 Using simulations of motion in an asymmetric potential, we've seen that anharmonic oscillations are characterized by two properties. First, the average distance between atoms increases with energy, explaining the phenomenon of thermal expansion. Second, as total energy increases, the fraction due to average potential energy decreases, while the fraction due to average potential energy increases. Therefore, at a given temperature, the total energy of the anharmonic oscillator is greater than the total energy of a harmonic oscillator. They have the same average kinetic energy, but the anharmonic oscillator has more average potential energy. We can use statistical mechanics to calculate the heat capacity of our anharmonic oscillator. Following the statistical mechanics video of this series, we write the total energy as a function of momentum and position. E equals p squared over two plus x squared over two minus bx cubed over three. For simplicity, we are taking the mass and spring constant to be one. We know that the probability that momentum and position will take on particular values at a particular temperature is given by a decreasing exponential function of energy. We write f of p x equals one over z e to the minus p squared plus x squared over two kt times e to the bx cubed over three kt. If b is very small, then for the x values of interest, the exponent of the last factor is very small. Then we can use the approximation e to the u equals one plus u plus one half u squared to write the last factor as a sum of three terms. The value of the partition function z is that which makes the probability summed or integrated over all p and x values equal to one. We find z equals two pi kt plus five pi b squared over three times quantity kt squared. Then we calculate the expected value of total energy as the integral over all p and x of f of px times e of px. To second order in temperature, we obtain kt plus five b squared over six times quantity kt squared, replacing kt with two times kinetic energy and adding a plot of this function to our previous graph. We see that it matches our experimental values quite well, except at very large energy. Those double integrals are a bit intimidating. However, all the calculations can be done with a computer algebra tool. This page shows how using the open source tool maxima. Given the average energy of a single oscillator, the molar heat capacity is Avogadro's number about six times ten to the twenty-three times the rate of change of energy with respect to temperature. This works out to the gas constant r times one plus five b squared over three times kt. Our prediction is that heat capacity should increase above the classical value r linearly with temperature. Here are the molar heat capacities of lead and aluminum as a function of temperature from roughly room temperature at 300 Kelvin up to 600 Kelvin. The lead curve is quite linear. At higher temperatures, the aluminum curve is also. The dotted horizontal line is the doolong peti value. The Dubai temperatures are 87 Kelvin for lead and 390 Kelvin for aluminum. This allows us to understand why the aluminum plot is curved near room temperature while the lead plot is not. 300 Kelvin is well above the Dubai temperature of lead, so its vibrational degrees of freedom are completely thawed and heat capacity follows the classical model with the anharmonic linear correction. But 300 Kelvin is below the Dubai temperature of aluminum, so its vibrational degrees of freedom are not completely thawed out. Only as temperature exceeds the Dubai temperature does the graph transition to the linear behavior we have predicted. We now have a complete explanation for the heat capacities of solids and gases. The classical prediction is the gas constant over two times the number of quadratic degrees of freedom. The horizontal orange lines in these graphs. Values below this can be understood as due to quantum mechanical freezing of vibrational and possibly rotational degrees of freedom. Values above this can be understood as due to the effects of anharmonic oscillation. Let's return to the central idea of the third law, entropy at absolute zero temperature. According to Boltzmann's principle, the entropy of a system is Boltzmann's constant times the natural logarithm of gamma, where gamma is the number of ways to arrange the system. So the third law is fundamentally a statement about the number of ways to arrange a system at absolute zero temperature. Thinking quantum mechanically, a system has various possible energy states. At absolute zero, it must be in its ground state, the state of lowest energy, since there is no thermal energy available to excite higher energy levels. Let's denote entropy at absolute zero by S0. If S0 equals zero, then gamma must equal one. There is only a single way to arrange the system, and we say the ground state is non-degenerate. Generally speaking, perfect elemental crystals, crystals of a single element with no crystal defects, have non-degenerate ground states. Other substances may or may not have degenerate ground states. This means that gamma equals an integer GGS greater than one, the degeneracy of the ground state. Then S0 is greater than zero. A classic example of non-zero entropy at absolute zero is the so-called residual entropy of ice. In the chemistry video of the quantum mechanics series, we discussed how the four highest energy level electron orbitals of oxygen tend to form a tetrahedral structure. Two of these orbitals are filled with two electrons, while the other two contain only one electron. The unfilled orbitals strongly bond to hydrogen atoms, creating the water molecule. In the common form of ice, the oxygen atoms form a crystal lattice based on this tetrahedral geometry. In between each pair of oxygen atoms lies a hydrogen atom. If these were positioned symmetrically, as shown here, there would be no degeneracy in the ice crystal. Instead, each hydrogen atom is chemically bonded to a single oxygen atom. The result is that two of the four tetrahedral directions will be occupied by chemical bonds with hydrogen atoms. The other two directions will form weaker so-called hydrogen bonds with an adjacent hydrogen atom, which itself is chemically bonded to a neighboring oxygen atom. The ground state degeneracy of ice arises from the different ways these two chemical and two hydrogen bonds can be oriented. Here are the six ways that two hydrogen atoms can be chemically bonded to an oxygen atom. Notice that for any one of the tetrahedral directions, a blue bar, half of the arrangements have a chemically bonded hydrogen atom and half do not. If this was our only consideration, then each water molecule in the crystal would have six possible arrangements. Therefore, a crystal with n molecules would have a degeneracy of six to the n. But the arrangement of each molecule has to be consistent with the arrangements of the neighboring molecules. If we place a chemical bond on one of the tetrahedral directions, the probability that the neighboring molecule will not also have a chemical bond in this direction is one-half. For two chemical bonds, therefore, the probability of being compatible with two neighbors is one-half times one-half equals one-fourth. Therefore, the average number of arrangements for each molecule is six times one-fourth equals six-fourths. So the number of ways to arrange n molecules of water in an ice crystal should be gamma equals six-fourths to the n, which is three-halves to the n, and S0 should equal K log three-halves to the n, which can be written Kn log three-halves. If n is Avogadro's number, then Kn is the gas constant R. In 1935, Linus Pauling used this logic to argue that the residual entropy of ice at absolute zero should be R log three-halves, equal to 3.37 joules per mole Kelvin. In 1936, Geo and Stout used measurements and calculations to establish the actual residual entropy as 3.4 joules per mole Kelvin, in excellent agreement with Pauling's prediction. Let's finish by attempting a rigorous formulation of the third law of thermodynamics. The origin of the third law was the heat theorem presented by Walter Nernst in 1906. We will express this as follows. The entropy change in an isothermal process approaches zero, as the temperature does. Mathematically, the limit as temperature approaches zero, of the entropy change of any process is zero. Suppose we represent the thermodynamic state of a system on an ST diagram, entropy versus temperature. Suppose u is some parameter, such as pressure or volume, or even say, magnetic field strength. Then u equals u1 forms a curve on the ST diagram, and u equals u2 forms another curve. At a given temperature, an isothermal process in which u changes from u1 to u2, while t remains constant, is represented by a vertical line between these two curves. The vertical distance between the curves is delta S, the change in entropy due to the process. The heat theorem implies that these curves get closer together as temperature decreases, so the entropy change decreases. Ultimately, the curves converge to the same value, S0, at t equals zero. This is the entropy at absolute zero, which is independent of the value of the parameter u. Moreover, the slope of the S versus t curve, dSTt, is equal to heat capacity over temperature. According to the Einstein and Debye models, this goes to zero as the temperature does. So the two curves not only converge to the same value, they converge to the same zero slope. Now, imagine the following strategy for cooling a system to absolute zero. The parameter u can represent any physical quantity, but as a specific example, assume u is volume, and u1 is greater than u2. We start with the system at volume u1, and in contact with a heat reservoir at some temperature. Then we compress the system to volume u2. The work done in the system will be converted to heat. To maintain a constant temperature, this heat will be transferred to the reservoir, which will lower the entropy of the system. Then, we insulate the system and allow it to expand back to volume u1. During the expansion, the system will do work on its environment. But this will lower its internal energy and enhance its temperature. Because no heat is transferred, the entropy will remain constant. Now, we repeat this process, starting at this lower temperature. Because the u1 and u2 curves converge, these steps will get ever smaller, and no finite number of such steps will ever get us to absolute zero. This type of reasoning led Nernst in 1912 to formulate the unattainability principle. No process can reach absolute zero temperature in a finite number of steps and in a finite amount of time. The heat theorem implies the following form of the third law, as stated by Charles Kittle in his textbook, Thermophysics. The entropy of a system has the property that S goes to S0 as T goes to zero, where S0 is a constant independent of the external parameters which act on the system. A special case was stated by Max Planck. For a perfect elemental crystal, the limit as T goes to zero of the entropy is zero. And this is true for any finite value of an external parameter u. In video five of this series, we derive the Sacher-Tetrot equation, which gives the absolute entropy of a monatomic gas. Experimentally, absolute entropy is determined by starting with the assumption of zero entropy at zero temperature, consistent with Planck's statement of the third law. Then experimental entropy increases or sum from zero to the temperature of interest. The excellent agreement between experiment and the predictions of the Sacher-Tetrot equation confirm the validity of the assumption of zero entropy at zero temperature. Planck's formulation of the third law, therefore, is firmly established experimentally. However, it is not a general statement, but only applicable to the special case of a perfect elemental crystal. In the early 1900s, Nernst, Einstein, and others vigorously argued the foundations of what we now call the third law. Einstein questioned if the concept of an isothermal process at absolute zero is possible, even in principle. He agreed that the heat theorem implies the entertainability principle, but he denied the converse that the entertainability principle implies the heat theorem. Einstein also argued that the entropy at absolute zero could not, in general, be a constant independent of any external parameters. Let's look at a cartoonishly simple thought experiment. Suppose we have N atoms. Half are of one type, with a larger mass pictured as a solid. The other half are of another type, with a smaller mass pictured as cross-hatched. Suppose the atoms do not form chemical bonds, but act like loose balls. Alternately, suppose both types have identical chemical properties. In either case, they will not sort themselves by some chemical process. Let our external parameter be a downward-pointing gravitational field. The system is held at a temperature high enough that individual atoms can diffuse throughout the system. The atoms will tend to separate with the heavier toward the bottom and the lighter toward the top. Then, assume the temperature is lowered very slowly to absolute zero, so the structure is frozen in place. The system will freeze in its lowest energy state, one in which all heavy atoms are on the bottom and all light atoms are on the top. Since each atom is indistinguishable from the others of the same type, there is only gamma equals one way to arrange this system. Now, repeat the process with the external parameter set to zero. No gravitational field is present. The system will freeze into an arrangement in which the atoms are randomly positioned. For large n, the number of possible arrangements is gamma approximately equal to two to the n. So, the entropy of this system at absolute zero depends on the value of the external parameter, the gravitational field. With a gravitational field, the entropy is zero. Without a gravitational field, the entropy is approximately kn log two. Therefore, Einstein argued, although using more sophisticated thought experiments, the heat theorem cannot be true in all conceivable cases. Indeed, it is stated by a recent paper in Nature Communications titled, A General Derivation and Quantification of the Third Law of Thermodynamics. One can easily find families of Hamiltonians, which satisfy or violate the heat theorem. So, the heat theorem and the associated statement of the third law is given by Kittle. Cannot be true in general, although they may be in situations actually observed to date in the laboratory. This has led some authors to argue that the third law is a statement about the kinds of systems actually encountered in Nature, excluding systems which are theoretically possible, but not actually observed. In this view, the third law, unlike the first and second laws, cannot be established from first principles. It does not have a solid theoretical basis. In the Reference Nature Communications paper, the authors use quantum information theory to argue that, even though the heat theorem is not absolutely valid, the entertainability principle is. When analyzed quantum mechanically, the entertainability principle does not depend on the heat theorem for its validity, as Nernst had argued. Consistent with this, to date, no experiment has reached absolute zero. And no one has yet put forth a sound theoretical strategy for doing so. As a result, the entertainability principle stands as the most widely accepted general formulation of the third law. In light of these considerations, a reasonable conclusion, given in a paper on the early history of the heat theorem and third law, is that the third law is of a different, less fundamental, and more experimental character than the other laws of thermodynamics.