 I have k B T L N Q to the power N is N L N Q then because there is Q to the power N and minus L N factorial this N comes because there is x 4 N here in Q N and this and this comes because N factorial then what is something is jumped here but I hope you will be able to do that I put now Q to the power this is Q my Q so Q to the power N has becomes more L N Q so these become what do you have in the last page this quantity you have to the N L N N by N e to the power N all these factors come here and these N becomes N K B T these factor this factor that is here with h square everything this thing that a L N V by e that L N V is not from there that comes that V comes from this V all Stalin's approach gives this this factor V by N to the power e that is play the very very important role as we will see right now so if this is the free energy which is following from this from there we get we do this this is a little unstable from this then we get this then we take the derivative the pressure minus d i d v t and thermodynamics then we get if we do that then all these things since they are log all of things these do not matter these do not matter I just get 1 over V because it is L N V 1 over V then gives N K B T by V and this P V N K B T so this is the ideal gas which was written down by people from Boilstein from 1830s or 1840s something which a used Maxwell used but there was no derivation P V equal to R T was derived kinetic theory of gas derived it by using remember P equal to one-third M N C square and then you put a C square there you get this quantity and but that is I told you kind theory of gas had this kind of strange set of assumptions within it it does not interact but at the same time it has a size and it is interacting with the wall having a elastic scattering like a billiard wall as a basic idea that it is a non-interacting collection of particles and that gives rise to this ideal gas law was proven so these are the as I told you in the beginning these are beginning of the applications of stress thermo mechanics so this is one of the result then this next result is that this is very very important thing that I told you succorted to the equation and the succorted to the equation follows trivial then you go now take with respect to entropy is negative temperature derivative of free energy okay and that when you do I take d dt what I I can see what can happen I can it can this is t to the power 3 by 2 here so that will become t to the power 5 by 2 so first term I take in d a dt first term will just take remove this term and have n k b l n 2 by k t 3 by 2 okay then second term n k b t remains n k b t I go and do derivative of this term when I do l n they are inside then all these things disappear they are locked term I do not have to take care of them I have to take care of 3 by 2 3 by 2 when I take a derivative it become 3 by 2 comes out and it becomes t to the power 1 by 2 because l n t 3 by 2 1 over t 3 by 2 and then you can combine that at the 1 t there that comes in and that hits you with the 3 by 2 so that again become 5 okay 1 by 2 the 1 by this 1 by 2 with by again t it again become 3 by 2 and when you do that ultimate result is 2 pi by I am not deriving it is fairly trivial but you can do it because it is just ordinary derivation so these quantity this is the expression of entropy of an ideal gas which plays a very important role we will show right now is called the circuitated equation this equation is given when the volume constant but I can use this equation within ideal gas I can derive more useful equation which is this equation which is we are not given in your textbooks you use volume p v equal to n k b t so if p v t is m k v by n v by n is equal to k b t by p and then k b t by p okay so this is the one that is used because most of the time we are working with n p t ensemble or n p t e a so in that case this is the circuitated equation so please ask me if there any problem if there any confusion in deriving this equation everything clear I am not doing it a by i line by line but I have explained to you how deriving taking derivative of that you get this term coming out log is preserved here because first is this term and then you have the a term everything remains log and k b t l n terms remain then that become 3 by 2 that is why 3 by 2 comes then it multiplied by that is absorbed the extra term that comes has been absorbed here remember a l n 5 by l n e 5 by 2 what is that what is l n e 5 by 2 exactly 5 by 2 so that term this were absorbed the 3 by 2 factor is absorbed okay and so that is this is very elegant and nice result which as I told you in the beginning that is widely used these many many cases is widely used amazing that such simple result can find so much use and what is the more useful than this which is not usually given in textbook is this quantity except one book who uses that that is why when researchers write books or give courses this little better the one place I found is not in Macquarie not in Hill but I found it is in Ben Wiedem's new little book for undergraduate statistical mechanics very popular book just about hundred not even hundred pages probably but he did actually only up to the atomic gas and a little bit of that not beyond that elementary level but he has this equation I needed this equation because when we are doing nuclear theory of nucleation in a constant this under this condition which is the experimental condition right much of our time then I needed that that's what I was very happy to be able to I did it myself as very happy the trivial step but sometime doing something new trivial is very nice okay so circuitator equation there many applications as I tell one is the evaluation of the entropy of ideal gas which is used in these let me write down this this is in total from my my stat mac book line by line probably the mistakes you find here will be six will also be there so please just go through the book once and let me know if there are any mistakes so the diffusion is called Rosenfeld scaling D or I sometime would be effective diffusion in an interacting system is a e to the power b a s x and then s x is entropy minus ideal entropy this is one more application which I have not written down because it was not written in the book so this is actually cut and paste from my book so this is a beautiful relation is called Rosenfeld scale this is big lot of papers are being written on that on the last five years or six years and it in a liquid or dense gases these equations works suffice in the well but so one plots this quantity against x so you can understand one thing s ideal is the idea at the maximum entropy that it can have so because of correlations and order entropy decreases so s x is negative more it becomes negative more the diffusion decreases and it describes the results exceeding a well and where I was saying that where we have to use it in a gas liquid nucleation there we needed this free energy calculation g many T s and when a low temperature gas is going into liquid then the free energy of the gas has both entropy and enthalpy you can make an approximate assumption that entropy of the gas is given this quantity and that is where in nucleation it is a widely used in a you can there is a book I forgot the name or several books on nucleation you will see that use now and beautiful result I think it is it is done very badly that from here specific it is the temperature derivative of entropy right ds dt specific heat t ds dt and I can do that now I can go I do ds dt do the derivative here and then ln there is no temperature here now to complicate things so just ln t to the power 3 by 2 plus other things are here they disappear when I take the derivative I just have ln t to the power 3 by 2 so that becomes just like before 3 by 2 and ln t becomes 1 by t and then I this it comes in denominator and then very interesting it comes in handy and gives me 3 by 2 term just like before then these t removes that t in the denominator is it clear and then I have 3 by 2 comes from this 3 by 2 I have a nkb so specific it is just 3 by 2 nkb when n is Avogadro number cv 3 by 2 R so this is the derivation of the ideal gas law so the 2 these are really beautiful stuff which everybody should know by heart because we you you will use them always always when you start thinking about interacting system start thinking with the ideal gas so PV equal to RT cv is 3 by 2 R so these are the 2 things that we so the things that you are under first year undergraduate physical chemistry first year undergraduate physical chemistry that comes out from a rather sophisticated thing but you of course understand when we explain introduce such a sophisticated thing this was not the goal of statistical mechanics this is kind of things on the way these are kind of rewards you get but you are that is not you started you didn't start to explain ideal gas you start to explain interacting systems we start by Newton's equation where there are forces between atoms and molecules but these are the way we got and it just makes you like happy you that you we understand what you are doing and we are on the right track okay now some things which are really very nice we should know so example translation interview for particle these are huge that's what I'm saying so neon this is 17.59 these are all given in that my my book see most of the statement books are done not with not with too much affection or love for this ideal gas and there's a strange language to use but you should feel good about it okay and so asymptotically I can get the entropy from a low density neon that's doable and these are very nice agreement of the why you chose neon argon krypton can you tell me one very good answer second exactly exactly so they are the good example of ideal gas so exactly that's a good answer yeah so agrees to the experimental values okay now if a canonical system gridded into then we can go to grand canonical which we did the other day this is a grand canonical sum over it is just mu n k b t right this z to the bar in actually fugacity so these are grand canonical partition function and then one can get the total number of particles and another expression for a chemical potential this is extremely useful in the density functional theory extremely useful so I can go from your grand canonical partition function now I take the grand canonical partition with respect to chemical potential the conjugate number a gets the average number of particles in the system okay and then I can write a log of and and make that into a density V I can be bring V here so n by V is density and I between your beta you then I take the log then this is extremely useful quantity where chemical potential is given in terms of logarithm and you guys know that in in a this is the expression undergraduate physical chemistry use that chemical potential logarithm of density to remember that of course you we didn't understand though castellan is the best book on physical chemistry even castellan that's the where you can understand but even then there was no meat to it why did we write chemical potential the first term in a non-interactive non-interactive signal l n o this is the the beautiful equation that comes out so chemical potential is given in terms of density as a long term oh lambda is this lambda is a deboglio wavelength 2 pi kbt m kbt by h square sorry thank you thank you very much so lambda is 2 pi m kbt by h square so there is the m m m m m l square by t square then everything else these also have m l square by t square so I I think this half that's the lens that's what I was trying to do here so it is it is a deboglio that comes out naturally when you do these things that a length is always sitting there I could have introduced and I should have introduced there but doesn't matter so this is a beautiful result a j m of a result that was that j m of a result that chemical potential so is the change in internal energy on addition of an extra particle when you enter a volume or k fits you know so all these things we we don't need to do at this point so we have got this is important result and then another important result we write it down here when I studied the reason I am so excited I am talking about it when I studied statistical mechanics nobody told us that this will be so important and we had to go back every time chemical potential has to have the dimension of and this has to be dimensionless and this has to be dimension of energy okay always skip a on dimension because you know if you cannot do analytical work I am telling you when you go to post doc you'll be miserable shape everybody expects because they are they do it from I have seen in my not so bright American colleagues my students they are very good in very basics because that has been they repeat these things from a seventh or eighth grade and all the way even after going to pH level I had to take thermodynamics twice and they are they take thermodynamics ten times so they just they just grind it into you you better know how to do very simple things okay yes that's what we'll do next micro canonical is that because for getting the micro canonical you need to need the energy levels or the micro canonical anyway I can do without a omega you can calculate by indication of the phase space when you do that exactly same result comes out for example I want to get the that's the one I want to do next in a very very elegant way the elegant way also allows me to establish connection with so what is omega omega is the volume of the phase space so that means given a volume v and energy e I have to know it's a little difficult because say dr1 and I'll do that in a very good question that actually brings out why micro canonical is difficult so I have the kinetic energy putting here now but when I have nve I don't have this term at all instead I have h has to be equal to be that constraint and that's a difficult constraint that means I am within a volume in phase space so I have to impose that constraint that becomes very difficult we just did it in our in one of the calculation well we did it also before many times but there is another way of doing it which is that in quantum mechanics gives you the energy levels where you can put this constraint much more easily that is particularly in a box and then you can get the micro canonical partition function of quantum mechanics and that gives you essentially the same result okay