Boltzmann's constant

let's talk about Boltzmann's constant it's named after first of all this guy Ludwig Boltzmann who was a genius he lived in the late 1800s and early 1900s and he was the father of modern atomic theory one of the big proponents early proponents that the world is made out of atoms and molecules the sounds obvious to us now but a hundred and twenty years ago it was definitely not obvious and some of the smartest people of his day vehemently disagreed with Boltzmann and Boltzmann had to defend these ideas over and over and what I mean by atomic theory is this if you had a container of say anything could be a cube of metal let's just say it's a gas let's say it's a container it's full of air well it feels like the air is continuous in here or like the gold if this was a cube of gold the gold is continuous but we know now and Boltzmann knew that it's really made out of atoms and molecules that wasn't obvious 120 years ago because you can't see the atoms and molecules if this was a container of steam let's say and you stuck your hand in here so I took my hand I put my hand in this container of steam I'd noticed that I know something was going on my hand would start to feel hot there's energy being transferred here but it wasn't obvious what exactly is the mechanism is this a new kind of energy is this one of our old kind of energies just in disguise Boltzmann's big claim and groundbreaking idea was that this gas if it's steam let's say is really made out of atoms and molecules these gas molecules are running around in here there's just little particles in here and what you're actually feeling are these particles striking your hands so your hands just getting bombarded by these particles but they're so small and there's so many of them you can't really tell that there's particles it just looks completely continuous so for Boltzmann this heat energy isn't really a new kind of energy at all all this is this heat energy that you're feeling is just kinetic energy and if it's steam it's just the kinetic energy of the h2o molecules flying around and here at some rapid speed and the faster they go the greater the impact with your hand which is going to transfer more energy so the faster they go the hotter it feels in here so for Boltzmann to say that something has a high temperature if you said that the temperature is large if it's hot outside that's kind of redundant we already had a word for that we could just say if it's a high temperature what we really mean is that the average kinetic energy of the gas molecules outside is large so if a gas as a high temperature the average kinetic energy of those molecules is large that's why it hurts when they impact on your skin because they're transferring kinetic energy to the molecules in your hand and when your hand absorbs too much energy these molecules move around your skin starts to get damaged you can get burned so this is often referred to as the kinetic molecular explanation of temperature and the details of this theory were one of Ludvig Boltzmann's biggest contributions to science but what does any of this have to do with Boltzmann's constant well doesn't get rid of all of this you've probably heard of the ideal gas law PV equals NRT so remember T is the temperature measured in Kelvin P is the pressure and I'm going to measure this pressure and I'm going to choose to measure it in Pascal's v is the volume I'm going to choose to measure it in meters cubed and n little n remember little n is the number of moles of the gas and if you forgot what moles are and the number of moles is defined to be capital n the number of molecules in the gas the total number of molecules in the gas divided by a constant and that constant is called Avogadro's number and if you forgot Avogadro's number Avogadro's number is 6.02 times 10 to the 23rd and there's that many molecules per mole so in every mole of a gas what we mean by one mole of a gas is 6.02 times 10 to the 23rd molecules and if you choose these units this are this gas constant R is called the gas constant and it has a value or has a value of 8.31 joules per mole Kelvin that's the gas constant R these units but these are pretty macroscopic quantities pressure and volume and temperature and moles even moles talking about one or two moles is talking about a huge number of molecules you're kind of glossing over some of the microscopic details so an alternate way to write the ideal gas law is P times V equals capital n so forget moles let's say we want to talk about how many molecules there are instead of writing little n let's write big n number of molecules we need a different constant because we're going to multiply by the same T so again this T is still temperature in Kelvin P is still the pressure and Pascal's V is the volume again in meters cubed and instead of being the number of moles is now the number of molecules and that means we need a new constant here we need a different constant that constants got to be really really small the rest of this stuff's the same P times V and T are all the same and all I did is I swapped out little n number of moles for big n number of molecules so there's gonna be a huge number we're plugging in in this spot now instead of plugging in like say too hard to plug in 2 moles right here the number 2 down here I'd plug in 2 times this so I plug in 12 point O 4 times 10 to the 23rd since this is a huge number I need a constant that's really small because it's got to balance out we know that n times R has got to be the same as capital n times this constant because the rest of this is the same this left-hand side is the same and the T is the same so if this is all consistent then n times R is got to be equal to n times this new constant and that new constant is Boltzmann's constant it's a lowercase K with a bee on it to denote Boltzmann's constant so what's the value of Boltzmann's constant we can find it pretty easily we know that little n times R has got to equal big n times Boltzmann's constant so if we just solve this for Boltzmann's constant we're going to get little n over big n times R but what's little n over big n just look up here we can figure it out little ant big n if I solve this for little n over big n what I'm going to get is if I divide both sides by big n I get 1 over Avogadro's number little n over big n right here is Avogadro's number or 1 over Avogadro's number so I get that one over Avogadro's number times the gas constant this 8.31 is Boltzmann's constant if you multiply that out the gas constant which is eight point three one joules per mole Kelvin and divided by Avogadro's number which is 6.02 times 10 to the 23rd molecules per mole you'll get Boltzmann's constant which equals one point three eight times ten to the negative twenty-third joules per Kelvin this is Boltzmann's constant this number right here is Boltzmann's constant why do we care about Boltzmann's constant well it allows us to write a more microscopically oriented version of the ideal gas law that focuses on number of molecules instead of number of moles and this number pops up all over statistical and thermal mechanics it's one of the most important constants and all of thermal physics in fact it was so important that on Boltzmann's own gravestone if you go to Boltzmann's grave there's a bust and a gravestone it doesn't actually look like a cross but there's a grave there with a big inscription and there's an inscription the big inscription is an equation s equals Boltzmann's constant times log W this was possibly his most important contribution and it says that the entropy of a system is equal to this case Boltzmann's constant we just talked about that log it says L og but nowadays we use Ln because really they meant to the natural logarithm here and nowadays it's conventional that L og is log base 10 but this equation is really referring to the natural log in WWE and of mysterious like entropy is w is the number of microstates so if you had a macroscopic system and you wanted to know microscopically what are all the ways I can arrange my particles with given speed and distributions and positions such that it looks identical the macroscopic state for someone standing out here they would look at this thing and they'd be like that's that's the exact same state but the particles are doing something different in here it's just on a macroscopic level identical how many ways are there to do that and still make the macroscopic view identical for this person out here that's what this is measuring the number of microstates and if you take Boltzmann's constant times the natural log of that number it gives you an idea of the entropy entropy is very mysterious and interesting has to do with the disorder or available energy in a system I don't have enough time to describe it right now but if you have time you should look into this this is mysterious and confusing and wonderful at the same time