In this side remark, I'm showing your additional evidence that things are made of atoms. From the kinetic theory of gas. The idea is, first we assume things are made of atoms. first we assume that, but then, from a number of derivations, we will show you that the atom has a finite size. Okay, as long as the size of the atom, the diameter of the atom is not shrinking to zero is a finite size. That's a self consistent evidence showing you that our starting point is right, that things are made of atoms. And the starting point of that is the Magdeburg hemisphere experiment hundreds of years ago. And what happened at that time is you put two plates together and assume that they can put exactly together, then you pump out of the air between these two plates and afterwards is extremely difficult to separate. Why? Because of air pressure. And this is actually an excellent piece of popular science practice that 16 horses, 16 horses in two teams are trying their best to pull these two plates away, and 16 horses do their best marginally. They can pull this two plates away, and there is a huge sound "bang!" and this to plates are separated. What that means? That means there is huge air pressure. And what is the huge air pressure? That air pressure p corresponds to 10 to the five Pascal. Okay, 10 to the five newtons per square meter, 10 to the five. A huge number. And what that huge number indicates. What is the pressure of gas? This pressure can be written as a third of nmv squared. Where n is the number density of the gas atoms or molecules. Here I use generalized the notation of atoms, including molecules and mv square is the kinetic energy up to a factor of two. And the factor of three is because air can move in different directions. this is pressure. And what is the density of gas? The density of gas rho. You are all familiar than that? Which is in nm. Okay. Just a number density, multiplying the mass per molecule. All right. And here, what do we know? Just divided P and rho. What we get is this v square is the ratio between P and rho. So as a result, this v is somewhere P over rho square root. And we have learned that what is the density of gas? The density of gas is not that large is about one kilogram per meter cube. And what that means is the velocity here is enhanced by this huge number of air pressure and the velocity is of order, 10 to the three meter per second. That means a kilometer per second. Okay, so the big "bang" of the two plates from the Magdeburg experiment tells us that the atoms if things are made of atoms, the atoms are moving pretty fast. All right, once we know that the atoms are moving pretty fast in gas. The next question is, do these atoms move freely in gas? Just very fast and freely? Or they frequently collide with other particles, which is the case? to answer this question, let us define a concept which is the mean free path. Which is on average, how frequently does a gas particle collide with other gas particles? What that means is assuming we have a gas particle, then the radius of the gas particle comes into the story, if there is no radius, then this gas particle doesn't collide with other particles. so there is an effective radius of the gas particle. And how do we calculate the mean free path? That on average, within a mean free path, one gas particle collide with another gas particle. So on average because it is random motion. how to calculate that? So imagine there is a tube along the random motion direction over this gas particle and on average just this gas particle moves along to the end of the tube. On average, it will collide with another gas particle. What that means? If the other gas particle is here or if the other gas particle is here. below about the tube is not going to collide only, some part inside the tube is going to collide. That means that so here we have if the length of the tube is L. And this L. By definition. So on average it collides, that means actually the mean free path. This L mean free path, multiplying the number of density of the particles n and multiplying the area here and the area here is of order the d squared, which is the diameter. That means the volume of the tube multiplying the number density is of order one that means there is of order one particle within this tube. And that means on average this particle, there is a number one probability to collide with another particle. so this is of order one. In other words, the main free paths can be expressed as the mean free path is of order one over nd squared. So this is indication that the atoms are not moving freely in gas. But there is a mean free path of order one over nd squared. And the next question that you're asking is, so here all you're doing is mathematics. Can you tell me the number? can you tell me the number what is the number of mean free path of gas? All right now, let me tell you the number and by telling you the number we are actually asking and answering another question. The question is, is gas viscous what do I mean by is gas viscous? What I mean is if I pull one layer of gas, for example, the upper layer of gas to move at a velocity u in this case is the lower layer of gas going to follow the motion? Or if just the lower layer of gas just dating here? just intuition, just think the lower layer of gas should follow the upper layer of gas. Is that right? And that is known as viscosity. And how do we define this viscosity? The viscosity is defined by the force. because if I move the upper layer, voscosity means the lower layer feels a force coming from this viscosity along this parallel direction of motion. So there is a force F parallel and this F parallel should be proportional to the area, proportional to the area between the two layers, and it should be proportional to the speed of motion of the upper layer and it should be inversely proportional to the distance between these two layers. Okay, divided by the distance between these two layers and there should be a proportional constant, meaning how efficient the upper layer can drag the lower layer. And the efficiency is known as viscosity, which is we put the number mu, and this viscosity mu is a pretty small number. This number is of order 10 to the minus 5 kilograms per meter per second. Pretty small number. All right. And now let us understand how this small number of viscosity is related to the mean free path of the gas particle. To understand that. Let us think about the microscopical origin of this viscosity. Okay, what is the microscopical origin? Let us consider the two gas particles in here and in here and their interaction. so this gas particle is moving with speed u along this direction. And is this the only motion of this gas particle? No. This guy's particle also have another piece of motion which is actually typically greater, which is the random motion. the random motion speed V which is a kilometer per second. Okay. And the speed v could be in any direction. So the component in the perpendicular direction of u would be of order That random motion speed. Okay. And this speed will also collide with this particle and give this particle a force. And what is that force? Let's call it F perpendicular. And what is F perpendicular? This f perpendicular per area is pressure. So this is simply pressure multiplying area. Okay, so this is F parallel, f perpendicular. And if we look at the system we realize that F parallel and F perpendicular, the ratio is simply because of the air motion in this direction and in this direction. So this ratio F parallel divided by F perpendicular should be parallel is because of u and perpendicular is because of V. So the ratio should be simply u over V. put this 1, 2, 3. Put the three questions together we have this u over V. The ratio equals to of order at least, mu A u divided by y and what is y? Y here is the mean free path between these particles. Since this is the nearest particle that this particle is likely to collide. So divided by y, y replaced by the mean free path. And then the pressure and area P and A. Here, very happily. We have a few things canceled. This u is canceled and this A is cancelled from here. We get that mean free path is of order mu and v divided by P. Okay. And what's this mu divided by P. Here the viscosity mu can be measured. We mentioned is very small. 10 to the minus five. And the V is pretty large, is 10 to the three but the pressure is even larger, pressure is 10 to the five. And that means that gives us a combination of numbers and the unit is meter. That gives us 10 to the minus seven meter. this is the main free part of the particle that on average the particle moves 10 to the minus seven meter before encountering another particle. The main free path is 10 to the minus seven meters. And I'd like to remind you the definition is of order one over d squared the size of the atom square and n air were n air is the number of density of air molecules. we have gone a long way and now let's ask the question. How big is an atom? There is one step away from that question and what is that step? That first of all, let us notice that now we have n air is of order this d squared L. And what his n solid? For solid, The n solid should be of order, the diameter of the atom cubed. And what that means? What that means is n air divided by n solid should be of order d square L and divided by this d to the cube, for the same type of particles. Okay. And we have the intuition what is n air divided by n solid? That will be of all the 10 to the minus 3. Okay. From our a lot of experiments. And then what is this? 10 to the minus 3? That is the ratio of L over d. And then what is the size of an atom? The size of the atom is of order 10 to the minus three L. Which is 10 to the minus 10 meters. All right. So that means after a long journey from the air pressure, from the velocity of motion of gas, from the viscosity of gas from the mean free path and from the ratio between air and solid. Eventually we get a number. That number is, the diameter of the atom is of order 10 to the minus 10 meters. And what that means is, although we have assumed at the beginning that things are made of atoms, gas made of atoms. However, then we have consistently, self- consistently derived a radius of the gas atom. And what does that mean is the radius is not vanishing. Once we assume there is such a radius, and more happily, this radius is the same order of magnitude compared to the oil film method. That is a strong indication then that things are indeed made of atoms of this size.
