 So, here's what we know about the harmonic oscillator so far. We know there's equally spaced energy levels starting with the ground energy level that has an energy of 1.5H nu, where nu is the fundamental vibrational frequency. We can calculate that if we know the spring constant and the reduced mass of the molecule. And the spacing between each one of these levels, regardless of whether it's the first pair, the second pair, the third pair, each of these energy levels is separated by the same gap, h times this vibrational frequency, or if we prefer k Boltzmann's constant times the vibrational temperature, where we have defined the vibrational temperature as h nu divided by k. So, we've also seen that for a typical diatomic molecule, this vibrational temperature is somewhere in the vicinity of roughly 1000 Kelvin or so. Depends on the individual molecule. Every molecule has a different mass, reduced mass. Spring constant, but it's somewhere in the vicinity of a few thousand Kelvin, typically. That raises the question, how big are these energy gaps? Of course, they're h times nu, but what is that frequency? With what frequency of light would we need to excite a transition from the state up to this state? Or, equivalently, so if that gap in energy is equal to h times nu, we would like to know what is that nu? Equivalently, if we'd rather have it in units of wave numbers, what would the wave number of the light be? Or perhaps we want to know what is the wavelength of light needed to excite that transition? And that'll tell us whether we should shine visible light, ultraviolet light, infrared light on the molecule to excite these vibrational transitions. So, as an example, let's again take carbon monoxide, where we know, we've seen previously, that the vibrational frequency is 6.5 times 10 to the 13th hertz. 6.5 times 10 to the 13th per second, number of oscillations per second. And then if we want to know, let's do the wave number first. So if the frequency is c times the wave number, then the wave number is the frequency divided by the speed of light. So that's relatively straightforward arithmetic. Speed of light is 3 times 10 to the 8th meters per second. Although, looking ahead, we typically want to talk about wave numbers not in units of 1 over meters, but 1 over centimeters. So I'm going to list the speed of light in units of centimeters per second. Light travels 3 times 10 to the 8th meters in a second, or 3 times 10 to the 10th, 100 times as many centimeters in every second. So if I do that math, this frequency divided by the speed of light, I can almost do that in my head. But to a few more sig figs, the units 1 over seconds cancels top and bottom. I'm left with 1 over centimeters. I'll write that as centimeters to the negative 1. And we find that the light needed to excite a transition from this state to this state or from the state to this state between any adjacent pair of states has a wave number of 2160 inverse centimeters. So if you think back to or remember the discussion about the different portions of the electromagnetic spectrum, that falls squarely in the middle of the infrared portion of the spectrum. So this is infrared light of this wave number needed to excite this particular vibrational transition for carbon monoxide. In terms of wavelength, the wavelength of light needed, so hc times new tilde is equal to hc times 1 over lambda. So the wave number and the wavelength are reciprocals of one another. So if I already know new tilde, I can just take 1 over that quantity to find the wavelength. So that's 1 over 2160 inverse centimeters, which works out to be in those units, 4.6 times 10 to the minus 4. If I invert 1 over centimeters, I get regular centimeters. So 4.6 times 10 to the minus 4 centimeters is the wavelength of light. If I find some photons with wavelength 4.6 times 10 to the minus 4 centimeters, that's the right energy of wavelength of light to excite these transitions. Now it's probably more convenient to convert that away from centimeters. If I divide by 100 to convert centimeters to meters, that many centimeters is 100 fold fewer meters. So 4.6 times 10 to the minus 6 meters, or we can convert that to using SI prefixes 4.6 micrometers. So the wavelength of light needed to make this transition is 4.6 micrometers. And again, that wavelength is in the region of the electromagnetic spectrum, electromagnetic spectrum that we call the infrared portion of the spectrum. So both that number and that number tell us that, in fact, the light we need is infrared light. So that's maybe something you know already from other chemistry courses, like analytical chemistry or organic chemistry. Very often infrared spectroscopy is what we use to identify specific molecules. And that's because carbon monoxide with its particular spring constant, its particular reduced mass has a very individual specific frequency of light required to excite these transitions. So if molecules absorb this frequency of light, that's a good clue that there might be carbon monoxide in our sample. And likewise for other organic molecules, each bond of which has its own spring constant and its own reduced mass, I suppose I should point out that it's not just diatomic molecules. So we've, let me draw a carbon monoxide vertically. So here's my carbon monoxide molecule, sorry, double bonded carbon monoxide molecule with a spring constant reduced mass that we can use to tell us this number for the frequency. Let's suppose we have another molecule with a carbon oxygen bond, a molecule like let's say acetone. So there might be the four carbon monoxide. We can calculate that reduced mass using the mass of the carbon, the mass of the oxygen. Let me go ahead and put the triple bond back there. That reduced mass, if we know the mass of carbon is 12, the mass of oxygen is 16, we can calculate that reduced mass. We could do something very similar for acetone. And then if we're talking about that CO bond stretch, the reduced mass that we need would end up using the, so that's the bond between the oxygen and this carbon CH3 2 unit. So the mass of this unit, the mass of this unit could give us the reduced mass for this quantity and that, that's a different number than it is for carbon monoxide. But it's not terribly different, it's in the same ballpark. So it's still going to be a frequency of infrared radiation, but it's going to be a different frequency of infrared radiation. So every individual molecule has its own signature, its own absorption energies in the infrared spectrum. So that's why we use infrared spectroscopy as sort of a fingerprint to identify specific molecules. So the other thing that this may, other reason you may know infrared spectroscopy is used to excite vibrational transitions is because the way we experience infrared light, infrared is not in the visible portion of the spectrum, it's off the red end of the spectrum. If you have any previous experience with infrared light outside of a chemistry lab, it might be, for example, in a fast food restaurant using a warming lamp, lights that don't give off very much light but are very good at heating the food underneath them, those lights give off infrared radiation. So we think of infrared radiation very often as simply heat. And the reason that's true is because the infrared photons that are coming out of an infrared light source excite these vibrational transitions, cause molecules to vibrate with more energy than they were previously. As those bonds vibrate, they cause lots of collisions within the material and then that heat turns into energy that's stored in other portions of the molecule so that infrared energy gets converted first to vibrational energy and then to other forms of energy within the molecule. So that's how infrared energy is used to heat molecules up. So we know how to predict now what types of light, what colors of infrared light to use to excite these vibrational transitions if we know the fundamental vibrational frequency for molecules that behave like harmonic oscillators. The next question is to ask which of these transitions are actually allowable and so we'll talk about the selection rules for harmonic oscillators next.