 All right, we've accomplished the goal of being able to write down, being able to extend the particle in a box problem from one dimension into three dimensions. So we've got an expression for the wave function for the particle in a box, including a normalization constant. We can also write down what the energies are for a particle confined to this three dimensional box. So we might ask ourselves, what real world problems could this be applicable to? Just as we did for the one dimensional particle in a box, it turned out linear conjugated hydrocarbons or electrons in those linear conjugated hydrocarbons were well described by the one dimensional particle in a box. We can ask ourselves, what problems the 3D particle in a box can help us with? And remembering the assumptions that we've made in deriving the particle in a box model, we assume that the potential energy is zero inside the box. We assume that the particle is confined to that box. And we assume, in this particular case, that the particle is three dimensions. So that's not much of an assumption. Any problem in the real world is going to be a three dimensional problem. So that is fine. We just need to ask ourselves what types of problems are there where a particle is confined to some region of three dimensional space. And within that region of space, the energy is the same everywhere within the volume, and it can't leave the volume. So I'll point out along the way that having solved the three dimensional particle in a box, the 1D particle in a box is actually a special case of this. If we have a problem that's confined only to one dimension instead of to all three dimensions, then all we have to do is ignore the Y and Z portions of the wave function, ignore the Y and Z contributions to the energy. And this much of the wave function with part of the normalization constant and this much of the energy is exactly the result we had for the 1D particle in a box. Likewise, if we find a particle confined to two dimensions, we can call those X and Y and only include the X and the Y contributions. So 3D particle in a box can be narrowed down to the solutions either for the 2D or the 1D particle in a box as we need. So if we want to understand what types of systems the particle in a box model applies to, we ask ourselves, what are the particles? And what is the box that they're confined to? We've seen an electron confined to a linear hydrocarbon molecule with some conjugation, linear molecules like this one. That would be a 1D example. So it's not exactly true, but we imagine that the particles confined to only one dimension within the box between the left and right edges of the box. There's relatively little difference in energy for a conjugated pi electron along the backbone of that molecule. So that was the 1D particle in a box. There's a very similar example. If I think of a conjugated molecule that's confined to two dimensions, I can draw organic molecules like naphthalene or if I had a third benzene ring, that's anthracene. If I draw a conjugated planar organic molecules like this one, polycyclic aromatic molecules where the pi electrons in these connected benzene rings can are delocalized over pretty much the whole molecule. I can think of the box in this case as a two dimensional box. It has length and height in this example, but it doesn't have any depth. So that would be like a two dimensional particle in a box model. So now we can cover a lot more molecules than just these linear molecules. There are also three dimensional versions of the same problem. If I ask myself, what would a three dimensional object look like? Where an electron somewhere in this three dimensional volume is free to move anywhere in the volume, can't leave the volume. Is it restricted to only being inside this object but is freely moving within the object? I could think of three dimensional chemical molecules that allow the molecule that allow the electron to flow freely within the molecule. But essentially what I'm talking about is a solid that is conductive. The electron can flow within the solid, can't leave the solid. So what I'm talking about in this case is a metal. If I have a block of metal that the electron can move freely within, that would be a particle in a box where the metal itself is the box and the electron is the particle. So that would be an example of a case where we could apply the three dimensional particle in a box or not just a metal, any conductive material. And as the fourth case we'll consider, all of these cases have considered cases where the particle is an electron and that makes a certain amount of sense because quantum mechanics deals with small objects and electrons are the smallest objects that chemists have to deal with routinely. So confining these small quantum mechanical electrons to a box is well described with these quantum mechanical models. But what if the object isn't an electron? What if the object itself is a molecule? What if I have a molecule again confined to a container? So here I wouldn't be talking about a molecule confined to a metal. Molecules don't diffuse within the inside of a metal. In order for a molecule to move back and forth inside this box, it has to be a little box, a container. So let's say, for example, I have a molecule inside this box that's bouncing around inside the box, able to move freely within the box, but the box has walls so it can't escape the box. Then essentially what I've described here is the case of a gas, a physical gas. So if I have molecules confined to a box, then those molecules, if they can fill their container, if they move everywhere within the box, then that is what we think of as a gas. And this would be another 3D example for which we could use the particle in a box energy levels and the particle in a box wave functions to describe how that system behaves. So it turns out we're going to get quite a lot of mileage out of using the 3D particle in a box model to describe how gases behave. But before we move on and do that, we're going to say one more thing about the energy levels for the 3D particle in a box that will need to understand how we use it for gases.