question that we will be asking today and next few lectures now as I just told you how do you go about it I told you that ideal gas doesn't have a phase transition because there is no interaction between them so phase transition is a consequence of intermolecular interaction actually everything is interesting is intermolecular interactions so let us first define what is what I mean by intermolecular interaction the simplest one or one of the simplest one is intermolecular interaction vr r is the separation between two molecules I have the simplest thing in mind they are two molecules and the r is the separation from center to center vr against r and I have this is the one a short term part of that this I call molecular diameter sigma this I call molecular diameter sigma this is harshly repulsive remember Pauli's comment this is because of electronic overlap of the electrons electron density this is an attractive path heart sphere itself is a very interesting system and statistical mechanics huge amount of study goes on just to study the heart sphere path whether heart sphere part is not really realistic like this is the this potential called renaissance potential explains properties of noble gas very well that's what the first kind of things that people developed so these vr is written as these are the things I will be using this called renaissance 612 potential this repulsive part that is very short range and diverging and this is the attractive part this is the part which is given by this minus sign here so this is the same one of the simplest interaction potential okay now it face transient due to interactions and if I this is the kind of interaction potential I am having and that's what I was mentioning to you even at a two-body potential I cannot solve Newton's equation I can reduce it to a quadrature and three-body I cannot solve at all that's why we need to go to statistical concepts I am now keep a little bit of that upstairs but then I am going to do the rest of the how do you go about doing statistical mechanics so I keep vr equal to let me just keep this that is enough okay so we cannot do to now I how do I start about it I want to consider a n number of molecules at volume v at temperature t I want to work in canonical ensemble and now I want to calculate the partition function of this quantity so statistical mechanics tells me if I want to calculate free energy I want to calculate the equation of state pressure versus density I have to work in one of the ensembles microchemical ensemble in this case is incredibly difficult that because the constant of constant energy does not let me any easy is not amenable so canonical ensemble we can make some progress grand canonical also we can make progress but this is the one historically as I told you before you work and canonical particle means I have a system with the total number of particles n which are interacting among themselves with this potential in I put them in a volume v and at temperature t now if I increase I decrease the volume or decrease temperature the system from gaseous state would go to liquid then to solid and how do I describe the sequence of phase transitions how do I decide that at what temperature or density a gas water for example you know at 100 degree centigrade and atmospheric pressure goes from steam to liquid water and then 0 degree centigrade goes twice what determines it what determines this temperature 100 degree centigrade or 0 degree centigrade what determines water goes into hexagonal open structure what determines iron goes to FCC and what determines sodium goes to the CC and what determines when you form nanoparticles what are the size of nanoparticles so these are the questions and you can understand this when you start asking why and how life does not remain that simple anymore it probably much easier to mix A and B or some sample together in a solution put in a boom I know one of my colleague puts in a boom I know what actually some kind of glass jar and they increase the temperature and they get nanomaterial and they get a paper which is not a bad way to live actually because here you have to think and hard but I told you always that is a famous story one guy was trying very hard to climb Mount Everest and then one of his colleague was lazy and he was saying why why why why do you want to climb this one you know you know you can be easily like me lying on bed why do you want to climb that that his great answer was that no it just it is there it is just it is there so I have to climb now I now start this climbing so you know we have already introduced the notation I am always bit A about 3N or 3 by 2 I think 3N then I have to integrate remember I have to integrate over all the positions then I have to integrate all the momentum if you have any problem you will please let me know because I have this constraint here is the Hamiltonian right and Hamiltonian is sum over there I assume all of them same mass I equal to 1 to N then I have interaction potential U N particle interaction potential that depends on R this is this is notation I told you which means is dependent N particles and U interaction potential then is depending on your V I I tend to lie like you because V most of the time is we reserve for velocity U I j R I j I less than this notation or you can put I j both from 1 to N if you do that you have to put half to avoid double counting so basically this is my system N 1 2 3 4 5 and so I am having interaction between 1 and 2 2 and 3 1 and 3 1 4 this all these interactions so this U is sum of all the interactions okay the kinetic energy is the kinetic energy and potential energy now we immediately note one thing that in a classical statistical mechanics one term is difficult but in classical statistical mechanics this integration of the momentum is a trivial path because these are nothing but Gaussian integral which we did before and it does not overlap with position at all so I can do the A integral part and momentum part and then write rest as another function which then would involve integration over the interaction potential okay this is the game so basically this is the let us little bit summarize fully my analysis of interacting many systems that we are trying to do starts with a given intermolecular potential in our case we are assuming it to be an adjunct does not matter you can do anything you want what we are going to develop is a graph theory of liquids and gases this was introduced by Joseph Mayer and then it went into quantum actually many of the quantum things that people do R cell expansion then the kind of many cluster expansion couple cluster and all the kind of things it was basically started in Statemac even before all the Feynman graphs and all these things happened it happened 1937 that Joseph Mayer formulated the theory that we are going to describe the very pretty theory very nice and beautiful theory and has tremendous insight into it. We will derive a term expression of the partition function we will derive the molecular expression for virial coefficients in terms of intermolecular potential we will go towards the radial distribution function and microscopic nature of gas liquid condensation it has been attempted by many people to go to liquid to solve it that does not work say why but it gave rise this theory gave rise to the most successful theories of liquids that we know today that are in all the textbooks so this is the partition function and what if we can evaluate that partition function I can get the free energy and if I can get the free energy I can get the pressure as a pressure as function of volume that was the that is called the equation of state and that should show the phase transitions in the presence of interactions and then this Hamiltonian which is to be integrated over with the Boltzmann factor is given by kinetic energy plus total provincial energy and total provincial energy is here and this is another notation that we use where you already put that I show you have a 1 2 1 3 1 4 then 2 3 2 4 you do not have 2 1 because you have this constraint that j has to be greater than i so the one that comes later has to be bigger than that is a way to arrange this kind of arranging we do in theoretical research repeatedly everywhere okay now so this is what I have been telling that we now do that the full thing in full glory and this integration is that I can now carry out that we did in monotomic gas ideal monotomic gas this is exactly so in ideal monotomic gas this part is 0 because molecules do not interact with each other then I only have this integral to do and that is what it comes here and that is why de Broglie wavelength by taking h yeah check me on that I always it is okay h but but okay will that go in there and de Broglie wavelength so in 1 over n factorial lambda to the power 3 n but main important thing that that set aside main important then the effect of interactions is taken in the Z n we are following in total mayors 1937 notation these called the configuration integral or from mayors configuration integral some things in the our science I get less credit and as I was telling you that in the folklore and the in the terms of the huge contributions another person we hardly talk about the wheel that gives who single-handedly do it so much similarly the one person that is considered to be we do not celebrate enough another person who did is enormous amount of contribution is Joseph mayer okay so now this is the then I am going to play certain tricks is the following I notice that these quantity which is some in the exponent now some in the exponent can also be written as a product right some in the here you understand know that this is you and you is the sum so some in the exponent is nothing but a product okay so now and the product is there all in all when all these things are n into n minus 1 by 2 correct okay so this is I can write that quantity as that quantity right okay now now comes a very important thing that look at this thing now e to the minus beta u and look at this my potential so in our goes to infinity then if you are goes to 0 then these quantity goes to 1 and this is I will now decompose that when I decompose these things I will find that a very large number of terms comes in so so this is a problem that's why I introduced a mayer a function but let me go through this so so you go to this now so you I get now this is the sum over all the particles which I am now writing as product of e to the power beta u ij and ij okay these then here the difficulty I face I tell you if I keep it like that this integrals here doesn't converge so mayer then did the following trick mayer and listen carefully what mayer did he said okay I this is the this is the difficult part I introduce a function af which is the deep function now if I do that function then this simple mathematical trick means that I have to replace e to the power minus beta u ij here this is a beautiful trick actually it a two different things that it does and as I told you this is essentially the the start of the graph theoretical language in physics and chemistry you go you bring this here now the simple trick allows the following advantage u ij goes to infinity here and e to the power u ij goes to 0 but however when u ij goes to r goes to infinity these goes to 0 these goes to however if I take minus 1 out then when r goes to 0 these are repulsive part of the potential u ij goes to infinity e to the power minus beta it goes to 0 like here and it goes to minus 1 but when r ij goes to 0 then these goes to 0 these goes to 1 then it goes to 0 so by this transformation you are removing the divergence okay and this is allows a systematic decomposition of partition function which is fairly trivial but we will see how it goes to graph theory