 So far, we have seen that the molar heat capacities of most room-temperature solids tend to cluster about the Doulomb-Petey value, 25 Joules per mole kelvin. This is the prediction of classical physics, based on the quadratic degrees of freedom of a solid. However, some values are considerably lower or higher than this. Plotting heat capacity versus the biotemperature reveals a strong trend of decreasing heat capacity with increasing Dubai temperature. The Dubai model explains heat capacities below the classical prediction as due to the freezing out of vibrational degrees of freedom. This same principle allows us to explain why the heat capacity of multi-atomic gas molecules correspond to different numbers of degrees of freedom at different temperatures. So quantum mechanical freezing explains heat capacity below the classical prediction. But we also observe values above the classical prediction. How do we explain this? The harmonic oscillator has played a central role in many of our heat capacity models. But modeling atoms joined by chemical bonds as masses connected by ideal springs might be too simplistic. So let's consider so-called anharmonic oscillators. Let's first review our classical model of a homogeneous diatomic molecule. We represent this by two equal point masses connected by a spring. Vibration causes the distance between mass centers to be a function of time, d of t. The most problematic aspect of this model is reducing the complexities of a chemical bond to a simple spring. The potential energy of an ideal spring is a quadratic function of d. One half a, quantity d minus d0 squared, d0 is the equilibrium bond link, and the constant a represents the stiffness of the spring. It's convenient to define the deviation from equilibrium as a variable x equals d minus d0. Then d equals d0 plus x. The potential energy is then one half ax squared. If the vibrational energy is e, then motion is limited to x values where the potential energy does not exceed e. The spring force is minus the rate of change of potential energy with respect to x, which equals minus ax. Let's examine motion in a quadratic potential. Here the red curve shows potential energy versus x. The five blue dots in the associated horizontal green lines correspond to relative total energies of 1, 2, 3, 4, and 5. For any value of x, the red curve is potential energy, and the distance from the red curve to a green line is kinetic energy. We start each oscillator with zero potential energy at x equals 0, and kinetic energy equal to the total energy. As the systems oscillate, we see that higher energies correspond to larger oscillation amplitudes, but in all cases the required time for one complete oscillation and therefore the oscillation frequency is the same. This is the characteristic behavior of a harmonic oscillator. Also by the symmetry of the potential function, the average position over one oscillation is always zero. Here is a plot of position x as a function of time for the e equals 5 case. It's a sine wave. For any other energy, we will get the same curve only with a different amplitude. Now we add a plot of potential energy versus time. This is proportional to the square of position and is a sine wave at twice the oscillation frequency, varying between zero and the total energy e. Finally, we add a plot of kinetic energy. This is the difference of the constant total energy and the potential energy. It is simply a vertically mirrored version of the potential energy curve, but it is also the potential curve shifted along the time axis. As a result, the average kinetic energy will equal the average potential energy. Now a shortcoming of the harmonic oscillator model is that no matter how much force is applied to pull the masses apart, the spring will only stretch a finite amount. But real chemical bonds, like real springs, can be broken. In the chemistry video of the quantum mechanics series, we use quantum mechanics to calculate the energy of the hydrogen molecule ion as a function of nuclear separation. We saw that the energy curve has a minimum, which defines the equilibrium bond length. For small deviations, the energy curve is approximately quadratic. But for large deviations, it is very asymmetric, growing much more rapidly for small separations and much more slowly for large separations. We expect that real molecular vibrations will be governed by asymmetric potentials of this sort. We can model this asymmetry by adding a cubic term to the potential, so that it becomes one-half AX squared minus one-third BX cubed. The added term increases the potential as the atoms approach one another and decreases it as they move apart. The corresponding force is minus AX plus BX squared. Let's examine motion in this asymmetric potential. Again, we have five oscillators with total energy one through five, and we start each oscillator with zero potential energy at X equals zero and kinetic energy equal to the total energy. As the systems oscillate, higher energies again correspond to larger oscillation amplitudes. But now the required time for one complete oscillation also increases with amplitude. Therefore, for this anharmonic oscillator, oscillation frequency decreases with increasing energy. For harmonic oscillators, the average value of X is always zero. But for this anharmonic oscillator, the average value of X is skewed towards larger positive values with increasing amplitude. As a result, the average bond length increases with increasing energy, hence with increasing temperature. For a solid, this causes the dimensions of the object to increase. So, the asymmetric potential of chemical bonds explains the phenomenon of thermal expansion. Now, let's look at the average kinetic and potential energies as a function of total energy. For E equal one, average kinetic energy is 0.491, and average potential energy is 0.509. For a harmonic oscillator, these values would be equal. But for this anharmonic oscillator, slightly more than half of the total energy is due to potential energy and slightly less than half to kinetic energy. As total energy increases, this effect becomes more pronounced. For E equal five, the average kinetic energy is just under two, while the average potential energy is just over three. We can understand this by looking at X versus time for the E equal five case. The curve is no longer a sine wave, but is skewed towards positive X values and away from negative X values. Plotting potential energy, we see that the oscillator spends relatively more time with large potential energy and less time with small potential energy. Kinetic energy is total minus potential energies. Its curve is a vertically flipped version of the potential energy curve. So the oscillator spends relatively more time with small kinetic energy and less time with large kinetic energy. This is why energy is not equally partitioned between kinetic and potential forms for this anharmonic oscillator. Here is a plot of total energy versus two times kinetic energy. The dots are the results of our numerical calculations for the anharmonic oscillator. For a harmonic oscillator, total energy is always two times kinetic energy and equal to kT. This is the dashed line. We see that the anharmonic values are greater than they would be for a harmonic oscillator. At a given temperature, the kinetic energy of either oscillator type will be the same, but the anharmonic oscillator will store more potential energy on average than a harmonic oscillator. Therefore, at a given temperature, the anharmonic oscillator will contain more total energy than a harmonic oscillator. And this difference will increase with temperature. Since our classical heat capacity calculations were based on the harmonic oscillator model, this explains how real materials can have heat capacities above the prediction of classical physics.