 Hi, I'm Zor. Welcome to Neuzart Education. We are continuing talking about relationship between the heat and as a form of energy actually. And in this particular case, we will consider ideal gas and its kinetic characteristics, which will be eventually related to heat and temperature actually. So, today's lecture is kinetics of ideal gas. Now this lecture is part of the course called Physics 14, presented on Unison.com. I do suggest you to watch this lecture from the website because it contains notes for each lecture and all lectures are actually presented in a logical sequence. Plus on the same website you will have mass for teens, which is a prerequisite course. So, I do suggest you to use the website rather than YouTube or any other source where you can just come up, come to accidentally basically find this particular lecture there. Now it's part of the course that's very important and the course should be taken sequentially. And by the way, the site is completely free. It has no strings attached. You don't even have to sign on unless you would like to take exams, which are also provided on that course, on that website. Alright, so let's talk about kinetics of ideal gas. Now we will talk about this in stages. First we will consider only one molecule of gas. Then we will consider an ensemble of molecules which are moving in a similar direction. And then we will go to a concept of ideal gas. So it will be stage by stage. And let's talk about one particular molecule which is enclosed. Let's say these are coordinates and we have somewhere a cubical reservoir with gas. Which has lengths of each h, l. So it's l by l by l. l is the length of the side of the h. Now let's consider one molecule. So this is x, this is y, and this is z. And let's consider one particular molecule which travels from this side, which is parallel to yz plane, to this side, which is also parallel to yz. So it's only within the direction of x coordinate. So it goes back and forth elastically reflecting from the walls. Well, in some way if you have one particular molecule that might be the case if it goes exactly in this particular direction. So this is one very, very simple model of one particular molecule of gas which is moving between two opposite walls of this cubical reservoir perpendicularly to the walls, to the opposite walls. It reflects exactly along the same trajectory. So what happens in this particular case with this molecule? Let's just think about it. Our purpose is to express the pressure which gas exerts on the walls of the reservoir in terms of kinetic characteristics like mass and the speed of the molecule. So what happens in this particular case? Well, the molecule goes into this direction and momentarily exerts certain pressure. Now the wall reflects it back and while it's traveling there is no pressure basically. Now, we used to have some kind of a concept of a pressure like for instance some kind of an object is lying on the table. It has certain weight and certain area on which it touches the table. And what we do, if you would like to know what's the pressure we basically divide the force which is the weight by the area. So that's the pressure each square meter or square inch or whatever matrix we are using. So pressure is basically the force per unit of area. Now in this particular case the force is variable. So during this moment of reflection the force basically is exerted because the way how the wall reflects the molecule which means acts against the molecule the molecule according to the third Newton's law is pressing the wall and that's what pressure actually is. So how can we actually talk about the pressure in this particular case? Well, a very reasonable suggestion is to average the pressure. So if during certain amount of time the pressure exists and then during certain other amount of time while the molecule is flying between the walls pressure does not exist all we have to do we have to average the force during that period of time. Now if we consider that there is certain average of this force what does this particular force cause? Well, it causes the reflection. So if my molecule of the mass m goes into this direction this is the vector then after reflection the same molecule goes with minus v. So if positive direction is towards the x then this is direction this is the moment of the molecule which goes towards this wall and this is the moment when it goes back. So let's establish a period of time. Let's consider the molecule is touching this left wall and from this moment it goes this way and then goes back. This is a period so to speak. It's like a pendulum goes back and forth, back and forth. So this is the period during which we would like to average our force because this is actually a repeated period because the next period it will be exactly the same. So that's why it makes sense actually to consider one period of time which is a real period it's a periodic function if you wish. The force is a periodic function of time and the period is the time during which the molecule goes back and forth. Let's call this period tau. Now what exactly tau is? Well, we know the speed, well we assume we know the speed. So it goes from one wall to another that's L and then back so it's 2L. So if we get 2L and divide by absolute value of the speed that would be our time during which the molecule does this repeated movement. It's a period. Now during this time again force is variable. However, let's just think about the variable force as acting to change the moment from this to this. So in absolute terms the moment is changed from mv to minus mv to minus mv which means the moment is changed by 2mv. Well I'm talking about absolute value right now not as a vector. So this is the change of the moment. So the force which was actually exerted during a very small amount of time but if you are averaging for the whole period this would be the change which this force actually did which means that the impulse of the force should be equal to the change of momentum. This is the impulse of the force again. This is not now a vector. This is absolute value of this force. So if we would like to know the average force then we have to multiply this average force by the time and that gives me the change of momentum. Now in practical life this force is actually zero while the molecule is flying in between the walls and it's something really strong during the reflection and then again it's zero while it's flying this way. But if you would like to average the force then the average force times average by time. I'm talking about the time. So averaging the force by time should give exactly the same result. So this is average force. Now why is it making sense actually to talk about the average? If we are talking about gas obviously there are more than one molecule and since each of them is doing something during some small period of time but then another molecule does something similar which means reflection then it does make sense because there are many molecules so that's why the average does make sense in this case. So this is the way how I can basically establish the force. So let's just skip these absolute value things so whenever I don't really use the vector it means it's a scalar. It's an absolute value. So 2m times the speed is equal to force times my time which is 2l divided by v from which f is equal to 2 is out so it's m v square divided by l. Now this is average force. Again not the momentarily exerted force because sometimes it's zero sometimes it's something which is the average force during this period of time and the period makes sense again because everything is repeated after this period. All right now. So what is the pressure? Well this force actually is exerted for the whole area. The whole area is l square so the pressure which is equal to f divided by l square is equal to m v square divided by l cube. Now what is l cube? That's the volume of the reservoir, right? So it's m v square divided by volume. So I used volume also v but the speed is v so it's much easier to distinguish them. So this is the speed this is the volume and it can be expressed differently. What is m v square? m v square divided by 2 is kinetic energy, right? So we will have 2 kinetic energy of the molecule divided by volume. This is actually a very interesting formula. The pressure which molecule exerts against the wall is equal to energy divided by volume. This is actually the energy of this cube. Okay that's my first step. Now I will try to go from this step to slightly more complex one and then to ideal gas. All right so let's just wipe out this. We will preserve our formula. The pressure equals m v square divided by volume or 2 e kinetic divided by volume. And the rest we will erase. Now my next level of complexity is when instead of one molecule which goes between these two opposite walls along the x direction I will have many molecules which are going exactly parallel course. All of them. So let's say we have n molecules. Okay now what happens in this particular case? All right each molecule will exert certain pressure, right? So the molecule number i will exact this pressure. Volume is exactly the same thing by the way. Now if I would like to know the entire pressure of all these molecules which are exerting against the wall I will have to summarize it, right? So I will have to summarize. So this is the total pressure. Now if m is constant which means all the molecules have the same mass then I can express it slightly differently. So I will put m divided by wall outside and instead of summarizing all my speeds in square I will rather do this and multiply by m. Now what is this? This is average of the square of the speed. So this is therefore n times m divided by volume and times average speed average. So square of the speed average. By the way, square and then average is not exactly the same as average squared. Just as an example if you have 2 and 4 their average is 3 and 3 square is 9 but 2 square plus 4 square divided by 2 is 4 and 16 is 20 that's 10. So they are not the same. So now I'm talking about not average speed squared I'm talking about average of squared speed. That's quite different. And if you wish in slightly different form it's n times, well actually I need to multiply 2 times n times average kinetic energy divided by volume. Because m and v square average is basically average kinetic energy of all the molecules. Now this form of the same basically equation has a slightly better way of interpreting this. You see if all our molecules are different then I have to put i here, right? In this case this formula actually still holds because this would be 2 times average energy, kinetic energy. And this one where m is outside of the averaging is only for cases where all the masses are exactly the same. Alright so we have come up to a conclusion this. So in this particular case when we have n molecules so let me put it here. If we have n molecules then the pressure is equal to n times m times square average of the squares of the speeds of the molecules or 2 and average of kinetic energy divided by volume. And now we are ready for the third step when we will introduce ideal gas and we will try to do very similarly to get the formula for a pressure in case of ideal gas. Now what is ideal gas? Well it's molecules which are absolutely chaotically moving in all the directions. All the collisions between the molecules are 100% elastic. They don't interact in any other way like magnetic forces or gravity forces etc. Nonexistent. Only kinetic aspects of their movement is taken into consideration. And obviously reflection of all the walls is also completely elastic. So the only difference is that instead of parallel moving we have moves in all different directions. So one goes this way, this way, this way, this way etc. All different directions. Now let's consider. We know that the speed as a vector can be represented as sum of 3 speeds along 3 coordinates. That's a vector equation. Now, if you have a movement which consists of 3 different movements along the x, along the y and along the z then if I would like to know what is the pressure on the, let's say this same wall which I was considering before on this side of the cube I only have to consider the pressure of the x components of all these molecules. So if they are moving in 3 different directions towards the wall, towards the upper wall and towards the side wall or a back wall, whatever it is this is the side, this is the back. So side wall I'm interested in and I'm not interested in top I'm interested in back. So I have to only take into consideration the x, which means if I would like to know the pressure on my wall, this one I have to take px which is equal to m vix2 divided by 2 this is the pressure of the molecule number i so molecule number i against the wall which I'm talking about this side wall along the x direction is this one. Now the total pressure on this wall would be a sum of px which is m let me put m inside so you will have the case with why did I put 2? It's my inertia, it's volume m vi2 divided by volume that's my total pressure which is equal to absolutely the same as before m times number of molecules times vx this is x again my problem square divided by volume right? So that's my pressure on the x on the side along the x along the x axis so I am dependent on the average of all the different molecules all of them but only on their component of the speed which is directed towards the x along the x axis now what is it? well that's not easy to determine basically because all the molecules are moving in all the different directions but don't forget that we are averaging this thing now let's talk about this equation again now from the solid geometry from here you obviously can it's a three-dimensional Pythagorean theorem okay so again if you have a vector you represent this vector as sum of this, this and this right? so along the... if you consider this triangle this is x, this is y, and this is z it's a right triangle this is the right angle, right? so it's a square of this is equal to square of this which is vz squared plus square of this and square of this along... according to the Pythagorean theorem it's vx squared plus v plus vy squared so that's why we have this three-dimensional equivalent of Pythagorean theorem now next consideration is all the molecules are moving completely chaotically in all the direction which means all directions are exactly the same so if I will take average of this it should be average of this plus this, plus this and since we are talking about averaging which means sum of all these different v-axis divided by the number of molecules they should be the same because all the directions are exactly the same from our perspective number of molecules which are moving into this direction is exactly the same as moving into that direction into this direction and their components which means if we are averaging them should be exactly the same which means it's the same as 3vx squared or if you wish 3vy squared or 3vz squared same thing which means in our case this average of vx squared average of vx squared should be equal to one third of v squared average right? what follows is that I would like actually this formula to be dependent not on the x component of the speed but on the speed itself so it's equal to 2m well, wait, it was 2 so far it's mn divided by 3v squared average volume that's what it is, right? because vx squared average is one third of v squared average or equals to in terms of kinetic energy it's n e kinetic energy divided by 3 volume so this is the real formula when all the molecules are moving chaotically now this, forget about this this is only an artificial case when all the molecules are moving in the same direction and this is not the case obviously so this is the real formula which basically tells what is the pressure of the ideal gas on its wall now it's really kind of understandable in many cases it's proportional to the number of molecules well, that's understandable the more molecules you put the greater pressure on the wall now, if the molecules are having more kinetic energy obviously it should produce more pressure either because of the mass because maybe the mass of the molecule is big but the same speed it hits harder which means the pressure should be harder if you're bombarding the wall with heavier molecules now, even if the molecules have a certain thick mass but their speed is increasing then, and this is an interesting thing now the pressure is increasing as a square of the average speed of all the molecules right? so that's what we came up with and obviously it depends on the volume because if you have N divided by wall by volume, this is a density of the molecule so you can have this new density of the molecules number of molecules per unit of volume so it's 2 it's 2-3 it's 2-3 N E kinetic average where N is the density of the molecules the number of molecules per unit of volume and again, obviously the greater the density of the molecules the stronger the pressure and with kinetic energy we already spoken about so this is the main formula this is the result of whatever we were trying to come up today now, this formula is a little bit more universal than if we will take as M and then average of the squares of the speed because this formula encompasses the case of different masses of the different molecules, right? so that's why this is a very important formula from which we will do at the next lecture we will do some consequences actually because kinetic energy obviously is related to a temperature as I was talking about in the previous lecture so my next step for the next lecture would be to connect the pressure the density or the volume or whatever and the temperature but that will be the next lecture so meanwhile I do suggest you to read a very detailed explanation of everything which I have talked about today in the notes for the lecture on theunisort.com so this is the very important part I was trying to explain basically what I was talking about in written form with certain formulas and I hope that after this explanation the written part would be maybe better understood so I do suggest you to read the written part of this of this lesson, of this lecture on unisort.com so other than that that's the end of it, thank you very much and good luck