 So there's another way in which the nuclei, the properties of the nuclei affect the chemical properties of a molecule. And that's not the nuclear spin, but the isotopic mass, the mass of the nucleus also affects properties of molecules. And to see why that's true, let's spend a little bit of time talking about bond association energies and how to understand those from the point of view of a quantum mechanical understanding of a molecule. Imagine that we have a diatomic molecule, an H2 molecule or a HCl molecule, some molecule with two atoms connected by a covalent bond. We spend a fair amount of time talking about that type of molecule, their potential energy. It has a shape like this. We've modeled that with a harmonic oscillator down at the bottom of the well. We need to treat it with n-harmonic terms in order to understand how it bends over and becomes flat when the molecule dissociates. So this axis is the bond length of the molecule and in the limit of stretching the bond for a long distance, that bond energy will plateau and become flat when the bond dissociates. So from a quantum mechanical point of view, understanding the molecule as a harmonic oscillator or an anharmonic oscillator, we understand there's a ground state and there's some excited states. There's a gap between these energy levels, some delta E between the energy levels. If it's a purely harmonic oscillator, then that ground state would have an energy of one-half h nu, one-half times Planck's constant times the fundamental vibrational frequency of the molecule, which sometimes it's more convenient to think of that as Boltzmann's constant times the vibrational temperature of the molecule. Likewise, the gaps between these states, E0 to E1 or E1 to E2, that difference in energy falls in the infrared portion of the electromagnetic spectrum for a diatomic molecule. That difference is h nu or k theta. Again, if we're a purely harmonic oscillator, if it's anharmonic, then these terms all have some corrections tacked onto them. For our purposes right now, it's enough to understand that that difference between energies is not much less than but much greater than kT. So kT is quite small compared to this difference in energies, which means that the population of the ground state at room temperature anyway is pretty close to 100%. Almost all the molecules live in the ground state, a very small fraction live in the excited state, even fewer of them live in the next excited state. Those populations are all significantly less than one. So when we think about how much energy it takes to dissociate a molecule to break the bond, the molecule never has this much potential energy. It can never sit at the very bottom of the well. The zero point energy of the molecule lifts it up above the bottom of the well. Most of the molecules, the overwhelming majority of diatomic covalently bonded molecules, have an energy of one half h nu above the ground state. So what that means is if I want to break the bond for this molecule, this bond dissociation energy that we call D0, is the energy it takes to get from the ground state E0 up to the dissociation limit, up to this energy that the molecule has when it's fully dissociated. We can contrast that to the energy that it would have taken if I were to dissociate all the way from the bottom of the potential energy well. So this energy right here, the difference between the dissociation limit and that energy, we call D sub e. For the same reason we call the fundamental vibrational frequency nu sub e. The east here standing for equilibrium. This is the dissociation energy from the very bottom of the well, from the equilibrium position at the bottom of the well. That never happens in real life, but it's still useful to be able to say the dissociation energy from the bottom of the well differs from the dissociation energy from the ground state by this zero point energy, one half h nu, or one half k theta. So what we can do with that information, that will begin to explain why it is that certain molecules have different dissociation energies than others, and in particular why certain isotopes of certain nuclei have different dissociation than others. For now, we'll just point out that these two dissociation energies, D e and D not, can be substantially different from each other. They're different from each other by this zero point energy, and that difference can be fairly substantial. And we'll plug some numbers in and work some examples of that coming up next.