Section 40–1 The exponential atmosphere

Idealizations / Approximations / Limitations

 

This section could be understood in terms of idealizations, approximations, and limitations. The core ideas trace back to Boltzmann’s (1868) first statistical paper, where he established that the probability of a state with energy E is proportional to e^−E/kT. The Boltzmann density distribution for an isothermal atmosphere was not derived by Boltzmann in its modern form, but it follows directly from his statistical mechanics framework. Historically, Laplace provided the first derivation of the barometric formula in Traité de mécanique céleste. In addition, Laplace expressed it in logarithmic form, since the exponential function, though known from Euler’s work, was not yet in common use among physicists. Well into the 19th century, logarithms remained the practical computational tool of choice for scientists, navigators, and astronomers.

 

1. Idealizations

“We limit ourselves for the present to conditions of thermal equilibrium, that is, to a subclass of all the phenomena of nature. The laws of mechanics which apply just to thermal equilibrium are called statistical mechanics… (Feynman et al., 1963, p. 40-1).”

 

The derivation of the Boltzmann density distribution for an isothermal atmosphere requires at least three key idealizations to simplify the complex behavior of real gases: (1) Thermal equilibrium: The temperature (T) of gases is assumed to remain constant at all altitudes h by neglecting the temperature gradients in the real atmosphere.

(2) Hydrostatic equilibrium: The atmosphere is assumed to satisfy dP/dh = −ρg, meaning the pressure gradient exactly balances the downward gravitational force, so there is no net vertical acceleration of the gas.

(3) Uniform gravitational field: The gravitational field g is treated as constant, even though it decreases slightly with altitude.

With these assumptions, the particle density can be derived as an exponential function of height, n(h) = n₀e^−mgh/kT, which is the Boltzmann density distribution.

 

The three idealizations—thermal equilibrium, hydrostatic equilibrium, and a uniform gravitational field—enable physicists to develop a simple model for molecular density in an isothermal atmosphere. The exponential decay of density with height can be understood intuitively as a self-diminishing process: the rate of loss depends on how much remains. A useful analogy is a crowd of hikers climbing a mountain: (1) At lower altitudes, where the "density" of hikers is high, many feel tired and turn back, so the dropout rate is large initially. (2) At higher altitudes, with fewer hikers left, fewer give up, and the dropout rate diminishes. The key insight is that the larger the quantity, the faster it diminishes—a hallmark of exponential decay.

 

In his textbook Statistical Mechanics, Feynman (1972) defines thermal equilibrium as follows: “If a system is very weakly coupled to a heat bath at a given ‘temperature,’ if the coupling is indefinite or not known precisely, if the coupling has been on for a long time, and if all the 'fast' things have happened and all the 'slow' things not, the system is said to be in thermal equilibrium. For instance, an enclosed gas placed in a heat bath will eventually erode its enclosure; but this erosion is a comparatively slow process, and sometime before the enclosure is appreciably eroded, the gas will be in thermal equilibrium (p. 1).”

 

2. Approximation:

“Now mg is the force of gravity on each molecule, where g is the acceleration due to gravity, and ndh is the total number of molecules in the unit section. So this gives us the differential equation P_h+dh−P_h = dP =−mgndh. Since P=nkT, and T is constant, we can eliminate either P or n, say P, and get dn/dh=−(mg/kT)n for the differential equation, which tells us how the density goes down as we go up in energy (Feynman et al., 1963, p. 40-2).”

 

Using the ideal gas law P = nkT instead of the van der Waals equation when deriving the Boltzmann distribution is an approximation method that treats real gases as ideal. The Boltzmann factor e^−mgh/kT, however, is universal—it applies to many systems in thermal equilibrium (or temperature is approximately constant). Furthermore, we must assume hydrostatic equilibrium and uniform gravitational field, which can be expressed by the equation: dP=P(h+dh)−P(h) = −mg n dh. However, the ideal gas law is sometimes described as an idealization because it rests on assumptions about “perfect” gases that do not exist in nature. At the same time, it can also be used as an approximation, chosen in place of more refined equations such as the van der Waals equation when solving problems. Specifically, the ideal gas law works well for gases at low pressures and high temperatures, where the effects of molecular size and intermolecular forces are negligible.

 

The Boltzmann distribution for the density of an ideal gas in an isothermal atmosphere can be derived in terms of the scale height, H = kT/mg, which defines a natural length scale over which density decreases. Thus, the differential equation dn/dh = −(mg/kT)n can be simplified as dn/dh = −n/H. Its solution, n(h)=n₀e^−h/H, shows that the density decreases by a factor of e (≈ 2.718) for every increase of H in attitude. This exponential behavior is analogous to radioactive decay, where a characteristic half-life determines how fast a quantity diminishes. We may define a half-height, the increase in altitude over which the density falls to half of its original value: n(h_1/2) = ½ n₀ ⇒ h_1/2 = H ln2 ≈ 0.693H. The scale height and half-height provide an intuitive sense of how quickly the atmosphere “thins out” with height.

 

3. Limitations:

“Therefore we would expect that because oxygen is heavier than nitrogen, as we go higher and higher in an atmosphere with nitrogen and oxygen the proportion of nitrogen would increase. This does not really happen in our own atmosphere, at least at reasonable heights, because there is so much agitation which mixes the gases back together again. It is not an isothermal atmosphere (Feynman et al., 1963, p. 40-2).”

 

The Earth’s atmosphere cannot remain at a uniform temperature with altitude because solar radiation interacts differently across its layers. For instance, the stratosphere becomes warmer with altitude due to the absorption of ultraviolet radiation by ozone layers. Generally speaking, dynamic atmospheric circulation—including winds and convective storms—redistributes heat vertically and laterally, disrupting any tendency toward thermal equilibrium. Phase changes of water vapor add further complexity: condensation releases latent heat in some regions, while evaporation extracts heat (cooling effect) elsewhere. These competing radiative and convective processes prevent the atmosphere from achieving isothermal conditions.

 

We should be aware of the limitation of the isothermal atmosphere model, whose applicability is restricted to special cases:

1. Low Altitudes: While the troposphere is never truly isothermal, the assumption may be reasonable over very small vertical scales (e.g., <1 km) where temperature variations are minimal.

2. Dilute Gases: In upper atmospheric layers, where molecular collisions are rare, the Boltzmann distribution can describe density decay—but only if temperature is nearly constant, which is seldom the case under solar radiation.

3. Short Timescales: The model ignores convection and radiative processes that drive temperature variations. However, over brief periods—before large-scale heat transport or weather systems intervene—a local isothermal approximation may hold.

While this framework cannot capture the essential physics of real planetary atmospheres or complex thermodynamic processes, but it is still useful for pedagogical purposes or intuitive understanding of exponential density decay.

 

“Suppose that we have a column of gas extending to a great height, and at thermal equilibrium—unlike our atmosphere, which as we know gets colder as we go up (Feynman et al., 1963, p. 40-1).”

 

Feynman explains that the gas gets colder as we go up, but it becomes warmer at higher altitudes. In the troposphere (near Earth’s surface), temperature decreases by about 6.5 °C per kilometer (Wallace & Hobbs, 2006), as rising air expands and cools under lower pressure. Above this, in the stratosphere, temperature increases with altitude because ozone molecules absorb ultraviolet radiation. The mesosphere follows, where temperature falls sharply due to decreasing air density and minimal ozone available for ultraviolet absorption. Thus, the isothermal atmosphere is a useful first approximation for explaining why pressure and oxygen density decreases with height, but it misses the “cold-warm-cold” (temperature flip) of real atmosphere (see figure below). This provides an instructive reminder that our intuition about atmospheric temperature at higher altitudes can be misleading, both for students and for physicists.

Source: Wallace & Hobbs, 2006

Review questions:

1. What key idealizations or assumptions are made in deriving the Boltzmann distribution for the density of an isothermal atmosphere?

2. How would you derive the Boltzmann distribution for atmospheric density under isothermal conditions?

3. What are the main limitations of applying the Boltzmann distribution for atmospheric density under real conditions?

 

Exponential Air Loss and Altitude Sickness: In the isothermal atmosphere model, air pressure — and oxygen availability — falls exponentially, not linearly. Each kilometer reduces a percentage of air, so oxygen pressure drops faster the higher you go. As partial pressure falls, your blood absorbs less oxygen, causing hypoxia. Symptoms can escalate from headaches to life-threatening edema — which is why climbers in Mount Everest’s 8,848 m “death zone” need supplemental oxygen.

Bottom line: The exponential model may be simplified, but it captures why high-altitude climbing is not just harder — it’s physiologically challenging.

 

The Moral (In Feynman’s Style):

The Boltzmann factor isn’t the whole story. Sure, most molecules move around average speeds, with fewer “fast and furious” ones. But real air? It’s got humidity (or mist*), turbulence, and sometimes a seagull flying through!

The distribution of money requires statistical mechanics too (Dragulescu & Yakovenko, 2000). For example, more folks have $5K than $5 million in the bank. But the economy isn’t some tidy lab experiment: raise interest rates, impose tariffs, or let the stock market tumble, and the whole curve wriggles and shifts.

The beauty isn’t that exponentials are right—it’s that they’re useful. Remember, the Boltzmann factor isn’t nature’s law—it’s her favorite approximation. The fun begins when you ask: “Okay—where does nature start cheating?”

 

*Many physicists drew inspiration from climbing mountains, seeing in the ascent a metaphor for the trials of scientific discovery. Among them was Werner Heisenberg, who in his memoir Physics and Beyond (1971) vividly recalled a trekking experience:

“If I think back on the state of atomic theory in those months, I always remember a mountain walk with some friends from the Youth Movement, probably in the late autumn of 1924. It took us from Kreuth to Lake Achen. In the valley the weather was poor, and the mountains were veiled in clouds. During the climb, the mist had begun to close in upon us, and, after a time, we found ourselves in a confused jumble of rocks and undergrowth with no signs of a track. We decided to keep climbing, though we felt rather anxious about getting down again if anything went wrong. All at once the mist became so dense that we lost sight of one another completely, and could keep in touch only by shouting.”

For Heisenberg, this moment of disorientation captured the essence of working on quantum theory. However, the mist on the mountain can be linked metaphorically (and even physically) to the isothermal atmosphere model and Boltzmann factor. In other words, the foggy alpine trek Heisenberg recalled can be read as a natural illustration of the Boltzmann distribution — the higher you go, the scarcer the air (and the harder the climb), just as the higher you push theory, the "thinner" the guidance from old frameworks.

 

References:

Boltzmann, L. (1868). Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten [Studies on the equilibrium of kinetic energy among moving material points], Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, 58, 517–560.

Dragulescu, A., & Yakovenko, V. M. (2000). Statistical mechanics of money. The European Physical Journal B-Condensed Matter and Complex Systems, 17(4), 723-729.

Feynman, R. P. (1972). Statistical mechanics: a set of lectures. W. A. Benjamin.

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.

Heisenberg, W. (1971). Physics and Beyond: Encounters and Conversations. Translated by Pomerans, A. J. New York: Harper & Row.

Wallace, J. M., & Hobbs, P. V. (2006). Atmospheric Science: An Introductory Survey (2nd ed.). Academic Press.

Section 39–5 The ideal gas law

Avogadro’s law / Equipartition Theorem / Degrees of Freedom

 

In this section, Feynman examines three concepts: Avogadro’s law, the equipartition theorem, and degrees of freedom. A more appropriate title might be “Theoretical Basis of the Ideal Gas Law” or “Avogadro’s Law and Equipartition Theorem,” as these concepts provide a statistical foundation for understanding the ideal gas law. The concept of degrees of freedom was delivered at the beginning of next lecture.

 

1. Avogadro’s law:

“Furthermore, at the same temperature and pressure and volume, the number of atoms is determined; it too is a universal constant! So equal volumes of different gases, at the same pressure and temperature, have the same number of molecules, because of Newton’s laws. That is an amazing conclusion! (Feynman et al, 1963, p. 39-10).”

 

The ideal gas law can be viewed as a synthesis of Newton’s laws of motion, Avogadro’s law, and the equipartition theorem:

(1) Newton’s laws provide the mechanical framework, describing gas molecules travel in straight lines (First Law), exchange momentum during collisions (Second and Third Laws), and exert pressure through impacts against container walls.

(2) Avogadro’s law establishes that equal volumes of gases at the same temperature and pressure contain equal numbers of molecules, allowing pressure to scale with the number of particles (or moles), independent of chemical identity.

(3) The equipartition theorem connects temperature to the average kinetic energy per molecule, giving a statistical definition of temperature grounded in molecular motion.

Together, these principles provide the microscopic basis of the ideal gas law: pressure arises from Newtonian momentum transfer and temperature from the equipartition of energy across degrees of freedom. The accuracy of the ideal gas law reflects not only the predictive power of Newtonian mechanics, but also the energy distribution suggested by the equipartition theorem and the statistical regularity expressed in Avogadro’s law.

 

While Avogadro’s law states that the volume of an ideal gas is directly proportional to the number of moles at constant temperature and pressure, this relationship is an idealization that real gases only approximately follow. In practice, gases exhibit intermolecular attractions and occupy finite volumes—factors accounted for in the van der Waals equation by the gas-specific constants a (which corrects for attractive forces) and b (which corrects for molecular size). The Redlich–Kwong equation offers a further refinement by more accurately fitting experimental data across a broader range of conditions. These corrections indicate that molar volume and particle count can vary with the physical properties of the real gas, thereby violating the assumptions of ideal gas. Therefore, Avogadro’s law does not strictly hold for real gases, particularly at high pressures or low temperatures where non-ideal effects become significant.

 

In the audio recording [48:50], Feynman remarks, “It is one of those famous gas laws whose names I do not remember—Boyle’s, Charles’s, and everybody else wrapped into one.” He could have referred to Gay-Lussac’s law, which he did not name explicitly. In 1834, Benoît Paul Émile Clapeyron unified Boyle’s law, Charles’s law, Avogadro’s law, and Gay-Lussac’s law into what is now known as the ideal gas law.

 

2. Equipartition Theorem:

“The equilibrium conditions are the same. No matter where the piston is, its speed of motion must be such that it passes energy to the molecules in just the right way. So it makes no difference about the spring. The speed at which the piston has to move, on the average, is the same. So our theorem, that the mean value of the kinetic energy in one direction is ½kT, is true whether there are forces present or not (Feynman et al., 1963, p. 39-10).”

Feynman’s statement—“the mean value of the kinetic energy in one direction is ½kT, is true whether there are forces present or not”—captures the essence of the equipartition theorem but it risks being misleading. The theorem assumes that the system is in thermal equilibrium, a condition that only holds when all forces involved are conservative. Dissipative forces, such as friction or viscous drag, violate this premise by continuously extracting energy from the system, thereby disrupting equilibrium. For example, in a gas subject to viscous damping, kinetic energy is gradually transformed into thermal energy, causing local temperature fluctuations. A more accurate formulation would state: “The mean kinetic energy associated with each degree of freedom is ½kT, provided the system is in thermal equilibrium and all forces are conservative.” This caveat is crucial for applying the theorem to real-world systems, where energy losses are not negligible.

“Incidentally, we have also proved at the same time that the average kinetic energy of the internal motions of the diatomic molecule, disregarding the bodily motion of the CM, is (3/2)kT! For, the total kinetic energy of the parts of the molecule is ½m_Av_A² + ½m_Bv_B², whose average is (3/2)kT+(3/2)kT, or 3kT. The kinetic energy of the center-of-mass motion is (3/2)kT, so the average kinetic energy of the rotational and vibratory motions of the two atoms inside the molecule is the difference, (3/2)kT (Feynman et al., 1963, p. 39-11).”

 

Feynman’s derivation of the average kinetic energy of a diatomic molecule as (3/2)kT, obtained by subtracting the center-of-mass motion, offers valuable insight—but requires qualifications. The equipartition theorem is not a universal law, but a classical approximation that holds only under specific physical conditions, such as quantum effects are negligible. For example, Einstein’s (1907) model showed that vibrational degrees of freedom freeze out at low temperature when kT << ℏω. At room temperature, diatomic molecules exhibit energy contributions from translational motion (3/2)kT and rotational motion (kT), while vibrational modes contribute minimally and become significant only at higher temperatures (See figure below). This underscores the importance of specifying the conditions under which the theorem applies—its dependence on temperature and molecular structure. Strictly speaking, the equipartition theorem is derived using Hamiltonian mechanics, applied within the framework of statistical ensembles that describe how systems behave on average.

 

(Source: Young, Freedman, & Ford, 2012) 

3. Degrees of freedom:

“These “independent directions of motion” are sometimes called the degrees of freedom of the system. The number of degrees of freedom of a molecule composed of r atoms is 3r, since each atom needs three coordinates to define its position (Feynman et al., 1963, p. 39-12).”

 

The degrees of freedom (DOF) in the equipartition theorem represent independent ways a system can store energy. Each DOF acts like a storage bin, holding an average of ½kT of energy. Translational DOFs correspond to motion along the three spatial directions, while rotational DOFs depend on its molecular structure. For a rotational DOF to contribute, the molecule must have a nonzero moment of inertia about the corresponding axis. Diatomic molecules like N₂ or O₂ can rotate about two axes perpendicular to the bond axis, but rotation around the bond axis contributes negligibly because the moment of inertia is extremely small. In contrast, monatomic gases have no internal structure—effectively all mass is concentrated at a point—so any rotation is effectively like a relabeling of coordinates (without change in rotational kinetic energy). In general, molecules possess 0 (monatomic), 2 (diatomic), or 3 (polyatomic) rotational degrees of freedom; in certain special systems like adsorbed large molecules, rotational DOF can be effectively reduced to just one due to constrained motion.

 

“Our theorem, applied to the r-atom molecule, says that the molecule will have, on the average, 3rkT/2 joules of kinetic energy, of which 3/2kT is kinetic energy of the center-of-mass motion of the entire molecule, and the rest, 3/2(r−1)kT, is internal vibrational and rotational kinetic energy (Feynman et al., 1963, p. 39-12).”

 

The application of the equipartition theorem to an r-atom molecule, as outlined by Feynman in one sentence, can be clarified as follows. According to this theorem, a molecule composed of r atoms in thermal equilibrium has an average kinetic energy of (3r/2)kT, corresponding to three translational degrees of freedom per atom. Of this total, (3/2)kT is related to the translational motion of the molecule’s center of mass. The remaining (3/2)(r−1)kT is associated with internal motions—rotations and vibrations involving the relative movement of atoms within the molecule. However, this classical model is only an approximation. It does not account for the quantum nature of molecular motion, especially the quantization of vibrational and rotational energy levels, which become significant at low temperatures or for light molecules. While some degrees of freedom may be thermally inaccessible, vibrational motion can contribute both kinetic and potential energy to the system.

 

Note: In Einige allgemeine Sätze über Wärmegleichgewicht, Boltzmann (1871) writes:

“If intramolecular motions are present, and if f is the number of degrees of freedom of a molecule, then the corresponding fraction of the average kinetic energy......”

 

Review questions:

1. Do you agree with Feynman that equal volumes of different gases, at the same pressure and temperature, have the same number of molecules, because of Newton’s laws?

2. Would you state the equipartition theorem as “the mean value of the kinetic energy in one direction is ½kT, is true whether there are forces present or not”?

3. How would you explain the degrees of freedom using an analogy?

 

The Moral of the Lesson (In Feynman’s Spirit): Your body is like a molecule—its energy depends on how freely it can move! A hunched posture isn’t just about looks; it’s like freezing out degrees of freedom. Just as a rigid diatomic molecule at low temperature can’t access all its rotational and vibrational modes, a hunched spine limits your mechanical range, restricting movement and reducing kinetic energy output. The result? Fewer calories burned, a slower metabolism, higher blood sugar, and—like a system trapped in a low-temperature state—a body stuck in a suboptimal equilibrium. But when you stand tall, align your spine, and move fluidly, you're not just defying gravity—you're activating dormant degrees of freedom and reclaiming your full thermodynamic potential.

 

In short, the best posture is changing posture by increasing your body’s degrees of freedom.

 

References:

Boltzmann, L. (1871) Some General Statements on Thermal Equilibrium. Wiener Berichte, 63, 679-711.

Einstein, A. (1907). "Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme" (Planck's Theory of Radiation and the Theory of Specific Heat). Annalen der Physik, 22(1), 180-190.

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.

Young, H. D., Freedman, R. A., & Ford, A. L. (2012). Sears and Zemansky's University physics with Modern Physics, 13th ed. Pearson Higher Education.