 Alright, so let's treat an example of a particle in a box with some quantitative calculations. And in fact, we're actually going to compare two different examples of particles in boxes. We've seen that we can use the particle in a box molecule, sorry particle in a box model to describe both electrons confined to a molecule, a conjugated section of a molecule. An example of that would be something like, I'll draw three fused benzene rings. That's an anthracene molecule. If we measure the frequencies of light absorbed by anthracene, anthracene absorbs light at around 375 nanometers. So that's the photon of light that has a wavelength of 375 nanometers. That value is at least approximately 10 to the minus 19th joules. That's the energy of a photon absorbed by an anthracene molecule. Remember that that corresponds to there's a pair of energy levels separated by this much energy. And then when I shine a photon with the same amount of energy, it can excite the electron from the lower energy level up to the upper energy level. So a photon with 375 nanometers has this much energy. It's enough to excite the electron between a pair of states that differs by exactly that much energy. And those correspond to states that we can predict reasonably well with the two-dimensional particle in a box model. Mention the energy just as a way of pointing out that this value compared to kT. Remember the value of kT at room temperature is about 4 times 10 to the minus 21 joules. So the 10 to the minus 19th joules that represents a photon on the red end of the visible spectrum is quite a bit, I'm sorry, that's the wrong direction, 10 to the minus 19th is quite a bit larger, several orders of magnitude larger than kT, which has a magnitude of about 10 to the minus 21 joules. So this energy level spacing between levels for the anthracene molecule is much larger than kT. We immediately know by comparing those energy differences to kT, several things. For example, we know that because it's so many kT, 100 or so kT to get up to this state, the upper state is not going to be very populated. Most of the electrons in this molecule, an overwhelming majority of the electrons in this molecule will be on the lower energy levels. We also know, for example, if we were interested in the partition function because the only states that are populated, the states low compared to kT, that the partition function is going to be relatively small for this molecule. Those factors both contrast with the other example we've talked about using the 3D particle in a box model for. If the particle is not an electron but an actual molecule, so let's take the example of a molecule like a molecule of nitrogen gas. Instead of confining an electron inside of a molecule, it's confined an electron inside of a physical box. So nitrogen as a gas will stay confined to a box that you place it in. And we can take, as an example, maybe the room that I'm standing in or perhaps the room that you're sitting in. Just as a rough estimate, let's say we're in a room, a cubic room whose length, width, and height are all about 5 meters, roughly 15 feet on a side. So if I have a gas molecule, one of the nitrogen gas molecules in the gas in the air that I'm in, confined to a room that's about 15 feet on a side, we can use the 3D particle in a box model to describe the energy levels of that system. So we can ask similar questions, what's the difference in energy levels for that system? To do that, we need to know for a cubic box, the energy levels look like h squared over 8ma squared plus the sum of nx squared and y squared and nz squared. So in order to calculate these energies, I need to know, I certainly know what Planck's constant is. I've assumed a value for the size of the box, the mass we need here. Remember that we're not talking about an electron anymore, we're talking about a nitrogen molecule, so we would need the mass, the molar mass of a nitrogen molecule, so we know that value. And we also need these n values. So if we go to perform this calculation, Planck's constant, 6.6 times 10 to the minus 34 joule seconds, that gets squared in the numerator. I've got an 8 in the denominator, nitrogen has mass 14, so n2 has mass 28, that's in units of grams per mole. Stop and ask if grams are the units that we want to have our mass in, grams per mole. And it's not, we want the grams here to cancel one of the kilograms that's hiding inside these units of joules. So we need to convert grams to kilograms. And also we need to get rid of this factor of moles in the denominator of the denominator. So Avogadro's number of molecules in a mole. So far I've managed to cancel moles and grams, so that takes care of the mass still in the denominator. We're going to use 5 meters for our box size. That number gets squared, and then we're going to multiply by this sum of n's. And I'll actually leave that unsubstituted for now, and I'll just say that regardless of whether we want, let's say I draw an energy ladder. Depending on whether I want to talk about that excitation or that excitation or this excitation, I'll use different values for the n's, but this number is going to be, if I'm going from the 1, 1, 1 to the 1, 1, 2 energy level, then I need a difference of 3. Or the 1, 1, 1 to the 1, 1, 3 energy level, I need a difference of 8. No, sorry, 10. But regardless, this number is going to be a relatively small integer. So the exact value is going to depend on which excitation level I'm interested in. But let's just calculate this first prefactor term first. And for that, I need a calculator, or I need to have done the calculation ahead of time. And what I find is that this number works out to be about 10 to the minus 42 joules. So that's the quantity for which I have numbers. And then again, without plugging in specific values for the n, I'll just say I'm multiplying by a relatively small number. I'm multiplying by a few. The important thing about this value, 10 to the minus 42 joules, and let me just point out that the units do in fact work. I've got kilogram meters squared in the denominator, second squared in the numerator. A kilogram meters squared per second squared is a joule, so that cancels one of my joules in the numerator, leaving me with just one joule. So those energies of 10 to the minus 42 joules, that's roughly speaking, the difference in energy between these states is a small number times 10 to the minus 42 joules. Compared to KT, remember KT is 10 to the minus 21 joules. So there's not just two orders of magnitude difference. There's 20 or more orders of magnitude difference between the energy level spacing for the nitrogen molecule in a room-sized box. And the value of KT. So in particular, the inequality works the other way around. This energy level spacing is considerably less than KT. In fact, saying just much less than is an understatement. 20 orders of magnitude is so small as to be negligible. So in fact, the energy level spacing for molecules confined to a box, that energy gap between levels, 10 minus 42 joules, is so small that we can't possibly measure it with any technology that we currently have. It's so many orders of magnitude below energies that we can measure. So those energy levels essentially are super closely packed together. So closely packed together, we can find an energy level almost at any energy we wish. So it turns out that significant difference, the fact that electrons confined to molecules have energy differences that are quite large compared to KT. Whereas molecules confined to boxes have energy spacings that are incredibly small compared to KT. That's going to turn out to be a very important difference that arises from two different factors. The masses are different and the box sizes are different. We've changed two numbers when we went from electrons confined to conjugated hydrocarbons to molecules in boxes. A nitrogen molecule is heavier than an electron by a couple thousand folds, so that's part of the answer. But an even bigger factor is the fact that I went from a box that was only less than a nanometer on a side when I'm confining an electron to a molecule to boxes that are meters on a side. So that's roughly 9 or 10 orders of magnitude. When I square that number, it becomes enormous. So that's the biggest reason why these energy level spacings are so tiny is because I've confined this molecule to a large box. It's not much of a confinement at all to confine a nitrogen molecule to a whole room, so the energy levels are not very discretized. So as we'll see coming up relatively soon, this very, very small spacing between the energy levels for a gas molecule in an actual macroscopic sized box is going to be convenient. It's going to allow us to calculate several properties of gases by using those energy level spacings as well as some statistical mechanics. So that's the next step.