 All right, so the next step is to combine two things that we've obtained so far and use them to learn something about ideal gases. So we found that the partition function for a one-dimensional particle in a box is a bunch of constants that are related to something we've called the thermal de Broglie wavelength multiplied by a box size. So we have derived an expression for what the partition function is for this one-dimensional particle in a box. We also know that because the energies for the x, y, and z dimensions add together, the partition function for a three-dimensional particle in a box is just the product of one-dimensional partition functions in the x direction and y direction and z direction. So let's combine those two pieces of information. The 3D partition function is an expression like this in the x direction. So that would be 2 pi m k t over h squared square rooted. So that's raised to the one-half power times a, the box length in the x direction. So this is the q sub x piece, q sub y looks exactly the same. So instead of writing out 2 pi m k t over h squared again, I'll just change the power instead of one-half. I've got two halves. And then when I include the third term, it's going to be three halves. So all together I've got these constants raised to the three-halves power. I've got a box length in x, a box length in y, and a box length in z all multiplied together. So our 3D partition function is just as simple as that. We can make it even simpler, of course, by noticing that the length of the box times the width of the box times the height of the box, the x, y, z dimensions, that's just the volume of the box. So if I want to, I can rewrite the partition function as 2 pi m k t over h squared to the three-halves times the volume of the box that the three-dimensional particle is confined to. Or remembering our definition for the thermal de Broglie wavelength. This quantity is one over the thermal de Broglie wavelength. So when I cube it, it's just one over the thermal de Broglie wavelength cubed. So any one of these three expressions is perfectly valid for writing the partition function of a 3D particle in a box. Notice that, or recall that this expression is true in the classical limit when we didn't compute the sum of the Boltzmann factors, but we computed the integral instead. So we're pretending that quantum mechanics doesn't exist and any energy level is allowed. So these expressions are also true for the 3D particle in a box in the classical limit, any one of these three expressions. So we could say they're true for a 3D particle in a box in the classical limit. Or another word for a three-dimensional particle confined to a box in the classical limit where it doesn't behave quantum mechanically but can have any energy we would like it to. That's just another word for what we'll start calling an ideal gas. So when we consider a gas molecule confined to a box, we use the 3D particle in a box model to describe it. If the energy levels are very closely spaced as they would be for a gas molecule in a box as we've calculated, then we can approximate the partition function with this expression. So it gives a good description of the partition function for a gas molecule in a box. But what we have so far, this is the partition function for one gas molecule confined to a box of volume V. The gas molecule has some mass M, it's at some temperature T. This is all for one molecule of a box. We solve particle in a box for one particle confined to a box, a three-dimensional box. When we're speaking about gases, we're almost never interested in a single molecule of gas confined to a box. We're much more interested in n molecules of gas confined to a box. Maybe a mole of gas confined to a container. So now we need to remember that the, let's use capital Q. If we have distinguishable particles, we've seen that the partition function for the full system of n molecules is the single molecule partition function raised to the m, n power. If they're distinguishable or if they're indistinguishable, we need to remember to include that factor of 1 over n factorial to avoid over counting the terms. So in this case, if we're talking about a gas, let's say the oxygen gas in this room. I've got some number of molecules confined to some volume. The gas molecules, any individual oxygen molecule in this room is indistinguishable from all the other oxygen molecules. So this is the formula that we're more interested in. So the partition function for a collection of n indistinguishable and identical molecules in a box would be this one-dimensional partition function. And let's use this variation, raised to the n power divided by n factorial. So that's going to look like 1 over n factorial. Little q raised to the n, this expression raised to the n. I've got the term in parentheses, 2 pi m k t over h squared, raised to the 3 halves and then raised again to the n. So all together that's raised to the 3n over 2. And I've also got a volume itself also raised to the n. So this expression, 1 over n factorial, 2 pi m k t over h squared to the 3n over 2 times volume to the n. This is the partition function for an ideal gas. Again, either thinking of the ideal gas as an actual physical gas. And what we mean by that is some problem that obeys the three-dimensional particle in a box, but with the energy levels spaced so closely together that we're OK using the classical limit. So under those conditions, this is our partition function. So what we can do with that is then go on and calculate the thermodynamic properties of a gas now that we have the partition function.