welcome to our first section in our unit on the kinetic theory of the idea of the mean free path is that as a particle tries to go through, um, Volume, it is constantly bouncing off of other particles or even the walls that constrain the volume. And because of this, it on Lee goes in a particular direction with a particular velocity for a certain amount of time before it bounces off something else. Elastic Lee and now has a new direction and possibly a new speed as well. So what we want to do is figure out. How long does it maintain a particular speed or velocity? So what we do then is we say I have a molecule and it's trying to travel through some volume, and within that volume there is possibility of running into other particles. That possibility is based on the number density of the particles in the volume. So, for example, here, if we think about particles that it might clip, they could be particles that it overlaps with, or just anything that's within a radius. It will just barely touch as it goes by. Because of this picture, we're able to say that our our mean free path for our particles through a particular number density volume of material is going to be this. You can say we have 1/4 times route two times pi times Here we go, the number density multiplied by R squared where r is the radius, the approximate radius of the gas molecules. So we have a whole lot of a whole lot of constant something that has to do with the gas and then the number density. And that's all there is to it. So conceptually. It's not too hard to understand, though the process in getting here is quite complicated, and so we're going to skip the derivation. But what you know now is that you have a mean free path, which is the average distance between collisions for an individual gas molecule. If we're talking about a monotone, Mick gas were to single atom, we can approximate ours, being 0.5 times 10 to the negative 10 m. If we're talking about a die atomic gas where there's two atoms next student bonded with each other for each molecule, then weaken double that and use this as our radius. Instead, we also need to think about how fast are these particles moving? So for this will use something called Maxwell's be distribution law, because not every particle is traveling at the same speed. In fact, they're all traveling in different with different speeds and in different directions. And this could make deriving anything very difficult. In fact, the true average velocity is very hard to determine. But using this model established by Maxwell. But you can see it's four pi times the Mueller Mass divided by two pi r t all to the three halfs multiplied by V squared times E to the negative MV squared over to Artie. Okay, so that's a lot to do there. So we're not going to run you through all the integration. But it does look like this so you can see there's a large peak, and then there's kind of this tale. This is known as the high speed tale, where you'll have ah, significant number of particles that are going much faster than the average that you would expect. In fact, this isn't even the average, because this shifts the average to the right. Now, when we say average and actually turns out to not be what we mean. Because when we calculate the average speed, remember, speed is going to be equal to the square is equal to V X squared plus V y squared plus the Z squared. And this can cause issues because we're moving in multiple dimensions here. In fact, what we find is that the average velocity is this value right here when we take the correct integral. But it turns out that if we're really looking for the average of how fast a particle is moving within here, we're going to use something called the root mean square velocity root mean square velocity is equal to the average of the square. So we're gonna actually take the square root of all the V squares that have already been average. So in other words, we take a V squared for an individual particle, we find its speed. Then we average all the speeds and we take the square root of all the speeds, and that gives you the root, mean square speed, and this is equal to the square root of three rt over him. This is the speed that we're really going to use a lot because it's the most meaningful one as faras telling us, Ah, majority of particles are going up approximately this fast Now. That being said, there's still something known as VP, which is the most probable speed. That is to say, right here, that is the highest number of particles. And this is like the mode of all the speeds. There are many, many particles traveling at the most probable speed, so it is not necessarily the average speed notice here. It's squared of to rt over m. So this is smaller than the R. M s and then the average is even smaller than VP. So v RMS is actually the highest value here. But it turns out to be the best approximation of, on average, how fast and individual particle is moving now, understanding that we have this distribution of many different speeds and many different velocities, we can begin to talk about what's going on here. So we have this picture in our head of of particles colliding with the walls of the volume and causing some force to be pushing outwards. This force comes from the particle coming in and then bouncing off and having a collision and remember when we talk about collisions, we're talking about changes in momentum. So we're going to say that the force on the outside wall is equal to the change in momentum of these particles divided by the amount of time in which that occurs. So if we take Delta P here to be the change in momentum of every single particle that bounces off the wall during the amount of time Delta T, then we can approximate it as being too multiplied by the average momentum. It has multiplied by the number of collisions happening here. Now, when we take this and plug it into our equation and do the right derivation, we can come up with an expression for forced. It looks like this, but we're using a kind of the wrong term here, which is V average squared. It's better, as I've said, to use V. R. M s squared or rather V RMS because that is much better calculation for the average speed off a particle. This is worrying, and this is still has some direction concerns inside it. V RMS has averaged out all of the directions very effectively. So, uh, here's our force. If we divided by area. We come up with the pressure and so pressure in terms of the speed of an individual particle can be measured like this. This is amazing. If we know the Mueller mass of a gas, the number density and the pressure, all things that we can calculate easily we confined the root mean square speed of that individual gas of individual particles within that gas. In fact, if we take the ideal gas law and plug it in here, all we really need is temperature and the Mueller Mass. Because that, combined with the universal gas constant, can give us the root mean square speed. So this is very cool. So we've managed to connect pressure toe how fast particles air moving. Well, now we're going to take energy, which is a little different from that and say, How can we relate energy here? Because that would be very helpful to So we take easy equal to one half MV squared normally. So that's gonna be one half MV r m s squared where m is our Mueller mass and notice here. If we multiply what we had for our P equation for our pressure equation by 2/2. We can actually write in this average energy here solving for it. Then we find that the average energy off a particle in an ideal gas is three halfs multiplied by bolts mons, constant times temperature. So simply taking the temperature of the gas gives us an estimate of the average energy. Now this average energy turns out to be very important, as we'll see later, because it is very much related to the thermal energy and the internal energy off the gas.

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## Video Transcript

welcome to our first section in our unit on the kinetic theory of the idea of the mean free path is that as a particle tries to go through, um, Volume, it is constantly bouncing off of other particles or even the walls that constrain the volume. And because of this, it on Lee goes in a particular direction with a particular velocity for a certain amount of time before it bounces off something else. Elastic Lee and now has a new direction and possibly a new speed as well. So what we want to do is figure out. How long does it maintain a particular speed or velocity? So what we do then is we say I have a molecule and it's trying to travel through some volume, and within that volume there is possibility of running into other particles. That possibility is based on the number density of the particles in the volume. So, for example, here, if we think about particles that it might clip, they could be particles that it overlaps with, or just anything that's within a radius. It will just barely touch as it goes by. Because of this picture, we're able to say that our our mean free path for our particles through a particular number density volume of material is going to be this. You can say we have 1/4 times route two times pi times Here we go, the number density multiplied by R squared where r is the radius, the approximate radius of the gas molecules. So we have a whole lot of a whole lot of constant something that has to do with the gas and then the number density. And that's all there is to it. So conceptually. It's not too hard to understand, though the process in getting here is quite complicated, and so we're going to skip the derivation. But what you know now is that you have a mean free path, which is the average distance between collisions for an individual gas molecule. If we're talking about a monotone, Mick gas were to single atom, we can approximate ours, being 0.5 times 10 to the negative 10 m. If we're talking about a die atomic gas where there's two atoms next student bonded with each other for each molecule, then weaken double that and use this as our radius. Instead, we also need to think about how fast are these particles moving? So for this will use something called Maxwell's be distribution law, because not every particle is traveling at the same speed. In fact, they're all traveling in different with different speeds and in different directions. And this could make deriving anything very difficult. In fact, the true average velocity is very hard to determine. But using this model established by Maxwell. But you can see it's four pi times the Mueller Mass divided by two pi r t all to the three halfs multiplied by V squared times E to the negative MV squared over to Artie. Okay, so that's a lot to do there. So we're not going to run you through all the integration. But it does look like this so you can see there's a large peak, and then there's kind of this tale. This is known as the high speed tale, where you'll have ah, significant number of particles that are going much faster than the average that you would expect. In fact, this isn't even the average, because this shifts the average to the right. Now, when we say average and actually turns out to not be what we mean. Because when we calculate the average speed, remember, speed is going to be equal to the square is equal to V X squared plus V y squared plus the Z squared. And this can cause issues because we're moving in multiple dimensions here. In fact, what we find is that the average velocity is this value right here when we take the correct integral. But it turns out that if we're really looking for the average of how fast a particle is moving within here, we're going to use something called the root mean square velocity root mean square velocity is equal to the average of the square. So we're gonna actually take the square root of all the V squares that have already been average. So in other words, we take a V squared for an individual particle, we find its speed. Then we average all the speeds and we take the square root of all the speeds, and that gives you the root, mean square speed, and this is equal to the square root of three rt over him. This is the speed that we're really going to use a lot because it's the most meaningful one as faras telling us, Ah, majority of particles are going up approximately this fast Now. That being said, there's still something known as VP, which is the most probable speed. That is to say, right here, that is the highest number of particles. And this is like the mode of all the speeds. There are many, many particles traveling at the most probable speed, so it is not necessarily the average speed notice here. It's squared of to rt over m. So this is smaller than the R. M s and then the average is even smaller than VP. So v RMS is actually the highest value here. But it turns out to be the best approximation of, on average, how fast and individual particle is moving now, understanding that we have this distribution of many different speeds and many different velocities, we can begin to talk about what's going on here. So we have this picture in our head of of particles colliding with the walls of the volume and causing some force to be pushing outwards. This force comes from the particle coming in and then bouncing off and having a collision and remember when we talk about collisions, we're talking about changes in momentum. So we're going to say that the force on the outside wall is equal to the change in momentum of these particles divided by the amount of time in which that occurs. So if we take Delta P here to be the change in momentum of every single particle that bounces off the wall during the amount of time Delta T, then we can approximate it as being too multiplied by the average momentum. It has multiplied by the number of collisions happening here. Now, when we take this and plug it into our equation and do the right derivation, we can come up with an expression for forced. It looks like this, but we're using a kind of the wrong term here, which is V average squared. It's better, as I've said, to use V. R. M s squared or rather V RMS because that is much better calculation for the average speed off a particle. This is worrying, and this is still has some direction concerns inside it. V RMS has averaged out all of the directions very effectively. So, uh, here's our force. If we divided by area. We come up with the pressure and so pressure in terms of the speed of an individual particle can be measured like this. This is amazing. If we know the Mueller mass of a gas, the number density and the pressure, all things that we can calculate easily we confined the root mean square speed of that individual gas of individual particles within that gas. In fact, if we take the ideal gas law and plug it in here, all we really need is temperature and the Mueller Mass. Because that, combined with the universal gas constant, can give us the root mean square speed. So this is very cool. So we've managed to connect pressure toe how fast particles air moving. Well, now we're going to take energy, which is a little different from that and say, How can we relate energy here? Because that would be very helpful to So we take easy equal to one half MV squared normally. So that's gonna be one half MV r m s squared where m is our Mueller mass and notice here. If we multiply what we had for our P equation for our pressure equation by 2/2. We can actually write in this average energy here solving for it. Then we find that the average energy off a particle in an ideal gas is three halfs multiplied by bolts mons, constant times temperature. So simply taking the temperature of the gas gives us an estimate of the average energy. Now this average energy turns out to be very important, as we'll see later, because it is very much related to the thermal energy and the internal energy off the gas.

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