 Welcome back and we are continuing our study of statistical mechanics of ideal gases. Till now we have done monatomic and diatomic and not only that we have done the statistical mechanical we derived the statistical mechanical expressions of free energy entropy and specific heat which for monatomic we did the translation that part remains invariant in the diatomic. But diatomic brings the two new ingredients one is the rotation and vibration and we have used the harmonic oscillator model of quantum mechanics to derive expression for those quantities for harmonic oscillator. And we showed that that this is one of the most respected used and time honored model applied to many systems of physics and chemistry and biology and then we did the rotation using the rigid rotator we worked out again the statistical mechanics. Now the most many of the systems that we are interested in like water ammonia alcohols they are all polyatomic molecules and polyatomic molecules have characteristics which are quite different from that of diatomic. The reason is that the polyatomic will have now in addition to translational degrees of freedom in addition to translational degrees of freedom at in rotational degrees of freedom it has vibration which is many more now as we all know that vibration number of vibrational degrees of freedom is more in case of linear because their rotation is only two but for non-linear molecules where you have three rotation we have three and minus six the vibrational degrees of freedom and this is what actually gives rise to when the large molecules like polymers form or when they form they are sitting together in a crystalline forming a solid amorphous then we lose the translational degrees of freedom that they have and that get transferred to vibration degrees of freedom and that showed up shows up as a normal mode which has played so important role in the specific age and thermodynamic properties of low temperature solids not just crystals but also amorphous materials and these are low temperature liquids they are very very important quantities. So but however the polyatomic molecules have many other peculiarities of their own that we are going to discuss today and they want to do a how to do statistical mechanics of polyatomic molecules so that a student knows how to start talking of entropy and free energy and specific heat the molecular component of the polyatomic molecule and that statistical mechanism gives you and we already have it but so we will have a slightly more generalized form of what we did in diatomic and so it will be it will be not too demanding a session we did do in a small way the polyatomic earlier but we are going to do a little bit better now and in certain sense also it is a review of many of the things that we are going to do okay and we wrote down before we are neglecting the electronic degrees of freedom because electronic energy level gap is much larger typically electronic energy levels are of the order of say 10,000 centimeter inverse 7,000 centimeter inverse which is huge so they do not enter at the room temperature the ambient conditions electronic degrees of freedom it does not show up in the thermodynamics it does show up in the spectroscopy when you are giving external source of energy and resonance condition and exciting but in the thermodynamic properties it is a ground state of electronic degrees of freedom that that is all that you need to worry about and we can take that energy as a zero so that just it does not change its temperature does not change its pressure problem we do not need to talk about it. So then I can write the partition function as a product of three as we already discussed that it is translation these are molecular partition function so this is a molecular partition function and these okay the molecular degrees of freedom the molecular partition function q small q is my notation for and capital Q is my notation for the partition function of n particle system so then this q translation q rotation and q vibration now translation is the center of mass so we do that then if I have a nonlinear molecular three rotational degrees of freedom I call it q dot 1 q dot 2 q dot 3 and then I have the all the other degrees of freedom that is given here and that is of course not just three if there are three vibrational degrees of freedom then it is 3 but otherwise that can be many more degrees of freedom that would be showing here so and that is shown here so since they are all these vibrations are a independent of each other the normal modes then q vibration is a product of all the vibrational degrees of freedom which is written here that I have a new stands for vibration in the notation and j is the different normal modes so then my total vibrational partition function is product of e to the power minus theta j by 2t and by t because the remember theta and this is the alpha is given here and theta uj is the vibrational temperature and this one is the one that comes from 0 point energy I to remind you and this is the one that comes from the detailed sum of this power series with x a 1 plus x plus x square and that becomes 1 for x less than 1 because 1 over 1 minus x so this is the one we derived it derived in the in the last or the one before last class so now if we know this so then we can set up the partition function now so vibrational part is fairly simple vibrational part is fairly simple because we have this thing and you give me the frequencies and give me the that gives me the vibrational temperature and I can I can give you the partition function so now we will concentrate on the the rotational part rotational partition function has is very important in polyatomic molecules and because there is a you know this kind of can be jittery and all the kind of motions that many times you face are basically some kind of restricted rotation now rotation has fairly different cases we need to consider one case where all the momentum inertia are the same because there are three direction in an all linear molecules and you can have three principal axes and you have three momentum inertia now if this completely spherical we call spherical top molecule like methane then we have no problem we have all the momentum inertia are the same which is like I a I b equal to I c the spherical top the degeneracy of the rotation as it becomes 2j plus 1 square if they can equal principal momentum inertia then all this and we can then write these 2j plus 1 the degeneracy of the rotational energy levels the equal principal momentum inertia is then you do instead of this product we will have 2j plus 1 square and then we get this expression for the rotational partition function and then we take the following limit at high temperature when temperature becomes very high then and a momentum inertia also not to small then the the ones that will be involved at very large j large quantum numbers will be involved and large quantum numbers then I can I can approximate j plus 1 by j that is a very neat neat approximation and then my rotational partition function becomes this become 4j square then it becomes like then it is a beautiful thing so in the high temperature limit I can do this integration now it is long that the difficulty I had but remember even in the diatomic case we could evaluate we could evaluate the large high temperature limit here also then the high temperature in in in the polyatomic case the high temperature limit can be worked out and again we get a very neat expression of the rotational partition function so continuing we now do symmetric molecules now symmetric top molecules are the ones who which are like symmetric top which are two momentum inertia are the same and third one is different it can be prolet in the prolet if it is a prolet then then in this v and c are the same momentum inertia and this a is different so momentum inertia ia is let is then ib equal to ic that is prolet and it can be oblate in the oblate case then it is these things so this direction c is different from in the in the spherical direction is a and b so prolet and oblate are two different things and in that in that case we can calculate the partition function by using the method similar to this one that remember that if you go to rotational case in the diatomic case the partition function is 8 pi square i by kbd 3 by 2 x square then the same thing happens here so the quantity is that two of them comes with 8 pi square ia kbd by a square half and half so when a equal to b when a equal to b then this half and half becomes just one and then you have this c half okay so whatever I do either prolet or oblate depending on a and b like for example in the case of prolet it would be a here and then okay b here and a here in the case of oblate the other way around a here and c here just just the same thing of course the values can be very different so but then one comes the most general case with the asymmetric top molecule in asymmetric top molecule all the principle moments of inertia and this is the general case by the case of water molecule case of methane they are all asymmetric top molecules where all the moment of inertia are different and this analytical expression might be levels cannot be obtained by solving equation that is the real unfortunate part so in the previous case part the this case these thing comes from solving the sorting equation the degeneracy factor of a spherical top molecules that it is 2j plus 1 plus whole square therefore a polyatomic molecule this is comes from sorting equation however this k in the case of completely asymmetric molecule we cannot do we cannot solve the sorting equation and as a result we cannot get those energy levels so sonic equation can be cannot be solved analytically so for the diatomic we get analytically for for symmetric top the spherical top we get in analytically with the different degeneracy now for the polyadenoblate we again get polyadenoblate when they are they they they they decompose separately and we can get the energy levels again and we can solve for the at least high temperature limit the partition function but in the case of fully asymmetric molecule where moment of inertia ABC are different it cannot be we do not have any close form analytical expression for the energy levels and that was the problem so in that case the sonic equation has to be solved numerically to obtain the energy levels this is very very difficult and these are laborious process these are laborious process but the we can we can obtain to this the classical limit the high temperature limit that the high temperature as I said the quantum goes over to classical partition function in the in the high temperature that means I can start partition function from classical Hamiltonian I can do from quantum Hamiltonian then I can show that high temperature the quantum goes over to classical it is very nice and has to be because there is nothing you know surprise there at all but in this case where you cannot do anything in the for the for the quantum case because energy levels are not known we can do anything other than numerically that is actually done many many cases numerically one solves and gets a but we we can make progress with the classical Hamiltonian which can be done and classical Hamiltonian of course can be just integrated I write down the moment of inertia I have ABC and write down the there is no there is no potential energy still it is just the rotating and it has the rotational kinetic energy and rotational kinetic energy I know how to do the rotational kinetic energy the rotational angular momentum divided by 2 I and then I can integrate that and that we will we know that that gives this kind of form so now I have one for a one for b and one for c so I have three and part total rotational partition function in product of these three which is just the one we got in classical from classical statistical mechanics or the high temperature limit of the quantum diatomic case with one moment of inertia but here also when they are written in terms of momentum inertia they are already decomposed and then we can get that and then we can write the rotational partition function I can combine these things and define a theta a theta b theta c in terms by using all these quantities because that quantity then becomes one over theta b sorry one over theta a and that becomes one over theta b and this becomes one over theta c then my partition function becomes this quantity root over pi by sigma t cube theta there is a way very nice and need and very elegant expression and remember the half is preserved now each comes with a half and that half shows up here so this is the rotational partition function that complete rotational partition function of a polyatomic molecule. So now we do we have this so we go back and we have the we have the we go back and we have this total partition function and now we want to do certain thermodynamics with this expression and what are the kind of values that you get okay. So let us now consider some specific cases now the temperature the rotational temperature and vibrational temperature gives you a measure is a very important thing they give you a measure of the quantumness of the system and the importance of that particular mode in my thermodynamic properties for example in case of water I have three vibrations what are the three vibrations one is symmetric stretching one is asymmetric stretching other is bending the three vibrations for water now look at them we already know one is 3650 centimeter inverse symmetric stretching a little bit higher is symmetric stretching that is 3700 centimeter inverse bending is half of nearly half of that that is 1885 centimeter inverse very nicely placed those three and that plays a very important role in vibrational relaxation there is some of the outstanding discoveries that made in recent times okay now if we do that then look at these in case of water so this is asymmetric stretching this is symmetric stretching this is the bending all right now if it is so then I can easily see that in thermodynamic properties my if temperature it will be relevant at 5000 Kelvin no so at room temperature I do not need to worry I do not need to worry at all for water but look at rotation rotational temperature are really very low the three momentum inertia of asymmetric that water molecules with three axis of rotation they come so it is temperature that means they are they are almost classical and I they they influence the statistical that influence the thermodynamics so I need to take into rotational degrees of freedom in understanding thermodynamics and these are the very interesting story so for some time quite some time people are trying to calculate the entropy of water this is actually big game now of last some 15-20 years what is the entropy of water because the entropy of water as I said plays a very important role in many many chemical reactions and what I give you an example when a DNA drug goes into a drug goes into DNA then of course the drug loses the entropy but the system gains entropy the way system gains entropy the water molecules which were inside where in biving where inside the inside the major and minor groups they are displaced so when they are in minor group they did not have any entropy there is some extra enthalpy but they are no entropy so when they go out they gain lot of entropy so we kind of take into account away that 10 molecules that are yet in this space that 10 molecules going from minor group of DNA to water so what is the entropy for water molecule if I know then 10 times that is the entropy gain in the process and then I can use substituted equation or something to get the but substituted equation still in a liquid water is not a great approximation but in that that they get you started very very interesting now when all these calculations were done in initially people are not getting very good agreement of the entropy of water in the and entropy of water can be obtained also experimentally in a laborious process by starting from all the way from zero kelvin ice then going into make it melt going to liquid then you know we can calculate at take the latent heat of fusion into and we know the entropy of water experimentally that was not working that well then we realized that we did not take into account the vibration and rotational entropy contribution but as I show you vibration is not important but rotation important translation is of course important so it turns out the rotational in contribution to entropy of water is about 30 percent is a huge contribution that comes from a rotation it is something like that one in the entropy unit rotational contribution is about 5 or 5.2 and the translation contribution is 17 so if total entropy is 22 you already had a table saying 17 the translation comes and then this this this another five comes from and you know to calculate that entropy I have to calculate I have to use I use this expression this expression I use to calculate the rotational partition function I get the free energy a equal to minus cavity l and q and I take the derivative of that and get the entropy that entropy now is an is extremely important contribution for water rotational entropy contribution at room temperature is 30 percent of the translation contribution this is a really very important that we missed you know very recently that we missed this thing this is just about I think 10 years old now another system interest is ammonia in ammonia has again this ammonia has four vibrations it is stretching anti symmetry stretching and and and bending mode now those are vibrational temperature the theta 5 that we define in terms of the frequency and that is pretty large so one is 4800 that is very large a vibration this another vibration these two are some kind of bending modes I do not remember fully the mode but they are they are certainly not one stretching modes but even then these are pretty high temperature the room temperature remember it will be e to the power minus the contribution comes as e to the power minus theta by t that would be theta vibration by t and since theta vibration is so large temperature at at at at 300 Kelvin even this guy is e to the power minus 4 and yeah 4 point something these are of course much larger so this contribution is minimum however the same e to the power minus theta rotation by t all in Kelvin look at that this theta is small so all these things play very very important role so all these three modes play very important role in the entropy or the thermodynamic properties of ammonia similar things you find in sulphur dioxide nitrous oxide methane everywhere you see the vibration temperature is very high so when it comes e to the power minus theta vibration by t the vibrational degrees of freedom do not contribute significantly at ambient conditions to the thermodynamic properties so molecular thermal molecular partition function and molecular thermodynamics the vibration is a silent thing we do not see it we find huge contribution of translation but then a significant 20 to 30 35 percent contribution comes from rotation so rotational degrees of freedom cannot be ignored in find what you are doing and this is a very very important thing because see most of the books and most of the studies that we hear everywhere that people are doing they they they as if the world begins at the ends with monatomic molecules maybe a little bit of diatomic like take take the phase transition theories and all the things that we will be doing something we we we start with sphere spherical molecules you know and we end with spherical molecules like about studies of glass transition theory do theory of liquids that we do they are all models are things who called in adsorbed models and others but they are all spheres but this is something so when we did statistical mechanics in 1970s and 80s we learned towards the end of 70s and 80s when I was doing PhD and post off we really never bothered about you know if the popular text books of statistical mechanics they do not have much of polyative molecules so our our emphasis and the book that I have written the statistical mechanics books for chemistry and material science that is why we have a a very big discussion of polyatomic molecules and a lot of emphasis given on polyatomic molecules because these days we are interested in these kinds of systems okay. So let us continue then then entropy of a polyatomic molecule is a just a wonderful thing as I just discussed to you so this is the thing the translational part so entropy since put a partition function is q translation q row vibration and q rotation and then free energy is minus kbtl and q so these that they decompose free energy become some of the free energy and entropy becomes some of the entropy this is trivial but sometimes it is good to spread out the trivial thing because to help the thinking that is then this is our circuit tetrodiquation this is the circuit tetrodiquation and then this is the rotational now I have arranged it such that most important thing in translation which water and ammonia has about 65 to 70 percent of the total entropy then the rotation which is classical limit but classical limit works out reasonably well as I showed you in these things fortunately for us this theta vibration is small so we are okay because the room temperature is much larger than this so we can rely doing classical mechanics this is a lucky break in quantum no way it is too much it is too quantum but this is nearly classical it is very very very interesting how lucky one can get some time so like for translation you get away by the classical mechanics for rotation for translation you get away rotation you also get away by doing as I showed you for many of the molecules vibration we are not that lucky but vibration with the harmonic is a good approximation so this is translation these rotation and these are vibrational contributions to the total entropy and as I said this is actually negligible so these two does the job perfectly for us so now where I have given you a table actually this I have preempted little bit I told you I already showed the table 70 and 5 which is I was talking from memory but there it is that water molecular weight and every detail is given and these are the three moment of inertia I have given you what are these are the things you do not get in the physics textbooks and or even earlier versions of stat mech but now these are so important in the present in search and present the calculations that polyatomic study of statistical mechanics of polyatomic molecule is important thing that is what I am discussing it though I discussed this a little bit before I am discussing it in more detail and as I tell there is no damage is done by doing it these beautiful things more than once okay so look at that to in water has this nice thing is the two of the moment of inertia very close to each other and these are these two a and b 1.09 1.91 and this is quite different now this what I was saying 17.41 is a translational part in the entropy unit in the KB and this is the one that rotation 5.45 but look at vibration contribution is negligible when you add up all these things you get this is the entropy of water beautiful very very nice now the other thing as a ammonia ammonia there are these things that two things that nearly degenerate and then this one is larger moment of inertia now a very similar story unfolds for ammonia translation and rotation now rotational contribution little bit larger that you could guess because go back you could guess that it will be larger because look at the rotational temperature rotational temperature of ammonia is lower than that observed so as soon as rotor temperature become lower its contribution becomes more to the partition function and to free energy and to entropy and right that is the one that happens so 5.45.91 this is slightly less but then at the end of the day we have these value of the entropy and as I said there are very important things in property to get the ideal entropy there are many expressions of entropy and for example connection between diffusion entropy there is a property called Desmond Rosenfeld scaling which needs the ideal entropy ideal gas entropy and what uses the value of ideal gas entropy of water molecules to predict the diffusion constant in liquid which certain other other terms involving but this plays a very very important role in in the property of liquid okay so this area I summarized it here that the this is straight from the book statistical mechanics book the Statemac book by CRC praise from that it is taken that a transfer entropy contributes in the most but also rotational contribution is this is my favorite I did not know myself about five six years ago the rotational contribution is 30% because we also grew up like doing as if the everything is land zones as if there is everything can be modeled that spheres but then you miss out all these things molecule vibration entropy is only only considers intermolecular model are not very important so by the contribution lies from vibration is negligible so this is very encouraged to carry out numerical calculations of polyunet molecules like chloroform this is a this is a homework that will set up to make you do something like methanol and you can calculate them by doing the homework so here ends the statistical mechanics of ideal gases we I just summarized what you do in one minute that we have done now in great detail monotomic gas which is spheres non-interacting spheres then with the diatomic like nitrogen oxygen they are rotating and they have a vibrational degrees of freedom vibrational degrees of freedom has beautifully comes with these harmonic oscillator and the flooding gets the solution and however in thermodynamics those vibrational degrees of freedom do not play in old but translation rotation place then we went polyatomic molecules and we did statistical mechanics of polyatomic molecules the very important case of water very important case of ammonium or other sulfur dioxide and all other important molecules that we need to do in everyday life and in a real problems like you know and that then at the end of this part so top here we will take a break and we will go over to the next class.