[MUSIC] Now let's look at another example, an ideal gas of diatomic molecules. Now let's consider n non-interacting diatomic molecules in a fixed volume v and temperature key. Now remember this is the canonical ensemble. Now what do you think about the gas molecules? Are they distinguishable or indistinguishable? Now the molecules in the gas phase are indeed indistinguishable from each other. And so the partition function is given by the single particle partition function raised to the power n divided by the over-counting factor n-factorial. Now from quantum mechanics we can analyze the diatomic molecule. We know that it consist of translational, rotational, vibrational and electronic quantum mechanical modes. Now, from quantum mechanics we found that there is a quantization of the energy modes associated with the translation, rotation, and vibration. The translational energy is simply given by the energy associated with a particle in a box. The vibrational energy is given by the particle in a quantum mechanical harmonic well. The rotational energy is given by the problem of the rigid rotor. Now owing to the fact that these quantum mechanical modes are decoupled, the single particle partition function is separable into quantum mode partition functions. Now we need to find the solutions for each of these quantum mode partition functions. Now let's first take a look the translational partition function. This partition function is simply a sum over three quantum numbers nx, ny, and nz, each running from one to infinity. Now from this analysis emerges a quantity known as the translational temperature, theta trans, which is defined in the following way. Now the translational temperature is in general very, very small, that is, very, very cold. For example, for an oxygen molecule in a box of one centimeter, the translational temperature turns out to be of the order, ten to the minus 15 Kelvin. This temperature represents the temperature when the translation nodes becomes excited. Now for any appreciable temperature that is any temperature that is above the translation temperature, we can approximate the summation of the partition function by an integral and after a little bit of calculus we can get the transitional partition function. The final expression turns out to be really simple. It only depends on the volume of the box divided by a quantity known as the debravely wave length. Now let's move on to the rotational part of the partition function and like the translational part, the rotational part has a degeneracy factor that needs to be accounted for. Now, it turns out that for each rotational quantum number l, there's a degeneracy factor associated with it, which is 2l plus one. Therefore, the rotational partition function is simply given by. Now, from this, analogous to what we had done in the translational case, we can define a rotational temperature turns out that the rotational temperature is also generally very small. However it turns out that it's larger than the translational temperature. For example, the rotational temperature for an auction molecule is about two kelvin. Now for any appreciable temperature, now the temperature being about the rotational temperature, the rotational partition function can be approximated by the integral. Now this yields a very simple expression that the rotational partition function is simply the temperature divided by the rotational temperature. Now let's shift gears and find out the vibrational partition function. Now remember the problem of a particle in a quantum mechanical harmonic well. From that analysis we can write the vibrational partition function as. Now once again we can define a vibrational temperature, and this temperature dictates when the vibration modes become active. Now it turns out that this temperature is usually large, for example the vibrational temperature for an oxygen molecule, is about 2,200 Kelvin. Using the property that the sum to infinity of Z raised to the power N is, simply, 1 over 1, minus z, allows us to evaluate this, the vibrational partition function, exactly. Now, note that this result is exact and valid over the entire temperature range, which is necessary because the vibrational temperature tends to be very large. In comparison, the approximate solutions that we derived for the translational and rotational partition function are only valid at temperatures T above the translational temperature and the rotational temperature. Now there's one additional piece to the total partition function, the electronic partition function. Now thermal excitation of the electronic states in a molecule leads to the electronic partition function. Now for simplicity, we will just account for the ground state electronic degeneracy genos. This must be found for the molecule or atom of interest. Now typically, these are tabulated for many simple molecules. For the oxygen molecule, the ground state degeneracy is the triplet state, that is G non equals 3. Now we've set up the whole machinery to analyze the statistical thermodynamics of an oxygen molecule. The partition function for the oxygen molecule is simply given by the single particle partition function raised to the power n, accounting for the indistinguishability factor of n factorial. That translational part is simply given by the volume divided by the terminal wavelength cubed. The rotational part is the temperature divided by the rotational temperature. The vibrational part is a result of the infinite summation that we carried out. And the ground state electronic degeneracy for the oxygen molecule is three, that is the triplet state. Now the [INAUDIBLE] energy can be found by the familiar expression -kBT log Q and this is found to be. Now from thermodynamics, we define the pressure to be the negative of the derivative of [INAUDIBLE] with respect to volume. Now, from this, we can actually evaluate the equation of state and this is the very familiar ideal gas equation of state. Now, what about the heat capacity? The heat capacity can be found using the following thermodynamic relation. Note that this expression is valid for temperature T where the temperature T is about the rotational and translational temperature but less than the electronic excitation temperature. The first contribution to the heat capacity of 3 over 2Nkb is associated with the 3N translational degrees of freedom. KB over two for each direction. The second term oft he heat capacity is NKB, associated with the two end rotational angles, each contributing KB over two. Now, as the temperature passes through the vibrational temperature, the heat capacity goes from 5/2 nkb to 7/2 nkb, and so a single vibrational mode contributes kb to the heat capacity as the temperature is raised the individual degrees of freedom get turned on in order to maximize the entropy. Since more active degrees of freedom contribute to a greater entropy. Now to summarize, what we did was to introduce the concept of distinguishability and indistinguishability. And then we proceeded to analyze the statistical thermodynamics of two example problems. First one was a two-level problem. That is a ground state and an excited state. And then we proceeded to analyze the statistical thermodynamics of an ideal gas of diatomic molecules. The partition function of an ideal gas consists of four components. One is the translational component. The second is the rotational component. Third is the vibrational component and finally the electronic partition function. Now, turns out that these modes are separated and decoupled so they can be written down as a product of these partition functions. From this, we can evaluate almost all thermodynamic properties of an ideal gas.
