 Again, I will briefly touch on what normally I would teach in kinetic theory. So, there are some very simple assumptions you make and then you realize why you get to something like the ideal gas law. And the key things that are normally put in kinetic theory is that you assume that molecules are equally distributed in a room. So, you say a gas is formed of molecules or atoms, they are equally distributed in a room or uniformly I would say distributed. And the other key thing is that we assume that all molecules are point like. That means by themselves they will not occupy any volume. And this is a key part and we realize maybe if I get time at del if we say that each atom or molecule occupy some volume then you will realize why you start deviating from something like the ideal gas equation. But we will say point like molecules. So, by themselves they will not occupy any volume. And the third thing that is most common and that is again a key thing which will define how kinetic theory comes up with this thing is that there is no interaction till you actually hit. What does this mean? It means that I will assume that there are just two balls which are of course point like and they do not have any volume, but they do not feel each other's presence till they are actually in physical contact. So, you a lot of times you just assume it is like some hard balls, billiard balls which only at the moment they touch each other they are aware of the other person's presence. And you will realize that this is definitely not true. And if I start relaxing it you will you know people come up with how why the Van der Waal equation of state is far more realistic. Because it considers this fact that you know interactions do take place. And one of the most common interactions maybe if you have studied in chemistry is a standard interaction which is normally called as a Lenard-Jones potential. Have you ever heard this term a Lenard-Jones potential? So, it tells you what the kind of potential energy is as a function of distance between two atoms. And people will normally plot something like that when they are very far off they do not see any kind of interaction. When they start coming in there is an attractive part to the potential which leads to the fact that they will start you know the potential starts decreasing. And as you bring it more closer and closer what really happens is that each atom is still made up of some nucleus and electrons surrounding it. And once you start at a particular distance they are good enough to form bonds or beyond a certain point it is like they are soft balls. And the electron clouds really repel each other and this is where you know they are heavily repelling each other and they just bring back. There is a repulsion force. So, if you go here the force is repulsive on this side if I take the derivative and here they are just. So, this is what is normally happening and you can say that even if they are at this distance there is an interaction. Of course, at this distance they are in interaction. So, there is an interaction at various distances normally and this is the most simple potential. It is not that the potential always looks like this. But this is a very easy way to tell how two atoms will interact. Whereas the model that we are assuming in kinetic theory is that we say that till they see a particular diameter there is no interaction it is just 0. And suddenly it is as if they see an infinite it is like two billiard balls fit and that is it at that moment at that diameter here when they hit each other that is when they see each other and that is infinitely they cannot come closer than that. That is the model we are putting here it is like a hard sphere model. So, if I go with this model that is what is normally put in kinetic theory and the last thing I normally do is I will assume specular reflection at walls. This is another key thing. So, you will say that if a molecule comes like this then it is reflected specular. So, this angle is the same as this angle which means the normal component of velocity is incident is just negative of normal component of velocity reflected. So, this is normal component this is the tangential component. The normal component is downwards it will be the same component upwards after reflecting or with a negative sign and the tangential component does not change at all V t i is V t i. So, this is what is normally a specular reflection. So, these are some assumptions and the last there is another assumption that is we assume that velocities are distributed isotropic. That means if I take any point here and I see all molecules then then all velocities any direction is equally possible. So, if they are moving it is not that you know unless I have put in some external force to push them in certain direction. If I just leave them as they are then every direction is just equally possible. I just leave a set of atoms here everywhere the you know. So, if 10 guys are going in this direction then are going in every possible direction all around. So, it is an isotropic distribution of the velocity. So, this is these are some of the standard things that we put in when we start with kinetic theory and then you go ahead and do various things. So, for example, if I want to calculate pressure then you realize that pressure is nothing but just some force per unit area and how will I calculate force if I take a surface I take a molecule which comes and hits and goes. So, I know that for every collision there is has been a change in momentum. Initially, the momentum was either tangential component is not changing. So, initially the momentum was m times v normal i which let us say if this is theta this is v tangential v normal and this is v. Now, if I take this angle to be theta then this angle is theta and the normal component is just v cos theta. So, it is m v cos theta going down and when it is going up it is m v cos theta, but in the opposite direction because now it is like this. If one collision has occurred then the change in momentum let us say downward is negative then minus m v cos theta minus m v cos theta I will get minus 2 m v cos theta. This is the change in momentum that has occurred per collision. So, that means this much you know. So, what is just the force it is just change in momentum per unit time. So, if I go to my regular definition of what a force is I need to know what has been the change in momentum in a unit time. Now of course, if one molecule went like this and collided and its change momentum it exerted a force on the other one the surface. And what you need to do is figure out in a unit time how many such collisions occurred and if you can find out how many collisions occurred you can get your force and if you calculate the force per unit area you get your pressure. So, that is how we would normally go about and one way to go ahead is take some kind of a surface area here. And say take a direction like this let us say which is theta here and find out how many people are rushing like this. So, now you come to each of the assumptions that we have listed in our kinetic theory. So, say molecules are uniformly distributed fine. So, that means what I will do is I will say consider everyone who is moving in this direction with this velocity. So, now how many people are going to hit this surface in a unit time. So, all I have to do is I will just take a projected area like this and draw one prism a rectangular prism. And let us say I have considered velocity V unknown quantity a velocity V then if I take let us say time dt then the length of the prism I take V dt and the area here is just dA times cos theta. So, the projected area is dA cos theta I have drawn a prism here. So, now everyone in this V dt in the next time dt is going to go and hit here. So, those many people in this entire. So, anyone close here would have hit in time dt and crossed anyone who is here definitely because its velocity is V also would have come here and hit. So, in the next time dt so many people with a collision V sorry with a velocity V would have come from this direction theta and come and hit. Now, all those guys would have undergone a change of momentum to Mv cos theta. So, this is something that you will know first. So, first you will find out how many people come from one direction. So, the normal way I will do it is I will take a surface I will choose a particular theta with respect to the normal and then I have 2 phi degrees the azimuthal angle. So, I will choose a particular theta and phi I will say now I will use the various. So, once I already said that I use the assumption that things are uniform next I will use the assumption that velocity distribution isotropic. So, I will say every direction is equally possible. So, how many people are moving in this direction that is just going to be the solid angle being subtended here upon 4 pi. 4 pi is a net solid angle and any particular direction is some d omega. So, d omega upon 4 pi is the probability that you are traveling in a particular omega. So, this is what I do that for a particular omega I know how many people are coming in and finally, what I do is I will just integrate over theta and phi. It is a very simple integration I do a integration over theta and phi at this stage I can do an integration over theta and phi there is no problem I will have only involved at this point you would have done only an integration over theta and phi and you would have just assumed a particular velocity v. Now, obviously all molecules do not travel with a particular velocity v they are distributed and a priori when you are teaching kinetic theory you never initially will discuss what is the distribution of velocities that is how many people have a velocity of 10 meters per second how many have a velocity of 20 how many have a velocity. So, the values you will not discuss you will just say that if I can put velocity on this axis and fraction with that particular velocity here I do not know what the distribution is. So, let us assume it is some distribution once I do an integration over all you know I do an integration over theta I will do an integration over phi then you will realize that the change in momentum involves m v cos theta I have assumed same molecules m is known I have integrated over theta I have to integrate over all velocities. So, at this point you would not know what the velocity is, but you can tell that if I integrate over all velocities and divide by the number of people I will have something called as an average velocity and that average velocity I will just call it as v bar and I am not going to do the derivation right now because of lack of time, but I will get that at any point if I want to calculate the number of people hitting which is normally called as thermal flux that will just be equal to n v bar by 4 per unit area and this n is number density and this v will be particular to a distribution is this fine and this quantity is called as a thermal flux. So, of course I have not done the derivation, but this is a derivation that I normally do and show that I integrate things show properly that you know this is a thermal flux that comes into it. So, once you know the thermal flux and I know if I know the average velocity let us say if I know the average velocity then I can get your pressure correct because based on the average velocity I know what is the change in momentum I know how many collisions are occurring per unit time and then if I know per unit time what is the change in momentum I know the force, force per unit area is the pressure. You will find out that if I use atmospheric conditions one atmosphere at you know 25 degree centigrade and if I use air normally you will realize that per meter square we are talking of close to 10 raise to 27 collisions per meter square per second. So, if I take 1 meter square of area this is the kind number of collisions that will occur if I know the mass of the molecule etcetera I will go ahead figure out what will happen. So, this is how I will go ahead with pressure and pressure I will get as some kind of a relationship with you know the number density and the velocity and then the velocity we will figure out that we can you know connect it to the temperature. The next thing that normally I would do is you know try to give some idea of what kind of distributions will exist for a you know sorry for the velocity and it is a slightly tough job to figure out exactly what kind of distributions will exist but you can show and I normally do a brief thing about it you can show that if you are at equilibrium then there is only one distribution possible which is normally called as a Maxwell Boltzmann distribution and the Maxwell Boltzmann distribution has only one parameter it is the temperature. If you know the temperature you exactly know how many people are with what velocity. So, the fraction with a particular velocity is known and that is normally derived. So, if I know the distribution of velocities I can calculate V bar because to calculate once I know the distribution to calculate anything regarding the velocity is known V bar would be just integral V times F V. How many people with a particular velocity multiplied by the velocity add up you will get there. If I want V squared bar then that is all I integrate V squared F V bar d and you will realize that if I do all these calculations if I use the Maxwell Boltzmann distribution V bar will come out to be 8 kT by pi m that this is in kg unit V square bar or I should say it is just 3 by 2 kT or square root of V square bar is square root of 3 by 2 kT sorry by m. So, every time you will get some factor here you get 8 by pi kT by m here you get 3 by 2 kT by m. So, something into kT by m you will always get, but I can do the integration and get you all these quantities and you would have probably seen these quantities earlier. So, this is the next thing which is normally done and following this what we normally talk of is the mean free path. So, mean free path is a very useful quantity when we are doing many calculations and you do the same thing you say I assume they are point particles, but when they hit each other there is some kind of a hard sphere diameter involved. So, if I have one atom here and one atom here and this is really where their points are you know the center is where I decide that the molecule exists. Obviously, if they are within r 1 plus r 2 I know that they are going to hit each other. So, I will say that as far as this point is concerned if this point is anywhere between this circle here it is going to have a collision. So, how much is this radius of collision is just r 1 plus r 2 and this circle area is just pi r 1 plus r 2 square this is what is called as the hard sphere collision model. That is there are two hard spheres this is roughly what is called as the hard sphere collision and this is the hard sphere collision cross section. So, I assume that they are all spheres. So, once I have this I will say that you know let us consider a sea of molecules here and one let us say they are all stationary and one molecule is just moving along in this. So, now obviously if I go to this previous case let us say this area I have chosen. So, I will say that if this is moving along let us say it has moved like this I will draw around it an area A and in I will say in one second it will have swept a volume which is equal to A times the velocity. So, it would be times one second that is the length of the prism it would have covered the base is A. So, this is the volume swept. So, A into V small is equal to volume swept in one second. Now, if I know that the density of these other surrounding molecule is n then the number of atoms or you know collision partners in this volume is just number density into volume collision partners is number density into volume is equal to number into area into. This is assuming that the other guys are stationary and you are moving with a velocity V. Obviously, it will be happening that everyone is moving you are also moving everyone is moving with different velocities you are moving with a particular velocity, but that is getting complicated, but it will change only things by a small factor, but once this is done you know that in one second you would have a set of collisions occurring for one molecule equal to. So, this is number of collision partners in one second that means that molecule would have undergone. So, many collisions in one second and most common way to write this is n sigma V where sigma is the most common symbol given for the cross section. So, this is what is called as the collision frequency. So, if I want what is the mean time between collisions it will just be reciprocal of this. So, it will be just 1 upon n sigma V what is the time between two collision and if I want to find out what is the mean free path then it is a distance travelled between two collision. That means I just see what is the time it takes to go between first collision and that collision which is this multiplied by the velocity. So, it is 1 upon n sigma V into V I will get it as 1 upon n sigma. So, this is what is called as the hard sphere mean free path I will know n I know sigma I can get what is the mean free path. If density increases mean free path decreases because you travel less between collision cross section increases again mean free path decreases because bigger cross section is more collision it will be equalized and this is the now it if you consider the fact that you know molecules are going with different velocities you will get some kind of a pre factor here, but it will be just a factor of the order one. So, you get the mean free path once you get the mean free path the next thing that we normally do is that if someone has undergone a collision. Obviously, the mean free path is just telling you what is the average free path, but if I say that if I have just undergone a collision and I am travelling somewhere here what is the probability that it will travel a distance x without colliding and then colliding. So, you will you will find out that the probability that things are not colliding again I am not doing the derivation here that if I set out with n uncollided molecules they will decrease exponentially you can show it is just like any other radioactive law if I travel so much distance then the proportion of collisions is just equal to the distance d x and the number of uncollided people just like any radioactive law it decreases exponentially and it will be of this form. So, this is the set of mean free path and things that I do the next thing that is normally done again I just mention it briefly is that once I have the mean free path in place once I say there is an isotropic distribution of velocities all this is done the next thing is to derive transport properties. So, this means viscosity and conductivity and diffusion constant everything that is required for diffusive properties the momentum diffusivity is new thermal diffusivity is k let us say I am calling it k here and number diffusivity is d and the most common trick that you do is that you take a surface let us say there is a variation of velocity like this you say there are so many molecule here you take a volume here you say how many collisions are occurring at any time in this volume because you know the collision frequency you know the number density you can get the number of collisions occurring here. Now you say that after collisions all velocities are equally possible. So, I will take a small area here and I will say out of all velocities how many are directed to this now since all directions are equally possible I will just say what is the solid angle subtended here that divided by 4 pi will give me how many are travelling here. Now once I know this I will say how many of these people who are travelling there will actually reach there without a collision and that will of course be will turn out to be just some kind of only this much fraction will reach there. So, now the key thing you will tell now is that these guys who are coming from here will carry information from this point similarly I will just I will I will see how many are happening here. So, I will just integrate over this entire space here with all these guys are carrying information from here. So, if there is higher momentum here all these are bringing higher momentum from up to down and you need to know how many are coming without further collision because you assume that if I came here underwent a collision here then it will carry information from this volume and this momentum and not from here. So, everywhere I will just do this take every possible volume figure out what is the net momentum crossing downwards I will calculate the net momentum which is crossing upwards from below I will find the difference in momentum upwards minus downwards per unit area that will give me force per unit area and that will be my viscous force or viscous pressure and you can show that whatever you derive from here matches very well for any of the gases within a constant of 1 or 2. So, I take oxygen argon nitrogen I can get a viscosity value just by saying this is the hard sphere model this is the distribution of velocities this is the probability that they reach this position and I will get my viscosity correctly. Then I will do the same thing I will say there is a distribution of temperatures I will get that if I have a particular molecule this is the amount of energy it carries I will get my thermal diffusivity correctly and similarly I will get my diffusion constant. So, this is what is normally done in a kinetic theory thing first list down the assumptions find out the relationship with pressure show what happens in a regular ideal gas with pressure then you know find out the distribution of velocities you know calculate average velocity mean square velocities define mean free path calculate transport properties So, this is normally what I would cover in a kinetic theory class. So, of course, I have gone through it reasonably fast, but you get the idea that this is the kind of thing, but it takes me close to at least 4 and a half 1 hour lectures to cover this I mean I have been trying to do it in 45 minutes, but you realize.