Worked example: Interpreting potential energy curves of diatomic molecules | Khan Academy

In a previous video, we began to think about potential energy as a function of internuclear distance for diatomic molecules. What do I mean by diatomic molecules? Well, we looked at molecular hydrogen, which is just H₂, which is just two hydrogens covalently bonded to each other.

At standard temperature and pressure, the distance between the two nuclei would be based on where there is the lowest potential energy. If you were to squeeze them together, you would have to put energy into the system and have a higher potential energy. Or if you were to pull them apart, you would have to put energy into the system and have a higher potential energy.

What I want to do in this video is do a little bit of a worked example. Over here, I have three potential energies as a function of internuclear distance graphs, and what I'm going to tell you is one of these is molecular hydrogen, one of these is molecular nitrogen or diatomic nitrogen (N₂), and one of these is diatomic oxygen (O₂).

What I want you to think about—pause this video—is which graph is the potential energy as a function of internuclear distance for each of these diatomic molecules. I'll give you a hint: look at the low point in potential energy. The low point in potential energy is what you would typically observe that diatomic molecules' internuclear distance to be at standard temperature and pressure.

This distance, right over here, is going to be a function of two things. It's going to be a function of how small the atoms actually are, how small their radii are. Smaller atoms, in general, have a shorter stable internuclear distance. But the other thing to think about is the bond order between these atoms.

I'll give you a little bit of a hint: diatomic hydrogen has just a single covalent bond, for diatomic nitrogen, it is a triple bond, and for diatomic oxygen, it is a double bond. The higher order of the bond will also bring the two atoms closer together, and it also makes it have a higher bond energy—the energy required to separate the atoms.

Remember, we talked about in the previous video, this right over here is the bond energy. So with that said, pause the video and try to figure it out: which of these is the graph of H₂, which is N₂, and which is O₂?

Let's first just think about it in terms of bond energy. If you look at it, the single bond, double bond, triple bond, here you would expect the highest order bond to have the highest bond energy. The highest bond energy is this salmon-colored one right over here. So just based on that, I would say that this is a good candidate for N₂.

This one right over here looks like diatomic nitrogen to me. Then the next highest bond energy, if you look at it carefully, looks like this purple one right over here. So just based on bond order, I would say this is a good candidate for O₂. And then the lowest bond energy is this one right over here. So just based on the bond order here (it's just a single covalent bond), this looks like a good candidate for diatomic hydrogen.

Let's also think about the radii of these atoms. If we get a periodic table of elements here, we can see that hydrogen only has one electron in that first shell, and so it's going to be the smallest. So that makes sense over here that your distance where you have the lowest potential energy is shortest for the diatomic molecule that's made up of the smallest atoms.

But then when you look at the other two, something interesting happens. Remember, your radius for an atom increases as you go down a column, but as you go to the right on a row, your radius decreases because you're adding more and more electrons to the same shell. But the Coulomb forces are increasing between that outermost shell and your nucleus.

If you just look at that trend as you go from nitrogen to oxygen, you would actually expect your atomic radius to get a little bit smaller. They're right next to each other; they might be close. But you say, okay, oxygen has one extra electron in that same second shell—maybe it's going to be a little bit smaller. So if you were to base things just on that, you'd say, all right, well, the internuclear distance for this salmon-colored one is a little bit shorter, maybe that one is oxygen and maybe this one is nitrogen.

But they would be close, and I would say in general the bond order would trump things. The bond order, because you see this high bond energy, that's the biggest giveaway that this is going to be the higher bond order diatomic molecule or N₂. They're close in atomic radius, but this is what makes all of the difference. We'll take those two nitrogen atoms and squeeze them together just a little bit more, even though they might be a little bit bigger.

So I feel pretty good with this labeling.