 So far, we've seen that the heat capacity of the solid approaches zero at very low temperatures. Classical physics cannot explain this, but Einstein was able to model this behavior qualitatively by treating each atom of the solid as a quantum harmonic oscillator. The predicted heat capacity of the Einstein model, however, approaches zero more rapidly than is experimentally observed. The quantitative shortcomings of the Einstein model were rectified in 1912 by the Dubai model, developed by Peter Dubai. We start as in the Einstein model with the solid treated as a collection of balls connected by springs. But we do not make the simplifying mean field assumption. Instead, we accept that the atoms are coupled with a force on each dependent on the motion of its neighbors. Let's see what kinds of oscillations are possible by numerically solving for the oscillations of a one-dimensional array of eight balls connected by springs. We set up several copies with the initial ball velocities corresponding to one of the possible modes of sinusoidal oscillation. We find a range of possible frequencies. In the lower frequency modes, near the top, many of the balls tend to move in the same direction as one or both of their neighbors, resulting in overall less compression and stretching of the springs. In the high frequency modes, near the bottom, neighboring balls tend to move in opposite directions, resulting in overall more compression and stretching of the springs. With coupling, we have a spectrum of oscillation frequencies instead of a single frequency as in the Einstein model. The additional low-frequency components of the Debye model correspond to smaller energy quanta. These should increase heat capacity at low temperatures, which is where the values of the Einstein model were too small. Calculation of the heat capacity is lengthy, and we simply present the result. c equals 3nk times 3 quantity t over td cubed times the integral from x equals 0 to td over t of x to the fourth e to the x over quantity e to the x minus 1 squared. Here, td is the Debye temperature, which is on the order of the Einstein temperature, is often determined experimentally. In the high temperature limit, c approaches the Doulomb-Petit value, 3nk, as does the Einstein model. In the low temperature limit, c behaves as 3nk times 4pi to the fourth over 5 times quantity t over td cubed, so heat capacity varies as tq near absolute zero. Here are both the Debye and Einstein models compared to the observed heat capacity of platinum. They agree with each other and experiment at high temperatures, but only the Debye model accurately tracks the variation of heat capacity at low temperatures. Here is a plot of the room temperature molar heat capacities of most solid elements versus atomic number z. The horizontal red line shows the classical prediction of the Doulomb-Petit law, 25 joules per mole kelvin. The actual values fluctuate about this line. The Debye model should explain values below the line. On the right, these heat capacities are plotted versus Debye temperature. There is a very strong correlation between increasing Debye temperature and decreasing heat capacity. Room temperature is about 300 kelvin. We see that all elements with heat capacities less than 24 joules per mole kelvin have Debye temperatures greater than 300. Let's look at three examples. Carbon, in the form of graphite, has a Debye temperature of 1550 kelvin. At room temperature, its observed molar heat capacity is only 8.5 joules per mole kelvin, far below the prediction of the Doulomb-Petit law. But we can understand this using the Debye model, which predicts a value of 8.6. Similarly, for silicon, with a Debye temperature of 692, observed heat capacity of 19.8, and a Debye model value of 19.3, and for germanium, with a Debye temperature of 403, observed heat capacity of 23.3, and a Debye model value of 22.8. Something that Debye model does not do is to explain heat capacities above the Doulomb-Petit value. Although, notice on the right plot that these are correlated with lower Debye temperatures. We will come back to this issue. Now that we have an explanation of how heat capacity due to atomic vibrations varies with temperature, let's see if we can figure out heat capacities of multi-atomic gases. We've seen that the molar heat capacity of monatomic gases is accurately described by the simple model of billiard balls bouncing around inside a box. The value is 3 halves r, which equals 12.47 joules per mole kelvin. However, this result does not apply to two atom diatomic molecules, such as nitrogen and oxygen. Those gases have molar heat capacities near 5 halves r, but the oxygen value is measurably larger than the nitrogen value. First, let's tackle an important issue for describing the heat capacity of a gas. The value is highly dependent on how we can find the gas. So far, we've primarily considered heat added to gases confined to a constant volume, but we can also confine a gas under constant pressure. Suppose we place a gas sample at temperature T1 in a cylinder with cross-sectional area A. The gas presses against the movable piston, creating a closed volume with height x1. One side of the piston is exposed to the atmosphere at pressure P0. The gas will expand or contract until the internal pressure is also P0. The volume V is the area A times the height x. From the ideal gas law, P0 V1 equals Nk T1. Suppose we add heat to the gas to increase its temperature to T2. If the volume were held constant, the gas pressure would increase. However, with the gas pressure held constant, the volume will increase. The piston will move to height x2. Now the ideal gas law reads P0 V2 equals Nk T2. The piston has moved a distance x2 minus x1. This requires work equal to force times displacement. The force is pressure times area P0 A, so the work W is P0 A times x2 minus x1, which is P0 times the volume change V2 minus V1. Using the gas law expressions, this is Nk times T2 minus T1. The heat capacity at constant pressure Cp is defined by added heat Q equals Cp times temperature change T2 minus T1. This does not equal the heat we would need to add to the gas at constant volume to get the same temperature increase. Cv times temperature change, or Cv is the heat capacity at constant volume. For the constant volume case, all added heat goes into increasing the internal energy at the temperature of the gas. For the constant pressure case, we need to add the additional heat that gets converted into work, Nk times the temperature change. Canceling the common factor T2 minus T1, we have Cp equals Cv plus Nk. If N is Avogadro's number, then Nk is the gas constant R, and the molar heat capacity at constant pressure equals the molar heat capacity at constant volume plus R. For an ideal monatomic gas, the molar heat capacity at constant volume is 3 halves R, so the molar heat capacity at constant pressure is 5 halves R. For a gas, we need to be clear which form of heat capacity we are dealing with. For liquids and solids, expansion with heating is a much smaller effect, and we will assume the heat converted to work in expansion is negligible. So for liquids and solids, we will take the heat capacity at constant volume or constant pressure to be effectively the same. Now, let's look at a fundamental principle in the theory of heat capacity, the concept of equal partition of energy. We considered a form of this in video two of this series. There we simulated the gas in which one half of the atoms have mass M1, and the other half, mass M2, equals 4 times M1. Setting the heavier atoms in motion, the lighter atoms eventually end up moving faster, on average, than the heavy atoms. This difference in speed is just that required to give the two groups of atoms the same, on average, kinetic energy. We say that the total energy of the gas has been equally partitioned among the atoms. So, one aspect of equal partition of energy is that at the same temperature, atoms of different masses will have the same average kinetic energy. Also in video two, we saw that the microscopic significance of temperature is that one half kT equals the average kinetic energy of an atom's motion in a single spatial dimension. One half M mean of Vx squared, or one half M mean of Vy squared, or one half M mean of V squared. The total energy of an atom is therefore 3 times one half kT. So another aspect of equal partition of energy is that each translational degree of freedom will, on average, contain energy one half kT. In video three of the mechanic series, we saw that for a mass oscillating on a spring, a harmonic oscillator, energy continuously transitions between potential and kinetic forms, with the average kinetic energy equal to the average potential energy. Therefore, another aspect of equal partition of energy is that harmonic oscillators will have the same average potential and kinetic energies, each one half kT. Now, let's see what happens when we mix single atom, the red and green balls, and two atom diatomic molecules, the blue balls. All balls represent point masses. The blue balls are rigidly attached so that their centers always maintain the same distance. We give the red ball an initial velocity and allow the system to evolve. All three molecules undergo a translational motion in two spatial dimensions. In addition, the diatomic molecule undergoes rotational motion about an axis perpendicular to the screen. Each molecule has two degrees of translational freedom. The diatomic molecule also has one degree of rotational freedom. Here, the top two plots show the total kinetic energy of the red and green balls over time. The third plot shows the translational kinetic energy of the blue balls, and the fourth plot shows their rotational kinetic energy. There are seven degrees of freedom in this system, two of translation for each of the three molecules, and one of rotation for the diatomic molecule. The labels at left show the average of each plot as a fraction of the total energy of the system. The red ball's two degrees of freedom contain, on average, 2.09 sevenths of the total energy. For the green ball, it's 1.87 sevenths. For the blue molecule's translational energy, it's 1.96 sevenths. Each of these values is very close to two sevenths, the number of translational degrees of freedom of the molecule over the total number of degrees of freedom. The bottom plot shows the rotational kinetic energy of the blue molecule. The average value is 1.07, very close to 1.7, the number of rotational degrees of freedom of the molecule over the total number of degrees of freedom. This demonstrates how yet another aspect of equal partition of energy is that each rotational degree of freedom will, on average, contain the same energy as a translational degree of freedom, namely 1.5 kT. More formally, in general, if the total energy, the so-called Hamiltonian of the system, is a quadratic function of m-momenta p and n-coordinates q of the form a1 p1 squared plus a2 p2 squared and so on plus b1 q1 squared plus b2 q2 squared and so on. Here the q's can be translational coordinates, distances, or rotational coordinates, angles. The p's are the corresponding linear or angular momenta. Then the number of quadratic degrees of freedom in the system is m plus n, and the molar heat capacity at constant volume is the number of degrees of freedom over 2 times the gas constant r. Note that degrees of freedom can be defined in different ways. Our definition is that a degree of freedom corresponds to a linear or angular coordinate or the associated momentum that appears in a quadratic term in the expression for total energy. These are sometimes called quadratic degrees of freedom. To emphasize this, we use the unusual notation, DOF, for degrees of freedom.