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The Maxwell–Boltzmann distribution

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Video transcript

so let's think a little bit about the maxwell-boltzmann distribution and this right over here this is a picture of James Clerk Maxwell and I really like this picture it's with his wife Catherine Maxwell and I guess this is this is their dog and James Maxwell he is a Titan of physics famous for Maxwell's equations he also did some of the foundational work on color photography and he was involved and thinking about well what's the distribution of speeds of of air particles of idealized gas particles and this gentleman over here this is Ludwig Boltzmann and he's considered the father or one of the one of the founding fathers of statistical mechanics and together through the molt Maxwell Boltzmann distribution they didn't collaborate but they independently came to the same distribution they were able to describe well what's the distribution of the of the speeds of air particles so let's back up a little bit or let's just do a little bit of a thought experiment so let's say that I have a container here let's say that I have a container here and let's say it has air and air is actually made up mostly of nitrogen let's just change this has only nitrogen in it just to simplify things so let me just draw some nitrogen molecules in there and let's say that I have a thermometer I put a thermometer in there and the thermometer the thermometer it reads a temperature of 300 Kelvin what is this temperature of 300 Kelvin mean well in our everyday life we have a kind of a visceral sense of temperature hey I don't want to touch something that's hot it's going to burn me or that cold thing you can do it's going to make me shiver and that's how our brain processes this thing called temperature but what's actually going on at a molecular scale well temperature one way to think about temperature this would be a very accurate way to think about temperature is that tempera I'm spelling it wrong temperature temperature is proportional to average kinetic energy of of the molecules in that system so let me write it this way temperature is proportional to average kinetic energy average kinetic kinetic energy energy in the system energy I'll just write average kinetic energy so let's let's make that a little bit more concrete so let's say that I have two two containers so it's one container whoops and two containers right over here and let's say they have the same number of molecules of nitrogen gas and I'm just gonna draw ten here I this obviously is not realistic you'd have many many more molecules 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 and let's say we know that the temperature here is 300 Kelvin and the temperature so the temperature of the system is 300 Kelvin and the temperature of this system is 200 Kelvin so if I wanted to to visualize what these molecules are doing they're all moving around they're bumping they don't they all move together in unison the average kinetic energy of the molecules in this system is going to be higher and so maybe you have this molecule is moving into that direction so that's its velocity this one has this velocity this one's going there this one might not be moving much at all this one might be going really fast that way this one went going super fast that way this is doing that this is doing that this is doing that so if you were to now compare it to this system this system you could still have a you could still have a molecule that is going really fast maybe this molecule is going faster than any of the molecules over here but on average the molecules here have a lower kinetic energy so this one maybe is doing this I'm going to see if I can draw on average they're going to have a lower kinetic energy that doesn't mean all of these molecules earnestly slower than all of these molecules or have lower kinetic energy than all of these molecules but on average on average they are going to have less kinetic energy and we can actually draw a distribution and this distribution that is the Maxwell Boltzmann distribution so if we let me draw a little coordinate plane here so let me let me draw a coordinate plane so I've on this axis I were to put speed if I were to put speed and on this axis I would put number of molecules number of number of molecules right over here for this system the system that is at 300 Kelvin the distribution might look like this so it might look the distribution let me just in a new color so the distribution this is going to be all of the molecules the distribution might look might look like this might look like this and this would actually be the Maxwell Boltzmann distribution for this system for system let's call this system a system a right over here and this system that has a lower that has a lower temperature which means it also has a lower kinetic energy the distribution of its particles so the most likely the most probable though you're going to have the highest number of molecules at a slower speed let's say you're going to have it at the speed right over here so its distribution its distribution might look something like this now so it might look something like that now why is this one it might make sense to that okay the most probable the the speed at which I have the most molecules I get that that's going to be lower than the speed at which I have the most molecules in system a because I have because these on average these things have less kinetic energy they're going to have less speed but why is this peak higher well you remember we're talking about the same number of molecules so if you have the same number of molecules that means that the areas under these curve need to be the same so if this one is narrower it's going to be taller and if I were going to if I were to somehow raise the temperature of the system even more let's say I create a third system or or I get this all right let's say we're to heat it up to 400 Kelvin well then then it would look then my distribution would look something would look something like this so this is if I if I heated it up he did heat it up and so this is all the Maxwell Boltzmann distribution is I'm not giving you the the more involved hairy equation for it but really the idea of what it is it's a pretty neat idea and actually when you actually think about the actual speeds of some of these particles even in the air around you the air around you might say oh it looks it looks pretty stationary to me but it turns out and the air around you is mostly nitrogen that the most the most probable speed of a if you picked a random nitrogen molecule around you right now so the most probable speed I'm going to write this down because this is pretty mind-blowing most probable speed at room temperature probable prabha probable speed speed of n2 at room temperature room temperature temperature so let's say that this was the Maxwell Boltzmann distribution for for nitrogen at room temperature let's say that that's let's say we make we call room temperature 300 Kelvin this most probable speed right over here the one where we have the most molecules the one we're going to have the most molecules at that speed i indictment guess what that is going to be before I tell you because it's actually mind-boggling well it turns out that it is approximately four hundred four hundred and Xu at 300 Kelvin is going to be 422 meters per second 422 meters per second imagine something traveling 422 meters in a second and if you're used to thinking in terms of miles per hour this is approximately 944 miles miles per hour so right now around you you have actually your the the moat the most probable the highest number of the nitrogen molecules around you are traveling at roughly this speed and they're they're bumping into you that's actual it's giving you air pressure and not just SB they're actually ones that are traveling even faster than that even faster than 422 meters per second even faster there's particles around you traveling faster than a thousand miles per hour and they are bumping into your body as we speak and you might say well why doesn't that hurt well that gives you a sense of how how small the mass of a nitrogen molecule is that it can bump at you bump into you 2,000 miles per hour and you really don't feel it it feels just like the MB air pressure now when you first look at this you're like wait a week 422 meters per second that's faster than the speed of sound the speed of sound is around 340 meters per second how can this be well just think about it sound is transmitted through the air through collisions of particles so the particles themselves have to be moving or at least some of them have to be moving faster than the speed of sound so not all of the things around you are moving this fast and they're moving in all different direction some of them are might might not be moving much at all but some are moving quite incredibly fast so I don't know I find that a little bit mind-blowing