 We have established the principle of Equipartition of Energy. Each quadratic term in the expression for total energy, what we call a quadratic degree of freedom, will contribute one-half the gas constant, R, to the molar heat capacity at constant volume. These terms can be quadratic in a linear or angular momentum, P, or in a linear or angular coordinate, Q. The total heat capacity is the number of degrees of freedom over 2 times R. This formula underlies the classical theory of heat capacity. Let's apply this theory to diatomic molecules. We model a diatomic molecule as two-point mass atoms, each with mass M, rigidly separated by a distance D. For simplicity, we only consider the case in which both atoms are the same element. Then the center of mass is at the center of this dumbbell shape. A molecule's center of mass can translate with velocity vj along any spatial dimension, with j equal to 1, 2, or 3. This contributes one-half 2M vj squared to the total energy. In addition, the molecule can rotate about its center of mass with angular velocity omega j, where j equals 1, 2, or 3 corresponds to one of the body's three so-called principal axes of inertia. This contributes one-half ij omega j squared to the total energy. With ij the jth moment of inertia, for a highly symmetric object like our dumbbell model, the principal axes of inertia can be determined by inspection. They are shown here as red, green, and blue cylinders. One rotational mode is about the red axis. A second mode is about the green axis. And a third mode is about the blue axis. More complicated rotations can be formed from a combination of these three basic rotations. For point masses M separated by distance D, the moment of inertia for rotation perpendicular to the molecule's central axis, the red and green axes in the previous video, is one-fourth Md squared. For rotation about the central axis, the blue axis in the previous video, the moment of inertia is zero because all mass lies on the axis itself. Therefore, there can be no kinetic energy associated with this third rotational mode. The result is that our rigid diatomic molecule model has five degrees of freedom, three translational and two rotational. The predicted molar heat capacity is therefore five-halves R, which equals 20.79 joules per mole Kelvin. This agrees well with the observed nitrogen value of 20.8. The oxygen value, however, is higher by a small but physically significant amount. We've seen how quantum mechanics can explain a continuous reduction with temperature in the effective number of degrees of freedom, and hence the heat capacity. But to increase heat capacity, we need an increase in degrees of freedom. We can increase the degrees of freedom by taking molecular vibration into account. Molecules are not rigid bodies. Instead, the atoms are held together by chemical bonds, which can be stretched and compressed about their equilibrium length. This motivates us to model the diatomic molecule as two masses connected by a spring. We know that vibration in a single spatial dimension introduces two additional degrees of freedom, one for the translational kinetic energy of the masses, and one for the potential energy of the spring. So our total number of degrees of freedom is now seven. This predicts a molar heat capacity of 29.1, which is much larger than the observed nitrogen and oxygen values. But now we can explain a reduction in the predicted value using quantum mechanics. At room temperature, the additional two degrees of vibrational freedom might be completely frozen for nitrogen and almost but not completely for oxygen. Following the Einstein model, let's define the vibrational temperature of a vibrational mode with frequency new as h nu over k. The vibrational temperatures for nitrogen and oxygen are 3521 and 2256 kelvin. Note that using a simple spring to model a chemical bond is an approximation that glosses over the complexity of real bonds. But for our purposes, it's a good one. Now we use the Einstein model's expression for the heat capacity of one mole of a harmonic oscillator, evaluating this contribution at room temperature. For nitrogen, we find 0.01 joules per mole kelvin. Adding this to the rigid body value 20.79, we get 20.80, which matches the observed value of 20.8. For oxygen, we find 0.25. Adding this to 20.79, we obtain 21.04, very close to the observed value of 21.1. Another manifestation of quantum mechanics is in molecular rotation. The kinetic energy associated with rotation about a single axis is one half i omega squared with i the moment of inertia about that axis. The corresponding angular momentum is L equals i omega. According to quantum mechanics, angular momentum about a given axis is quantized to integer multiples of h bar, Planck's constant over 2 pi. If we set L equal to h bar, then omega equals h bar over i. Substituting this into the kinetic energy expression, we get h bar squared over 2i. Then, we can set this energy equal to Boltzmann's constant K times a temperature, T rotational. This defines the rotational temperature as h bar squared over 2 Ki. Let's examine the implications for the three diatomic molecules, hydrogen, nitrogen, and oxygen. If we take the mass of a hydrogen atom to be 1, then the nitrogen and oxygen atomic masses are 14 and 16. If we take the hydrogen molecule bond length to be 1, then the nitrogen and oxygen bond lengths are 1.5 and 1.6. Since moment of inertia is proportional to m times d squared, the nitrogen and oxygen moments are about 30 and 40 times the hydrogen values. Calculating the actual moments and rotational temperatures, we find temperatures of 87.6, 2.88, and 2.08 Kelvin for hydrogen, nitrogen, and oxygen. The hydrogen value is about 30 times the nitrogen value and about 40 times the oxygen value. Let's look at the corresponding boiling points at standard pressure. If the rotational temperature is well below the boiling temperature, then as temperature decreases, the gas will condense to liquid form before we see significant effects of freezing of its rotational degrees of freedom. This is the case for nitrogen and oxygen. But for hydrogen, the rotational temperature is much larger than the boiling temperature. So when the low-temperature heat capacity of hydrogen gas is measured, we can see the rotational freezing effect. Near the boiling point, the heat capacity of hydrogen gas corresponds to just the molecules 3 degrees of translational freedom. The rotational degrees of freedom are frozen. Only near room temperature do the rotational degrees of freedom thaw, and hydrogen heat capacity approaches that of a classical diatomic molecule. We now have enough theory to sketch out, conceptually, the behavior of diatomic gas molar heat capacity at constant volume with respect to temperature. Near absolute zero, the substance is a solid, and heat capacity is governed by the Debye model. As temperature increases, the solid transitions to the liquid phase. Liquids are difficult to model, and we don't have good, generally applicable models for their heat capacity. At still higher temperature, the liquid transitions to the gas phase. If, as for hydrogen, this occurs below the rotational temperature, gas molecules will have only three translational degrees of freedom, and the heat capacity will be that of an ideal monatomic gas, 3 halves r. As temperature increases further, the two rotational degrees of freedom thaw, and heat capacity smoothly transitions to 5 halves r. As temperature approaches the vibrational temperature, the two vibrational degrees of freedom thaw, and heat capacity approaches 7 halves r. Based on the theory we have developed so far, after this transition, heat capacity should not change further with increasing temperature. Let's compare the behavior of hydrogen gas to our model. At its boiling point, heat capacity divided by r is 3 halves. As temperature increases, this smoothly increases to 5 halves and levels out. As temperature increases further, it smoothly increases to 7 halves. So far, the behavior agrees with our theory. But at even higher temperature, heat capacity does not level out at the 7 halves value. Instead, it continues to increase. We do not yet have an explanation for this, but we will revisit the phenomenon later. Let's see how we can extend our diatomic molecule results to more complex molecules. We will use water as an example. The water molecule consists of one oxygen and two hydrogen atoms. Treating this as a rigid body, there will be three translational degrees of freedom for the center of mass. And if, as for water, the atoms do not lie on a single line, there is no axis of zero moment of inertia, so we have three additional rotational degrees of freedom. Here is the first rotational mode, the second, and the third. For vibration, we can reason as follows. If the n atoms in a molecule were unbound, there would be three n translational degrees of freedom. We have already associated three degrees of freedom with translation of the center of mass and three degrees of freedom with rotation about the center of mass. This leaves three n minus six translational degrees of freedom to describe the vibration. Each of these will have an associated potential degree of freedom also. Here is one vibrational mode called the symmetric mode. Here is the asymmetric mode. And here is the bending mode. For water vapor, the quadratic degrees of freedom are three translation and three rotation of and about the center of mass and six of vibration for a total of twelve. So classical physics predicts a molar heat capacity of twelve over two equals six times r. However, the observed value near the boiling point of water is only about three r. But we know how to explain this as due to the six vibrational degrees of freedom being frozen out. The vibrational temperatures for the symmetric, asymmetric, and bending modes are 51, 58, 50, 44, and 27, 12 Kelvin. We can use these to form a simple model of water vapor molar heat capacity. We have three r for the six degrees of freedom for translation and rotation of and about the center of mass. Then for each of the three vibrational modes, we add an Einstein model term at the corresponding vibrational temperature. Comparing our model to the observed values, we see that it gives fairly accurate results except for very high temperatures. There the model flattens out towards the value six r, while the observed values pass through six r at a bit more than 4,000 Kelvin and continue to increase beyond that. So we have developed a reasonably good explanation for the heat capacity of a solid or gas based on the classical numbers of degrees of freedom combined with the concept of quantum mechanical freezing of vibrational and possibly rotational modes. But we've seen that our model has a major failing. It does not explain how heat capacity can exceed the prediction of classical physics. Quantum freezing can only explain values below the classical value. How then do we explain heat capacity above the classical limit?