 All right, so now it's time to understand what isotope masses have to do with bond association energies, based on our understanding of these covalent chemical bonding wells and the difference between the dissociation from the zero point energy, the dissociation from the ground state, versus this hypothetical dissociation from the bottom of the well. So first, if we use some actual numbers for real molecules, let's say we take a hydrogen chloride molecule for which I can tell you a few facts. I can tell you the vibrational temperature of an HCl molecule is 4,230 Kelvin. That number being much higher than room temperature tells us that's a very quantum mechanical molecule. The zero point energy is relatively large. The light mass of the hydrogen makes this a very quantum mechanical molecule. I can also tell you, let's say from experimental observations, if we dissociate, if we find out how much energy it takes to dissociate the covalent bond in an HCl molecule, dissociating it from the ground state of the molecule, that energy can be measured, and that energy is 432 kilojoules per mole. All right, those two pieces of information are enough to tell us what d sub e is. So the zero point energy, first of all, the energy of the ground state, 1 half h nu, or easier in this case, 1 half Boltzmann constant times the vibrational temperature, which we already have. If we were to calculate that, 1 half Boltzmann's constants, the same thing as the gas constant, which I'm going to use in this case because the units of joules per mole are convenient for measuring these energies at the moment. If I multiply the gas constant by the vibrational temperature, units of Kelvin cancel, and I'm left with just 8 in change times 4,000 is 30-some thousand joules per mole. If I cut that in half, it ends up being, specifically, it ends up being 17,600 joules per mole or 17.6 kilojoules per mole. So that's the zero point energy. That's the height of this ground state above the bottom of the well. So what that tells us is if we do want to know d sub e, the dissociation energy from the bottom of the well, that's the 432 kilojoules per mole from the ground state plus an additional roughly 18 kilojoules per mole worth of energy that the ground state is above the bottom. So 432 plus 18 d naught plus the zero point energy, 432 plus 18 works out to 450 kilojoules per mole. So as just a quick example, there's clearly a difference, 432 kilojoules per mole dissociation from the ground state, 450 from the bottom of the well, those two numbers are significantly different from one another. They're different by a few percent. So one thing that means is when you go to look up a bond association energy from someone else in a table that's already measured it, be sure you know whether you're looking up d naught, the dissociation from the ground state, or d e d sub e, the dissociation from the equilibrium position at the bottom of the well, because they can be substantially different. But that's not actually the most interesting thing about this difference between these two types of bond association energy. It gets even more interesting when we consider the effect that isotopes have. So now, instead of talking about HCl, let's suppose we're interested in an isotopic variant of that molecule. So not H, but d, d for deuterium. So d here means the heavy isotope of hydrogen, a mass two isotope of hydrogen. So quick digression to make sure we understand what that means. It's still a hydrogen atom, so it's still got one proton in the nucleus. So it's got a proton. One proton in the nucleus means the nuclear charge is still plus one. It's still a neutral atom, so it still has one electron in an electron cloud surrounding that nucleus. The difference between this and the lighter mass of isotope, since it has a mass number two, it has one neutron in the nucleus as well as the proton. So the only difference about the molecule is it's got a heavier nucleus because of the extra neutron. The charges are still the same, the number of electrons are still the same, the number of protons are still the same. So what that means about the covalent bonding well is actually absolutely nothing. If you remember from thinking about the hydrogen atom and quantum mechanics, the potential energy that we're talking about here that describes the interaction, the potential energy is given by the interaction between the electrons and the nuclei. We haven't changed anything about the electrons or the nuclei in this hydrogen atom to turn it into a deuterium. We've only made it heavier, we've only changed the mass. So the potential energy for a DCL molecule, exactly the same as the potential energy curve for the HCl molecule. There's no difference in the potential energy, there's no difference in this bonding well. However, if we think about the zero point energy, one half h nu, the height of the zero point energy, that does depend on the isotopic mass of this molecule because the frequency for a harmonic oscillator, remember, is 1 over 2 pi square root of the force constant, the spring constant for that covalent bond divided by the reduced mass. So here's where the masses come into play. If I change the mass of one of the atoms in that molecule, it's going to change the reduced mass of the molecule. In particular, if I compare the reduced mass for HCl and for DCL, we'll go ahead and do HCl first because that's the more familiar molecule. Reduced mass is always mass of one atom multiplied by the other divided by the sum of the two masses. So for HCl, that works out to be rounding these off. Masses of hydrogen and chlorine, let's say we've got the mass 1 isotope of hydrogen, let's say we're dealing with the mass 35 isotope of chlorine, so the masses will be something close to 1 times 35 in the numerator and 1 plus 35 in the denominator. So mathematically that works out to a little less than 1 gram per mole, 35 divided by 36.97 grams per mole. If we compare that to the same thing for DCL, here we're going to be using the mass of a deuterium rather than the mass of a light isotope of hydrogen. Multiply it in the numerator, add it in the denominator, so I've got 2 times 35 divided by 2 plus 35. That works out to be a larger number of course because of this 2 in the numerator, 70 divided by 37, 1.89 in units of grams per mole. So not exactly twice as large, roughly twice as heavy a reduced mass for the DCL than for the HCl. So we could, as the next step, if we know the zero point energy, 17.6 kilojoules per mole for HCl, we can work backwards and figure out the force constant, use that force constant because the bonding well is exactly the same for HCl and for DCL, and the new reduced mass of DCL, we can work out what the zero point energy is for DCL. I'll skip the details of that calculation. I'll give you a number in just a second as if we had done that calculation, but the important thing to notice is that when I change an isotope, for example, HCl to DCL, when the reduced mass goes up as it's done for HCl turning into DCL, the effect that has when the reduced mass goes up, it's in the denominator of this square root. So that makes the frequency go down. The vibrational frequency of a DCL molecule is slower, lower frequency than the vibrational frequency of an HCl molecule. We could think of that either in terms of the vibrational frequency or if we prefer in terms of the vibrational temperature. So when the vibrational frequency goes down, the vibrational temperature goes down as well. If either of these quantities goes down, then of course the zero point energy goes down as well. The zero point energy drops, and now the important consequence when the zero point energy drops, so if the zero point energy right here I've drawn is for the HCl molecule in DCL, the zero point energy will be lower. If I've modified the isotopes in a way that lowered the zero point energy, even though I haven't changed the dissociation energy from the bottom of the well, the dissociation energy from the zero point energy in the heavier isotope starting from a lower point, so I have to give it more energy to get it up to the dissociation limit, and that means the dissociation energy of the molecule goes up, at least when we're measuring it from the ground state. So I told you I would give you some numbers. For DCL, if we had continued with this calculation, calculating vibrational frequencies, zero point energies, the zero point energy for DCL, smaller than the zero point energy for HCl by this ratio of 1.89 over 0.97 in the bottom of a square root, so if we had done that calculation, that zero point energy works out to be 12.6 kilojoules per mole, roughly 1 over square root of 2 as big as the 17.6 kilojoules per mole for HCl. Equivalently, if we wanted, we could talk about the vibrational temperature. That's also smaller by the same ratio. It's only 3,000 Kelvin. DCL with a lower vibrational temperature is not quite as quantum mechanical as a heavier molecule. It's less quantum mechanical, more classical behaving. And then lastly, the important property from our point of view right now is what happens to the bond dissociation energy for this DCL molecule. Again, because it's heavier, zero point energy is lower. It takes more energy to dissociate it. That we can calculate since the dissociation from equilibrium is 450, dissociation from here is higher by around 13 kilojoules per mole. So that's going to work out to be 437 kilojoules per mole. Again, just like we decided over here qualitatively, the dissociation energy for DCL is a larger number than the dissociation energy for HCl. It's about 5 kilojoules per mole more difficult to dissociate a DCL bond than it is to dissociate an HCl bond, not because the potential energy is any different. Because a deuterium attracts a chlorine or the electrons any more strongly than do the hydrogen or the chlorine purely because of this quantum mechanical effect that the zero point energy is lower because the DCL is a more classical molecule. That fact that the bond dissociation energy is larger for a heavier isotope has important consequences on lots of other properties of molecules as well and we'll consider a whole list of those next.