 Welcome back and hope that you got some time to spend on the last class which is Mayer's theory which is a formidable class. We are this is the second time we are doing it, first time we do it little bit more detail but it is a very difficult thing and we I decided to review it once more along with the revision of diatomic monotomic and diatomic and polyatomic gases because these are fairly new things to students of chemistry for whom this course is designed and so in chemistry we do a large amount of body of work on phase transition from the beginning of chemistry from Van der Waals theory of gases from phase transition gas to liquid transition. See in our even undergraduate chemistry there is a big, big chapters and on phase equilibrium phase transition and precipitation in chemistry, physical chemistry, organic chemistry laboratory we are all the time doing melting as a taste of purity and identification. So phase transition varies dear to physical chemistry and physical chemistry is mandated to explain this. Now there is always this curiosity to why how the rain drops form, how gas goes to liquid then of course how ice melts and water goes into ice. What are the molecular processes? These are only present they are all around us but it turned out explanation of them understanding of them challenged the intellectual ability of man to a very high degree and the people and the scientists who are involved in doing these are some of the best minds of that we have produced in entire civilization. So I already described and I hope you spend some time and I encourage you to spend more time in understanding Mayer s theory. How intermolecular interactions entered the picture to Mayer a function which gives us real series and also tells us how one can envisage or picture gas to liquid transition as small, small clusters in the in the in the group like for instance in this room small, small clusters coagulate together to form a gigantic big cluster almost the size of the total number of particles of the system which is the liquid phase which is the correct picture is that exactly what happens that when a gas goes into liquid then many, many clusters of smaller size quite small size actually can be clusters of the size of only maybe 20, 30 molecules not more than that but many, many of those clusters most of them still monomer and dimer they coagulate together in a very short temperature span to form a gigantic cluster of avocado number of molecules. This happened in so shortly and such a short temperature range that we call it a singularity we call it that the properties some of the properties diverge for example you calculate specific heat it diverges. So phase transition has then characteristics of the divergence that things just fantastic things happen but then how do I explain these fantastic things? Mayer's told us okay think in terms of small, small clusters and they suddenly come together to become big clusters but I told you the Mayer's theory however beautiful it is it breaks down because we do not get the cluster integral three level cluster integral we told us how to do these things but those things are very difficult we cannot do even now today we cannot calculate irreversible cluster integral for the inner zone systems of something more than the size of the so for example 10 particles together there is a huge number of graphs that cannot be which cannot be evaluated and 10 particles doubly connected will be 30 dimensional integral which just is not possible actually people in inner zones have done only up to 6 that by he and Hoover who did this beautiful work you know landmark work in about 30 years ago but nothing more has been done after that. However at the 1937 38 and the same time when Mayer was developing his game changing theory of in terms of graph theory in terms of clusters in terms of cluster integral one brilliant physicist called Landau Landau was developing a simple phrenological theory to describe the phase transition and this is called outer parameter theory that went on to become the mainstream of phase transition study that went on to develop many languages and went on to giving probably 10 or 15 Nobel prizes in this area of phase transition. So we will briefly describe Landau theory we did it once before but I am going to do it again probably somewhat from different perspective and as I go on this way I will go I will repeat things course will go a little advance then go back and then again go forward so in that way as I said painting is a way to learn things. So this is a this our picture of Landau the great Landau and did many many things and this is kind of a phrenology diagram which came was inspired by Landau that if gas to liquid high temperature in the temperature large it is gas where molecules are far from each other but in a liquid the molecules that which are so this is the kind of droplet the small clusters from Mayer's theory this one gigantic cluster that forms at the phase transition but look at that there is a barrier it has to go over the barrier and this free energy by going around the free energy barrier is also very difficult process but the appearance of this double minimum is the essence of the Landau theory and this when plotting free energy against the appropriate quantity on the x axis free energy y axis is called Landau's land energy landscape. So we will now do so this is kind of thing from the book that phase transition ubiquitous it manifested large and very important thing of phase transition large experimentally detectable change large change macroscopic change so macroscopic change but the macroscopic change takes place in point a small change that means I change something very small but something very big happens that means this is a small change in a control parameter needs to huge change like the temperature at 100 degree centigrade I just make it I reach 99.9 and make 100 that 0.1 degree centigrade needs to huge change similarly I 0 degree centigrade I make it 0.1 degree centigrade I it makes so the real thing to emphasize that note that that infinitesimal change infinitesimal change leads to a macroscopic change in specific it enthalpy entropy of parts of the system this is the subject of phase transition this is in great great importance in most branches of natural and biological science like it helix called transition in DNA and protein protein folding these are kind of a cooperative phenomena so what is the language here the degree with that the classification there is that many many molecules kind of acting together collectively to give rise to this act of macroscopic change so this is one of the most fascinating field of condensed matter science the natural science that how molecules talk with each other the molecules collectively change over one phase like gas through another phase like liquid phase so how do I understand that may I try to understand from intermolecular potential that is detailed that gives us some picture how things are happening but that fails to give us to fail to describe the generality of the phenomena so we need description which is simpler but which is general which captures the essence of this macroscopic change in a in a thermodynamic property due to microscopic change in infinitesimal change in a control parameter like temperature or density so that is what Landau did so I described it before but I will describe it a little bit more physically this time so phase transition are very common liquid liquid solid magnetic transition ferromagnetic paramagnetic transition ferroelectric and all these things well known examples are all these are a many many phase transitions here so then we study then we ask with three questions what is what why and how what are the characteristics of the phenomena what is the phase transition why does it happen that is a why how does it happen so how essentially brings a dynamics which we will do later not today and that we will describe why how do I describe in terms of phase transition like I go back to in years in years it has come together but how does it happen manifested in how do I describe in free energy this is what Landau did and so Landau said so there is a hint that I this will be given to you there is a large number of examples paramagnetic ferromagnetic phase transition liquid solid gas liquid order disorder in very common in solid phase like alloys binary mixtures again they phase separate in polymers in superconductivity in super fluidity is all in elix coil there is just phase transition is all around us and you have to give proper importance to phase transition not just because it is all around us but phase transition brings in introduces some of the most common and most useful language to physical this is something we learn from like the concept of order parameter the concept of singularity and these languages free energy expansion free energy landscape the all the language you routinely use they started in the field of phase transition so I am talking a very elementary language but those are the newcomers this is a I must emphasize is a very very important thing so these are the examples phase transition let me go through so the all these examples this is my favorite is huge is characterized by this continuous characterized by this continuous change of macroscopic variables that is finite change of a macroscopic variable due to infinitesimal change so the thing to know finite large scale change due to infinitesimal change small change in some variable and this is the hallmark of phase transition so if you say what is the phase transition then this is the phase transition what this is the definition of phase transition what so the I accident why now these are the examples we already did pressure against density in mayors theory you know this then we always ask this in the our interview of one decade new students and then we ask pressure versus a plot pressure versus a temperature plane and this is solid liquid and gas gas here and in our classroom we find really very interesting things pure students put gas here solid here and liquid here and all kinds of things they do but this is the phase diagram so this pressure temperature when these are triple point and these are critical point you know all these things then the another representation so there are three representation these temperature against density then this another with empty so this is the gas liquid persistent then this beautiful way goes so this is the solid so I can describe the same phenomenon three ways it says something very interesting and new like here it says when this kind of van der Waals loop or the kind of spin order or instability here it because these are the coexistence line these are the layer lines along with gas and liquid coexists liquid and solid coexists solid and gas sublimation across that that coexists and here it says how gas liquid crystals are showing together in a beautiful way including the this this parabolic shape inverse parabolic shape which we showing you the bell shape which showing you the critical point and all these things so these are the three representation so phase transition deals with these kinds of graph this is for gas liquid this simple thing I can draw identical thing for almost magnetic systems now as I said one a very important contribution of the phase transition that goes on to many different fields of condensed metastime or even protein folding or any other things is order parameter what is an order parameter and order parameter is chosen as a quantity which is zero before the transition but becomes non-zero afterwards a clear example is point by liquid for the transition where is the order also magnetic transition that is magnetization is an order parameter so a system in a zero magnetization non-magnetic before the transition after the magnitude when it becomes ferromagnetic in magnetization goes up large so gas liquid or liquid solid the entropy is so this is solid this is liquid entropy diverging then you look at specific it that is diverging so order parameter is a quantity that is zero before the transition becomes non-zero is solid we will have some more examples there for example if I so liquid solid so there is a random system then that becomes periodic and periodic I can expand it in the reciprocal that is vectors you know this expansion and this is a fractional density change and these quantities are the connected removal of factor but these are the order parameters so in this example phi naught phi g are the order parameters then in the gas liquid transition is zero liquid minus rho gas that is the order parameter magnetic and ferromagnetic transition magnetization is order parameter gas liquid this difference liquid solid we find out phi g order parameters solgents because it is order parameter so it is a quantity which is zero before the transition becomes non-zero after the transition this was land out deflation now why did land out define it why did land out define it okay land out defined it because land out had in mind to describe a phase transition so land out was looking for a quantity see in in in physics when you want to describe a change we want to quantify the change why you want to quantify change why because if I have to describe a free energy then I want a smallest parameter this smallest parameter is the one that I can expand free energy about and that smallness parameter is the order parameter it is something which is zero and then become non-zero so I can do a Taylor expansion in the smallest parameter so order parameter becomes the smallest parameter in my Taylor expansion and that was the idea of land out and that is a very beautiful idea that now there was when this was done it is another brilliant this is name is Erenfest Erenfest came up with it he noticed something very interesting he noticed that that the there are some phase transitions where entropy undergoes a jump like melting and then liquid solid transition away from the critical gas liquid transition away from the critical point there is a jump like you know you know in water pipe 40 kilo kcal per mole kilo kcal per gram for gas liquid 80 kilo kcal per gram for calorie gram or kilo I do not forget that but they are fine actually however when you go to critical point then specifically diverges a not change in that entity and then magnetic transition paramagnetic free magnetic transition if in the absence of a magnetic field it is a transition which is again continuous but has this divergence peculiar behavior super fluidity has this peculiar transition then order disorder transition solids like brass can show these characteristics this shows these characteristics where second order properties of free energy undergoes divergence but in the melting is the first order property is entropy is the first derivative of free energy so first order property means when I have an order parameter and I control parameter I change that control parameter first derivative of free energy becomes discontinuous so in interest say let me now define capture this universality across the phase transition that first order phase transition or those phase transitions where first derivative of the free energy is dismantled and second order with the second derivative of the free energy they expect order parameter or control parameter become discontinuous suddenly everything kind of falls in place oh I now I say when I say first order phase transition I know the characteristics of first order phase transition whatever that the material is is a gas liquid transition I have from the critical point or melting or magnetic transition depends on magnetic field all have the same characteristics similarly in a second order phase transition is second derivative of free energy but most of the second order phase transition actually continuous transition like the critical point and super fluid transition except one there is one through the second order phase transition in the sense the second derivative shows that it is continuous and that is the most important termination is a superconductivity but however there are all these little qualifications here and there but in interest introduced a very powerful paradigm saying the order of a phase transition following order definition of order now Landau theory comes in beautifully Landau says okay I now know I know I can describe the free energy and I can describe the so this is the hysteresis and metastability then what shows in the presence of a magnetic field so h is the magnetic field and is the magnetization m I change that however be behold that I am switching the site but even after switching it does not reverse immediately it hang on and then go well in while again the other way around it is a long time so this hanging on on the old phase is called metastability and hysteresis and this is a trademark of first order phase transition second order phase transitions never so metastability or hysteresis this is very very important and this is what net Landau to develop beautiful theory so again come back why does the system so such has changed what is the origin of discontinuity then how can we calculate how can we calculate how can we evaluate the transition parameters like like an attendee what are the order parameters then why liquid sodium goes to BCC and while argon goes to FCC these are the questions one would like to answer this is where Landau now came in Landau these are the thing is stability analysis that when does one system become stable and one talks of convexity and concavity like for example I draw the line between the two minima then if the curve then it is convex however if it lies below then it is concave so from stability to metastability the stable phase to metastable you will have everywhere this convexity but however the other early you will have the concavity so these kind of things analysis done in terms of first and second derivatives as the energy is done here but that is very interesting but not the stability analysis and it will essentially go back the specific it has to be and I told you that specific it nothing but delta E square 0 is the average of a fluctuation and square of fluctuation fluctuation can be positive or negative but square has to be positive always the specific it has to be always always positive quantity and that is its stability condition so minus cpt must 0 that means cpt must be positive these are very this concave function and concave function that I describe here you can read it and this and we have done it little bit before but you can do it yourself but now I go to Landau theory and in 5 or 10 minutes that I have I will just describe Landau theory Landau theory is a beautiful theory based on the order parameter Landau theory is a free energy expansion in the order parameter Landau theory is the basis of almost all theories is a phenomenological theory but it is a beautiful theory and so is a mean it is sometimes called mean field theory because some kind of fluctuations are neglected but it so main merit of the Landau theory that it taught us to think in terms of order parameter talk of the minimize free energy and free energy landscape so this is the Landau free energy expansion so Landau said once it will find the order parameter Landau said okay now I know I can expand free energy so you went ahead and expand free energy in terms of this order parameter eta is the order parameter he said okay I am just a Taylor expansion I want to describe this is free energy of the new phase this is free energy old phase this is new this is old and he wrote down the Taylor expansion that all of us know the Taylor expansion right very simple Taylor expansion now we know Taylor expansion we know now Taylor expansion if I have to give alpha I will take DFD DFD eta at the old phase and I get alpha the way we do the a extrapolation interpolation of function but in my case that first derivative of order parameter free energy but free energy has to be minimum because it has to be stable that is we did all the stability analysis then what does it mean that means this quantity cannot be there alpha has to be zero because the free energy of the old phase must be minimum but how do I but that does not apply to this so I can now get the secondary AT I can get AT by taking this second derivative of free energy and then evaluate it in the old phase that is what we tell our expansion we take the derivative but evaluate it in the old phase OATB and all these quantities that temperature dependent and they are obtained in terms of the old phase free energy and now so this is now land of free energy by introducing this condition and eta is my order parameter which is used as a small next variable this is just beautiful now what do you land I do now okay this again I am writing this thing he then said okay let us make think of meta stability and let me think in terms of our hysteresis so if there has to be hysteresis then I must be that when I am hitting this there should be a minimum so it is getting stuck in the minimum and though I am this new phase is more stable old phase still remain a minimum so that explain meta stability what does that mean that means I have this kind of structure of free energy that means I have one deeper minimum one less deep minimum how do I get certain minimum I can get certain minimum by keeping the odd terms here however in critical so this is the first order phase transition where I need to have these odd terms I will not talk of these things I will just talk of this fourth up to fourth power however in a critical phenomena there is no meta stability there is no such no hysteresis so to minimum odd then how does second order phase transition the so important transition in a critical phenomena then magnetic transitions happens that then he is argued very nicely he said okay that happens in the following fashion he said in that case the one minima that becomes flat and it becomes increasingly flat and then it just becomes two phases same so if I go to this critical phenomena point temperature against density in gas liquid or binary mixture I am coming down like that then the minima in the gas phase there is a gas minima in the gas phase is becoming flatter remember second derivative of free energy is the compressibility that diverges it becomes infinite and that then then suddenly when you go below the critical point two minima appear that of the gas of the liquid look at the symmetry is called the symmetry breaking transition so how do I describe that now okay in the second order phase transition I go back to my Landau theory of this expansion I will come back this little bit later and then I say okay there is a the new phase I am going to describe as the symmetry that at plus eta and minus eta free energy is the same this is called parity then this is going to when I said that odd terms these appear I it remain only given term and in the even term there is a very important quantity now this is the free energy old term the a is the one that is the spring constant because this is the harmonic if I forget for a minimum of this thing this is the thing that gives rise to this two minima but before two minima my this is becoming flatter my getting flatter who describes that this quantity describes the flat that means this quantity which is the spring constant which holds the system together which is second derivative of free energy is becoming smaller and smaller how do I describe that and this the brilliance of Landau Landau said okay let us assume that happens linearly they are happy wounds that these a temperature dependent goes through zero at critical temperature approach and that goes so he assumed t minus pc and this is given by the time derivative of this temperature derivative of k now it changes sign in this quantity now changes sign at tc this becomes this is the second derivative of the order parameter but first there will be this temperature now because this quantity is compressibility so it that is why it is a second order position we are talking so this quantity now is made to change sign so as soon as they have changed sign this mine it become minus and become plus that gives rise to this so this is the Landau essentially then one goes on and doing okay this then one solve for eta we take the if I these are two minima then that satisfies the minimum condition that means derivative of df d eta has to be zero and then I do this this term this quantity this quantity and this a I put a equal to p minus cc and I take the derivative and I get this and eta equal to zero is the maximum which is this solution the trivial solution but then I get two values one is for this one and this one equal plus and minus however the beauty is that this order parameter these value the value from here to here where the minima scales in this funny way tc minus t to the power half this is the critical exponent or exponent of the a um called critical exponent or Landau exponent or mean field exponent that is something more needs to be done so this is Landau theory of critical phenomena then one can go on second order phase transition one can many many things are like that magnetic transition weekly first order phase transition then superconductivity isotopic nematics make it this weekly first order almost second order phase transition order disorder is a second order phase transition gas liquid at critical point use second order phase the huge number of things that happens in the second order phase transition so sorry I have to now go back and do little bit so this is the way second order phase first order phase transition is done like van der waals now we have we have we cannot ignore the cubic term we cannot ignore this term because that gives rise to metastability and one can again go back and do one thing van der waals gas we can do that so this is the van der waals equation the free energy starting from this equation I can integrate because free energy is dvp and put this then I can get the free energy and that is this quantity now I do what Landau told us to do I had one expansion in density density is the order parameter I told you gas liquid transition then I get this beautiful thing which is nothing but the real series and that now describes this phase transition and now and described the free energy that one can show that this is essentially very much is the ideal gas part this is the part that describes this free energy with one deeper than the other the asymmetry in the first order phase transition from land learning from land now how to do it go back to van der waals and we can describe that okay so I think this is we will stop here today this beautiful area of phase transition I have done it little bit before I did more now many things I forgot into say in the first time to give you students more physical insight but now I have done it here a little bit more so from now we will go on to doing few more things in phase transition in the next class one or two more class will be on phase transition then we go back again to do statistical mechanics of what the what we will do in phase transition essentially will be simple will be phenomenological landau theory because of the generality it is so general the things of phase transition that it is better to have a very general description that means we need not talk of intermolecular interactions like me at the time when you talk of intermolecular interactions we are demanding much more we are doing so much more work we need to then get the exactly the values of parameters like latent heat with the what are the order parameters then it is okay justified to do a very detailed calculation but then each calculation has to be for each phase transition I want to do a melting transition I have to do theory for melting well still landau theory in the background but I have to freezing that will be different from melting because the phenomena though they are thermodynamically or reverse of each other flat mag they are very different because liquid is going to crystal is different from crystal is going to liquid but if I want to do in terms of thermodynamics the general parameter then again landau theory comes in the mayors theory as is so such a detailed theory so the goal is different and our work is different our these things are extremely connected to each other and a scientist a physical chemistry physicist knows what is time and then develops the theory particular to the need of the time the need of the moment okay thank you we will see you in the next class