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. 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.