 So, if we attempt to use the echopartition theorem to predict the heat capacities of some larger molecules, then we'll discover that we can use it not only to predict the properties of gases, but also of condensed phases as well. So, we've seen that it works pretty well, the echopartition theorem works pretty well to predict the heat capacity of monatomic molecules, diatomic molecules, so let's go one step larger and consider a triatomic molecule. So, we'll use water as our example, gaseous water to begin with. So, let's do what the echopartition theorem tells us to do, which is to count the number of quadratic degrees of freedom. We can break those up again into translational, rotational, vibrational degrees of freedom. We know for the triatomic molecule n equals 3 atoms, the total number of degrees of freedom is going to end up being 3n, so that's 9 for a triatomic molecule. As always, there's only three ways the molecule itself can move around translationally in x, y, and z. To count the rotations, we have to ask ourselves, is that a linear or a nonlinear molecule? The structure of water of course has the bent geometry, so it's not a linear molecule, the atoms are not in a straight line, so nonlinear molecules have three different rotational degrees of freedom. I can rotate that molecule just to make sure we understand those rotations. If this is my HOH molecule, I can rotate it around the z-axis, I can rotate it around the x-axis, or I can rotate it around the y-axis. So there's three different ways I can rotate that molecule. That leaves, in order to add up to 9, that leaves three different vibrational motions. So notice we don't have to identify what the vibrations are, they might be OH stretches, so they might be, if you know something about the spectroscopy of water, you may be able to name those degrees of freedom, but we don't have to know what they are. We know that there must be three of them because it has to add up to the total of 9 degrees of freedom. So the next step is to count how many of the degrees of freedom have quadratic contributions to the energy. So we can make sure to distinguish between the kinetic and the potential contributions to the energy, and we'll just jump right in this chart to noting down the contribution to the heat capacity either from the kinetic energy or from the potential energy. Kinetic energies are always quadratic, whether it's 1 half mb squared or 1 half i omega squared, kinetic energy is always quadratic. So these three degrees of freedom contributes 1 half r, each of these rotational, and each of these three vibrational degrees of freedom, they all contribute 1 half r with the caveat that the vibrational degrees of freedom only contribute if we're hot compared to the vibrational temperature, if the vibrations can be treated classically. The potential energy, when a water molecule translates around a box of gas molecules, there's no contribution to the potential energy, when it rotates there's no contribution to the potential energy, so those are not quadratic degrees of freedom and they don't contribute to the heat capacity. Vibrations do contribute. So remember the harmonic oscillator says the potential energy is quadratic, so each of these three degrees of freedom does contribute 1 half r to the heat capacity if we're hot enough to treat the vibrations classically. So if we add all those up, the result we find if we're cold compared to the vibrational temperature, then we only count the translations and rotations, so 3 plus 3 is 6 times 1 half r or a total of 3 r, if we're hot, and we don't actually have to be very, very hot compared to the vibrational temperature, if we're a bit above the vibrational temperature then we also include 3 from here and 3 from here and additional 6, so a total of 12 factors of 1 half r gives me 6 r for the heat capacity, so I predict that my heat capacity is going to be either 3 r or 6 r depending on the temperature, and as usual I predict that in between the two temperatures it would make a gradual transition between them, so we can see how that prediction works with actual data for water, so if I bring up a graph here of the heat capacity for water, gaseous water, we see that the equilibrium there is indeed working pretty well at relatively cold temperatures, down around 500 Kelvin or so, hot for us but relatively cold for a water molecule compared to its vibrational temperatures, the heat capacity is not exactly but fairly close to 3 r, at high temperatures if I heat up to 2, 3, 4,000 degrees Kelvin the heat capacity is approaching 6 r, so it's beginning to plateau at the value predicted by the ecopartition theorem, so far the ecopartition theorem looks like it's doing a pretty good job, we can ask ourselves what if we continue to colder and colder temperatures do we indeed approach this lower limit of 3 r as predicted by the ecopartition theorem, but we run into a problem which is if we look at the units on this graph, the temperature at which I've stopped plotting the data here, that's down a little bit below 500, a little bit below 400 Kelvin, in fact this point is 373 Kelvin, which once I phrase it that way that may give you a hint why we don't have data for gaseous water below 373 Kelvin, below 100 degrees Celsius when we're at one atmosphere, we don't have gaseous water condenses to form a liquid because that's the boiling point for liquid water, so if I were to continue plotting data below this point on the graph I'd be plotting data for liquid water not gaseous water, so that's the next thing we can do, we can repeat this exercise and ask ourselves what we would predict before I show you the data for the heat capacity for liquid water, and we'll learn something interesting about the heat capacities of condensed phases like liquids, before I do that though we actually know what the heat capacity of liquid water is, heat capacity for liquid water if I asked you for that value in units of calories per gram, sorry not calories per gram, yeah that's right calories per gram Kelvin, that's not a molar heat capacity that's a specific heat, and by definition that's the definition of the calorie, water has a heat capacity of one calorie required to raise the temperature of one gram of water by one degree Celsius or Kelvin, so in order to convert that to units that are the same as the units we're traditionally using for R, I can convert calories to joules, I can convert grams to moles, and that will leave me, so after I do that that will leave me with a heat capacity in units of joules per mole Kelvin, the way I convert grams to mole of course is with the molecular weight of water, so if I calculate 4.184 times 18 times one calorie per gram Kelvin in SI units, heat capacity of water is 75.4 joules per mole Kelvin, on this graph that's somewhere up here, we've got a sneak preview of what the answer is going to look like in a minute, but the heat capacity in units of R, as a multiple of R, if I divide 75.4 by 8.314 joules per mole Kelvin, I'm going to get a value very close to nine, so indeed what we see on this graph is the heat capacity of liquid water, one calorie per gram Kelvin, 75 joules per mole Kelvin, that's just about exactly nine times the gas constant, it varies a little bit with temperature as you can see here, but the heat capacity is nine times, somewhat surprisingly perhaps it's larger than the heat capacity of the gas, and it jumps up to about nine R. We can actually use the Equal Partition Theorem again to tell us why the heat capacity is nine R, so let's do a chart that helps us understand that, so again we'll count degrees of freedom, identify how many of those are quadratic, and use that to sum up our prediction for the heat capacity, we still have three translational, three rotational, three vibrational degrees of freedom for a total of nine, the kinetic energy always contributes to the heat capacity, so three times one half R for translation, three times one half R for rotation, three times one half R for vibration, as with gaseous water, if we're hot enough to be classical, the vibrations will contribute three times one half R via the potential energy to the heat capacity, the difference comes when we think about the potential energy, so think about what a box of liquid water, a sample of liquid water looks like relative to a sample of gaseous water, rather than one gas molecule far away from other gas molecules in the liquid, what we have is, let's say, a water molecule, and we know that's hydrogen bonded to another water molecule, hydrogen bonds connecting these molecules, so this water molecule is surrounded by another water molecule and surrounded by another water molecule, it's a condensed phase so the molecules are close together, so when I ask the question, do the translational degrees of freedom contribute to the potential energy of the molecule? In the gas phase that was no, when this molecule translates around, its potential energy doesn't change, but if this molecule changes its x or y or z position, it does change its potential energy because it begins to break these hydrogen bonds, it begins to bump into other nearby water molecules, so as that molecule moves around, its potential energy was relatively low at the position where it was, that if it moves to the left or if it moves to the right, the potential energy goes up, so even without knowing what the exact equation is for how that energy varies as its position changes, I know it's going to go up as it moves to either direction because it was at a relatively low energy position where it had chosen to be, so this does, it may not be a perfect quadratic, but near the bottom of this potential energy well, it's going to behave quadratically, so those are indeed at least approximately quadratic degrees of freedom and each of those three translational degrees of freedom contributes one half hour to the heat capacity. Likewise for rotational motion, if this water molecule rotates, that also is going to break the hydrogen bonds, increase the energy, if I change the angle of the molecule away from where it is, it's also going to increase the energy, so if we treat those quadratically, we get three additional factors of one half hour, and summing those up, we see that either, depending on whether the temperature is cold and I don't include the vibrations or what temperature is hot and I do include the vibrations, the heat capacity will be predicted to either be 6R at cold enough temperatures, 9R at hot enough temperatures, and if we compare that to the data we actually see for liquid water, we see that the heat capacity of liquid water is in fact 9R, so the actual experimental observation is that even at room temperature, colder than the temperature where gaseous waters vibrations had stopped behaving classically, liquid water is behaving fully classically, the translations, the rotations, the vibrations are all contributing to the heat capacity and we see heat capacity of close to 9R, so that tells us two important things, tells us number one, for some reason the vibrational temperature for liquid water has dropped, the vibrational temperature for liquid water is lower than it is in the gas phase, and we'll have reason to explore why that's true when we talk a little more in detail about the capacities of solids a little later on, the other important thing that we've noticed is the potential energies of translation and rotation do contribute to the heat capacity for substances in the condensed phases like liquids and solids, because translating or rotating those molecules in those condensed phases causes changes to their interactions with their neighbors which are now nearby, so the potential energies do have a contribution, so if you're careful about keeping track of what temperature you're doing these calculations at and careful to keep track of what phase you're in, whether you're solid or liquid or gas, you can make a relatively good prediction about the heat capacities of a variety of substances using the Equipetition theorem.