Section 40–6 The failure of classical physics

Maxwell’s (1860) paper / Freezes out / Harmonic oscillator

 

In this section, Feynman discusses Maxwell’s (1860) paper, which first revealed the limitation of classical physics in explaining the behavior of gases. He then extends the discussion by introducing the quantum model of the harmonic oscillator, showing how the “freezing out” of certain modes of motion at low temperatures resolves the discrepancies found in the classical kinetic theory pertaining to the specific heat ratios. Thus, the section could be titled “A Limitation of the Kinetic Theory of Gases (or Classical Equipartition Theorem),” highlighting one of the key failures of classical physics; other, historically significant failures include the ultraviolet catastrophe, the photoelectric effect, and the discrete spectral lines of atoms.

 

1. Maxwell’s (1860) paper:

“The first great paper on the dynamical theory of gases was by Maxwell in 1859. On the basis of ideas we have been discussing, he was able accurately to explain a great many known relations, such as Boyle’s law, the diffusion theory, the viscosity of gases, and things we shall talk about in the next chapter. He listed all these great successes in a final summary, and at the end he said, “Finally, by establishing a necessary relation between the motions of translation and rotation (he is talking about the ½kT theorem) of all particles not spherical, we proved that a system of such particles could not possibly satisfy the known relation between the two specific heats.” He is referring to γ (which we shall see later is related to two ways of measuring specific heat), and he says we know we cannot get the right answer (Feynman et al, 1963, p. 40-8).”

 

In his 1860 paper Illustrations of the Dynamical Theory of Gases, Maxwell considered only translational and rotational motions when applying the equipartition theorem to molecular gases. He derived a necessary relation between these two types of motion and concluded that non-spherical molecules could not satisfy the observed ratio of specific heats at constant pressure and at constant volume. However, Maxwell’s analysis did not include other possible degrees of freedom—vibrational, torsional (rotational), and electronic—which later proved essential for understanding the specific heat capacity. These contributions were developed later, e.g., through the work of Boltzmann and quantum theory, which clarified that at ordinary temperatures many of the molecules still move and rotate, but do not vibrate (“frozen out”).

 

To be precise, Maxwell’s 1860 paper is titled “Illustrations of the Dynamical Theory of Gases.” (The paper was published in 1860, but it was read at the Meeting of the British Association at Aberdeen on September 21, 1859.) This is different from “On the Dynamical Theory of Gases,” a title that belongs to Maxwell’s later 1867 publication. This distinction matters because the earlier paper introduced the foundations of the kinetic theory of gases and the Maxwellian velocity distribution, while the 1867 work further refined and extended these ideas especially different properties of gases (diffusion, viscosity, and conductivity). While Feynman’s pedagogical insights remain undiminished, his potentially misleading description serves as a gentle reminder that the history of science is built on a meticulous attention to detail.

 

“Ten years later, in a lecture, he said, ‘I have now put before you what I consider to be the greatest difficulty yet encountered by the molecular theory’ (Feynman et al, 1963, p. 40-9).”

 

Feynman (1963) remarked, “Ten years later, in a lecture, he said, ‘I have now put before you what I consider to be the greatest difficulty yet encountered by the molecular theory.’” However, Maxwell’s actual words were slightly different. On February 15, 1875, in his lecture On the Dynamical Evidence of the Molecular Constitution of Bodies delivered before the Chemical Society, Maxwell stated: “I must now say something about these internal motions, because the greatest difficulty which the kinetic theory of gases has yet encountered belongs to this part of the subject.” Feynman’s paraphrase captures the spirit but not the precise wording of Maxwell’s statement. Moreover, this lecture was delivered roughly fifteen years after Maxwell’s 1860 paper Illustrations of the Dynamical Theory of Gases, marking a shift in focus—from establishing the foundations of kinetic theory to confronting its conceptual challenge: the role of internal molecular motions.

 

2. Freezes out:

“How can we understand such a phenomenon? Of course that these motions ‘freeze out’ cannot be understood by classical mechanics. It was only understood when quantum mechanics was discovered (Feynman et al, 1963, p. 40-9).”

 

The concept of the “freezing out” of vibrational modes is based on the foundational work of Planck and Einstein, who recognized that the classical equipartition theorem fails at low temperatures. Planck (1900) introduced the idea that oscillators (or resonators) can only exchange energy in discrete amounts of hν, explaining blackbody radiation. Einstein (1907) applied this concept to the vibrations of atoms in solids, proposing that each atom behaves like a Planck’s resonator. He showed that when kT << hν, most resonators remain in their lowest energy state, causing the specific heat to drop below the values predicted by Dulong–Petit law. In short, Einstein explains the peculiar decrease in the specific heat of solids at low temperatures using Planck’s theory of radiation.

 

“About 1905, Sir James Hopwood Jeans and Lord Rayleigh (John William Strutt) were to talk about this puzzle again. One often hears it said that physicists at the latter part of the nineteenth century thought they knew all the significant physical laws and that all they had to do was to calculate more decimal places. Someone may have said that once, and others copied it. But a thorough reading of the literature of the time shows they were all worrying about something. Jeans said about this puzzle that it is a very mysterious phenomenon, and it seems as though as the temperature falls, certain kinds of motions ‘freeze out’ (Feynman et al, 1963, p. 40-9).”

 

Lord Rayleigh and Sir James Jeans contributed to the understanding of limitations of classical physics through their work on blackbody radiation, but neither said that certain kinds of motions “freeze out.” However, the word “freeze” or “frozen” is potentially misleading because it implies that motions stop entirely—like water turning into ice, but molecular motion never truly stops, even at absolute zero. More important, the molecules still have translational kinetic energy and rotational kinetic energy, but their vibrational kinetic energy is reduced. What actually “freezes out” could be the vibrational motion (or rotational motion), which is its capacity to absorb and store additional thermal energy in that particular mode. The atoms still remain in motion, but the number (or probability) of atoms that are vibrating is dependent on the available thermal energy or temperature.

 

3. Harmonic Oscillator:

“Now it turns out that for a harmonic oscillator the energy levels are evenly spaced… Now let us see what happens. We suppose we are studying the vibrations of a diatomic molecule, which we approximate as a harmonic oscillator… All oscillators are in the bottom state, and their motion is effectively ‘frozen’; there is no contribution of it to the specific heat (Feynman et al., 1963, p. 40-9).

 

Feynman idealizes a diatomic molecule as a harmonic oscillator (linear vibrator) captures the essential concept of discrete energy levels. However, this simple model comes with physical limitations. Strictly speaking, molecular vibrations are anharmonic; the restoring force weakens as atoms separate, which could be more accurately described by the Morse potential. As the molecule vibrates, its fluctuating bond length alters the moment of inertia, thereby shifting the rotational energy levels, which are also quantized. This vibration-rotation interaction produces the distinctive fine structure observed in molecular spectra—a phenomenon which the harmonic oscillator fails to predict. Despite the limitations of the model, it illustrates that vibrational degrees of freedom "freeze out" at room temperature—a key insight that cannot be explained by classical physics.

 

“If we change the temperature but still keep it very small, then the chance of its being in state E₁ = ℏω remains infinitesimal—the energy of the oscillator remains nearly zero; it does not change with temperature so long as the temperature is much less than ℏω. All oscillators are in the bottom state, and their motion is effectively “frozen”; there is no contribution of it to the specific heat (Feynman et al., 1963, p. 40-9).”

 

The Boltzmann factor (e^{-energy gap/kT}) provides the probability of a molecule being in a higher-energy state relative to its ground state. When the energy gap is large compared with kT, this exponential factor becomes very small. (Instead of saying all oscillators are in the bottom state, it is possible that some are vibrating.) In other words, the chance that a molecule actually occupies an excited vibrational state is significantly lower because the energy is quantized. As a result, the vibrational modes do not really contribute to the molecule’s heat capacity. In essence, the process of thermal excitation is inherently statistical, governed by the Boltzmann factor, which determines probabilities. The thermal energy available from random motion is proportional to kT, but there is also a temperature fluctuation among the large number of molecules. (This could be related to the concept of random walk and Brownian movement, which will be discussed in the next chapter.)

 

 

Key Takeaways:

The classical physicist treats energy like a generous bartender with an endlessly adjustable tap: you could pour any amount—a sip, a half-glass, a full pint—into a resonator (like a molecule). However, the quantum physicist enforces a strict house rule: energy must be taken in whole quanta, like drinking beer only in full pints. No half-pints. No sips.

This single rule changes everything:

• The "Freeze Out" as a quantum problem: At low temperatures, the available thermal energy kT might only be enough for a few “sips.” In this context, quantum mechanics doesn’t allow sips. It cannot accept the partial energy, so the vibrational mode remains dormant, "frozen out," and doesn't contribute to the heat capacity.
• The Boltzmann Factor as the Club Bouncer: The probability that a molecule has enough energy to “buy a pint” is given by the Boltzmann factor e^-DE/kT. If the energy gap DE is large compared with kT, the bouncer almost always turns you away.

In short, “freeze out” does not mean the atoms stop moving, but it is about a degree of freedom becoming thermally inaccessible due to quantum energy gaps. In other words, the vibrational states are quantized and need a certain minimum of energy before they can be excited.

 

The Moral of the Lesson (In Feynman’s spirit):

In his Nobel lecture, Feynman (1965) remarks:

“The harmonic oscillator is too simple; very often you can work out what it should do in quantum theory without getting much of a clue as to how to generalize your results to other systems. So that didn’t help me very much, but when I was struggling with this problem, I went to a beer party in the Nassau Tavern in Princeton.  There was a gentleman, newly arrived from Europe (Herbert Jehle) who came and sat next to me. Europeans are much more serious than we are in America because they think that a good place to discuss intellectual matters is a beer party. So, he sat by me and asked, what are you doing and so on, and I said, I’m drinking beer.”

      However, the real lesson is not "drink beer to solve physics," but rather: "Seek out relaxed, interdisciplinary, and social environments where unexpected conversations can happen." Valuable insight emerged not from alcohol itself but from an open, spontaneous exchange with a thoughtful colleague who happened to ask the right question at the right moment. It is also important to resist romanticizing alcohol as a catalyst for scientific creativity. Using beer drinking as a form of “self-sacrifice” for solving physics problems is not only misguided but potentially harmful. Any amount of alcohol consumption has been associated with an increased risk of various health problems, including liver disease, cardiovascular issues, certain cancers, and neurological damage.

 

Review Questions

1. Why is it important to distinguish between Maxwell’s “Illustrations of the Dynamical Theory of Gases” (1860) and “On the Dynamical Theory of Gases” (1867) when interpreting Feynman’s reference to Maxwell’s work? Explain how each paper contributed differently to the development of kinetic theory and why misattributing them may obscure the historical evolution of the subject.

2. What common misconception can the term “freeze out” create, and what is the more accurate physical interpretation of the phenomenon? Clarify why the term might imply that molecular motion stops, and describe what actually happens to a molecule’s ability to absorb energy in a given mode.

3. What are the main physical limitations of modeling a diatomic molecule as a simple harmonic oscillator? Discuss how real molecular vibrations deviate from perfect harmonic behavior without using the term “freeze out.”

 

References:

Einstein, A. (1907). Planck’s theory of radiation and the theory of specific heat. Ann. Phys, 22, 180-190.

Feynman, R. P. (1965). The Development of the Space-Time View of Quantum Electrodynamics. The Official Web Site of The Nobel Prize.

Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

Maxwell, J. C. (1860). II. Illustrations of the dynamical theory of gases. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 20(130), 21-37.

Maxwell, J. C. (1867). On the Dynamical Theory of Gases. Philosophical Transactions of the Royal Society of London, 157, pp. 49-88.

Maxwell, J. C. (1875). On the dynamical evidence of the molecular constitution of bodies. Nature, 11(279), 357-359.

Planck, M. (1900). On the theory of the energy distribution law of the normal spectrum. Verh. Deut. Phys. Ges, 2(237), 237-245.

Section 40–5 The specific heats of gases

(Monatomic gas / Diatomic gas / Polyatomic gas)

 

In this section, Feynman discusses the specific heat ratios of monatomic, diatomic, and polyatomic gases to expose the limitations of the classical equipartition theorem. His main aim is to show where classical physics breaks down—how the theorem, which assumes the sharing of energy among all degrees of freedom, fails to account for experimental observations of specific heat ratios. In essence, the specific heat ratios depend on temperature rather than remains constant. Thus, the section could be titled “The inconstancy of specific heat ratios” or “limitations of the classical equipartition theorem.”

 

1. Monatomic gas:

“We may compare these numbers with the relevant measured values shown in Table 40–1. Looking first at helium, which is a monatomic gas, we find very nearly 5/3, and the error is probably experimental, although at such a low temperature there may be some forces between the atoms. Krypton and argon, both monatomic, agree also within the accuracy of the experiment (Feynman et al, 1963, p. 40-8).”

 

For a monatomic gas—such as helium, argon, or krypton—the specific heat ratio is theoretically 5/3, a value derived from the presence of only three translational degrees of freedom. In these gases, their internal energy is entirely translational, resulting C_v​ = 3R/2 and C_p = C_v​ + R = 5R/2, and thus γ = C_p/C_v​ = 5/3. This ratio, however, is not strictly constant under all conditions. In helium, quantum effects become significant at temperatures below its boiling point, leading to measurable changes in its effective heat capacity. For argon and krypton, deviations from the theoretical value may arise from weak intermolecular interactions that become significant at higher densities or lower temperatures. Moreover, the variation of γ can be explained by the “frozen” electronic degrees of freedom when atoms remain in their grounded states (Schwabl, 2006, p. 235).

 

“We saw earlier that if U is the internal energy of N molecules, then PV = NkT = (γ−1)U holds, sometimes, for some gases, maybe (Feynman et al, 1963, p. 40-7).”

 

The expression PV = NkT = (γ−1)U can be understood through the thermodynamic identity C_p = C_v​ + R. The relation between the heat capacities at constant pressure and constant volume follows from the First Law of Thermodynamics, which states that the change in internal energy ΔU of a system equals the heat added Q plus the work done on the system PdV: ΔU = Q + pdV. (Feynman did not explain (γ−1)U possibly because the First Law is introduced later, in Chapter 44.) The ratio γ = C_p/C_v expresses how much energy goes into raising the internal energy in comparison to the work done on the system. Furthermore, the relation C_p = C_v​ + R helps to explain how thermal energy not only increases the microscopic motion of molecules but also accounts for the macroscopic work associated with pressure–volume expansion in an ideal gas.

 

2. Diatomic gas:

“We turn to the diatomic gases and find hydrogen with 1.404, which does not agree with the theory, 1.286. Oxygen, 1.399, is very similar, but again not in agreement (Feynman et al, 1963, p. 40-8).”

 

Theoretically, the specific heat ratio g for hydrogen was expected to be 9/7 (» 1.286) because a diatomic molecule has 7 degrees of freedom: 3 translational, 2 rotational, and 2 vibrational. From this, C_v = 7R/2, C_p = 7R/2 + 1 = 9R/2, and thus, γ = C_p/C_v = 9/7. However, at ordinary temperatures, the vibrational modes of hydrogen are not significantly excited because the quantum energy spacing between vibrational levels is large compared with kT. With this in mind, only the 3 translational and 2 rotational degrees of freedom contribute, giving g = 7/5. Experimentally, the observed value of g » 1.40 reflects the “freezing out” of vibrational motion at moderate temperatures, whereas the theoretical value of 9/7 would emerge only at sufficiently higher temperatures where vibrational motion is possible.

 

The concept of vibrational degrees of freedom in molecules is primarily credited to Boltzmann and Planck, building on earlier insights by Maxwell. Maxwell (1860) introduced the idea that molecules possess translational and rotational motions and that their energies can be distributed statistically. Boltzmann (1876) proposed the inclusion of vibrational motions* and applied the equipartition theorem to explain the Dulong–Petit law for the specific heat capacities of solids. Later, Planck incorporated quantization of vibrational energy in his study of blackbody radiation, but it was Einstein who first explained why vibrational motion can be “frozen out” at low temperatures. This is one of the limitations of equipartition theorem, which will be discussed by Feynman at the end of the chapter.

 

*In his paper On the nature of gas molecules, Boltzmann (1876) writes: “… on the basis of his earlier results generalized by Maxwell and Watson, that then the ratio of the heat-capacities of a gas must be 1 2/3 when its molecules have a spherical form. The ratio of the heat-capacities becomes equal to 1.4 if the molecules have the form of rigid solids of rotation which are not spheres, and 1 1/3 if they are rigid bodies of any other form whatever. These numbers appear to accord at least so far with those found by experiment, that it cannot be said that experiment furnishes any confutation of the theory thus modified. It is also pointed out that the values found experimentally for the heat-capacity of gases on this hypothesis are in satisfactory accordance with the heat-capacities of solids. It is self-evident that gas molecules cannot be absolutely rigid bodies; this is disproved by spectrum-analysis. It may be that the vibrations which give rise to gas-spectra are only brief agitations lasting during the collision of two molecules, comparable to the sound- exciting vibrations which ensue when two ivory balls strike one another (p. 320).” It helped Boltzmann to formulate a more general version of the equipartition theorem—one that recognizes vibrational motion as an additional degree of freedom.

 

3. Polyatomic gas

“Let us look further at a still more complicated molecule with large numbers of parts, for example, C₂H₆, which is ethane. It has eight different atoms, and they are all vibrating and rotating in various combinations, so the total amount of internal energy must be an enormous number of kT’s, at least ½kT for kinetic energy alone, and γ−1 must be very close to zero, or γ almost exactly 1. In fact, it is lower, but 1.22 is not so much lower, and is higher than the 1 1/12 calculated from the kinetic energy alone, and it is just not understandable! (Feynman et al., 1963, p. 40-8).”

 

Feynman's reference to a specific heat ratio of “1 1/12” (≈1.083) is based on a classical calculation (e.g., g = 1 + 2/f) for ethane. In a more refined model, the eight atoms contribute 3 translational, 3 rotational, and 18 vibrational degrees of freedom. According to the Equipartition Theorem, each vibrational mode contributes R to the molar heat capacity at constant volume (Cᵥ)—½R from kinetic energy and ½R from potential energy. This leads to a total of Cᵥ = 21R. For an ideal gas, C_p = C_v​ + R = 22R, resulting in γ = 22/21 ≈ 1.05. The slight discrepancy between this and Feynman's 1.08 arises from a variation in the inclusion of vibrational degrees of freedom in the calculation. Importantly, the core of the puzzle is that both classical predictions (1.05 and 1.08) are far lower than the experimental value of ~1.22. However, in reality, vibrational modes of ethane remain “frozen out” at ordinary temperatures, making its specific heat ratio also temperature dependent.

 

“In fact, it is lower, but 1.22 is not so much lower, and is higher than the 1 1/12 calculated from the kinetic energy alone, and it is just not understandable! (Feynman et al., 1963, p. 40-8).”

 

It is not entirely clear what Feynman meant by “it is just not understandable.” His puzzlement could be related to the role of torsional modes—the hindered internal rotations in ethane, particularly twisting about the C–C bond. These torsional motions partially contribute to the molecule’s heat capacity, a concept more commonly covered in chemistry than in physics. The experimentally observed values of the specific heat arise from the combined effects of all degrees of freedom—translational, rotational, vibrational, and electronic—along with the partial activation of internal (torsional) rotation. In a sense, Kemp and Pitzer (1937) identified a potential barrier that hinders the internal rotation of the methyl groups about the C-C bond, providing the missing link that resolves Feynman’s “not understandable” discrepancy.

Source: (Gupta, 2007). 

 

Historical Note: John James Waterston deserves recognition as a pioneering figure who anticipated the equipartition of energy including translational, rotational, and vibrational degrees of freedom. His 1845 paper, however, was rejected by the Royal Society and remained unpublished for more than forty years. When Lord Rayleigh rediscovered it in 1892, he remarked: “The history of [Waterston's] paper suggests that highly speculative investigations, especially by an unknown author, are best brought before the world through some other channel than a scientific society, which naturally hesitates to admit into its printed records matter of uncertain value. Perhaps one may go further, and say that a young author who believes himself capable of great things would usually do well to secure favorable recognition of the scientific world by work whose scope is limited, and whose value is easily judged, before embarking upon higher flights (Lyttleton, 1979).” Waterston’s story thus stands as both a cautionary tale about the conservatism of scientific institutions and a testament to the vision of a solitary thinker who grasped the essence of molecular energy distribution long before it became accepted.

 

Review Questions:

1. Feynman noted that monatomic gases closely follow the theoretical specific heat ratio of 5/3. How does the concept of “frozen” electronic degrees of freedom clarify whether classical equipartition appears to hold for these gases at ordinary temperatures?

2. Do the vibrational degrees of freedom in a diatomic molecule like hydrogen remain “frozen out” at room temperatures?

3. How do hindered internal rotations (torsional modes) contribute to the heat capacity of ethane, and why did their consideration help resolve Feynman’s “not understandable” discrepancy?

 

Key Takeaway: The variation in the specific heat ratios of monatomic, diatomic, and polyatomic gases can be attributed to the activation of internal modes of motion, e.g., electronic modes in monatomic gases, vibrational modes in diatomic molecules, and torsional (rotational) modes in polyatomic molecules.

 

The Moral of the Lesson (In Feynman’s style): The equipartition theorem suggests Nature is fair, giving each degree of freedom its due. Don't believe it. Nature distributes energy like a biased landlord: translation gets a free ride, rotation is dependent on its twist, but vibration's utilities are locked to a minimum temperature. It's not a democracy; it's a thermodynamic hierarchy.

 

References:

Boltzmann, L (1876). "Über die Natur der Gasmoleküle (On the nature of gas molecules)". Wiener Berichte (in German). 74, 553–560.

Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

Gupta, M. C. (2007). Statistical thermodynamics. New Age International.

Kemp, J. D., & Pitzer, K. S. (1937). The entropy of ethane and the third law of thermodynamics. Hindered rotation of methyl groups. Journal of the American Chemical Society, 59(2), 276-279.

Lyttleton, R. A. (1979). The gold effect. In R. F. H. Duncan & M. Weston-Smith (Eds.), Lying truths: A critical scrutiny of current beliefs and conventions (pp. 57–65). Pergamon Press.

Maxwell, J. C. (1860). II. Illustrations of the dynamical theory of gases. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 20(130), 21-37.

Schwabl, F. (2006). Statistical Mechanics. New York: Springer.