 So we will continue initially with review of monatomic statistical mechanics of monatomic gas and then we will do diatomic gas today itself we did it once before but we need to redo many other aspects because these are fairly detailed things so you can call it as a revision but it is more likely many new new information will come in in today's and so in the study of statistical mechanics and study of any subject which is with mathematical and involved in a lot of equations it is worth going through this many times like when you used to derive equations when you are a student one of the things that we are told and we did is that we derived the same thing many many times that is the way to you do not memorize it but you derive it many times you go through it many many times it is like painting when you in order to understand something and get get it inside you deeply you do like we do painting of the one you know you do several quotes so we will briefly and quickly go through few things and then we will go to the new things today we will be in a monatomic gas we will do two new things which we missed before then we go to diatomic gases and we will do some old and some new now we made that in the last class we discussed first the partition function and the partition function is the thing that we I have here on my board the qn and in the circle rate that qn coming from integration over the momentum and for a integration gives volume real space integration gives volume and then these essentially then what happened that q small q that we derived has contain this quantity and multiplied by the volume so n particles we have now q to the power n and then we went and we derived that that the free energy so all these details were done in the previous page so we did that that we wrote down qn is q to the power n by n factorial and then we have the volume given to the n factorial and q is this quantity that 2 pi m kBT by h square to the power 3 by 2 and the volume V that was quantity when that qn with this q is used in the statistical mechanical expression of free energy statistical expression of free energy in minus kBT ln qn qn partition canonical partition function then we get this and then one get an expression of free energy in terms of n total number of particles in the system temperature of the system and the volume of the system this is really beautiful and exact expression a microscopic expression what is hemorrhage free energy and that has is a function of a is a function of n VT and I have those dependence and we have now some very fundamental quantities like I have Boltzmann constant KB I have mass of the particle and I have the Planck's constant okay this is a beautiful really beautiful expression these in the box now then I can calculate by using thermodynamic relation from these free energy I can now calculate pressure and then another beautiful thing comes in that is equal to n kBT you all know the ideal gas law so then the next what we did we went on using that partition function and the and the hemorrhage free energy we get the entropy and the entropy then is obtained by d addtv the thermodynamic relation and now we get a beautiful expression for entropy again the expression of entropy includes the n the number of particles temperature T and when I remove V by n by using the ideal gas law then it is in pressure so entropy suddenly is a exact expression of which in depends on total number of particles in the system the pressure of the gas and the temperature T again I have the mass of the particle is a monatomic gas monatomic gas or the one gas this can be generalized to binary mixture or three particle strings almost trivially then what will happen I have to add say I have argon and krypton then I have n argon here number of argon will come in here and then mass of argon will come here everything same pressure remains the same then I will say n of the krypton and then again mass of the krypton again remains the same so it is ideal gas law you know follows Raoult's law so you add up the entropy multiplied by the number of them and if I say entropy per particle then it become mole fraction so the things to remember things to really note that we have a number of particles n and we have the temperature T here and we have the pressure P here if I have the volume they are following this is the famous circuit temperature equation which I discussed is used in many many many many applications but I before I go ideally you want to impress upon you the beauty of this equation that this is where the temperature comes as 3 over 2 so if I combine temperature with this with it becomes T to the power 5 by 2 and these are much less known than and much less emphasized but these are the elegance but these are beautiful things so then we commented that we use this circuit temperature equation in a drug DNA interpolation we discussed this thing that in that this drug is going to go into a minor or major group and this is a very important problem in chemo chemotherapy or many other drugs and then in doing the calculation there is a beautiful paper by Charles then there is a paper beautiful paper on drug DNA interpolation which you can do the name of Casey Heinz, J T Heinz, Jax I think 2000 I think probably 13 or so and with that this thing was used the circuit temperature equation was used to calculate the free energy of the intercalation and is very important that we get the free energy of intercalation because that allows us to choose or select drugs you know very very important then we use it extensively in the formation of water droplets in the cloud and that you know the entropy when the water goes to form a droplet above water droplet which has to become rain that is lot of loss of entropy because instead of moving separately they are stuck together and with mass and so then the entropy decreases but that thing is used in many many other in euclidation many many other things to continuing now that I gave some a number you should always think in terms of numbers and in entropy number is always in terms of this KB unit that is called entropy unit in a one order Boltzmann Boltzmann constant. Now this is under ambient conditions that they are respective liquid state their gas state neon is the entropy is this and argon is this so they are pretty close to each other this entropy and they are with the whatever experimental value we have by integrating but on thermodynamic variables they least exactly with the experimental values. So now we will do two simple calculations which we have not done before and they are very nice calculations they are simple calculations but they are used some mathematics. So we now derived expression for pressure we derived an expression for entropy we derive an expression for free energy we derived an expression specific heat 3 by 2 R and 5 by 2 R 3 by 2 R comes just as I showed you from a and then 5 by 2 R because you add R to it which comes from PV terms H equal to E plus PV because PV is the derivative of enthalpy and enthalpy comes with the PV extra PV term. So that PV then we replace PV by RT and you take the derivative then one R comes which adds to 3 by 2 R and you get 5 by 2 R that is the reason that Cp is 5 by 2 R and Cv is 3 by 2 R with PV term the extra PV term that goes over to becomes RT in the for a more one more for ideal gas law. Now we will do something we have discussed the grand partition function but we have not used too much of grand partition function we will use a grand partition function later quite a bit but now we are just somewhere about in the beginning of the statistical mechanics course and but we already getting many many nice results but we will get one more nice results the result that we used extensively in your freshman chemistry or in so-called chemistry but the in solution chemistry or in undergraduate chemistry at least I can personally say myself I found it boring and even now teaching it sometime fairly boring so there are lot of very nice books have come to make it easier but the problem of those physical chemistry equations is that they are introduced much of the time they are not derived and example says the solution the concept of chemical potential why the chemical potential goes at logarithmic term that comes as a kind of derivation as a definition but what we will do now it is not a definition we will derive it from statistical mechanics more fundamental why chemical potential goes at logarithmic of density it is very important very important remember in your under guide which study that we start working on chemical potential and it is a very important quantity and in describing the flow of matter but that is proportional to overall density so if we keep low density high density together then matter flows from low density to high density and the driving force will be given by the logarithmic of density and that goes into the diffusion equation in fixed law and all this kind of stuff which we will do later as we go on anyway in a little bit of time dependence statistical mechanics which is just a wonderful subject. So what we did before that we did the grand partition function and grand partition function is this quantity e to the within that is can be I wrote it also like that this quantity and here the z is the fugacity and mu is the chemical potential so z equal to e to the power beta mu so this relation between chemical potential and fugacity z mu is the chemical potential and z is the fugacity. So this is the and remember that this z or beta mu comes these in terms was a Lagrangian multiplier or Lagrangian undetermined coefficient and mu came as the conservation condition that number of particles must be conserved. So we form from micro canonical ensemble to when you go into canonical ensemble we introduce conservation of energy of my super system and that conservation of energy or the super system that means there is energy exchange within the system. But in the super system which has been put by contact putting them in contact with each other so that extent energy and can attain a temporary routine then the however the conservation of the total change was maintained conservation of total energy of my super system and that conservation condition gave rise to beta the temperature beta the temperature term beta is 1 over kb it came as undetermined coefficient but it came as a conservation condition. Similarly, when you go to the canonical partition function then the chemical potential or fugacity came again as a undetermined coefficient but to take care of the conservation of the number of particles. So this is very very important that they come very naturally from a very fundamental condition that is conservation of energy conservation of number of particles that means they are not at all. Now then with that going the way just like q n was sum over energy levels e to the power minus h i k b t we get grand partition function as e to the power beta mu n and q m v t which is written here in another form. So now we will do something very interesting with that before you do that if I now talk of ideal gas in ideal gas I know q now q is a quantity which goes as q to the power n by n factorial. This is the ideal gas of monotomic gas that means I have an exact expression for q. Now those of you like mathematics you can initiate this immediately see that this is a at something to the power n and this is also something to the power n z z e to the power minus beta u. So this is z to the power n this quantity z. So I have one to the power n and that is n so z and q multiply and to the power n and divided by n factorial and what does that remind you that remind you that becomes nothing but an exponential function. So e to the power z q it becomes e to the power z q. So now and that so now we know in we know in so I have an expression for grand partition function that is e to the power z q and I know q. So this is the my grand partition function beautiful. So now I also know that that chemical potential and number of particles are conjugate of each other and the exact relation between them is just like pressure is dv dv entropy is ds dt dv dt and everything like that number is related to this grand partition function through a of chemical potential. Now when I put q and all these things together here because this is e to the power beta z is again e e to the power beta mu I add them here put them I take a log term and then this beta mu comes in front and then a ln v will comes it and then I take the derivative of this quantity then e to the power z q then it becomes it comes out as z q and q is a proportional to volume and z is e to the power beta mu then when I do that simple algebra I get the derivative the derivative exponential is exponential brings me beta down and when I do these things together I get just this beautiful relation. So now you see the volume v the volume if n becomes this quantity so now I can take this v on that side or I can take yeah I can take v on that side I can take n by v that becomes density and lambda q so this whole part is taken there so giving rise to me this term and e to the power beta mu and then I take logarithmic term and beta becomes 1 over kbt so you can easily see that I get this beautiful relation this beautiful relation I like so I use one more color so this beautiful relation now that means chemical potential is logarithmic of density many times we write ln rho which is not correct because you cannot put a dimensional quantity within logarithmic or you cannot put dimensional quantity in exponential so something is missing in these equations but in classical physical chemistry we just write ln rho we cannot not there is always that function lambda cube there and what many times what we do we do chemical potential with respect to one another space each on another state then delta mu term then I have you know ln minus minus ln and then I that becomes ln a minus ln v ln a by v and then lambda q cuts out you get ln rho 1 by ln rho 2 ln rho 1 by rho 2 sorry so that is why lambda q is always kind of a silent player in all these things but I really want to impress upon you this this beautiful relation that mu mu goes as ln rho this is something which is hugely used so when I talk of non-ideality so this is the ideality chemical potential is dependent on linear logarithmic dependence on density but when I talk of non-ideality now the non-ideality will just introduce other activity coefficients and other quantity that is where this is where it enters all these things so there is a very solid foundation of these things these were actually these terms were actually done by Gibbs himself so that is why in the thermodynamics of solution theory many many things that gives equation gives them equation and all this stuff because this was worked out by Willard Gibbs himself. Now so there is a little bit of writing here now that which led me through chemical potential is the change in internal energy on addition of an extra particle in the entropy and that is given here this this thing however an extra particle would result in more number of arrangements in the same volume as a result entropy increases. So basically what one trying to tell here the chemical potential in and that is how it is done in undergraduate textbooks the chemical potential in ideal gas when their molecules are not interacting with each other each other is an entropic order is very very important understand that chemical potential in ideal gas is an entropy. So ideal gas law gives us a fantastic way to talk of entropy because you all you have is entropy and what we do then later when you add interactions many times we try to preserve the entropic term the ln row term like in the member binary mixture ideal entropy of mixing what is that that xi ln xi so essentially rho i ln rho i which is has the same same basically the same structure of ln row term coming and then multiplied by the relative weight of that ln row term is just just beautiful things that that the way they fit together. But this is a very important thing is taught here that it is the entropy that is the important player here because in ideal gas law we do not have interaction but by entropy alone we get a huge number of phenomena and as again I am repeatedly is part of entropy we take we try to take over when you do interacting systems the real systems these are not real but very important just like part in a box in quantum mechanics important harmonic oscillator is important rigid rotator is important and then you go and start doing after doing these things you start doing at hydrogen atom energy and molecules here also we will go over the complex systems. So now we now know we want to go to the and very important very important second calculation of the monotomic gas ideal monotomic gas then with and the result of prediction consequence a result that is used extensively in for example solid state physics and in spectroscopy in conductivity of solids you know the electron gas and that kind of problem where we need to know how what is the number of density of states how many number of quantum states between energy e at energy e between e and e that is called density of states and is very important how the density of states changes with energy and there is a simple trivial calculation but very nice calculation that later go through that now you have done particle you have done quantum mechanics part of the box and you know that the energy is given by this quantity given by this quantity that e is a h square by h m square in x square in x in y of the quantum number and their values are from 1 2 3 4 like that these are the a yeah yeah we discussed at length y this quantum number starts from 1 and they are integer numbers 1 2 3 4 5 6 7 8 9 10 like that now this is a beautiful equation now I realize if I can be little bit messaging of that then in x square in y square in z square I can write it at n square then that n square and I can take now this quantity this quantity on the side then I can write and this is going to be very important 8 ml square by h square I go there so and then n square becomes equal to 8 ml square by h square now this is I can call these as any x square n square all these things like you know x square plus y square plus x square equal to a square that is the equation of sphere so this is the equation of sphere and then I will have something very interesting thing to do because this is centered at origin in 3 dimension now that become the radius now but I know the radius I know the radius that radius in term is given by that okay now one plays a very very smart game and is that we now want to consider that we have a sphere hypersphere let me consider to like that and now these this is sphere now in that sphere I have nx ny nz all has to be in the opposite they I do not have nx ny nz negative so if I say n ny nz and nx now I want to calculate now I say they are all I point and then they these are the point which are correspond to nx ny nz in that one a one part you know one octet of that in the in the hyper in the sphere three-dimensional sphere so now I can now play this smart game I said okay the total number of states would be you know when these are very large in order to calculate the total number of states all I need to do and this is their place unit one and this is the real the crux of the matter that nx ny and nz increases by one nx goes one two three four ny goes one two three four so there is this grid that I am forming I have all spacing want so that allows me to if I get the volume because of them in this unit thing the volume gets the total number of density total number of states and that is what I do now and since the one eighth of it so I know the total number total volume and total volume is this four pi four pi by three n cube n is the radius and n is nx square plus ny square plus nz square and then I have these total number in all these regions all the one is one eight four pi by three and two that is just just beautiful so now I also know what is n I just did in last class by n is n square n square is 8 ml square by h square that I did in the last slide so n cube is then 8 ml square h square by 3 by 2 this is just wonderful so total number of states is increases as e to the power 3 by 2 so it increases fairly rapidly and also what is very interesting that it depends not only on the energy it also depends on the mass and it depends on the length of the box remember al is the length of the box in this in this in this particular in a box model so now but this is not the density of states I am interested in density of states it is not the density of states this is the total number of states now how do you do density of what is the definition of density of states density of states is that you take the the energy the number of states in e and e plus there is a density of states and so you do it by taking derivative you know this this number so it is it is it is it is a increase from e to e plus daily is of course will be proportional to daily in a linear way so now we do that little thing and this we get that density of this is omega is the density of states fairly universal notation and then I get the derivative and daily I can divide by daily and this daily here when I do that then 3 by 2 comes out and this 3 by 2 these 3 by 2 comes here and that 3 cancels this and I get a 2 and 2 is here so I get 4 so I get pi by 4 here then 8 ml square by square remain untouched 3 by 2 and 3 by 2 become root e e to the power 3 by 2 daily e to the power half root e so this is a beautiful result which shows that omega e goes as square root e density of states in a particle in the box goes and and this is this is very important this is the result as I told you the result is used in many applications now there is one more interesting result very interesting result that comes now we do not have one particle we have n number of particles so we do not have a sphere we have a hypersphere and in hypersphere now you have the volume that is going as 3 to the power n by 2 and then you can do exactly same derivative I say 3 by 2 becomes minus 1 and these in the volume these gamma to the power 3 by 2 and this is gamma n plus 1 is nothing but our n factorial and this comes in the volume of the 4 pi by 3 that factor comes like that and then and this is coming again from the particle in a box energy so you now have you now have density of states of n particle systems so this is the density of states of n particles in a cube in a cube of length l so you have a cube in a hyper of length l and this beautiful thing now is omega e daily you can be cast proportional daily but daily is not the important quantity important quantity is these two n factorial that comes from the same logic as partition function but this is the one comes from the factor factors that come in the volume and e to the power 3 n so now you can see a very very important thing that we made already to in the statistical mechanics in the initial part that there is a huge number of states when you particularly when you go to the high temperature so number of states or density of states of n particle system scales as e to the power n and that is a very important result and this means there are huge number of states up there and this is an exact result in a ideal gas law exact result but I have again and again telling much of it goes over even in interacting system and this is the one which is used now in the conductivity problem and many other calculations of solid state physics where electrons are used as free particles and that is a very important model of solid state physics that free electron gas and when you want to talk of free electron gas we take the mass of electron m there and then we calculate this we use this these things in the particle okay so this is what we wanted to talk of this beautiful thing of in a in a way of classical and quantum the beautiful interaction between class quantum and statistical mechanics and then so we will take a we will take a short break and we will start on the diatomic