 We saw in the last part that the wavelength of a microwave is macroscopic, we can actually see the wavelength, but they're all nice, they're that big-ish. We also saw a little bit about the fundamental relationships between wavelength and frequency and energy. But what about the physics of microwaves? To cut to the punchline, microwave radiation induces molecular rotations in anything that has a permanent dipole. That is any molecule where the electrons are being pulled asymmetrically by electronegativity. Now, these rotations can be split over three axes, one, two and three, and they're also quantized. They have quantized energy levels just as with molecular orbitals and atomic orbitals. Now, this sounds truly bizarre, after all, you would expect these things to be able to rotate quite freely, however they like, in any direction, at any energy, and locking them up into quantized energy levels also feels a little bit bizarre. How does this become quantized, for instance? But we were to break it down to the simplest models, the diatomic molecules. I think we need to make a lot more sense. While the details of this are best left to a dedicated quantum mechanics course, we start with confining a particle to a one-dimensional box, like a wire or a string. This model tells us where quantization of energy comes from, only certain ways fit in a box, and therefore only certain energies do. We can then expand this model by adding new dimensions to show a 2D or 3D space. We can add a potential energy well, more on that one later, or we can fiddle with the constraints a little. Instead of locking off the box at both ends, we can connect both ends to form a ring. This model is often used to develop quantized angular momentum in electrons, but it can really work for any particle that can be said to orbit another. That includes atoms, and some molecular rotations are also quantized. To help justify this, we're going to dig into our physics toolbox and pick out the reduced mass. For particles with a big difference in their size, you can see that the center of rotation lies very close to the center of the largest particle. We can model this system as if it's just a smaller particle rotating around a fixed point. For particles of a more similar size, the reduced mass tends towards a value of about half their average mass at the same distance. But it's this orbiting around a fixed point that we can model as a particle constrained to a ring. In a quantum mechanical world, that can be modeled as a wave. It doesn't matter in the case of molecular rotations that this is an atom instead of an electron. It's still enough to be a quantum system. It's small enough to have quantized energy levels, a quantized wave form going around it. And it's this rotation, this sort of energy level that's manipulated by energy in the microwave region. To see what we can now find out from a rotational spectrum, we want to start with a bond length r and then the reduced mass mu. We then combine those together to produce a quantity known as the moment of inertia. Inertia being a measure of how an object can resist changes to its motion, and a moment of inertia being the rotational version. Do be sure you can follow the units and dimension analysis of this formula. It's essential to work in meters and kilograms rather than atomic units. At this point, I just want to add that I'm being deliberately vague with where these formulas come from. There's a deeper dive into the mathematics behind them in one of the pre-lab lecture videos if you're interested. This moment of inertia, I, also pops up in the solution to the Schrodinger equation for the energy of a system just like this. This equation has all the usual suspects in it, Planck's constant for relating frequency to energy, and pi because we're dealing with rotations. The J in this case refers to the quantum number of the energy level. For rotations, this begins at zero. We can also reformulate it in a few different ways such as using the reduced Planck's constant or adding in the speed of light to convert the energy to wave numbers. For any particular molecule, most of these things are constant, so we can wrap them all up in something called the rotational constant, B. Because B is constant, the rotational structure of every heteronuclear diatomic molecule will be very similar just with varying energies depending on the constant B. And because B represents an energy, we can read it directly off the rotational spectrum, which we'll see in the next video.