 So, now that we know something about the energy levels for a rigid rotor, one of the interesting things we can talk about is the frequencies or wavelengths of electromagnetic radiation required to excite molecules from one level to another. So remember how light interacts with molecules that occupy these energy levels. If there's some molecule in one of these particular states, then perhaps it can absorb a photon with some frequency or with some wavelength, and if the amount of energy in that photon matches the difference in energy between two of these states, then that molecule can absorb the photon and jump up to the higher energy level. Alternatively, the molecule can fall down to a lower energy level, and in this case it would give off a photon with its own different frequency or wavelength. So now that we know the energies, we can calculate the differences between the states, and those differences tell us something about the types of light that can be either absorbed or emitted by molecules that behave as rigid rotors. So first let's take a look at these energy differences. So we know when we've written the energies either in terms of a rotational constant or sometimes if we prefer a rotational temperature, the energy levels are either 0 or twice the rotational constant or 6 times or 12 times or 20 times and so on. So the gaps between those energy levels, this delta E would be twice the rotational constant, 2B E minus 0. This one would be 6B E minus 2B E is 4. This one, 12 minus 6 is 6. So each of these gaps is different. They get larger as they go up. The size of the gaps goes from 2 to 4 to 6, the next one will be 8 and so on. And in general, so if I want to know, let's say, for the L to L plus 1 transition, the gap between the L energy level and the L plus 1 energy level, that's going to be the energy of the upper level minus the energy of the lower level. We have an expression that tells us how to calculate each of those. So the L plus 1 level has an energy of a rotational constant, not times L and L plus 1, but when I am asking about the L plus 1th level, L has the value of L plus 1 and L plus 1 now is 1 higher, so that's L plus 2. So that's the energy of level L plus 1 and level L is still L times L plus 1. So that simplifies a bit. So both of these terms have a BE and an L plus 1 in them. What's not included out front, I have an L plus 2 minus an L and the L's cancel, leaving me with just a 2 and I can say that the gap, this energy level difference in going from L to L plus 1 is twice the rotational constant times L plus 1. So it's no coincidence that these gaps were going up as even numbers, twice the rotational constant, four times the rotational constant, six times the rotational constant. That pattern will continue going up as even numbers multiplying the rotational constant. So now if we want to know for some reason the gap between the 21st and the 22nd energy level is just going to be twice the rotational constant times 22, the upper level. So that's enough to tell us what the gaps are. We can, let's go ahead and use some actual numbers. So let's ask again about the carbon monoxide molecule for which we know or I'll tell you the rotational constant, 3.82 times 10 to the minus 23rd joules. So we could now calculate, let's go ahead and say for the L equals 0 to the L equals 1 transition, this gap between the levels, so that delta E is going to be twice the rotational constant, which will be twice this number, 7.6 times 10 to the minus 23 joules. So that's a number in joules. More interesting than knowing how many joules, what the energy level difference is, is knowing something about the light required either to induce that transition upwards or the light that's emitted when a molecule falls down from the L equals 1 to the L equals 0 level and a photon gets emitted. So if that energy difference is equal to either H times nu or HC over lambda, we can ask ourselves what would that frequency be? The frequency would be rearranging this expression twice BE over H, so twice this rotational constant divided by Planck's constant. So thinking briefly about units, joules on top from the rotational constant, cancel the joules that are hiding inside of Planck's constant. The units leave a second in the denominator, so the units are going to be 1 over seconds, which is correct for a frequency. And if we do the math, that frequency turns out to be this number, 1.15 times 10 to the 11th per second. So the frequency of a photon used to excite this rotational transition from a non-rotating molecule up to a slightly rotating L equals 1 rigid rotor state would be 1.5 times 10 to the 11th per second. That number perhaps is not terribly intuitive. Let's try instead the wavelength. We might get more information out of knowing what the wavelength of that photon would be. So if I rearrange 2B equal to HC over lambda, I'll get lambda is equal to H times C divided by twice the rotational constant. So again, Planck's constant multiplied by the speed of light, 3 times 10 to the 8th per second, and divide all that by twice the rotational constant. The units in this case, again, I've got joules from Planck's constant, which cancel joules in the rotational constant. Planck's constant has units of second, which cancel 1 over second in the speed of light, and the units I'm left with are just meters, which is correct for a wavelength. Numerically, that answer turns out to be 2.6 times 10 to the minus 3 meters, or if we prefer 2.6 millimeters. So that would be the wavelength of light, the wavelength of a photon needed to be absorbed by a non-rotating carbon monoxide molecule to make it rotate the smallest amount that a quantum mechanical carbon monoxide molecule can rotate. If that number had come out in units of several hundred nanometers, then we could start talking about whether it was in the red portion of the visible spectrum, or the blue or the green portion of the visible spectrum. But that's clearly nowhere near the visible portion of the spectrum. This frequency is also nowhere near the visible portion of the spectrum, of course. So what that means is these photons are not visible photons, they're not photons of visible light, they're photons in a different portion of the spectrum. So to make more sense out of that, we'll spend a little bit of time talking about the different portions of the visible spectrum next. This is of the electromagnetic spectrum.