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Video transcript

in other videos we touched on the notion of kinetic molecular theory which i'll just shorten as kmt and it's just this idea that if you imagine a container i'll just draw it in two dimensions here that it contains some gas you can imagine the gas as being these particles where their collective volume is much smaller than the volume of the container and the temperature we're dealing with is related to the average kinetic energy of the particles these particles are all moving around zooming around and they have some they each would have some kinetic energy remember kinetic energy you calculate that as m v squared over two so each of these particles would have some mass and some velocity but they could all have different velocities for sure even if they're the same type of particle and if they're different types of particles they can have different masses as well but the average of these kinetic energies across all of these particles that is proportional to temperature when measured in kelvin and pressure the pressure remember pressure is nothing but force per unit area and so you can imagine this surface of our container this could be some type of a cube so i can draw it in three dimensions here so there's some area over here and you have your particles let me do this in a different color these particles are constantly bouncing off of it and there's way more particles than what i have drawn here so at any given moment you're having some particles that are bounced off bouncing off of this side of the container actually all sides of the container and these are perfectly elastic collisions they're preserving kinetic energy and so they're applying some force collectively on this area so the pressure is because of these particle collisions on the surface now what i want to do in this video is take these ideas that we conceptualize in kinetic molecular theory and to understand why the ideal gas law pv is equal to n rt makes sense when we conceptualize the world here just a reminder p is pressure v is volume n is the number of moles of whatever gas we're dealing with the amount of that gas and then t is the temperature in kelvin and r is just the ideal gas constant that's just whatever constant you're doing so that the units all work out together so let's first think about how pressure relates to volume if we were to hold everything else constant well the ideal gas law tells us that pressure times volume is going to be equal to this if we hold it constant i could even just write a k here for a constant but that would also mean we could divide let's say both sides by v we could say that pressure is equal to some constant over v another way to think about it is is that pressure is proportional to the inverse of volume you could also write this if we divide both sides by p is that volume is proportional to the inverse of pressure does that make sense from a kinetic molecular theory point of view pause this video and think about it well imagine we have our original cube right over here and i have the same number of particles they have the same average kinetic energy but let's say i were to increase the volume so if i were to make the volume go up so i would somehow expand this or maybe put the exact number the same particles with the same temperature and a larger container and then in any given moment you're just going to have fewer bounces of particles off of the container because they just have more room to go in that volume and even the surface area of the container is going to be high as well so it makes sense that if the volume goes up the pressure is going to go down and you could think about it the other way if you make this smaller that same number of particles with the same average kinetic energy there's going to bump into the container that much more often and that's going to increase the pressure so volume goes down pressure goes up and this relationship that pressure is inversely proportional to volume or vice versa if you hold everything else constant that's often known as boyle's law now another relationship what if we were to hold volume and the number of moles constant and we want to think about the relationship between pressure and temperature well if this is constant this constant this is constant the ideal gas law would say that pressure is going to be proportional to temperature or that temperature is proportional to pressure does that make sense well let's go back to our original container if you were to increase the temperature that means that the average kinetic energy is increased that means that these particles when they hit the side of the container they're going to hit it with more velocity that means that they're going to have at any given moment you're going to have more pressure exerted on the sides of the container and you could go the other way think about lowering the temperature so the kinetic energy goes really low then these particles are just slowly drifting and the speed with which they are hitting the side of the container is going to go down and so the pressure would go down so it completely makes sense if temperature goes up pressure goes up if temperature goes down pressure goes down and this is known often known as guy lusak's law now another relationship and i'm really just going through all of the combinations over here what if we were to hold pressure on the number of molecules constant so we're really looking at the relationship between volume and temperature so once again if p n and r is always constant if those are constant the ideal gas law would tell us that the volume is proportional to the temperature once again holding everything else constant well to think about that you can go through that same thought experiment we just had we want to if we increase the temperature if these things are moving around faster if you want to have the same amount of force per area on the container on the side of the container you're going to have to increase the volume so this relationship which is completely consistent with kinetic molecular theory is often known as charles's law now another one is the relationship between volume and the number of moles if everything else is held constant the ideal gas law would tell us that volume is going to be proportional to the number of moles of our particle or of our gas that we are dealing with and that makes sense because once again you're holding everything else constant you want pressure to be constant temperature to be constant if i were to double the number of particles here but i don't want to change the pressure of the temperature it makes sense that i would have to double the volume likewise if i wanted to double the volume here and i didn't want to change the pressure of the temperature i would have to put twice as many particles in there so i still have sufficient number of interactions of bouncing of the particles with the sides of the container so that i have the sufficient pressure and this notion is called avogadro's law last but not least let's say i have two identical containers if i have two identical containers that's one there that's one over here and actually i'm going to draw that same container a third time and let's say over here i have gas 1 and it has some partial or in this case it has some pressure due to gas 1. we're going to assume the volume and the temperatures are the same across all three of these and let's say we have gas 2 and it is exerting pressure to if i were to take all of the gas in both of them and put them both into this third container so this third container is going to have all of the original of gas 1 and all of the original of gas 2 but we aren't changing the volume and we aren't changing the temperature in any given unit area on the surface of the container you're going to get the collisions from particle 1 which would give you p1 in that that force per unit area and you're going to get the collisions from particle 2 which would give you that force per unit area so it makes sense that the partial pressures would add up to be equal to the total pressure on the container and this is known as dalton's law but the whole point of this video is just to appreciate that everything we've talked about with the ideal gas law actually makes a lot of sense i would argue it makes the most sense when you think about it in terms of kinetic molecular theory
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