 So, now I will go and start take you through the ok. So, we now calculate the partition function of the system and you can kind of you just look at it that we wrote down this ideal gas. So, ideal gas is the total Hamiltonian is the just kinetic energy ideal gas means a particle does not interact with each other. So, we do not have any potential energy if you do not have any potential energy then it is just kinetic energy and kinetic energy means it is just sum over the here as we shown in this equation here is just that now I want to get the so why that means that I want to get the partition function and as I have shown the partition function here is 1 over n factorial and then 1 over n factorial and h to the power 3 n then d r 1 d r n then d p 1 d p n e to the power minus beta h. So, this would be the partition function now look at this partition function the there is no interaction so position of the position of the atom. So, I have a monatomic like I have taken a monatomic gas of spheres and they do not interact with each other like an argon or your noble gas in load density and then interestingly if I do that then I can this particle they do not interact with each other. So, they are completely disjoint if they are completely disjoint I can separate them out that means this integration d r 1 and d r 2 d r 3 are not coupled d p 1 d p 3 are not coupled. So, I have essentially n n integrations identical n integrations where the product like these are just product. So, then I have to the power n and then I have h to the power 3 n here and so this is then my partition function where q is now given by this integration of d p x d p y d p z d x d y d z e to the power minus beta h this is the partition function of a single particle and then that I can now do because d x d y d z there is no in beta h there is no position. So, I can just integrate over d x d y d z and I can get volume v this clear that I this volume v comes out and then I have to evaluate this integral which is this Gaussian integral and I can do that Gaussian integral this integral I can do because because just bear with me for a minute this is I know that minus infinity n v d x e to the power minus a x square is root over pi by a I know this integral that integral is used here then if you do that this integral then becomes there are there are 3 integrals there are 3 because d p x d p y d p z are independent of each other because kinetic energy p once one particle is p x square plus p y square p p z square. So, they are again uncoupled. So, a 3 Gaussian integrals and that integral I can do. So, then I know root over this. So, this is 3 of them and each comes with a root then and there are 2 m. So, 2 m goes over 2 pi m k b t and k b t is there beta is 1 over k b t. So, when I do that I get this. So, this is the partition function of a single molecule in ideal gas this is the single particle partition function this quantity single particle partition function. So, that thing is done. So, then we can now go back and construct the next part next. So, now we got the partition function and the partition function q n is 1 over n factorial and then 2 pi m k b t by h square 3 n by 2 this is the partition function. So, now we know that from canonical ensemble we have done that in lecture I think 4th or 5th that free edge equal to minus k b t l n q this quantity. So, now we can put the q that we this q here. So, then we get minus l n this become minus l n n factorial and this become n this is q to the power n this is q to the power n. So, n l n q. So, and this is q n equal to q to the power n by n factorial that thing is used here. Now, we put the value of q there which we derived before. So, we get the l n n factorial starlings approximation is made. So, l n n factorial equal to n l n n minus n if we do that this minus n actually shows up here as f a and we already have v. So, q is already have this n 3 n by 2. So, I bring out that. So, n a I have n here. So, this only 3 by 2. So, so n comes out. So, n k b t here this k b t and q to the power n capital q is q to the power n that brings it n. So, n k b t l n then 2 pi m k b t h square 3 by 2 v this is multiplied by whole thing here v by n because I they separated out l n v and the o 1 is this comes out. So, this comes from these things you can just work it out the basic things which are you will find out what is fine. So, important thing to understand now that everything is in logarithmic. So, I have a logarithmic here. So, there will be logarithmic v term because it is v to the power n. So, that comes from the volume integral and so in l n that part is l n v to the power n. So, that becomes in l n v. So, now I can calculate from here the pressure I can calculate the pressure from here and that is by thermodynamic relation pressure minus d a d v k b t and then you can do that look at the let us do these things here. So, this is l n. So, then I break it out I like l n this term plus v by l n. So, it is just l n v d a d v then that n is there this just goes out and I get the 1 over v term l n v. So, d d v l n v 1 over v. So, n k b t in front. So, n k b t by v by doing the derivative then p v by n k b t. So, this is the ideal gas law. So, now I have a microscopic derivation of p v equal to n k b t which was surmised by doing experimental enormous amount of experimental studies and also what comes out from Maxwell Boltzmann kinetic theory of gases, but here these no kinetic theory of gases no collisions with the wall no cross section that we have done you have to do the collisions of the wall with the kinematic description. We did not take do any kinematic description anywhere we did have a kinetic energy of course, but we do not have a kinematic description, but this is very important to understand that this is the pressure of the system of molecules which is contained in a volume v this is really magical this derivation. Next what goes on something even more magical and that is the yeah. Now, we have been something really very very important and this is something one of the major reason of reviewing this class once more and also we will do that in the in the in the in the in the in the diatomic gas also there are certain very far reaching consequences and we do not actually this kind of jumps that it comes from almost nowhere because we did not expect such a beautiful relation. So, early in our derivation or study of a such a major area of research sort of formidable area of research like in like in statistical mechanics which is actually as I told you again is one of the two major disciplines of our thinking our theory or understanding of huge number of phenomena very very first class I spent fully on describing that thing ok. So, now we have the partition function and now we have the free energy because we have the free energy a equal to we have this beautiful relation in now I know entropy because I know entropy by this I know entropy as equal to minus dA dB t I have a qn I know the qn because I know qn is 1 over n factorial h to the power 3n small q to the power n and small q I know is 2 pi m kB t 3 by 2 and volume v I know everything. So, I know qn and small q is the single particle and then I know free free energy exactly and then I know that I can now calculate entropy I can calculate absolute value of the entropy it is very important it could summarize cannot give me. Now, yes you can say I can get it from ideal crystal all these things but it is ok but you know everything have to be done numerically. Now I can do that and the beauty of this thing is now that if I do I take now look at this I show I take qn I take ln of this thing now look at that. So, if free energy has temperature in two places one is in temperature t one is temperature t this is one another place it has is in temperature t here in small q here. So, when I have to take these derivative I have to take care of this temperature and I have to take care of this temperature ok. Now, I do that first one is easy d a d t just this goes off I have kB t ln qn t v and this my qn t v great that is why 2 pi m kB t by h square all these things come in wonderful now I have to do the second derivative that now kB I do not hit kB with d t the kB t I come here and do the derivative here ok. Now, as I told you it is not q to the power n. So, n comes out I have a small q and I have to take time a temperature derivative the small q ok then ln q. So, then I do ln q then ln 2 pi m kB plus ln t I have a kB t sitting in front when I do that ok. So, now I can take that derivative I can do this derivative on this quantity and I can then I combine the terms this you can do yourself and it is left as an exercise you can do it yourself and you will get that entropy just do it yourself it just takes a couple of minutes then you get that this quantity this entropy. So, this part temperature part is ln q the ln is here and q is here 2 pi m kB t and that is a volume v this volume v is here v by n and so everything remains the same except e becomes 5 by 2 because one extra term comes in it 3 by 2 n comes because this derivative this derivative this 3 by 2 that comes out. So, then that t 3 by 2 here. So, when you take the derivative of that then 3 by 2 comes out and t becomes half the t becomes half then you have a t to the bar t it is become t to the power half you have the t here. So, you combine that t with the t that becomes half here after taking derivative. So, that become t to the power 3 by 2 again. So, you combine get 3 by 2 you are left with 1 3 by 2 term that your one term you combine and get e to the power 5 by 2. So, the main thing is to combine and you can get that term. So, this is the then this is an exact expression for an entropy a high degree non-trivial equation which is this thing that entropy of n number of ideal gas law is n k b l n 2 by m k b t 3 by 2 k b t by d. Now this v v by n one can use now p v equal to n k b t ideal gas law p v n p v n k b t and then you remove v by n bring it here. So, it becomes k b t by p this is more useful because most of the time we want at a constant pressure not at constant volume and we know the number of particles and if we know and at temperature constant temperature constant pressure constant n. So, this more of an experimental thing then this is an exact expression of the entropy of the system beauty of it is the following the v by n that comes here when you do the calculations is a very important thing is very important to understand from that that what makes everything an extensive quantity. So, the entropy that I have here is just entropy is proportional to n as it should be everything else inside is an intensive property the temperature is intensive property pressure is an intensive property. So, the entropy is extensive because it is proportional to n, but this is a beauty of an equation it is called circuit tetrodéquation circuit tetrodéquation called a it is often called S t equation this is equation which is one of the. So, you are now the basic idea of my telling all these things again in great detail is to give you an idea how the statistical mechanics flows just like in quantum mechanics you do particle in a box you learn that how you solve Schrodinger time independence Schrodinger equation by putting boundary values and when you do that low and behold this sketch energy levels appear and then immediately find an use of that particle in a box explaining the spectra of butadiene or conjugate polymers. Then you go harmonic oscillator and low and behold you can read some idea of the vibrational spectroscopy rigid rotator you get rotational spectroscopy those are very preliminary very simple things that we solve time independence Schrodinger equation, but we get back huge amount of information as if without doing any work that you are getting such formidable and very fundamental things this is essentially analogous to particle in a box or harmonic oscillator we will use harmonic oscillator immediately now, but important thing to know that we are getting we are we are getting p v equal to n k b t we are getting an expression a beautiful expression of entropy and absolute value of entropy although ideal gas, but it will find lot of use in even today is research this equation finds a lot of use. So, next what we will go we will do the so one can now calculate the translational entropy this is called translational we have taken on the translation account so entropy the values of the absolute value of the entropy see in thermodynamics cannot give you absolute value of the entropy absolute value of the entropy this is the neon argon and krypton in atmospheric pressure, but then atmosphere pressure of course is probably not a but whatever condition is not written here I do not remember, but the experimental condition that we can calculate this translation into the particle then we can we can an experimental way of finding the entropy and it agrees agrees with the experimental values of this entropy of these three gases they are very similar to each other. So, now I just want to tell you why these things are so important and so what are the circuit retro equation one of the important applications of circuit retro equation of modern times is this quantity which is the very recently this has been used by several people that you want to calculate the you want to calculate that how a drug goes into mind order major groups of DNA and that is very important to know because of many chemo therapeutic drugs they go and bind to many other drugs in mind to DNA is the target of many therapeutic studies and so we need to know what is the binding energy of this molecule drug molecule this is now done in the following way that you consider your drug molecule is a drug molecule is going to go and bind to mind order major group of the DNA. So, now this has a lot of entropy in the ideal when it is in solution now in the solution we can treat it as an ideal gas if it is a single DNA molecule of a low density sorry single drug molecule or very low density drug molecule that goes and combines to these DNA. So, that we need to know the entropy loss of entropy of the drug when it binds there so we need to know the what is the entropy of this molecule in solution that is why circuit retro equation is used then you want to do for example gas liquid nutrition in a low temperature gas is going to go to bubble to form a bubble like in atmosphere to form a nucleation then one need to calculate how much entropy how much entropy the gas particles are losing in order to form the bubble because you know what is from the bubble it moves very slowly and it moves as a one particle. So, n number of gas molecules like water forms and goes from the water droplet then it is a huge amount of loss of entropy so that actually resist formation of the droplet formation of the cloud bubbles and in particular that is why high temperature when the water molecules are still there but they have lot of entropy and they will not form the bubble they will not nucleate they will not nucleate to form a water droplet which can form as a ring. So, entropy of these water molecules in the gas phase it is not just translation entropy this is a rotation entropy there is a vibration entropy we will talk of that so all those entropies this is a very important role so this is done by this gas liquid nucleation is a huge area of research and in the study of this nucleation against this entropy this in the nucleation theory again this entropy that we derived today finds out you. Now, finally to end this part that specific heat we know 3 by 2 R and that is very trivially derived now if you can go T it is C v T ds dT you go to circuitry equation do the ds dT then you will find that it just becomes 1 over N k B T by T that T cancels and you get C v equal 3 by 2 R and that is one of the P v equal to RT along with the ideal gas law C v equal to 3 by 2 R and C p is 5 by 2 R and that is because another way to derive is E equal to 3 by 2 RT that you know then dE dT is C v this is C v and that is 3 by 2 R that is the derivation that is the you know from ideal from the theory of gas but this is the derivation here coming from entropy directly with the entropy. So, it is the alternate derivation same result and is since equipartition theory is involved I cannot say it is more fundamental because equipartition theory is as fundamental as the gate but this is a statistical mechanical derivation with starting from partition function. So, this is about the ideal gas law what monotonic gas and ideal gas law that I wanted to do again in order to tell you the take you through a second time the things.