 is done we are now going to go to rotation. Again what the game we play? We have a diatomics which is rotating and then we go and calculate the, solve this Schrodinger equation assuming as a rigid rotator. Remember rigid rotator of quantum mechanics? Okay and what are the energy levels? Important thing is that I is moment of inertia in the denominator just like in translation M would be in the denominator. Moment of inertia more energy levels get quashed together. Okay so this is the A we usually use this notation but I do not swear I use this one. So Ej. So now important thing one very very important thing in rigid rotator that given J is the quantum number like N there was the quantum number in harmonic oscillator. J is the quantum number however here it comes with the degeneracy. If you remember rotational levels come with a degeneracy. Anybody remembers Ray's hand? Who remembers the rotation as a degeneracy? Okay one guy remember two guys remember. These two does not, oh you three remember others do not. That so and when there is a degeneracy then it is trivially, partition function is written trivially. So this factor is the degeneracy factor. Degeneracy of an energy level. Okay so it is just the weight. You know G is the universally used as a weight factor. It is the density of states. G is used as a density of states and so it is essentially weight of that like in radial distribution function again G used. So G comes we use G as a is almost universal thing in a density of states and so this is the same thing. So this is the density of states 2J plus 1. So I put it there and this is the energy. So I put the energy H square JJ plus 1. Now something very interesting here did you notice that? That if I take derivative of that I get that quantity. This one beauty. So then I write this quantity but if I take the derivative of that what happened? J square plus 1 become 2J plus 1 but I also get out H square by 8 pi I square I get which I should then compensate for and that is compensated here. Okay so now write the partition function in a very neat way in this in this in the following form. Pretty neat very nice but unfortunately we cannot evaluate it. Nobody has been able to evaluate this. So our rotational partition function remains little bit stuck. This has to but of course these days is trivial. Before people used to have tables and people tabulated tables for all by rotation and different molecules. Our times we used to have tables but now you do not need tables, tables have disappeared because you can put one line in computer. So you see now in this case there is a very strong dependence on momentum inertia I. Okay very strong dependent on momentum inertia I. As I told you it is very important. So low temperature but we can though we cannot evaluate this one. We can evaluate a high and low temperature because at low temperature there is one case. High temperature one case. A high temperature this becomes small. Low temperature this becomes very low temperature this becomes so large. This becomes large and so negative it goes to 0. So low temperature and only first term would be enough. Okay so and then low temperature I can just evaluate this j equal to 1 becomes 1, j equal to 0 become 1 then I keep I need to keep only one term j equal to 1 that is 3. Okay and then I get this term and that is pretty good. Okay so this is the partition function at low temperature. High temperature now gap between energy levels become very small. So I can now replace this sum by an integral. Okay is that clear? One things become very close like translation of partition function. I can replace the by an integration and this integration I can evaluate. Any way I know that this is d by dj and d by dj is trivial to integrate because you just evaluate in the two limits. This integration is done for you and then that is this quantity. So in the high temperature your rotational partition function is t by 8 pi square i by k B T. Now we define and since this quantity has a dimension 1 over temperature then I define this temperature as a rotational temperature. These things are important because this can tells you that what is the relative rotational temperature tells you below this temperature you have to be a low temperature limit above the temperature you have treated as high temperature. Similarly in vibration theta vibration tells you this is a crossover temperature. Below that and above that things start of course it does not change at that temperature but it kind of goes through this swing. So it gives a very good estimate of doing those quantities. Now we go again and do the partition function minus k B T L and Q. Again I have a bunch of rotators which are on interacting I have N k B T L N Q rotation. Now this Q rotation is L N 8 pi square by A k B T. This is I am doing now the high temperature and as I told you many of the cases high temperature is reasonably okay because it is 50 80 centimeter inverse. You are drawing say 300 Kelvin to 1 centimeter you are not great but still you can get away. You are better than low temperature limit and then that is this quantity N comes out because Q is Q rotation to the power N that comes out minus N k B T L N 8 pi square I k B T by A square and then I use my theta R again. So I have this neat expression of free energy N k B T L N T by theta R okay. Remember that now I play the game again I find the what is the rotational entropy that is a term turned out to be extremely important and modern days. What is the rotational entropy? It was not even our time when we studied these things we really thought circuitated equation. Translational partition function only good for P B equal to k B to the beautiful derivation. Then vibration we liked it very much we knew because side by side with soil state physics the soil state chemistry was taught so we knew that rotation we did not pay any attention to. However it turns out I will tell you why it is so important. So this is the entropy now I take this derivative from here this quantity neat and clean and very simple. So I can calculate the rotational entropy right we go there you take the derivative one term comes from here another terms comes from here and when I do that I get this is the first term this is the second term. So this is the beautiful expression for rotational entropy then I take the specific it just I get in vibration and then I get derivative of that this term does not carry exactly parallel to vibration. So then this ln become 1 over t by theta r then A and it becomes 1 over t and that cancels the t cancels the 1 over t minus minus and all these things become nkv. So rotation with this is beauty is that rotational specific it is nkv vibration you already know. So it is becomes r now we will just talk of some of the thermodynamics. So we have now derived all the things vibration translation I did in the morning we did vibration and rotation and I have to come back and do little bit of monotomic but that I will do in later because there is still some amount of way to go and as I told you that I wanted to start fresh and do this good job to diatomics because that is not done in a unified way. I have not seen even the all the stat mech books have not done a good job in bringing the whole picture clearly to A. Now let us see some examples again. So ideal we go to now before we do that ideal polyatomic molecules we did ideal monotomic ideal diatomic then polyatomic for example linear we have 3 translation 2 rotation these are the vibration these are standard questions in oral some type even in phd interview of course it has to be even your job talk even job talk sometime we ask the suddenly the faculty candidate also ask these questions. So if we are nasty we ask that and many times we find that by that time they are giving a job talk they forgotten these basic things ok. So you know all of this that in a non-linear 3 rotational degrees of freedom so 3 and minus 6 ok. Vibrational partition function total partition function rotation not vibration total partition function then is translation non-linear means 3 rotation then 3 vibration. So this is the total partition function this should be total partition function. So this is the total partition function in all its glory. So vibrational partition function is this quantity rotation and translation we just described. The reason of writing separately it was not written before. So, this is the rotational partition function of a spherical top. Spherical top is like a methane is we are now doing polyatomic we have done monatomic, diatomic, polyatomic and this is where much of the, well diatomic already lot of interesting things are there like oxygen, nitrogen then you have carbon dioxide not carbon monoxide, oxides, cyanide all these things are diatomic but polyatomic is where water, methane, ethane and ammonia all these things. So, but you have a symmetric molecules then all the moment of inertia are the same that is the definition of a spherical top then it is very easy this is it. These things do not matter in vibration but it matters in the rotation. Then there we can do these calculations here that these integration as we did just before is the same thing again done. However, if they are symmetric top not spherical top then the condition is that two are equal but third one not equal moment of inertia. In that case I write the rotational partition function in the following way clear. Two of them that is why half goes away from here third one the asymmetric one comes here as the half. So, this is the partition function of the rotation of asymmetric top molecules and that can again be calculated by the kind of approximation. Now, if it is completely asymmetric then you have to be work much harder you have to write IA, IB, IC are completely different from each other then you have to write them explicitly with their moment of inertia. There is a symmetry number that comes to take care of the symmetry of the Hamiltonian here which is typically 2 or 3 but I forgot the exact origin of that and anybody remembers this is classical mechanics otherwise you have to clarify in the next lecture. In terms of theta r this is the rotational partition function. So, all these things work done now leaks through this is the table that I wanted to talk of that characteristic temperatures and then and we now beginning to see what is the why what is the low temperature high temperature approximation. Look here water so this is with the remember with respect to KBT h nu by KBT and this similarly theta rotation is here is the theta rotation. So, it is essentially the moment of inertia everything else is constant. So, then look at here how high these things are in vibration rotation they have come down drastically. Water is an there is no symmetry and all the 3 vibrations are different all the 3 rotations IA, IB, IC 2 of them are close to each other in vibration and by rotation all 3 are different. Here is ammonia and these are the ammonia vibrations and these are the rotations of vibration then sulphur that's all these things. Now these are given in my book and they have been one of the important thing that say all these things are taken from the book. Now this is entropy of polyatomic molecules. So, this is trivial because they are not interacting so entropy is the sum of the entropy. So, you have the rotation this is the translational entropy circuitator equation this is the vibrational entropy in the general form for anisotropic molecules and this is the vibrational entropy. So, translation rotation vibration 3 things added. Now most of the time because of vibration water I think this is so high that in there is only one term that is important and here is this the table for water translation is this is the absolute value of entropy that it is 17 but look at rotation rotation at 5.45 that is rotational entropy is about 30 percent of translational entropy. This is an extremely important result for ammonia is again but look at vibration vibration entropy is negligible. So, if I think of water ideal entropy of water whether I told you is a significant actually real entropy of water still these 22 makes a huge contribution. Entropy goes down to maybe a value of 10 or 12 but these still makes the lines contribution. So, thermodynamic properties of water and ammonia lines contribution comes from the ideal this is something I had no idea myself when I I was taken away completely by this interaction and all these things but ideal plays a very very important role. Again repeat then case of water rotation vibrational entropy makes no contribution rotational entropy is about 30 percent of total and this is the decomposition but you can see though these are fairly different molecules there is a near constancy of yes that is correct chemistry of the molecules will come through electronics and so that that is a good question. So, but the ammonia and water is different chemically because of their electronic property also there is a polarizability which is again electronic properties and so we are doing ideal and because of electronic polarizability and the shape of the molecules interaction between two water molecules and two ammonia between water and ammonia they are very different those things are not here. So, we are not having any interactions. So, when I am doing of thermodynamic properties ideal gas that we should not be carried away you are right when it comes to chemical properties then the electronic features play an important role. So, the translation entropy contributes most but rotation is 30 percent of the total actually very recently till about about 5 to 10 years ago when you are calculating entropy of water we are neglecting rotation contribution and we are writing a paper which will be submitted in the next few days. I told you are that you are doing a ice water interface and ice water interface the ice goes into water in the ice you know I never have translation entropy not of much rotation but when it goes to liquid translation and entropy you do not gain too much which is still highly constrained but rotational entropy is the one that dominates ice water interface for any molecule sorry rotational entropy is very easy to recover because rotation is not a conserved quantity but translation is a conserved quantity is a very important. So, I can destroy the rotational order very easily you know I can take one out of a rotational degree of freedom but I cannot do in real space orientation is not a orientational density is not conserved conservation is a spatial density or number density yeah but you see it can be anywhere in this sphere. So, again it is a very fundamental concept coming from a little bit different field but it is a number density okay if I I don't I had to do that because I think most of you will not be able to follow it but let me tell let me try I have a density which is space dependent and I have a orientation dependent okay I write it as a spherical harmonic expansion that is what I didn't want to I should not have said what I said because it is just going to confuse you guys now this density is conserved where does the conservation come if I integrate over and I say that's a conserved quantity that will then these conservation will be will be constrained only on a 0 0 isropic part orientation this you have to think yourself I think I should not have got into that and you don't want to get into that but that's a very interesting thing I mentioned in the context of rotational entropy that many cases you recover rotational entropy but you do not recover translational entropy okay this is the path of I now want to go back and five minutes do the okay these we have done in the morning up to this day chemical potential gran canonical take the derivative the now we'll do what the question you asked that what is the way to do the micro canonical ensemble for ideal gas right and I said the following it is very difficult in a classical but it's trivial in quantum so if I want to do if this is the ideal gas monotonic ideal gas this is the Hamiltonian so partition function should be dr1 I just do one particle dr3 dp1 dp2 dp3 then delta h-c so nve constraint means Hamiltonian has to have constant energy okay so you pick up a hyper surface you pick up a hyper surface and these of course gives me just the volume as before but this has to have this constraint so I cannot have any momentum and any a so see what they become trivial in the canonical partition function because I have just a Gaussian integral because I love energy fluctuations in by bringing temperature but in the constant energy surface this is you can do only numerically okay however in quantum it becomes easy because in quantum I have to sum over energy levels and I can solve particle in a box and this is my volume here right a length of the box this is a cubic I can give a cubic box then it is like that okay now this is nx ni nz all I introduce as n square then that n square here if is the constant energy is there then I can go there is a reason I am going to do that this is very cumbersome way I will first define the energy then I would say nx ny nz can be 0 to infinity and then it is amazing that students even after 5th year PhD students they have no idea many times how to write equations of course that is the problem will come in chemistry I remember I worked with a very distinguished physicist who is now a chair professor of something university of urban shampoo and he has a very famous book on statistical mechanics called Yoshigo Ono and I used to work with Yoshig at one state in Chicago and as a postdoc and Yoshig always used to tell me Viman why do you write equations in such cumbersome form and a physicist has certain affection and love for mathematical equations which shows in their writing here also see how cumbersome it is written okay nx ny nz are all of them from 1 they are not from 0 remember okay if that is then if I now I should introduce that these quantities n square because they come together remembering that nx ny yz can vary each of them independently then I put them equal to E now I am going to use something the because I this nx ny nz make a make a cubic system I vary nx along x axis ny along yn is there they are discrete points in a in a digit in a disc in a grid so however since mine is 1 to infinity I populate because of my energy levels 1 1 1 1 8th of that okay so however the n square I can write at as this quantity a vector in that space is n square and that is given by e so that is this 8 ml square h square by e now in three dimensions with the radius n so I can calculate the density of states by calculating the number of points is this clear so these are all so so my total number now nx ny nz are all positive okay so they will be on the one side of that that will be I need to calculate okay it is not the whole space so if I do that then it is 1 8th of the whole average so total number of points is in the radius radius is n is nx square ny square nz square this is really lucky so then it is 4 pi by 3 pi n cube this is the volume oh so 1 8th of that n I know n I know because that is connected to energy e I have a envy envy ensemble ideal gas in envy e they are non interacting so my volume of the box comes through l energy comes through e total n will be when I put them to the n because n number of them so envy each play a role each plays a role okay is a really very pretty so this is your density of total number of states omega of that gives you the partition function sorry this is your partition function ln of that gives you the entropy so this is the way you know this goes on doing this is the what soil state physics the density of states and all these things but we do not need it right now for statistical mechanics this used in some cases but we if we need later we will do but this is where correspondence between quantum and classical statistical mechanics of a monatomic ideal gas this is the capital omega these total number of states capital omega which depends on I should have put it into capital omega again so this is the depends on e this is depend on v of course because n can be one to infinity particle in a box remember the quantum numbers quantum numbers from one to you remember that are to forgotten okay so then no between doing so you are using so these that's why many parts of statistical mechanics we cannot teach unless quantum is done like when you particle in a box we need harmonic oscillator we need rigid rotator so that's why in everywhere stat mac 1 starts in the second semester first semester quantum 1 quantum 1 and thermodynamics so courses that I have taught in many places it was always the second semester which was very bad because everywhere quantum stat mac used to be full winter when I taught at Harvard taught Wisconsin everywhere it was always the month of January February March April which is horrible thermodynamics is the first semester which is the called fault here also in at our a in a list of science ss 201 sorry our sorry state ss 201 used to be thermodynamics ss 202 used to be statistical mechanics same in physics everywhere but by the time we are doing ss 201 side by side ss 204 or 5 is the quantum mechanics so quantum is done parallel to thermodynamics then comes as okay but that's a different thing so this is what completes I said I'll take 10 15 minutes more is done so next in the book one usually goes doing both science and condensation and both statistics I'm not going to do that that will do later because this is a non-interacting system I think next what we need to do is the interacting system because that's where much of our applications are but if you do not do the ideal gas or ideal case then you cannot do the interact interact resistance no you take the center of mass center of mass is always well defined yeah the center of mass total mass and then total mass then for rotation you will have mass of the you will have momentum inertia along each axis there are three principal axes so in non-linear molecules we have three moment of inertia are you on i to i3 similarly when you do vibration each vibration we will have an effective mass so translation will have one mass which is the central mass and then three masses for vibration three masses momentum inertia for rotation anything else okay