 So here's a collection of things that we know so far about the energy levels for rigid rotors and for diatomic molecules that we can approximate with the rigid rotor model. We know that there's a sequence of energy levels described by an equation like this where the rotational constant helps us simplify a lot of these constants. And then the difference in energy between any pair of these states can be calculated. The difference between the L and the L plus 1 energy levels can be given by this expression where actually there's a typo in that expression. It's just 2 times BE times L plus 1. So that's the gap between any two energy levels. For typical diatomic molecules, so we could model anything we want using the rigid rotor model if we wanted to. We could take an object like this marker and model its rotational motion with the rigid rotor model if we wanted to. All we need to know is it's reduced mass and the length of the object. But if we use the rigid rotor model to model diatomic molecules based on the typical reduced masses for diatomic molecules and the typical bond lengths and diatomic molecules, we get a range of different energy differences partly due to those different chemical properties of the molecules and partly also due to the fact that each one of these delta E's is a little different than the others. So those transitions for diatomic molecules typically occur at energies that can be spanned by photons with, if we want to talk about their frequencies, it's in the range of roughly 10 to the 9th to 10 to the 11th inverse seconds or hertz. So again, depending on what molecule we're talking about and which gap we're talking about, those can span a pretty large range. The wave numbers of those photons are typically in the range of a 10th to 10th or a few tens of inverse centimeters. So particularly light quantum mechanical molecules like an H2 or an HCl or any molecule involving a hydrogen atom will have a particularly small reduced mass. It's going to have a wave numbers for those transitions that are on the higher end of this range, very heavy molecules, heavy diatomic molecules like maybe a Br2 bromine or iodine molecules, those will have wave numbers of only tenths of an inverse centimeter or so will take to span the differences between these energy levels for those molecules. And since we're frequently going to be talking about those energy differences in terms of the wave number of the photon that's required to generate that much energy, it's convenient if we rewrite some of these equations in terms of wave numbers rather than in units of energy. So for example, if I take this expression, the gap between any two successive states is twice the rotational constant times L plus 1, that's for the L to L plus 1 transition. I know that an energy is related to the wave number as delta E is Hc times the wave number. So if I just divide through by Hc, wave number is equal to an energy difference divided by those two constants, Planck's constant and the speed of light. So on this expression, if I divide by Hc on both sides, then now that equation, instead of being in units of energy on the left and energy on the right, I've divided by Hc, so I've got units of wave numbers on the left and on the right. So I've got units of inverse length like wave numbers are. So anytime we write something in units of wave numbers, just like when frequencies got converted to wave numbers, we put this tilde on top, we do the same thing for many other variables as well. I write this difference in energy, but I don't want to express it in units of energy, I want to express it in units of inverse length, then I can just write that as delta E with a tilde on top. And we know what that means is take the delta E, convert it to units of wave numbers, and we do that by dividing by Hc. And on the right, I can write twice B to B with a tilde on top. And that tells me that I take whatever the rotational constant is that has units of energy in order to convert it to units of wave numbers with the tilde on top, I just divide by Hc. So this is a more compact version of the equation, it's exactly the same as this equation, but this one's written in terms of energies, this one's written in terms of wave numbers. So that suggests that when we're doing spectroscopy, when we're talking about the wavelength or frequency or wave number of light required to cause transitions between energy levels for rotating diatomic molecule, we're not so much interested in the rotational constant as we are in the rotational constant with units of wave numbers. So just take the rotational constant, divide it by Hc. So H squared over 8 pi squared mu R squared with an extra Hc in the denominator. So an even more complicated collection of constants is what our rotational constant is when we want it in units of wave numbers. So let's work one quick example to see how that works. If we take the carbon monoxide molecule, for which I'll tell you we already know, the rotational constant is in units of energy, 3.82 times 10 to the minus 23rd joules. I'd like to know what is the rotational constant in units of wave numbers. So just divide the rotational constant by Planck's constant, joule seconds, also in the denominator, divide by the speed of light, 3 times 10 to the 8th meters per second. So let's think quickly about what those units are going to work out to be. In this fraction, I've got joules on top and joules on bottom. In the denominator, I've got a second and I've got a one over seconds that cancel. And what I'm left with is just nothing in the numerator and meters in the denominator. So the answer is going to come out in units of one over meters. And what it works out to be is 192 inverse meters. As we've seen, we tend to like wave numbers in units of inverse centimeters rather than inverse meters. So if I divide by 100 to convert meters into centimeters, that's going to work out to be 1.92 inverse centimeters. So the rotational constant in units of wave numbers is 1.92 inverse centimeters. What does that mean? That means that if we don't want to think about the energy ladder, if instead we want to think about the frequencies or wave numbers of light you needed to excite these transitions, if I just stick a tilde on top of each of these quantities, the gaps between these energy levels are either two times the rotational constant in units of inverse centimeters or four or six or eight times that quantity. So if I pull up a picture here of what the spectrum absorption spectrum is for carbon monoxide in the microwave portion of the spectrum, then what we see is that carbon monoxide, so here's some data for carbon monoxide. Each one of the lines on the spectrum is at a particular frequency given in units of wave numbers and if there's a line it shows that the molecule absorbs light at that frequency or at that wave number. So what we'd expect from this calculation is that carbon monoxide should absorb light with energy of two times the rotational constant. So twice 1.9 would be 3.8. That's going to be down here somewhere, or four times or six times or eight times and so on. So let's see, this line is at about 7.6 inverse centimeters, so 1.9 times 4. This line, a little bit larger, comes at, so this one was four times the rotational constant, this one is at six times the rotational constant. I've also got a line here at 8 times, 10 times, 12 times, 14 times and so on. So I've got lines corresponding to many different transitions more than I've drawn on this diagram here. In fact, we can tell from the heights of these curves what the height tells us is how many molecules are sitting around waiting to absorb light at a certain frequency, so the peak with the highest intensity. So 2B, 4B, 6B would correspond to the transitions beginning at L equals 0, L equals 1, L equals 2. We can counter away 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13. This would correspond, if I haven't miscounted, to the L equals 13 going to L equals 14 transition. And because that peak is taller than any of the others, it tells us that under the conditions that the spectrum was taken, the L equals 13 energy level had a lot of molecules sitting at that energy level, sitting around waiting to absorb a photon. So that's why it absorbed so much light at this frequency, more than the energy levels with lower quantum numbers, because there aren't nearly as large a degeneracy down there, also more than the ones at much higher quantum numbers, because those are so high up that they're not very well populated. So this curve emphasizes what we've already seen about the intermediate levels being most populated. So this tells us several things. Number one, we've seen that rotational transition for these diatomic molecules tend to be in the microwave portion of the spectrum, a few tens to a few tens of wave numbers, depending on the molecule we're talking about. We can calculate those quantities quantitatively based on what we know about the energy levels and what we know about these rotational constants, and use that to predict or to understand the spectrum of frequencies of light that these molecules absorb in the microwave portion of the spectrum.