Well, let's review the concepts from Week 2 that are particularly important. We discussed gases, and in particular, real gases this week. But we did start with the ideal gas equation of state. And recall that that's PV equals nRT. Where P is pressure. V is volume. n is the number of moles. R is the universal gas constant. And T is the temperature. It's a little bit more convenient to work with molar volume which is an extensive property, not an intensive property. And that would be expressed as PV bar equals RT. And indeed, the ratio of PV bar to RT itself has a name. That's called the compressibility and it's usually indicated by capital Z. Gases are in fact rarely ideal. That is, they do not obey the ideal gas equation of state except at very low pressure. The van der Waals equation of state, on the other hand, is a better predictor of non-ideal gas behavior, and there are other equations of state that do well, as well, and they include the Redlich-Kwong and the Peng-Robinson equations, which we looked at briefly. All of those three equations of state for non-ideal gases are cubic in the molar volume. And as a result, they are able to describe the liquid and gaseous regions of pressure volume isotherms. Recall that, on the pressure-volume isotherms, there are some important points. One of them is the critical point. At the critical point, the van der Waals equation of state has only a single root. A single value, value of molar volume that satisfies the cubic equation. And, once we know that critical volume, we can determine the van der Waals constants, a and b from that position, the critical volume, the critical temperature, and the critical pressure. The Law of Corresponding States says that all gases have the same properties if compared at their corresponding states. Where under corresponding conditions that is, where corresponding conditions means that the conditions for that gas, relative to its own critical conditions are the same as the conditions for a different gas relative to its critical conditions. So that is a corresponding state or corresponding conditions. We also saw another equation of state, the virial equation of state. And the virial expansion that appears in that equation of states has coefficients that are directly related to intermolecular interactions. The second virial coefficient, B2v, measures the deviation of the volume of a real gas compared to an ideal one under the same temperature and pressure conditions. So, if B2v is a positive number, that describes how much more volume the real gas occupies than an ideal one would under the same conditions and vice versa. When B2v is negative, that means that the gas is occupying less volume than an ideal one at the same conditions. We looked at the sorts of attractive interactions that could occur between two different molecules in a real gas. And in particular, we looked at the Lennard-Jones potential where the intermolecular u is expressed in terms of the strength of a molecular interaction epsilon and the size of the molecules themselves sigma. And the Lennard-Jones potential has an attractive term that drops off as r to the 6th, that is, as it goes as r to the minus 6, and a repulsive terms that goes as r to the minus 12. So at very short distances, the repulsion rises very, very steeply. The Lennard-Jones parameters themselves, epsilon and sigma, can be determined through analysis of experimental second virial coefficient values. We also talked about the physical underpinnings of the r to the minus 6 attractive term and the most dominant contribution to it, namely dispersion. We talked about London's development of a quantum mechanical explanation for dispersion. And his equation that produces its magnitude based on atomic or molecular ionization potentials and the polarizabilities of the individual molecules. Finally, we looked at some simpler potentials than the Lennard-Jones potential that allowed us to solve for the second virial coefficient analytically in order to gain some physical insight. One potential was the square-well potential, where there is a region of attractive interaction followed by a hard wall. And determined that the predicted second virial coefficients are not bad. And compare reasonably favorably to the full Lennard-Jones potential over reasonable ranges of temperature and pressure, and I showed an example for nitrogen gas. The simplest potential, the hard-sphere potential, where there is no attractive force, and at the point of two molecules touching, there is an infinitely repulsive potential. is not very appropriate under most conditions, but at very high temperatures, it does describe reasonably accurate gas behavior, and it says that the second virial coefficient is positive and independent of temperature. So it's that region in a plot of the second virial coefficient versus temperature, where the, the plot has turned over and is almost a straight line horizontally. The square-well potential that has a free parameter lambda in it, includes the length over which a constant potential is effective, and this emphasizes again, that we can get good agreement with experimental data. All right. Well, that completes the summary of the key concepts from this week. you'll have an opportunity to explore the concepts in more detail on the homework associated with this week. Good luck with that. And I look forward to continuing as we move to next week. And, to talk about what we will be looking at next week, we're going to start with the concept of Boltzmann probability. See you then.