 Welcome back to our continuing exploration of statistical mechanics. We have been studying interacting systems, the systems that allow us to explain the natural phenomena and though we actually model the different natural phenomena by certain approximations and certain pictures and things which then goes into theory of become matured into a theory over the many decades and years. So, in the theory we just were doing osmotic pressure we are doing binary mixture and which is a phenology that came from the experimental observation that we need to put the pressure to stem the stop the flow of solvent pure solvent to the solution and that of course is a consequence of interaction between solute and solvent, but we did not have any experimental estimation of the magnitude of this interaction. And osmotic pressure allows us the magnitude of the osmotic pressure allows us to do that, but how we did not know. So, that was the theory of Macmillan and Mayer who developed a detailed theory which is extension of Mayer's theory of real gases and phase transition in terms of Mayer's cluster expansion that we have done in great detail and that is why the Mayer's theory played a very important role because it allowed us to think in terms of interaction in terms they decomposed into 2 particle, 3 particle, 4 particle interactions. Then Macmillan Mayer took that to theory of solutions and then they explained that these could be the expression of osmotic pressure. In the process they derived the osmotic pressure of the ideal gas law. Then it exactly become osmotic pressure equal to RT just pi becomes CRT and C is the concentration of the solute or RTx solute mole friction of the solute which is just the ideal gas law. So, that so the Macmillan and Mayer could derived the ideal gas law just like you derived the ideal gas law from elementary statistical mechanics. Then what they did was what Mayer did they derived a virial series, but in the process they introduced an effective interaction. Now that immediately give you the idea that then I can if I could really play around with that play with that I would be able to get an estimate of the effective interaction very short after quantity between 2 solid molecules from osmotic pressure the way Lennard-Jones build his potential from second virial coefficient. But when you try to do that in the Lennard-Jones way the second virial coefficient he used what Mayer gave that B2 is the integration over the Mayer F function FR which is exponential minus UR by KBT minus 1. So, that is the effective interaction. So, WAAR and that I can also explain in terms you know and the experimental another verification of that is the radial distribution function how 2 molecules A and A are connected to each other which we get from Newton scattering we can estimate of molecules arranging with respect to each other. That is one another way another tool to get into the intermolecular correlations how molecules are arranged to this to each other. So, that the radial distribution function how A in a binary mixture A molecules are placed around A molecules B molecules are placed around B and A molecules are placed around B. These are the radial distribution functions or partial radial distribution function in pure case we call it just radial distribution function GR but they are dependent on intermolecular interactions. They are dependent on this effective interaction of McPillan Mayer or which is a flow parameter as we discussed. But how do you go about it? I told you and will show that experimentally we get an estimate of that directly through structure factor which was first time become available after second all to one through Newton scattering and Newton beams become available. These days complex system we also get through X-ray solid we always got through X-ray but represented development allows random systems also through X-ray because X-rays have become now very intense from synchrotron. So, these are more recent developments in the last 10-15 years that high intensity X-ray and getting structure of random liquids from X-ray. But before we did only through neutron. So, now we need several theories we need to understand how do I discuss radial distribution function which I tell you gives you a arrangement of the molecules one the all the molecules around one one tag molecule one specified molecule one chosen molecule and then the Newton scattering. So, this is the what will be the topic of today's lecture and we will be able to do it again it has been done in a quite great detail in this book and we probably will not do a great justification of this beautiful subject of pure liquid but it is done you know much more in this in this book. However, we will be able to take you through the basic understanding the basic physics basic phenology the basic aspects of the field to give you an intuitive feeling and also certain amount of workings of the theory. Now, this picture here is an interesting picture which we obtained from one of J. D. Barnel's co-worker he was a wonderful person Professor J. D. Barnel and he was the first to really map this the obtained structure of polymers proteins many of the things from X-ray he is called the father of you know could be considered one of the father of the biological crystallography. So, this person J. D. Barnel is doing here is he built a structure of liquid water and he then playing around it and seeing how does this many of these things work with but what is famous for this J. D. Barnel is this following experiment and this has a the first thing of theory of liquids. He took a big jar and then he put lot of marbles lot of marbles spherical marbles and he jam packed them then he did something really interesting then he put he poured some sticky paint like ink or sticky ink and that goes inside these all pervades through all through but then he could take out and see around a sphere where are the other molecules because that is where the this ink has not gone through. So, you will find on the surface of molecule there are patches like that from that you would find out what is the arrangement of molecules around that marbles from that you calculate you found a radial distribution function and when long, long later in computer simulation hard sphere potential radial distribution function was indeed calculated by computer simulation is very difficult to do this things analytically. Now, we have some methods and radical methods it found that the radial distribution function the number of molecules number of molecules around is g r you found is nothing of course there could be nothing inside here 0 then there is certain these arrangement goes like that and like that first peak second peak Bernal got it almost exactly correct. So, that was the beauty of first experiments of JD Bernal. So, JD Bernal is the father rightly called the father of the theory of liquid structure he is the one who experimentally measured by this very smart simple experiment the radial distribution function around hard spheres. Let go around now statistical mechanics at that time it was still at infancy because we had started having some equations very Bernal did that in 1935 and our beginning of the theory of liquid started in 1939 or 1940s when the Yoban in France, Kirkwood in United States born in Max born in Germany and they started putting a theory of liquids together by using the methods of statistical mechanics. It probably not surprising that all of these things followed the pioneering water Fowler and Guggenheim in 1920s and early 30s and then the pioneering work of Joseph Mayer in 1937. So, this work of Fowler Guggenheim and Mayer opened really a floodgate and people started seeing how to do a theory of liquids in terms of the interaction potential. So, basic aim is I give you an interaction potential and you give me a structure of liquids and what do you mean by structure of liquids we mean the radial distribution function we mean that water the probability of having one molecule and another molecule and that we will go directly into the radial distribution function we go back and forth in these things because sometimes the structure of a book the way it is developed it does not do a good justice to the neither the historical development nor the the driving force in our Indian field. So, so microscopy thermodynamic properties partition all these things the statistical mechanics that we have all done with many many times. Now, Newton scattering experiment gives you molecular arrangement this is what I am saying. So, Newton scattering gives you a cross section called SK this structure factor that is S comes from that gives you molecular arrangement and molecular arrangement is nothing molecular arrangement is nothing but gr. So, Newton structure give me gr radial distribution function that I am going to talk about now then all the thermodynamic properties can be in terms of gr and there are lots of equations that have been derived which are of some importance if for a theory of liquids but on a course of this kind we might not spend too much time on that because you know it is this lecture should be viewed as a stepping stone or intermediate between a book of physical chemistry like Atkins Castell and Moore Blaston from that book the contents into the real world of research and statistical mechanics and computer simulation. So, this is kind of connect these many many things. So, you should take this course in that spirit that we are going to give you an understanding and a motivation of what we do. So, now this is again schematic representation of interaction we are talking that here you can see 2 particles interacting then 3 particles interacting but by 3 particle interaction I do not mean a 3 body interaction potential I just mean the 3 particles are there. So, the interaction potential is could be represented there is additive some of these interaction these interaction these interaction. So, now any given microscopic configuration configuration of a given at a given time a snapshot of a liquid is a very primitive and approximate because everything spheres one component atomic all these things. But if that captures such huge amount of phenology and huge amount of microscope phenomena that it is really is the kind of a demodel the benchmark model of statistical mechanics that we play our theories and gain our understanding and then extend complex system. So, basic idea is that if I can understand and describe this system I can describe complex system very safely. So, these kind of interactions now are present I know how to interact consider this interaction in terms of the Reynolds potential. I have a basic idea of how this structures are though I do not have the microscopic information. Now, I want to know that if I give you the density temperature pressure and the interaction potential can you give me if I give you are can you give me gear. If I give you interaction potential can you give me the radial distribution function that is the whole question and how to get that that is the aim of this and this lecture. Let them go ahead. So, the basic flow chart again I am very much a fond of flow chart but unfortunately for you this flow chart will not be too much of a useful but still let us go through it that we start everything which I have not told you is a equation it is called the Louisville equation and that is the Louisville of n particle that means you have n particle hold n particle distribution in position and time this is what mulchman tried to do these are n particles an equation of motion of that is a Louisville equation by Louisville operator we have not told you about that that can be reduced essentially into an equation called b b g q i equation which goes to equilibrium we have one boundary equation. Now, what we want to do that the radial how do you talk of radial distribution functions. So, radial distribution functions as I said that I have this kind of molecules and I tag one molecule and now I want to tell the liquid is homogeneous there is no heterogeneity anywhere when we say homogeneous what do we mean we mean that if I follow a local I make a cubic grid I make a grid this is something worth for we spending a little time that I have this big thing and I make it into small grids and now a grid is such that it has 10 12 molecules that is kind of size that means it is two molecular diameter here two molecular diameter here and two molecular diameter in a cube and that has a 10 12 but maybe 20 molecules if I system a grid like that then now I can mentally just like we construct a transomber and our got a hypothesis all this stuff we can look into it with they say I have a microscope and I counting number of atoms and molecules into that grid when I say homogeneous I say that this grid over a time averaging long time has the same number of molecules of a grid of the same size that sort of homogeneous means that they are the same. So, there is no heterogeneity that means no permanent long term one region has more density than the other region. So, we are talking of homogeneous system now in such a homogeneous system I now want to describe what is the probability of I want to describe the structure and as I have repeated the saying the way of getting the structure is how many molecules are there at certain distance are. So, I have a molecule here then I say okay I give you a distance r and I can now give a small shell around that my picture similar to grid and what is the probability or how many molecules will be in that grid if I know how many molecules in the grid at a distance center that it is just r I construct a shell then I will be knowing about the molecular arrangement about my central molecule I know how many molecules are there and that really tells me many many things that will tell me what is the cage what is the cage around my central molecule this is called cageing or cage structure in a liquid because of high density as we discuss the density of liquid remember by introducing quantity rho star as rho sigma cube and sigma is molecular diameter and that is very large is about 0.8 typically while crystal is 1. So, the liquid is a very dense system but it is random but I need to know how many molecules are around my and at what distance they are if I know that then I will be able to calculate quantities like diffusion I will be able to calculate quantities like viscosity and many other properties and that will have come very handy when I want to do chemical reactions because how to two reactants come close together where are the placed that will give me an idea of the rate of the chemical reaction or they will give me about the salvation structure how one die molecule is solvated how we reactant is solvated in water is very important to know that in order to know the many properties. So, what we are beginning to do is to beginning to take a shot at those kinds of things coming back again continuing with my picture here this picture of how many molecules are there in a shell these I now say okay number of molecules in a shell if this volume the volume is 4 pi if at a distance r then volume of that shell is 4 pi r square and width is delta r of my shell or delta this is the volume of the shell. So, now if I say probability of finding a molecule at r is this quantity g r whom I called radial distribution function then that gives me number of molecule at a distance n. So, these are wonderful relation that number of molecules at in a shell at a distance r total number is volume of the shell which is 4 pi r square delta delta is the width of the cell 4 pi r square delta and g r then g r gives me the probability of finding a molecule at a distance r and g r is the radial distribution function which is the central quantity of molecules. So, now how do you go about it how do you go about calculating the cage okay we do what we have done before we are taught to do we write down Hamiltonian this is the momentum term and the total amount of potential energy and total potential energy should be small u again u r i the probability of in this case is the so I want now the probability what are the locations what is the probability where are the molecules distributed with what momentum complete distribution complete equilibrium distribution of all the n part I do not know what I will do with it but let me think about it that will be p r n p n that is very trivial by I know it is a 2 to the power minus beta h. So, I give you one particular configuration that my molecules first molecule is in r1 second molecule in r2 third molecule is in r3 fourth molecule in r4 fifth molecule is in r5 or like that n molecules in n positions I give you that. So, I give you locations of each molecules you can exchange them classical systems nothing much nothing changes here but given the positions I can now calculate the interaction potential these quantity. Now, I have the also momentum position all the momentum gives me kind of energy. So, this is what actually the equation does is the time dependence of things I am suddenly bringing the equation because we had one slide on the equation we will talk of the evolutions slide be not much because it is a more complex and the level which probably we are not going to take this course 2. So, this is my Boltzmann distribution in a general sense with all the n molecules and this is the partition function. Now, let me see if I do things correctly here the part then I go to configuration integral that we have done in Mayer's theory then this is the configuration integral with beta u this is just kinetic energy term is taken out and kinetic energy term goes into on this debug you have built. So, now I can be little bit more I have removed the I have factored out momentum taken out because of classical system I can take momentum taken out then I have this equation is the same thing I am just saying probability of molecules at given positions 1, 2, 1, 2, 3, 4, 5, 6, 7, 7 molecules say I give their positions with respect to coordinate system which can be corner of the box or any external different systems I know all the positions and if I know the position of each particles I know the interaction energy and then this is the configuration integral. So, this is the complete PRN Pn put wherever you want is the complete n particle distribution is a complete distribution this is all the information that we need but it is very difficult to get is a hugely difficult quantity to get and now we are going to simplify the things because we most of the time experiments and theory and all our applications we did not we do not need the full n particle distribution I do not need to knew where each particle is there I need to know much reduced restriction I need to know how many molecules around one molecule that is the radial distribution function not the complete distribution well complete distribution is very good quantity very noble quantity but we do not need that in our theory and applications and equations that we derived that quantity does not come of course if I know that then I know GR but we need GR we do not need the full thing. So, now it is defined in the following way just as you could expect that the that this is the full distribution function. So, in order to get a reduced distribution lower dimension distribution of fewer particles what I need to do I need to do integrate out I need to okay I do not need you. So, I need all the other so I need smaller number. So, instead of P and N in position of n particles all the n particles of all the molecules of n particle system. So, one is the number of particles in the system other is the order of the distribution function I do not need that. So, but however I do I am continue with classical mechanics that means I am saying okay in this huge number of particles there I want this this and this 1 2 3 where are they I am not yet I am now still specifying n and so then I would integrate out all the other ones I will integrate out right I will integrate out that I will integrate out that. So, what I will do I will just erase these guys I will I have integrate them out or I will fill them up in our earlier in may have theory thing. So, these are the things I have integrate out all others I have integrate out all others I have integrate out but I have to keep the Boltzmann factor. And then I say okay how many ways I can get these my reduced number of particles out of n particles the how many ways I can pick them up then that is n factorial by n minus n factorial not that small n factorial not denominator because we still have the distinguishable particles in their locations which these locations are given by this Rn. We have not yet made that assumption that that positions can be interpreted I am not doing that yet okay then I of course if I indicate over all of them then I should get equal to 1 okay that 1 is missing here but I think probably is there in the textbook. Now, so now I want to now go to the reduced description I want to know that I have only one here if I only one particle I want to know in this case let us consider anomalous system there is a gravitational field external field. So, I want to know what is the probability one molecule here then what I do as I said I integrate over everything else. So, I integrate over all the from 2 to n and from here I have n factorial by n minus because I now have small n equal to 1. So, I have n factorial by n my own factorial that is n by n Zn but I have integrated over everything else d2 d3 all these things have been done what do I get next you can of course guess what you get next. So, when you do that for a homogeneous liquid I can anyway integrate over because I can choose one molecule as this central molecule and everything else I can integrate over. So, then I integrate over the last one. So, these dR2 dR3 with respect to the central tack molecule then the last I integrate over the tack molecule that give me a volume v if I do that they are the same then in a homogeneous liquid I get density single particle density is the number density there is a trivial exercise but important exercise that shows that I am formulating it correctly.