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.

Section 40–4 The distribution of molecular speeds

Molecular collisions / Independent of direction / Relativistic effects

 

This section can be examined from the perspectives of molecular collisions, directional independence, and relativistic effects. Strictly speaking, Feynman is not explaining the distribution of molecular speeds, but the distribution of molecular velocities. For example, Figure 40-5 depicts a velocity distribution function with a Gaussian, bell-shaped form, which differs from the speed distribution function (chi distribution). However, the section could be titled “The Maxwell–Boltzmann distribution,” which refers broadly to the probability governing molecular motion in an ideal gas at thermal equilibrium.

 

1. Molecular collisions:

“Now we return to the question about the neglect of collisions: Why does it not make any difference? We could have pursued the same argument, not with a finite height h, but with an infinitesimal height h, which is so small that there would be no room for collisions between 0 and h. But that was not necessary: the argument is evidently based on an analysis of the energies involved, the conservation of energy, and in the collisions that occur there is an exchange of energies among the molecules. However, we do not really care whether we follow the same molecule if energy is merely exchanged with another molecule. So it turns out that even if the problem is analyzed more carefully (and it is more difficult, naturally, to do a rigorous job), it still makes no difference in the result (Feynman et al., 1963).”

 

Feynman emphasizes that neglecting collisions in deriving the molecular velocity distribution does not alter the final result, because what matters is the total molecular energy, not which molecule carries it. Even though collisions continuously redistribute energy among molecules, total energy is conserved, so the statistical distribution of velocities remains unchanged. In Illustrations of the dynamical theory of gases, Maxwell (1860) writes: “the mean distance travelled over by a particle between consecutive collisions, = 1/447000th of an inch, and each particle makes 8,077,200,000 collisions per second (p. 32).” This shows that Maxwell did not simply ignore collisions; rather, he derived the mean free path—the average distance of a molecule traveled between collisions—without tracking individual trajectories. However, Maxwell did not elaborate on how collisions drive gases toward thermal equilibrium.

In contrast, Boltzmann placed collisions at the heart of his model: they enable the transfer of energy among molecules, gradually reshaping the distribution of energies and velocities until the equilibrium distribution is established is reached. This is related to Boltzmann’s H-theorem, which shows that when collisions between molecules are allowed, such distributions tend to irreversibly seek towards the minimum value of H. Molecular collisions are therefore essential for explaining not just the form of the distribution, but also its dynamic emergence. Feynman’s discussion, which downplays collisions, is better seen as a pedagogical simplification to focus on intuition and the final distribution. However, it is worthwhile to recognize the distinct contributions of Maxwell and Boltzmann in shaping our understanding of the molecular velocity distribution.

 

Note: In Further Studies on the Thermal Equilibrium of Gas Molecules, Boltzmann (1872) writes “This is essentially the result already obtained in another way by Maxwell: once this velocity distribution has been reached, it will not be disturbed by collisions (p. 263)……. As a result of collisions, many molecules will acquire larger velocities and others will come to have smaller velocities, until finally a distribution of velocities among the molecules is established such that it is not changed by further collisions. In this final distribution, in general all possible velocities from zero up to a very large velocity will occur (p. 265).” It shows that Boltzmann explicitly identified collisions as the mechanism by which gases approach thermal equilibrium.

 

2. Independent of direction:

“So far we have, of course, only the distribution of the velocities “vertically.” We might want to ask, what is the probability that a molecule is moving in another direction? Of course these distributions are connected, and one can obtain the complete distribution from the one we have, because the complete distribution depends only on the square of the magnitude of the velocity, not upon the z-component. It must be something that is independent of direction, and there is only one function involved, the probability of different magnitudes (Feynman et al., 1963).”

 

In Illustrations of the Dynamical Theory of Gases, Maxwell (1860) assumed “the existence of the velocity x does not in any way affect that of the velocities y or z, since these are all at right angles to each other and independent (p. 22).” This independence assumption (or intuition) allowed him to factorize the probability distribution function. However, his contemporaries were uneasy with his approach: while isotropy could be accepted as a consequence of spatial symmetry, statistical independence of perpendicular components seemed a less obvious claim (Brush, 1976, pp. 177–179). In On the Dynamical Theory of Gases, Maxwell (1867) responded to their concerns by providing a more rigorous justification, but it was not a direct use of Newton’s laws of motion. With this statistical proof, Maxwell strengthened the foundations of kinetic theory despite its limitations of applicability.

 

It is important to recognize that isotropy (independence of direction) and the statistical independence of velocity components are distinct concepts. Isotropy means all directions of motion are equally probable in an ideal gas and the distribution is rotationally invariant. Statistical independence means the probability distribution for one component of velocity (e.g., v_x) is independent of the distributions for the perpendicular components (v_y, v_z). Currently, the Maxwell–Boltzmann distribution is derived from the canonical ensemble, where the Gaussian form of the velocity distribution leads to both isotropy and statistical independence as results rather than assumptions. In a sense, Maxwell effectively inverted the modern approach by assuming statistical independence right at the beginning. However, directional independence and statistical independence are not universally valid features of all velocity distributions.

 

Note: Maxwell’s derivation of the distribution of molecular velocities rested on two key assumptions: (1) Isotropy (independent of direction) and (2) Statistical Independence (Brush, 1976). Maxwell later recognized that the second assumption was precarious. The assumption of isotropy does not necessarily imply the statistical independence of the variables along different directions (Walstad, 2013). To be precise, the Maxwell–Boltzmann distribution was derived under the assumptions of an ideal gas with no external force fields or significant gravitational effects.

 

Of course these distributions are connected, and one can obtain the complete distribution from the one we have, because the complete distribution depends only on the square of the magnitude of the velocity, not upon the z-component (Feynman et al., 1963).”

 

The word connected used by Feynman is potentially misleading, since in an ideal gas at thermal equilibrium the velocity components are statistically independent. One possible clarification is that the connection is mathematical rather than physical. Mathematically, the Gaussian form of the full distribution f(v_x, v_y, v_z) implies the probability law for any single component (e.g., v_z), so the “connection” follows from probability theory and symmetry. Physically, the velocity components are uncorrelated—random collisions do not somehow connect v_x, v_y, and v_z of moving molecules together by some unknown forces or influences. Recognizing this distinction prevents the potential misconception that the distribution of  molecular velocities arises from some inherent physical linkage between the velocity components, rather than from statistical symmetry.

 

3. Relativistic effects:

“Since velocity and momentum are proportional, we may say that the distribution of momenta is also proportional to e^−K.E./kT per unit momentum range. It turns out that this theorem is true in relativity too, if it is in terms of momentum, while if it is in velocity it is not, so it is best to learn it in momentum instead of in velocity: f(p)dp=Ce^−K.E./kTdp (Feynman et al., 1963).”

 

Feynman emphasizes that the Maxwell–Boltzmann distribution can be expressed in terms of momentum rather than velocity. In the non-relativistic case, momentum and velocity are proportional p = mv, so the probability distributions in either variable are equivalent, with the distribution proportional to e^−K.E./kT. In the relativistic case, the proportionality between momentum and velocity does not hold, so the velocity distribution cannot remain as a simple form. However, expressing the distribution in terms of momentum, f(p)dp=Ce^−K.E./kTdp, preserves the exponential form even at relativistic speeds. Therefore, Feynman suggests that the distribution in terms of momentum is more fundamental, which is applicable beyond classical speeds. This approach highlights that momentum-based distributions provide a consistent description of thermal equilibrium for particles moving at classical and relativistic speeds.

 

Feynman’s statement requires careful interpretation in the context of special relativity, where momentum and velocity are no longer simply proportional and directional effects become significant. The correct relativistic generalization is the Maxwell–Jüttner distribution. Unlike the Maxwell–Boltzmann distribution, which is isotropic in the rest frame of the gas, the Maxwell–Jüttner distribution exhibits apparent directional dependence when viewed from a moving frame because of relativistic transformations. In this respect, Feynman’s explanation oversimplifies the relativistic case and risks leaving the impression that the classical Maxwell–Boltzmann form remains valid in special relativity. However, the Maxwell–Jüttner distribution reduces to the Maxwell–Boltzmann distribution in the non-relativistic limit, thereby unifying the description of thermal equilibrium for both classical and relativistic gases.

 

Difference between distribution of molecular velocities and molecular speeds:

The distribution of molecular velocities gives the probability that a molecule has a specific velocity vector, i.e., that its components lie between v_x and v_{x +} dv_x, v_y and v_{y +} dv_y, as well as v_z and v_{z +} dv_z. By contrast, the distribution of molecular speeds gives the probability that a molecule has a certain speed, regardless of direction. In this case, the probability of finding a molecule with speed between v and v + dv is obtained by summing over all velocity vectors whose magnitude is v. Geometrically, this corresponds to integrating over the spherical shell of radius  in velocity space. Thus, the velocity distribution is directional and vector-based, while the speed distribution is scalar and direction-independent (See figure below).

Source: Maxwell–Boltzmann distribution | tec-science

 Analogy: To see the difference between the distribution of molecular velocities and the distribution of molecular speeds, imagine throwing darts at a flat target. The velocity distribution is like asking for the probability of a dart landing in a tiny square at some coordinates (x, y) on the target—it tracks direction as well as magnitude. The speed distribution, by contrast, is like asking for the probability that a dart lands at a given distance from the bullseye, regardless of angle; what matters is the radius, not the coordinates. There is only one point lies at the bullseye (speed = 0), but entire rings of points exist at larger radii (higher speeds), so higher speeds could be more likely. The velocity components follow Gaussian (bell-curve) distributions (see figure below), while the speed follows a chi distribution that rises from zero, peaks, and then tails off asymmetrically.

Source: Derivation of the Maxwell-Boltzmann distribution function | tec-science

Historical Note:

In his 1867 paper On the Dynamical Theory of Gases, Maxwell reflected on his earlier mistakes: “I also gave a theory of diffusion of gases, which I now know to be erroneous, and there were several errors in my theory of the conduction of heat in gases which M. Clausius has pointed out in an elaborate memoir on that subject (p. 51).” Later in the same paper, he addressed a key assumption in his derivation: “1 have given an investigation of this case, founded on the assumption that the probability of a. molecule having a velocity resolved parallel to x lying between given limits is not in any way affected by the knowledge that the molecule has a given velocity resolved parallel to y. As this assumption may appear precarious, I shall now determine the form of the function in a different manner (p. 62).” His remark reveals an early awareness that the independence of velocity components was not self-evident but required justification. The fact that he attempted to resolve the potential problem within the same work shows both his humility and scientific thoroughness.

 

Key Takeaway:

The behavior of gas molecules can be described only statistically, not deterministically, with equilibrium distributions emerging from symmetry, randomness, and conservation laws. Maxwell’s work shows that assumptions such as isotropy and statistical independence—when carefully justified—lead naturally to the Gaussian form of the velocity distribution (the Maxwell–Boltzmann distribution), which successfully explains macroscopic properties like pressure and temperature. His treatment also highlights the importance of making assumptions explicit, testing their validity, and, where possible, providing multiple lines of justification, as he did with the independence of velocity components. Importantly, the Maxwell–Boltzmann distribution is not obvious—without Maxwell’s insight, one might wrongly expect velocities to be uniformly distributed, missing the statistical order in molecular chaos.

 

The Moral of the lesson: Maxwell’s willingness to acknowledge weaknesses in his earlier paper, and to replace them with a more rigorous derivation, reflects both intellectual honesty and scientific integrity.

 

Review Questions:

1. Should molecular collisions be neglected when deriving the distribution of molecular velocities, and what role do they play in reaching equilibrium?

2. Are molecular velocities truly independent of direction in the context of the Maxwell–Boltzmann distribution?

3. Does Maxwell’s distribution of molecular velocities remain the same or valid at relativistic speeds if it is expressed in terms of momentum?

 

References

Boltzmann, L. (1872). Further studies on the thermal equilibrium of gas molecules. In The kinetic theory of gases: an anthology of classic papers with historical commentary (pp. 262-349).

Brush, S. G. (1976). The Kind of Motion We Call Heat: A History of the Kinetic Theory of Gases in the 19th Century. Amsterdam: North-Holland.

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). V. Illustrations of the dynamical theory of gases.—Part I. On the motions and collisions of perfectly elastic spheres. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 19(124), 19-32, 281–291.

Maxwell, J. C. (1867). On the dynamical theory of gases. Philosophical Transactions of the Royal Society of London, 157, 49–88.

Walstad, A. (2013). On deriving the Maxwellian velocity distribution. American Journal of Physics, 81(7), 555-557.