 high temperature solid you have done in your high school and that you know that specific heat goes CV goes as 3R from this is from classical vibration that was done in quantum mechanics. And this is called Doulomb physics law however when one calculates it measures its specific heat, specific heat against temperature one before people had up to this and that is you know Doulomb fatigue law 3R. But when you go to low temperature that means these the experimental results become available after low temperature studies were becoming available in 17, 18, 19, 20 and around the time this remarkable departure from classical statistical mechanics though this is just one of those beautiful things that happened starting with the blackbody radiation, photoelectric effect, bummer series, quantum all the all the beautiful things. So breakdown of specific heat at low temperature immediately due to huge amount of atmosphere and people could not do, people could not solve it, yeah because they did not have energy levels. However these running the solution of the harmonic oscillator and energy with N plus of h nu they came available in 1926, 1927. And then at that time these beautiful departure from classical mechanics it was done by Einstein. What did Einstein do and then corrected by there and that became to known as the Dubai theory. So what Einstein realized that the solids has many, many low frequency modes, they are low frequency in current compared to this is a very, very interesting, they are low frequency with respect to atomic vibration but they are not that low frequency that you can treat them classical mechanics. So there are frequencies which are normal modes. So basically when I have to say consider one dimensional model you do Ketial or Ascot Mermin or all these. So we have this chain linear chain then the Hamiltonian is the vibration this vibration this vibration this vibration this vibration, N number of vibration let us say. Now these modes are this position of these molecules X1 minus X2 square plus X2 1, 2, 3 they are in terms of atomic position goes like that, right. Now I can make so when I expand that the X1 square, X2 square, X1, X2, X2 square, X3 square then X2, X3. Now I can make a coordinate transformation which is same as diagonalization where I go to representation where since X1, X2 is a linear coupling and this is a very important and I do not need to go to it but non-linear I cannot do that but linear coupling I can diagonalize this and I go a new set of vibrations those vibrations are now is collective vibration that means it is like in water we have a collective symmetric stretching or asymmetric stretching where oxygen remain the same but two hydrogens are moving together, okay. Similarly here all the molecules atoms are moving together that means there is a kind of this vibration all of them moving together they are like sound waves and they are compression and very position and they are some of them are like this others are going like that. So just like symmetric asymmetric stretching in water there are many, many 3n minus you know if it is linear case 3n minus 6 vibrations are coming. So these are the collective modes when you get by diagonalizing the Hamiltonian and where collective the molecules are moving together and these are low frequency modes and then these low frequency modes as I told you they are low enough frequency to contribute enormously significantly to entropy and to specificity but they are high enough such that you cannot contribute them to classical method. When you go to high temperature then these modes low frequency modes can become treated as quantum mechanically but when you go to low temperature when you go to low temperature to low temperature this is the temperature I am talking of 0 kelvin temperature in kelvin. So in this region so maybe a 300 kelvin or 200 kelvin you can still have the Doulomb Petit's law but when you go to 10 kelvin or 5 kelvin then those low frequency modes has to be treated quantum mechanically and they make a very important contribution and that is now the Debye-Einstein famous theory which first Debye did it Einstein did it and has a exponentially to the power minus 3 dependence what Einstein made he did not this do all this diagonalization or anything he just said all the frequencies from the solid he got the total number correct all they have the same frequency constant frequency that called Einstein frequency then this is very easy then I go to Einstein frequency and you can now calculate the specific it from there all of them same by ds dt Debye did it tightly Debye took care of the what you call frequency dispersion that all these modes are of different frequency they have lowest frequency and highest frequency when he did that he got the real value it is as e to the power cube famous to the cube log but that came out essentially from just what we are doing here okay so so this is specific it dds dt and then when I do this from entropy then I get this expression h nu by kb square so this is the entropy I take derivative I get one term from here I get two terms from here I can combine them and when I combine them you get this expression we then check it out was it real from entropy this very trivial things I am not going to do it but let me that this is the expression that comes in this is the specific heat of the one harmonic oscillator one one harmonic oscillator with frequency nu now we go to solids so it is one by as I read in Einstein data by Debye this expression of the harmonic oscillator plays an important in specific heat of solids and low temperature so high temperature will go to do long physics law but low temperature we now in order to calculate make it as a frequency dependent city nu and then we have to understand integrate with the density of states the nu then see the nu and this is what Einstein need and Einstein said okay I can have only all the frequencies are one at the same identical then I have n number of vibration into then CV nu vibration nu Einstein and then I put that nu Einstein here and I get exponential dependence as Einstein showed in this video. However all these vibrations do not have the same frequency they have a maximum frequency because the wave is essentially waves that you can accommodate within this thing and so that extra consideration important becomes becomes important at a even low temperature that is so this is a really interesting thing that happens in 1930 to around the time that Einstein did and then Debye corrected Einstein so the statistical mechanics of harmonic oscillator plays such an important role in doing this so now continue so the next we will do down the rotational partition function this is the vibration and as I said in the very recently even today all many many people are doing those entropy those specific heat to consider amorphous solids these are low temperature liquids for example we ourselves did a work as you told you last year with the one collaborator courses in the site where long paper came out in the all of chemical physics we used this thing to calculate the specific heat and entropy of low temperature water and that simply explain several things so it is many many people are using these things so they are not just so just like in quantum mechanics the vibrational spectra this by the starting the solution and the harmonic oscillator plays a very important role in it is something really surprising that such a simple model and kind of artificial model plays so important role but that happens in physics again and again that is why harmonic oscillator model is very much cherished that is very loved and liked by scientists because they can use in many different situations similarly now it goes was like that translation like that vibration also all these things are very very important now rotation that why we are repeating it now we do the rotational statistical mechanics of rotational mode of rotational motion with the translation with the vibration we said electronic is not important and but the rotation is important now rotation has a place within vibration and translation in translation it got a without quantum mechanics in vibration we had to do quantum mechanics but in case of rotation it is in between so this rotation energy levels if translation energy level is facing like this very continuous then and vibration will be like this vibration is like this so this is where we are somewhere in between quantum and classical so we will see that in low temperature these energy descriptors the quantum nature of the energy levels the descriptors plays an important role but in a high temperature it you can be like one like like like classical mechanics so let us do now again sweating that is all for a rigid rotator the molecule is rotating with the momentum inertia I momentum inertia plays the role of mass you know so the momentum inertia I this is the energy levels that again the again that Schrodinger gave us E j is j square j plus 1 but there is a very very important thing now which we never faced before that these energy levels are degenerate so now before we are talking of individual energy level they are not degenerate in particular box they are not degenerate in a particular box they degenerate in a different sense that means I can x y z I can 0 I can have 1 1 1 and there are or I can have 1 2 1 or 1 1 2 but we just sum them up because as we they decoupled actually if you take up that it becomes more complicated but in rotation even for a single rotation we have the degenerates which is 2 j plus 1 then partition function goes over you sum over this energy states which are given by energy E j and then we have g i e to the power minus E i by j d t so instead of a energy each energy level we count energy states that means okay this energy state is E 1 but I have 2 j plus 1 of them so then this thing is 2 j plus 1 and this E j is this quantity so my partition function now my partition function now become this quantity my partition function now become this one and I will tell you that again why there is an important role this case also so now q rotation is 2 j plus 1 e to the power minus E j by 2 k by k B t and then I put the E j here and then I get these now I notice one something very interesting till now I have been doing very mundane and very simple substitution but now I can be little smart I realize the 2 j plus 1 is nothing but a derivative of j into j plus 1 so 2 j plus 1 if I take d by d j j into j plus 1 then this is d by d j j square plus j and that is 2 j plus 1 so this quantity is nothing but a derivative of j plus 1 say then I okay be little smart I do d by d j but however that brings out many other quantities the h pi square i k B d by h square so I have to cancel them so I put them in font 8 pi square i k B d by a is this quantity this quantity is the this quantity is the same as this quantity but that is been now very nice okay I made some progress I cannot evaluate this sum exactly which is a pity really pity but we cannot do it many times you will see some fairly simple things we cannot do maybe some of you can be smart and do a better approximation in this case but we have not been able to do so then I can write 8 pi square by from here 8 pi square i k B d by h square equal to d r j and I define the function r j as because it is not no point of writing anything and again so I point r j as this quantity so this is my partition function rotational partition function that means this then is so rotational partition function this is my rotational partition function let me see what I can do it is as I said for many many molecules this particularly important for physical chemists that why we have the rotation and I remember when I was doing postdoc at University of Chicago with James Frank Institute and there are some fantastic group of people there and there are some guys who made history and there are people like Stuart Rice, Don Levy and they were doing spectroscopy in the GAC and there was a lot of excitement to detect the rotational level dependence of electronic and vibration relaxation and this was the reason they have the partition function they know how to they know this thing is there so if a molecule rotate very fast then they will modify the coupling between different energy levels and then we would be able to see if the temperature dependence of relaxation vibration relaxation rotational level dependence that you would not be otherwise if the vibration rotation coupling is not there okay so let us go ahead and do some little bit more of them now okay so important thing is that is a very important comment here they are strong dependence of partition function moment of inertia as I said this plays a very important role very very important role in energy relaxation in vibration and energy relaxation of A so then low temperature we go to T going to 0 limit then this quantity if I go to low temperature then J by J plus K this term low temp T is small and then J equal to 0 of course survives but J equal to 1 is this 1 into 2 then 2 by this K B T survives and then next one when J equal to 2 that becomes 2 into 3 6 but then T is small so the exponent is large so larger J terms gets they gets cancelled when you do that term then you find low temperature is enough to keep one term that is J equal to 0 comes with 1 that J equal to 1 is enough so you get this quantity so at low temperature temperature going to 0 Kelvin I do have a simplification which is this term now the partition function is 1 plus 3 by this a similar simplification happens in the high temperature also now high temperature energy levels are continuous like in classical mechanics that means J into J plus 1 by K B T at temperature is very large they get high temperature they get squeezed they are very close to each other now I can replace this sum by an integral if I replace this sum by an integral then I can do that integration because I already have that interesting thing that this guy is a derivative of this guy so it is just an integration dx D by dx this is the integration I can do that integration and then it is trivial and I get this quantity 8 pi square i K B T by 8 square by 3 by theta so this is the this theta is theta r so I have now a very nice partition function high temperature that that is proportional to and I define a rotational temperature just like before that is h square by i K B T because this quantity partition function is dimensionally must be dimensionless so this quantity is the inverse of temperature then just like in vibrational I will define rotational temperature this is something one has to be bit careful I said a very important thing here that partition function is dimensionless and so dimensionless Q because it is sum over energy levels so this is number exponential is a number so Q is a so when we did the translation of vibration function the quantity that makes this dimensionless is lambda so you have to be very very careful lambda de Broglie wavelength and de Broglie wavelength comes again and again and that comes through this Planck constant and all this combination h square by lambda K B T and that together for the de Broglie wavelength that plays very important role in when in the quantum classical analysis so in the when the Boltzmann and Gibbs beat they did not have the Planck constant so they left their partition function with a volume term and that then they were reconciled as you see the volume of the of the cell in the phase space and all these things but you do not need that really you have to for bookkeeping you do not need the physical interpretation but because there is an exact expression well you can go ahead doing a physical interpretation like it is a tall man did lot of it and that probably helps in certain cases but you do not need that so now I have the beautiful expression of the rotational partition function which 8 8 pi square i K B T by h square this is my rotational partition function so I have the transmission of my partition function I have the vibrational partition function I have the rotational partition function now I go the rotational partition function can calculate the free energy at in the particular high temperature limit so this is the high temperature limit high temperature limit I do K B T L and Q and then K B T L and Q rotation and then I get this quantity and then I get the free energy as I look at that then it is T by theta r and because the T is the numerator theta is the denominator theta you can like that that has to be careful about that and then minus K B T then this theta heart is T by theta heart so there is a minus or you can write theta r by T okay so free energy is this is the free energy then we can calculate the other quantities like entropy and entropy and this is important as I told you that when you are going to talk of low temperature and up in the cloud the entropy of water molecules entropy of linear molecule the still now we are doing diatomic we will do polyatomic very soon but we get the these are the quantities we need so entropy is now a function of moment of inertia entropy of molecule moment of inertia I and the and the expression is by taking the derivative of the we get this quantity because this quantity before that and this quantity there is a temperature here there is a temperature here so two terms come out now entropy and these are the two terms beautiful expression of that that entropy is N K B which is the entropy unit N of course the extensive N number of particles in non-interacting rigid rotators so K B 1 plus T by theta r ln theta r is the entropy now we get the specific heat and the specific heat T T STT we take the derivative of that there is no temperature here there is only temperature here when you do that ln drops Q by theta r in below and the theta theta get cancelled and you take a derivative and T this T 1 it becomes 1 over T and this T cancelled to 1 over T and you get a beautiful expression just like an ideal gas that specific heat is N K B and so this is the thing that we really wanted to tell you that now next in the next lecture we will start with the polyatomic case we finish the diatomic completely we calculated entropy of rotation we vibration translation so but as I again and again telling these are very very important things this even though they are non-interacting limit they go on the backbone on each interaction is built in like virial equation so we start with ideal gas law and you add into attraction and that the virial equation of things okay there we start with the harmonic oscillator model we put an harmonic and so we add the correction term to that comes from ideal gas law this is the way stress stream mechanism is built up actually that is why quantum mechanics also built up that pure perturbation theory that you want to talk of harmonic oscillator but the transition vibration you need a perturbation otherwise you know they are eigen states they do not talk with each other so so the interacting system comes with the perturbation sometimes when the perturbation does not work is to have things are really interesting and so so the next lecture we will start here in the polyatomic system and then we will go on doing some very very interesting stuff so stay tuned so thank you