 So, we know harmonic oscillators have equally spaced energy levels. We know the selection rules tell us that we can make transitions that only change our level by one. We can gain one from zero to one, one to two, two to three. But the one missing piece of the puzzle we have now is which of these states are occupied in the first place? Can I expect a molecule to be occupying the n equals two state and absorb energy to go into n equals three? Or does that state not have a particularly high population? I shouldn't expect that transition to occur. So, we can talk about which of these states are populated or not? Which amounts to the question of whether the energies of these states are large or small compared to kT? Is there enough thermal energy at a particular temperature to populate a few or very many of these states? So, we know that the size of this quantity h nu or k times theta, the gap between these energy levels h nu or k times theta, that corresponds to the energy of photons in the infrared portion of the spectrum. Infrared photons are what we use to excite these transitions. So, we've seen that theta vibrational is in the vicinity of several thousand Kelvin, equivalently the fundamental vibrational frequency in units of wave numbers, that works out to be a few thousand inverse centimeters, typically. So, those are photons in the infrared portion of the spectrum. If we want to answer this question of which of these states are populated or unpopulated, we would like to know, let's say, for example, what's the population of one of these energy levels relative to the one below it, how populated is E3 relative to E2 or E2 relative to E1? And it turns out, doesn't, that answer is going to be the same regardless of which N I'm talking about, because each of these delta E's is the same. We know that this is equal to E to the minus difference in energy between those two states divided by kT, and all the delta E's are exactly the same amount. Those are all equal to h nu, or if we prefer, as we do in this case, k times theta. If I write that delta E as k times the vibrational temperature divided by kT, those k's are the same k, those are both Boltzmann's constant. So that's just E to the minus theta vibrational over T. So the population of an upper level relative to the population of the level below it is this exponential. So up in the exponent is this ratio of the vibrational temperature, several thousand Kelvin typically, divided by the actual temperature, maybe 298 at room temperature. So for example, if we stick with our carbon monoxide molecule, where we have seen that the vibrational temperature is about 3100 Kelvin. Again, that doesn't mean the molecule is hot. That means this property of the molecule, h times its vibrational frequency divided by Boltzmann's constant, that constant that describes the properties of the molecule has units of Kelvin and it has a value of 3100. If we are at a temperature of 298 Kelvin, and let's say we want to know the probability of being in state one relative to the probability of being in the ground state, the zeroth state, that's going to be this formula that we've just calculated, e to the minus delta e over kT, or e to the minus vibrational temperature divided by actual temperature, e to the minus 3100 Kelvin over 298 Kelvin. Doing the arithmetic in the exponent first, 3100 divided by 298, that works out to be a little over 10. So the vibrational temperature is a little more than 10 times as large as the actual temperature. So that means our ratio of populations is going to be e to the minus something a little larger than 10. That's a pretty small number. That's 3 times 10 to the minus 5. So what that tells us is the population, the fraction of molecules that occupy the n equals 1 state is 3 times 10 to the minus 5 as much as the population in the ground state. The population in the n equals 2 state, that's going to be even lower. That's going to be another factor of 3 times 10 to the minus 5 smaller than the n equals 1 state. Every state I go up, the population drops by this factor. So what that means is in the ground state, something like 99.997% of the molecules live in the ground state, only .003%, 3 times 10 to the minus 5, or so occupy the energy level 1, an even smaller factor, that number squared, 3 times 10 to the minus 5 of this number. So some very small percentage, so what's that going to be? Something like 9 times 10 to the minus 10 as the fraction of molecules occupy this state. So a very large majority of the molecules, 99.997% of the molecules at room temperature for a carbon monoxide molecule are in the ground vibrational state. So most of the carbon monoxide molecules are not vibrationally excited at room temperature. The few that are are only excited into the first vibrational state. Almost none of them are in the second vibrational state, and the populations get smaller as we climb that ladder. And that just like we know that most vibrational temperatures are in the same range of a few thousand Kelvin. Any time we're at room temperature, this number is going to be, if it's a few thousand Kelvin, many fold higher than room temperature. So most diatomic molecules, the same fact is going to be true, the vibrational population in vibrational states is overwhelmingly in the ground state for diatomic molecules. Most diatomic molecules are not vibrationally excited at room temperature. You'd either have to go to very high temperatures closer to 3,000 Kelvin before you begin exciting the vibrational excitations, or you'd have to find a molecule whose vibrational temperature is much lower than 3,100 Kelvin. So we've figured out how to calculate these populations. What we'll do next is see how the fact that these molecules mainly occupy the ground state affects their thermodynamic properties.