Hello. In this video, we're going to discuss dense gases. Dense gases are gases where the density is high enough that molecules are close enough together so that they feel the electrostatic field from other molecules, at least the molecules that are reasonably close to them. We're just going to look at the case for a relatively small departure from the ideal gas limit. We'll start by considering in the general form of the partition function, and that will lead to an integral called the configuration integral, that contains the interaction potential energy terms. So if we start with the canonical partition function, this particularly simple form, then for a dense gas, if we calculate U, there's an internal energy term, a translational energy term, and then there's an interparticle potential energy, which is written symbolically as Phi of r1, r2, and so on, where r1, r2 are the coordinates of all the molecules in the system. If we factor out the internal energy, then it comes out, and so we note that we can write U of the internal energy separately. Then we have that the sum of e to the minus Beta U over the internal energy can be written in terms of the so-called internal partition function, or molecular partition function for internal motion, which we introduced at the beginning of our discussion of ideal gases. So if we plug that into the summation, we get q internal to the n, where n is the number of molecules in our system. So that the partition function becomes q internal to the n, times the sum over e to the minus Beta U translation, plus the potential energy term. So it pays to replace this sum with an integral, and this is okay because translational energy states are very closely spaced. So it turns out that one can show, and you can see this in McQuarrie, I'll give you the reference in a minute. But you can write that sum in terms of the integral expression on the right, where H is the classical Hamiltonian. The Hamiltonian is the sum of the kinetic and potential energy. So it turns out we can separate this integral into the product of two integrals, one over momentum, and the other over space. But the derivation for all of this can be found in these references, the McQuarrie statistical thermo textbook Chapter 10, Section 7, and the papers by Kirkwood, which were the formative papers back in 1933 and 1934. So the first term is just the translational energy. The second term, we call the Configuration Integral. So it's the integral over all position of all molecules of e to the minus Phi over kT. It's called the configuration integral because it depends on the configuration. In other words, where are all the atoms and molecules located. So if we put all this together, the Canonical Partition Function becomes as shown in this equation. So you have the product of the internal partition function to the one over n factorial, times this term, 2pi mkT over h squared, to the three-halves N times the configuration integral, I'm sorry, which was denoted as Z sub Phi. This result is identical to an ideal gas, except for the presence of the configuration integral in place of the volume. So Phi is zero, the integral, in fact, of Z is equal to the volume to the nth power. So you can then write the canonical form of the Massieu function as shown here, and you can then get all the properties from this, if you knew Z, of course. So the problem devolves to evaluating Z. Z is taking into account the effect of inter particle forces that we did not need to deal with for an ideal gas. So we require some simplifying assumptions, and these are typically that the orientation of molecules is unimportant. So the force between any two molecules is a function of distance only. This is actually not true for non-symmetric molecules. Another simplifying assumption is that the total potential energy is the sum of the pairwise potential energies between molecules, and that's only strictly true for a not too dense gas, or just a relatively small departure from ideal gas behavior. So if we do that, we can write the potential energy term, Phi, is equal to the sum over Phi for rj interactions, and then we have to multiply that sum by one-half because otherwise, we'd be counting each interaction twice. Or we can write it as shown on the right side of this equation, where we've got some restrictions on the indices that we're summing over. Here, rij is the distance between molecule i and molecule j. Now, we can make a simplification if we introduce this new function f as e to the minus Phi, all minus 1, and if you're astute, you would think that if Phi is small, then there are some great simplifications to be made, and that is the case. So it turns out that we can write e to the minus Phi over kT as equal to this product of e to the minus Phi ij, which is the potential energy between molecule i and molecule j over kT, and that in turn can be written in terms of this function f that we just introduced. So what we've done is we've eliminated the exponential in this product. If we assume that the gas only weakly interacts, that means f is going to be much, much less than one. This is where the simplification comes in, and if we drop all the higher-order terms, then e to the minus Phi over kT is just equal to 1, plus the sum over all the fij's. Then plugging that into the integral expression for the configuration integral, we get the equation on the top, and if we integrate the first term, we get V to the N, and then the second term we're going to integrate over the sum of the f_ij's. Since each f_ij depends only on r_ij, it turns out we can write this integral as V to the N plus one over 2N squared times the integral, from 0 to infinity of f of rdr. This is where we've assumed that all of the f_ij are the same function. That is that the intermolecular potential between any two particles is the same. This obviously is not true in the general case. If you have a mixture, for example, the potential is going to be different between different types of molecules. So this is what we call the weakly interacting configuration, and it is then typical to write Z. This first order approximation to Z, as V to the N times 1 minus N squared, times something called B over V, and B is a function of T, it's got f in it. It happens to be called the second Virial coefficient, and is equal to minus two pi, the integral from 0 to infinity f of rdr. Now, Virial means force in Latin, and we'll come back to that idea that it's force in Latin. So more generally, we can write that Z is equal to V to the N, all times one minus N squared B over V plus higher-order terms. Of course, people have explored those higher-order terms and so on, we will not do that. But it provides a context for the discussion of so-called Virial equation of state, which are equations of state which have terms in the increasing order of one over the volume to the power. So that's it for this video. Thanks for listening, have a great day.
