 Welcome back to the course, continued course on statistical mechanics and I hope you are having fun and good morning or good afternoon wherever you are. And so we continuing with the basics and we did the monotomic gas in a very detailed form. The reason of that I wanted you to go through the working of monotomic gas and it is not just monotomic gas is the simplest system, but it also you get the beautiful wonderful stuff that is used repeatedly which is kind of unbelievable that giving that it is such a simple thing. But then I always draw the analogy quantum mechanics in quantum mechanics you have simple systems like particle in a box, harmonic oscillator, rigid rotators, but those three models play enormous role in quantum mechanics, physical mechanics, quantum science and field of chemistry. The particle in a box you know goes over to be used in explaining spectra of conjugated polymer like butadiene and then dependence there is a hml square term then it goes on to use as I showed in the last class to the density of states and that is to the model of free electron gas a well known model and it is root over energy dependence of density of states and that plays very important role. So there is amazing that how so if you read the book that is given or run by Feynman statistical mechanics you will find that and also Landau-Litz's statistical mechanics these two are the best books in the field earlier things dealt with foundations and earlier parts of then you will find that they are amazing that how much importance these very simple results find in those master books of those masters. So now I want to spend some more time with that because we have again done it little but we did not little in the hurry now I want to review and now again it is kind of you can say that we did fast you know as my my analogy of painting painting of a wall we do it in not in one shot you paint it once and then you paint it again and wait for a sign that dry then you paint it in the final course usually the third course that comes and through that you know the way our mind also works very much like that we get to this. So now we will do the ideal diatomic gas look at the term that is ideal you know ideal and that it is diatomic. So now we have not monatomic sphere but we have molecules like nitrene we have monoxygen you know these kinds of molecules then we will do polyatomic also which will particularly be talk of water and we will talk of ammonia and some other molecules. So first we will do diatomic then we will do polyatomic and the equations you might have seen before but it will be done more aggressively now and more connection to the real world okay so let us start then so we now have in case of molecules that these molecules are more realistic systems and and that they have they they continue to have the if this is the diatomic molecule they continue to have translational motion of the center of mass but they also vibration they have vibration and they have rotation. Now we have to translational part we have done and we can now you have done a lot of these things that you go to a reduced mass description where I can consider translation is that of the center of mass with m1 m2 by m1 by m2 the reduced mass of that and if they are the same then it becomes half but we do have to take care of now rotation and we do have to take a vibration and as I repeatedly say we will again go back to quantum mechanics and find the energy levels from the vibration and rotation and we will use that we can also do it with classical we can also find out the rotation rotational kinetic energy and vibrational kinetic energy and we can integrate them the same way we did in the translation however in translation we did both classical and quantum see we basically did classical when you evaluate the partition function but when you went to density of states we did quantum and one can show if I evaluate the partition function from the quantum particle in a box in a level then high temperature limit I will go over to quantum mechanics that you can work it out yourself we will do somewhere down the line however in the rotation vibration things are little bit dicey particularly in case of vibration in vibration vibration frequency many times in thousands of centimeter inverse like oh it is stretching sorry 600 centimeter inverse and you know one cavity a 300 Kelvin is 206 centimeter inverse. So now thermal energy so the energy of these molecules are 15 times more than thermal energy so that means gap between two vibration energy levels is 15 times KBT and that you cannot treat by classical mechanics so energy levels are discrete and you have to do it quantum mechanical not and the situation rotation is not that at adverse to classical mechanics but that also need to be treated quantum mechanical particularly in low temperature we will see high temperature you can still get away by doing classical mechanics but not at low temperature. So then also as we said that many many many many we need to talk of vibrations many things and again what we learn from the simple non-ideal diatomic and polyatomic gas will go over to very complex system that is my reason to do this system repeatedly because in the this you know in chemistry in physical chemistry we need this much more than a few system please please do not need them that much this is very easily go to interacting systems and they do of general phenomena like phase transition but we in physical chemistry we think of the molecules they are flowing from one part to the other they are undergoing a chemical reaction and when they undergo a chemical reaction then entropy changes the amount of entropy changes in a chemical reaction or the vibrational degrees of freedom that play a role in chemical reaction or spectrum vibrational spectroscopy rotational spectroscopy so we are more in the microscopic world and we explaining our mandate is to explain this property and that require much more detail understanding of the spectrum. So let us continue then with this after this motivation so that let us consider just a diatomic if you consider nitrogen oxygen then we have to have a translational degrees of freedom of the center of mass that is 3 then it rotate like this you know in a molecule it can rotate like this it can rotate like that this rotation is not does not change anything so we have two rotational degrees of freedom and we have this stretching vibration and the vibration degrees of freedom total number of degrees of freedom is 6 and then there is interesting things that come in which at the experimental observable things and gives a very very important properties. So one of the reason we are so interested in physical chemistry so interested in a vibration and rotation is that there is a through vibrational spectroscopy and through rotational spectroscopy we this some of the things which we can access to like an anharmonicity of vibrational frequencies experimentally we get through a vibrational spectroscopy then the molecules are not harmonic we will assume your harmonic but they are an harmonic otherwise they would not undergo a bond breaking event as all of you know that information about the anharmonicity how when you put puts them away how the bond weakens is very important quantity that we get through vibrational spectroscopy then rotation when molecule rotation they rotate very fast they get couples to vibration and called Coriola's coupling that is a very important thing in the bond breaking event in a atmosphere high up in the atmosphere when molecules rotate very high temperature rotate very fast so those are the things that the mandate of a physical chemistry and that we are you know and so those very important information of the anharmonicity or rotational coupling constants we get it from this study that I am going to tell you today and these are the things I did not tell because I did not want to burden you but now we have to do these things and get the what they call the out of these real fruits of our study. Okay so we are talking of ideal gas we are talking of molecules right now not interacting there will be the kind of things I said here that the interact the the coupling coupling between that but we are going to ignore that now and we are going to do with the non-interacting so your ideal gas and the rotation translation vibration are not interacting with each other and there are actually these interactions here so very very important but we will do that later so if I do not consider the coupling then I get a very very nice very very nice decoupling because energy levels are not coupled to each other so remember my total partition function the molecular partition function of a single particle that is why I use the notation small q m stands for molecules then my q m sum over all the energy level not energy states each individual energy level we will we have to change that later as we go along and then the energy is sum over rotation vibration so energy is sum over E E I rotate a plus E vibration plus E rotation vibration translation and plus E electronic that is so then these these are separate sums separate sums so they go into a product so total molecular partition function is product of partition function of electronic the translation rotation vibration now as I said much of these studies at room temperature ambient conditions the electron is in its ground state unless there is electronic transition optically excited which is again a very good branch of ideal chemistry if not then we are we are looking at a large separation between ground and excited electronic states and that does not affect the thermodynamics most of the properties so we can ignore that so these these we do not need to take top of electronic right now then my molecular partition function become product of translation rotation vibration we already have done translation now we will go into rotation and vibration and we will do the vibration first and then we will do the station and you will see a lot of very very interesting things that I will talk to you today which I have not talked before and they are amazing amazing particularly the the amount of results we get and the natural phenomena we can directly access to by the vibration part just we translation I told you that it is very application but here you see even more dramatic things okay so now we start we can write the vibrational Hamiltonian before we go to partition function with the Hamiltonian and then we need the energy levels so vibration is this half mu square you can write it in many ways you know you can write half many times we we are and we write half omega square that is my my favorite vibration and okay so all right so now now the quantum partition function is when you have almost the atomic molecules this is very important as I was taking that for example for a vibration of frequency 300 3000 centimeter inverse the energy h nu h nu is a vibrational frequency energy 15 kb and water OH H the symmetric vibration they are together going like this symmetric that is 36 and 1 kbt 1 kbt 1 kbt equal to 300 Kelvin is 206 centimeter inverse so we are talking of 36000 centimeter inverse versus 206 centimeter inverse so you are 18 kbt now then this bending is about 2000 centimeter we are talking of 10 kbt so these are very very large numbers that means wide way by in vibrational energy levels now vibration energy levels so if I talk about vibrational potential in surface then vibration energy levels this gap is say 15 to 20 kbt for water stretching and for bending it is bending it is you have above 10 k 10 kbt where T is 300 Kelvin so this gap is large this gap is large 0.5 so this is the scenario we are looking at and you have to we cannot do classical mechanics because classical mechanics what does classical mechanics says this say energy levels are continuous so I can go from one energy into next one this small amount by 100s of kbt 100s of kbt or something but that is not true that is true in transmission where I have the next energy level just sitting next to it of the but not in a not not in vibration and not even in rotation here the energy levels are spaced so wide that I cannot consider it to be continuous so then I cannot do classical mechanics this is a prime case where I have to do quantum mechanics and the quantum mechanics aspect plays extremely important role that we see very quickly and most of the results that we do vibrational degrees of freedom even thermodynamics using classical mechanics by using half kx square so I write h half kx square then I write my Hamiltonian is vibration plus then p square by 2m so and then I do the integration by writing the partition function molecular partition function and doing integration there I am just one dimension one particle and dx dp e to the power minus beta h and then this double integral I do and I have both x and p as a Gaussian I can do the integral that is a trivial actually 2 Gaussian integral so one again is the half kx square no longer volume because the particle is now constrained particle is constrained by this potential so then I get root over pi k by m over pi m by k and then this is again 2 pi m kbt that 2 pi kbt 2 pi kbt by k from this term and 2 pi m kbt from this term that is a classical partition function which is trivial and that I telling you it does not work it does not work in most of the cases so we have to do the quantum mechanics so this kind of classical approach which is trivial because my I have such a simple Hamiltonian and I can integrate and get the molecular partition function then go on and put the capital Q and then a equal to minus kbt l and k and I can go on doing my rest of the stuff but that does not work so now we have to do a little bit more so now in quantum mechanics we know the harmonic oscillator so these kind of you know the levels that we have you know starting at solve the system for us like you also solve particle in a box the one of the reason in quantum mechanics textbooks I have told you before that you do not see any name all the way in hydrogen molecule because it is starting to solve all of them there is a very nice book collected works on quantum mechanics by studying that and where you see that he has solved everything particle in a formulation of starting a equation then solving for particle in a box then solving for harmonic oscillator solving for rigid rotator by hydrogen atom and hydrogen molecule all the way despite problem I I got that book very cheap in there Calcutta patience got it bookstore by paying couple of couple of rupees and it is a wonderful position that I had an old book somebody bound it and there are some kind of notes here and there but that book was eye-opener for me that that it was and I felt much more much more understanding of the how quantum mechanics developed is very important historical statement however now coming back to you starting that is a result for classical mechanics and here are the energy levels of harmonic oscillator that energy levels this is a 0 energy level the 0 1 2 and this in the energy of this is half h nu and and then and this is 1 plus half so it is half h nu this is 3 by 2 h nu this 5 by like that goes I have to add it up now so the vibrational partition function is sum over this n 0 to infinity I have up all the way n plus half h nu you want to give by k d t this is my canonical single vibration partition function canonical and I have this from the tax wise the temperature t is coming so that means my my this harmonic oscillator which is oscillating here this guy is oscillating but this is in a temperature bath with a temperature t and temperature t plays a very important role because that allows the relative because of the energy gap relative weight of the different energy level you can easily see how why the many of the cases these particular 0th order energy level plays such an important that is why in vibration on such a big deal to 0th vibration level has been such a big deal for many other reasons but in thermodynamics that is very important but even then even in 0 vibration level it plays a very important role so now I can do this calculation because you know I take the half out so I am going to do that this h nu by 2 k b t so q vibration is e to the power minus h nu by t then I have 2 k b t here missing here I have 2 k b t then I bring here this is correct in book but the when it was transferred from there it was this mistake was met so now e to the power minus h nu by 2 k b t so this is the sum that I have to do and this is very easy right because if I put x equal to e to the power minus h nu by k b t if I make that into x this is my x then I have partition function becomes e to the power minus half h nu by k b t n equal to 0 to infinity x to the power n good that it starts from n equal to 0 to infinity and I know that this series when x is less than 1 this is nothing but 1 by 1 over x all right and good so now this is become my so this h nu by 2 k b t that is in front that comes h nu by 2 k b t and the sum this sum which is geometric series becomes 1 minus x and x is e to the power minus h nu by k b t so this is my partition function beautiful and neat expression and actually in this case Schrodinger did all the work giving us this beautiful energy levels and now we will have lot of fun with this expression so this is now my partition function and I am going to do some thumb diaries with that now because I now know how to do extract free energy from this I know how to extract entropy I know how to extract specific heat and specific heat transfer then entropy of this transfer we wrote a paper last year long paper and water and the whole thing of calculation of entropy of the vibration of water but those are in intermolecular vibration modes but low temperature solid this thing we will discuss now and we will probably discuss more later but we will discuss this something definitely discuss it now so now this beautiful expression that I have here I am going to use that okay so again we write this expression because we love this expression beautiful e to the power minus h nu by 2 k b t 1 minus h nu by 2 k b t so characteristic temperature now I introduce one thing I realize that that h nu by k b remember k b t is energy the dimension of k b t is energy and remember m r g m l square mass by t square the dimension mass length and time k b t is energy h nu is also energy h nu is also energy both are energy so now I realize then h nu by k b h nu by k b has the dimension of temperature this is the dimension of temperature that is now that because this must be dimension of temperature because it has to be dimensionless because it is upstairs now I define a temperature very important characteristic see it is very interesting you have one universal constant plank constant you have another universal constant k b and you multiply it by by the frequency and you get a temperature it is a beauty plank constant Boltzmann constant and the frequency so together define a temperature and that temperature is very important role that temperature is denoted by theta vibration is a vibrational temperature that is how hot a vibration is now I can rewrite introducing this notation that e to the power minus theta by 2 t 1 minus theta vibration by t so this is now my new my rewrite my partition function remember that very beautiful thing but remember that 2 a factor of 2 is there so now I go and do the partition function so this is so this is the 2 vibration now I want to go the free energy so free energy I know that there are n number of them so in number of n number of non interacting identical harmonic oscillator that is the thing so then my q n is q to the power n and and then I write free energy a equal to minus k b t l n q and that n comes out this n comes out in front giving me n here and then k b t remains here so n k b t and l n give me l n e to the power by 2 so this is just using the one previous slide of previous speech and so now this is the I will go back to theta vibration again so free energy now is I can do something more I note that remember that we have to keep track of this minus I have to keep track of this minus so this part has two part one part is the numerator part denominator part and numerator part then l n exponential l n e to the power minus x is l x so h nu comes out minus and minus becomes plus k b t the k b t comes denominator here that cancels this k b t I get n by 2 h nu so this is the first term that comes from there now this comes in the denominator the minus sign that makes this minus plus so I get a plus here then I can n k b t this n k b t here and then l n 1 minus h nu by k b t remember that 2 is not here in the denominator so this is the free energy so free energy of a number n number of harmonic oscillator is given by this quantity okay this is well and good there is no need to spend too much time here now I want to calculate the entropy this fun starts so I do that so my free energy is here I take derivative with respect to temperature remember that has no temperature this term has no temperature so that goes to 0 and then I have here here I have two terms one is that one is this temperature and this temperature here so when I take the derivative I get two terms and that they are shown here one term is this so one term is just this term I take the derivative with respect to temperature this disappears and I just get this term other term n k b t will remain I go and take derivative with respect to that this with a complex because first l n x is 1 over x so that will go in the denominator then I have to go inside and take a derivative and when I take a derivative I this term goes to 0 then this minus comes with the minus but then I take derivative of that exponential comes down minus minus become plus however I have t in the denominator so another minus come minus t square so that minus will be carried through minus but that 1 over t square one of the 1 over t square get cancelled by t so I get 1 over t remain and Boltzmann constant gets cancelled that what happened and then when I do all that so I told you one so one term is a first term that comes and here because this is after taking derivative of exponential d by dx e to the power minus x is minus e to the power minus x so that remains from that term and as I told you one goes to the denominator then goes to the denominator and there is one more negative sign here so what happens in that derivative we have the in the derivative we have the following thing there is one negative here and that get cancelled by this negative another negative comes from here third negative but then fourth negative comes from here so I get a positive term so this is my definition of the entropy this place a very very important role in many many calculations so let me so this is the entropy of the of a harmonic oscillator n number of harmonic oscillators this is look at that is a very strong dependence on frequency very very strong dependence it is in e to the power minus h nu and this is over e to the power minus h nu. Now when let us consider some things first the strong dependence on there is one term then the enzymes and many many molecules DNA and many cases the bond vibrations this is something is amazingly interesting but we will talk of it little bit little bit now this this particular issue before I do that let me consider the following thing that what happens when nu is small there are some cases where you can be small like you know iodine in iodine vibration is about 500 percent there are some vibrations which are even smaller in the molecular vibrations they can be smaller of the order of 100 percent or even less some collective vibration and I will explain what is the collective vibration collective vibration when you put many molecules together then the their translation moves their constant their translation moves are got but number of degrees of freedom must be maintained right so say I have n number of molecules put together they are 3 n vibration then we know that they become 3 n minus 5 in linear 3 n minus 6 in a non-linear degrees of freedom that come so these degrees of freedom go over to the in we call them intermolecular vibrations so they could be rotation of the respect easier like consider see still be then they are rotations of this kind or even long polymer chain there are many many motions which are bound motions but they are very low vibration molecules and so then they can be low vibration or low frequency modes and then then you can start playing certain games you can say okay if nu is small let me consider nu is small when nu is large then we can easily see when nu is large temperature kept fixed then e to the power minus h2 by kb goes to 0 and I have ln 1 and ln 1 goes to 0 and then I said I have okay I go to nu very large nu very large is this quantity would go to 0 and then I have this quantity goes to 0 I have only one compared to one this is small h2 by kbt is large again h nu by kbt large means this term goes to 0 that means if I look at the limit large frequency limit that h nu by kbt much greater than 1 if I look at that limit then this quantity goes to 0 because of this small quantity in front but h nu by kbt will kill it this will go to 0 so this term will go to 0 h nu by kbt will go to 0 will become very small I have left with one and that is already kept in it one so entropy is going to go to 0 so entropy contribution of a large vibrational mode is negligible that is why many many times in molecular calculations we do not consider the contribution of entropy of vibrational modes these are very important very important statement that I am making however that is not the case in this case of collective modes that I said you know the collective modes now let us see that low frequency modes that coming what will happen low frequency mode now h let me consider the other limit that h nu by kbt is much less than 1 believe me there are cases like that in collective modes that in water on many cases that 10 centimeter inverse 20 centimeter inverse and in solids you know you know less than 10 centimeter inverse modes are there and they play very important amazingly nobody thought that they will play certain important modes but they play important mode and I will come to that so now when now nu becomes very large then I can expand that I now say h nu by kbt is small I can expand e to the power minus x and that e to the power minus x I can write as 1 minus x this one cancels this one and this become plus x so I have n kb h nu by kbt kb kb gets cancelled I have h nu by t sitting here which is nothing by my I have done it before h nu by t sitting in a come in front now let us look at this quantity now nu h nu by k this is small so again I can expand that I can expand that so this will become 1 minus x 1 gets cancelled I have get h nu by kbt coming out from here with a plus sign in the denominator this h nu by kbt I said 1 minus h nu by kbt but since nu h nu by kbt must be less than 1 I can neglect the next term so now h nu by kbt that so I have one only from numerator denominator I have h nu by kbt and that now cancels h nu by t and I am left with a kbt term here and that goes upstairs so it has I get a just a kb contribution from this entire things and I get a contribution from here h nu by t from here so high low frequency have a very different contribution to entropy they come up with a significant contribution so while high temperature high frequency modes make no contribution to entropy because h nu by if they are all in exponential however they are all in exponential like this term like this term but in a low frequency as I just discussed a significant nonzero contribution comes and that play extremely importantly and that is where these seemingly trivial calculation that you are doing play the enormous the important role in understanding in a in a very large scale large important phenomenon that is nothing but specific heat of solids now specific heat of solids specific heat of solids low temperature solids is an extremely interesting thing