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.