 Hi, well, I'm Stephen Nesheva, and I'm here with part two of this couple of lectures. And it's still about Boltzmann's distribution when we're dealing with quantum mechanical molecules, molecules under which quantum mechanics is valid, which means that there are discrete energy levels. And the idea is this, that we are told that we're at a certain temperature, therefore kT has a certain value, and we would just like to know what is the total energy of the system, and then we'd also like to know, given that we are told that there's a certain fraction of molecules in that state versus that state, does it conform to the predictions of Boltzmann's theory? So in this case, we're told that kT has a value of 0.417 epsilon. Now, according to my diagram here, that's not even quite half of one of these steps here, so there's what kT is, and therefore we would guess that the partition function, Q, was about, you know, somewhere around 1. It looks like most of the molecules will be in this state, a few will be in that one, and even fewer upstairs. And so, and now, then that looks sort of more or less consistent with this picture here, so I've given an occupation set here, I've said the number of molecules in that ground state there is 121, and then in the second state we have 11, and in the third state we have 1. And so, well, one thing that's good to do just to kind of get you warmed up on this is to say, well, what's the total energy of the system? Well, I just count the number of molecules that are in the ground state multiplied by its energy, which is 0. Then there are now 11 molecules in the first state, which has energy epsilon, and add that to one molecule, which is in the second state, which has an energy of 2 epsilon. And if you add those all up, it looks like you get 13 epsilon. So that's the total energy. What about the ratio of populations? Well, according to this given configuration, the ratio of populations in state 2 versus state 1, I would just say that must be 11 divided by 121, which if you calculate it, it turns out to be 0.919. Okay, that's fine. Now, so the final question is, does that conform to Boltzmann statistics? Well, in order to answer that question, we have to say, what does Boltzmann predict? Okay, and for that one, we use the ratio of populations according to Boltzmann, and that would be P2 divided by P1, okay. I've already drawn in here the degeneracies of the first two energy levels. There's just one state with that energy and one there, so G is 1 for both of those, like for 1 over 1. This is the energy of the state 2 divided by KT and state 2, as you can see, has an energy of epsilon. State 1 has an energy of 0. So really, we just have a ratio of 2, you know, we really end up with just that exponential factor because that denominator is 1, and if you go through that calculation, you get the same number, 0.0909, and from which we conclude, yeah, indeed, the given configuration of populations is consistent with the predictions of Boltzmann's theory. Now, if there had been some other distribution, then you would have to go through this whole calculation again, okay.