 In the morning we did the monotomic gases and I will come back there is one part we did not do which is lot of fun, we will come back to that but I just want to start fresh with the diatomic gases which has a huge number of very important results that we use in everyday in statistical mechanics in liquids and gases and solids and you will see then monotomic gas results are used also in the diatomic gases that is one of the reason I want to straight go there and get those things done to get started okay. So there are diatomic gases of this additional degrees of freedom not just translation we have rotation and vibrational degrees of freedom important difference now is that while translational degrees of freedom barion large except at any low temperature can be treated classically vibration cannot be treated classically vibration degrees of freedom except some cases and we will discuss like normal modes of very soft modes and low frequency modes and the vibration that we see in molecules that cannot be treated classically rotation is on the border line and you are better off doing it quantum mechanically and they are the most realistic things because as we say our diatomic gases are essentially noble gases but most of the other molecules that we nitrogen oxygen and all these things are diatomic. So this discussion today we will do will be extended to polyatomic molecules like water and ammonia we will discuss lot of things for water and ammonia which will be very very important and then applications we will do specific of solid and tropics okay these are applications we will not do too much of that but we will mention them how these things go in it is equations we will derive today are used in understanding glasses understanding proteins and DNA in a very very well known examples and these are what are the true many body systems which we usually in our chemistry vocabulary which is so much saddled with individual atoms and molecules and quantum chemical calculation of them we hardly do many body phenomena but when you do many body phenomena where many particles are interacting with each other to give such phenomena as phase transition we need to do this interacting many body systems however before you do the interacting many body systems we need to understand non interacting many body systems in the morning we did monatomic gases and now we are going diatomic gases and diatomic gases introduced many profound important concepts which will again will be used in many many cases many many applications these are just some of them which we will now go to okay as I said it is very important after pretty many things things will get more mathematical but nothing very difficult so remember that if you have such a diatomic molecule then degrees of freedom there is a three transitional degrees of freedom we model by the center of mass then two rotational degrees of freedom and one vibrational total is the 6 2 n so then when you have a high temperature when the molecule rotating very very fast there is a coupling between rotation and vibration that comes in what is the name of that coupling you have done that you are undergraduate remember the qr branch of spectroscopy huge amount of things done by mood and castellan you know have some interest for God's sake these are very nice important things because they give huge amount of molecular information okay that because these are called Coriolis coupling and also the other forms of coupling that comes when for example the molecule rotates very fast then the bond another is centrifugal so there are two things that come in we rotate very fast then the bond can get stretched and that those are the called anharmonic effects but at low temperature we can low temperature these things can be ignored so we can have approximate low temperature by rigid rotator which is much of the time okay in low temperature by the 300 Kelvin room temperature you are little bit on the border line you know we start seeing signatures like in pqr branch of where one sees in vibrational spectra the rotational level dependence and that has been enormously important study in 1980s that to see the rotational level dependence of vibrational or electronic luxation that was a big big thing in those days you know you know just want to see that to be so remember the Nobel prize that Ahmadzuel got in 1998 experimental result was done 1987 used to many of these many of these things to establish certain reaction pathway okay so if we now assume that these things are independent of each other rotation vibration electronic independent of each other that is an approximation yes rotation there is a very good question that can also come but that is can come by the way for example if you are large scale collision then you can have a coupling between rotation and translation but in usually that is weaker than in the rotation vibration rotation coupling okay now so if they are in non interacting and they are independent then total molecular partition function qm starts for molecule molecular partition function is a product of the individual partition functions that because if they are independent the energy levels of each are different and partition function is some of the energy levels you understand that okay q is sum over e to the power minus ai by kbt and this part I think I will probably remind you few times because this we have not done properly or in great detail so partition function if I say q notation then e to the power minus ai by kbt movement constant now sum over i now see if ai depends on e i electronic plus ai vibration plus ai rotation plus ai translation then you go some then you separate them out because these goals in the exponent these all becomes all becomes a product so this is then a non interacting molecular molecular partition function now the important thing is that electronic energy levels now I have asked you a question that what is the typical electronic energy levels in atoms right sorry in molecules and how does it compare with the vibration rotation and translation let us start with translation what is the energy gap between two if I do quantum mechanics then I get that that particular in a box give me a number what is the typical number of energy levels spacing between two translational energy levels these can kind of tells you whether you think at all or not so you have done all of these things you are doing statistical mechanics and you should be able to tell what is the energy level what is gap in translation say loudly area so many of you should make at least one guess it is okay to be wrong but you should have some guess still okay I give you one number room temperature 1 kbt is 206 centimeter inverse then how many centimeter inverse that is what experiment is do and I worked a lot with experiment is how much okay 10 centimeters that is a good guess how much your rotation how much more that is actually correct no much more than about say 100 or maybe 50 on the meter above depending on the momentum inertia now vibration what is the energy levels in vibration molecular vibration this you guys should do because this we did a lot in undergraduate what is the vibrational gap between vibration energy levels say for nitrogen or oxygen exactly 3600 centimeter inverse one is symmetric stretching is 36 I think 60 and asymmetric stretching is 3700 centimeter inverse how much is bending OH bending you said something correct next one very important the OH bending that plays a very important role in your body so I will you I will ask you so I will I will continue I will come back to it so translation is much less than kbt so at normal temperature translation is like a continuum rotation is borderline low temperature it also discrete or high temperature is continuum vibration if it is asymmetric stretching 37 bending is 1885 then that is about 10 bending is 10 but others are typically 15 to 20 times so that is absolutely quantum how much is electronic energy levels see you cannot you you will not be able to talk with an experimentalist if you know don't know these basic things that's what they're doing all the time in IR FTIR NMR these are all the things how much is electronic energy levels more than 10 something 10,000 centimeter inverse so 10,000 centimeter inverse means you have something 50 kbt or so so gap between ground state and excited state is that huge amount that's why it does not affect that there is no population at normal temperature in the excited different state everything happening in the ground state then we don't need to carry this around this is also the reason when the solid state physicists let not talk about solid state I don't know how much they really follow things solid state physicists never bother about temperature they don't need to do statistical mechanics for much of the time the reason because that when they talk they are talking of the excited electronic excitation much of the time so they are talking of this huge energy gap okay let us now do vibrational so we have to do it quantum then we have done translation translational part remains the same as we did in the morning then we do vibration then you do rotation okay so this is a half nu square x minus x square so this is m l square and this frequency 1 over t square so this is energy m l square by t square it is reduced mass of the system in this case says that you know I am going to 1 degree reduction of degree of freedom that means the my a is this is my reaction my coordinate now we have to solve this now tell me how we will go about it now here I say 3,000 15 k v t all the numbers are there now tell me one thing how do I go about it now okay the way I go about it is I solve the quantum problem Schrodinger equation and this is the energy levels this part all of you have done many many times okay and then vibrational partition function there is just sum over these things with some n from 0 that is a big difference 0 to infinity okay if n equals 0 to infinity then this quantity half h nu by k v t there is certainly here and there are some mistakes we have to find the mistakes my I asked my student to this there is a 2 missing here why they went to write these things I don't know you should have just hear that okay there should be n plus half h nu so e to the power minus half h nu by k v t would come out I remember this slide we need to change okay and then this goes out and then we sum n equal to 0 to infinity e to the power minus n h nu by k v t now this sum can be done and tell me what is this sum this is the energy level so I have to revive k v t like e i by k v t this the thing that is here so e to the power minus half does not have any n it comes out that comes out here actually that should come in front actually okay but we can leave with that because that has no n then I have brought it here then I have to sum n equal to 0 to infinity minus h nu by k v t so this now n equal to 0 so n equal to 0 is 1 then this now e to the power minus h nu infinity is the x okay so it is 1 plus x plus x square plus x cube that is the geometric progression and then that is equal to 1 over 1 minus x and x is equal to 1 minus h nu by k v t so it is 1 minus no h nu cube and this quantity comes here so we have a but this is a very beautiful result very nice result the vibrational partition function e to the power minus h nu by 2 k v t 1 minus very pretty okay and this is the one we will discuss more about it okay good that we corrected it so again the full glory equation now this defines temperature which is also the one actually done by d by so now h nu by 2 k v t this kind of game I divide by a theta by b so I see this quantity h nu by k v has the dimension of temperature because exponential has to be dimensionless so I define as the vibrational temperature and vibrational temperature as you said would be 10 15 or 20 times the normal temperature okay so it is theta vibration by 2 t 1 minus e to the power theta vibration by g okay now we from this now we are going to go this is the form which this is the form what is used by d by when I was doing the specific derivative we might come back to that now so free energy I know this is a canonical partition function free energy now let us again slightly go back let me repeat how we did this calculation we are going to calculate the canonical partition function we have a system that characterized by n v t and then the quantum mechanics giving us the energy levels in a classical system quantum mechanics gives up the force field that what you guys use that the particle interact with it with Elena Jones. Elena Jones parameters were initially obtained from experiments by fitting they found the small a and small b of interval parameters and the Lena Jones Lena Jones actually found it what b s is named so either experiment or quantum calculations give me the potential force field here quantum calculation will give me these quantities and this mass we know but these are the quantum one's quantum gives us that with the assumption of we get the energy levels once we get the energy levels we are now going to get the vibrational partition function and here the temperature t is there okay now and then this follows so now this canonical partition function has the temperature in it we are now going to calculate the free energy so n v t now the vibration this is a single one vibration but if there is n number of vibration then what will happen we have discussed in many many times you should be able to tell me if I have n number of non interacting harmonic oscillators exactly will be the same and they will be to the power n right so capital Q vibration n v t is q vibration okay so if that is the case then l n q that q n comes in in form n k b t l n this quantity now a minus sign is there minus sign is here so now these are two logarithmic terms I take that four in the first term this one I get n by 2 h nu k b t k b t cancels second one n k b t remains since in the denominator minus and minus gets cancelled then I get 1 minus into the h nu k so this is the canonical partition function of n number of harmonic oscillators now I get to go to entropy s minus d a d t and s n minus from there I take the derivative d a d t this guy has no temperature dependence and by the way what is where this term coming from please tell that where is this term coming from it is very very important where this term coming from it is the 0 no it is 0 vibration level the famous 0th vibration energy which saves the uncertain principle so even at 0 Kelvin this motion remains this amazingly important thing so with the 0 vibration energy level okay that is this term that is no temperature dependent so that term drops out and so now let us do this this I have to do temperature derivative one is just n k b l n this term next term n k b t then just like we did in translational case we take then that comes as 1 over minus this thing then I go inside and take derivative of that and that then get give me one I recover again into the normalization by k b t minus is here plus another minus from 1 over t derivative so that minus 1 over t square okay and now I take that t out here and there was a t in front here so that t and t square here gives me this t here h mu by t because that comes from derivative and then this is the quantity because I said I will bring that here and three negative makes it negative that negative comes here okay so these then when you do this derivative come here you do again the derivative you say this is just n k b l n this term and the derivative this term comes in denominator so then this term plus this term okay so this is the expression of entropy so one important very important thing is that there is a very strong dependence on the frequency of the harmonic oscillator because it is in exponent everywhere entropy is strongly dependent on the vibrational frequency which is a very important thing because since nu is large in the atomic and molecular cases this does not contribute much to thermodynamics so many times in our calculations we do not take vibration into account when you are doing simulation of water you do so s p c e at tip 4 p but you do not have vibration there or any other systems we just do not that because this is the reason however when you have big proteins there are many vibrational modes which are low frequency which are conformational vibrations DNA enzymes proteins or even you know when you are looking of low temperature glasses there are these so low temperature for solids this becomes classical approximation of course on solid but quantum nature of vibration leads to decrease in the specific heat this is the famous they remember lowering of specific into the temperature what is the asymptotic what is the dulong pettit's law dulong pettit's law tells the specific heat of a solid is equal to 3r very very good 3r 3 by 2 comes from a and translation divided by vibration okay we will derive that in a minute but what Einstein tried to explain that high temperature it goes to dulong pettit's law 3r but low temperature it deviates significantly that was among many other things signature of quantum mechanics Einstein did one more signature of quantum mechanics what is that photoelectric effect it started of course with planks right blackboard irradiation blackboard irradiation also these vibrations are involved because that's electromagnetic heat okay now I will do the specific heat specific heat this quantity just we did before so here we have to take the derivative of this quantity now it is getting little complicated they are not as simple as translation but we would be able to take that like ds dt you know again this guy comes below then I take the derivative and that becomes then e to the power minus h nu by kbt and then and outside comes minus h nu by kbt square minus another minus comes here as I said you minus minus minus it becomes at the minus 4 minus it becomes positive and then you take derivative of this thing there is fairly complex you have to take derivative of this that means minus 1 over t square and then you have to take derivative of this guy then again the same game minus and minus plus h nu by kbt square and then you have this quantity which you also have to take the derivative that then it becomes a square here because it is a denominator then you take the derivative of that when you do all these things you get that square comes here and so this is finally combining everything this beautiful expression comes in so this is a specific heat of a bunch of harmonic oscillators which is so important in solids because solids remember normal modes now what are the normal modes in solids what are the normal modes in solids how do you get normal modes in solids right absolutely that is called dynamical matrix you are like that x1 minus x2 square plus x2 minus x3 square so x1 minus x2 x2 minus x3 x4 x3 they are all coupled but x1 and x2 are the positions instead if I make the transformation that is x1 minus x2 is the coordinate so it is called displacement coordinate then it you can decouple and it is formally done by introduce what is the dynamical matrix at transformation then it becomes a bunch of harmonic oscillators and the eigenvalues are the harmonic frequencies and eigenvectors are the modes so solid state much of solid state can be described as thermal properties not the electronic properties thermal properties of solids can be described and that is what Einstein and divide it so shown by Einstein let me divide this expression specific heat of harmonic oscillators plays an important return by specific of solid it continue to play very important role you know and the guy who great who got Nobel Prize for other things Phil Anderson has a famous theory of showing that at a very low temperature this one shows an exponential dependence that does not work out Debye corrected it by saying that Einstein assumed only one high there is a beautiful work he said whole solid has one frequency that is Einstein did yesterday this expression Debye said no that is not possible because there is a distribution and also there is a upper limit when you introduce that fact he got the t-cube log so specific heat of solids so it goes like that this is 3R the long credits law here it you find some signature of exponential but here it goes as t-cube that is the one that Debye corrected by by by improving on Einstein's assumption of single mode however from now look at that you take the case that I go to high temperature in this region this region here this one I am going to look at now okay and that is by when temperature is large then you can look at that take the large temperature then it goes to 1 you expand this one because this is a h mu by kb t small okay I expand that I call we already called it x once before you can call it x once or more so so then e to the power minus x I expand 1 minus x minus x and plus x square so the one cancels one I keep the lowest order term which is x there is a square here it becomes x square and that then cancels this because h mu by kb t and then cb is 3 nkb okay so which is nothing but 3R so high temperature this beautifully goes over to Dulong Petit's law this is an amazingly very petty very very pretty part of statistical mechanics that as I told you in the morning that we get ideal gas law we get the pv equal to RT we get cv equal to 3 by 2R so all these things we get we get the circuited equation but now we are seeing that it is a huge thing in we are explaining the properties of the specific heat of solid which is enormously important and experiment loves that