Boltzmann's constant

- [Instructor] 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. This sounds obvious to us now, but 120 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 and 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 'cause 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 notice it, I'd 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 hand, so your hand's just getting bombarded by these particles. But they're so small and they're 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 in the H2O molecules flying around in here at some rapid speed. And the faster they go, the greater the impact with your hand, which is gonna 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 has a high temperature, the average kinetic energy of those molecules is large. That's why it hurts when they impact on your skin 'cause 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 Ludwig Boltzmann's biggest contributions to science. But what does any of this have to do with Boltzmann's constant? Well, let's 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 gonna measure this pressure in, I'm gonna choose to measure it in Pascals. V is the volume, I'm gonna 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've forgotten what moles are, n, 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's called Avogadro's number. And if you've forgotten 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 R, this gas constant, R is called the gas constant, and it has a value, R has a value of 8.31 Joules per mole Kelvin. That's the gas constant R with 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 kinda 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'd need a different constant 'cause we're gonna multiply by the same T. So again, this T is still temperature in Kelvin. P is still the pressure in Pascals. V is the volume, again, in meters cubed. N, 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, and that constant's gotta 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 was I swapped out little n, number of moles, for big N, number of molecules, so this is gonna be a huge number we're plugging in into this spot now. Instead of plugging in like, say, two, if I were to plug in two moles right here, the number two, down here, I'd plug in two times this. So I'd plug in 12.04 times 10 to the 23rd. Since this is a huge number, I need a constant that's really small because it's gotta balance out. We know that n times R has gotta be the same as capital N times this constant because the rest of this is the same. This left hand side's the same and the T's the same. So if this is all consistent and n times R has gotta be equal to N times this new constant, and that new constant is Boltzmann's constant. It's a lowercase k with a B 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 gotta equal big N times Boltzmann's constant, so if we just solve this for Boltzmann's constant, we're gonna 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 n over big N, if I solve this for little n over big N, what I'm gonna get is, if I divide both sides by big N, I get one over Avogadro's number. Little n over big N right here is Avogadro's number, or one 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 8.31 Joules per mole Kelvin, and divide by Avogadro's number, which is 6.02 times 10 to the 23rd molecules per mole, you'll get Boltzmann's constant, which equals 1.38 times 10 to the negative 23rd 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 in 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 k's Boltzmann's constant, we just talked about that. Log, it says l-o-g, but nowadays we use ln because really, they meant the natural logarithm here and nowadays it's conventional that l-o-g is log base 10 but this equation is really referring to the natural log. And W, W's kinda 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 speeds 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 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's 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.