 Hi, I'm Stephen Esherbad. I'm here to tell you a little bit about Boltzmann's distribution of the energy that molecules could have given that energies have quantized energy levels. So that comes from quantum mechanics. So just to contextualize it a little bit, we just imagine, say we have some molecules and let's suppose they can vibrate, quantum mechanics tells us that the energy levels that that molecule could have are quantized there. It can't take on any energy level. It's discreet. So I've just kind of laid out a possibility here. So here's energy on the vertical axis, the energy levels on the labeling 1, 2, 3, and so forth. And just to reference those energies, if they were all equally spaced, and I arbitrarily said that the lowest energy, the ground state energy, had an energy of zero, well the first level would have an energy of say epsilon, 2 epsilon, and so forth. So an important idea here is what's called the thermally available energy, and you just imagine that you've got a molecule here and another molecule might come along and hit it, and therefore how much energy, you know, typically, you know, might this molecule that was initially, you know, quiescent, how much energy might it pick up? And the answer, well, you know, from translational, from the equal partition theorem that that's about three-halves kT for translational energy, and we're just going to kind of round that off and say, you know, more or less kT is the amount of thermally available energy. And so an important idea as well now is how many states are thermally accessible. That's given by this symbol Q, okay, a different meaning of the symbol Q. It's called the partition function, and it just counts the number of thermally accessible states. So going back to this diagram, we would imagine something like this, at kT, you know, had a value given by, say, that, then you would say, well, roughly, looks like Q must be equal to about two, because that states thermally accessible. And so is that one, you know, on the other hand, if kT was bigger, then, you know, kT were that big because the temperature were bigger than we would say, oh, looks like there's about one, two, three thermally accessible states. Q would be three, in that case, okay? So that's the partition function. Now, a little wrinkle on that is what's called the degeneracy, and it goes something like this, as long as each energy level, you know, has only one state associated with it, well, that's fine, we just count them like that. But in this case, the degeneracy is two, we would say g equals two here. That's because there are two states with the same energy, okay? So that's the degeneracy, and that kind of equips us, then, to talk about Boltzmann's function that describes the fraction of molecules in any given sample that, you know, that have a certain amount of energy, or put it differently, we'd like to know how many molecules in any given sample have the i-th energy, which I've indicated here as e-civil i. And it goes by this formula, I say, well, this is the fraction we're talking about, the number of molecules in state i divided by the total number of molecules. Well, that's the degeneracy of that level, okay, and times this exponential factor, e to the minus the energy divided by kt of that state, divided by q, because that's sort of, that counts the total number of thermally accessible states, okay? Now, what we often, you know, the way we often use this is actually as a ratio. I would like to know the ratio of populations in state two versus state one, okay? And to do that, I would just take this formula with i here, and then divide it by the same thing, but with j, and when you do that, of course, we'll get ni over nj. That's what that is. Total number of molecules then cancels out. The degeneracies in this exponential terms say, but now the partition function also cancels out. And so, if we were talking about n2 divided by n1, then we would have something like g of state two divided by g of energy level one times the ratio of exponentials. Now, in this particular case, since the energy of the jth level, the first energy level is zero, the net term goes away, and I just have times e to the minus epsilon of the second state, which we've already said is just epsilon, so I'll just say epsilon divided by kt. And so that's how you would calculate that ratio of populations of these two energy levels according to Boltzmann statistics. This would be the prediction that Boltzmann statistics would give you for the ratio of those two populations.