Section 42–1 Evaporation

Idealizations / Approximations / Limitations

 

In this section on evaporation, Feynman uses a simplified kinetic-theory picture: he treats the liquid as if each surface molecule occupies a definite area A and volume V_a; he assumes a single, well-defined binding energy W that must be overcome to escape, and the molecules behave like nearly independent particles in a liquid. He estimates the escape time crudely as D/v (a molecular diameter divided by a average speed), ignores angular distributions, surface structure,  collective effects, and temperature dependence of W and V_a, and assumes W>>kT so that the exponential dominates all prefactors—hence the model captures the essential exponential temperature dependence of vapor density and evaporation rate but cannot provide quantitatively precise coefficients or account for detailed molecular interactions.

 

1. Idealizations

“Let us say that n equals the number of molecules per unit volume in the vapor. That number, of course, varies with the temperature. If we add heat, we get more evaporation. Now let another quantity, 1/V_a, equal the number of atoms per unit volume in the liquid: We suppose that each molecule in the liquid occupies a certain volume, so that if there are more molecules of liquid, then all together they occupy a bigger volume. Thus if V_a is the volume occupied by one molecule, the number of molecules in a unit volume is a unit volume divided by the volume of each molecule (Feynman et al., 1963, p. 42-1).”

 

Feynman’s use of 1/V_a for number density may seem indirect, since he is expressing a “number per unit volume” as the reciprocal of a volume rather than as a direct count. Importantly, V_a itself is not the literal geometric volume of a molecule; it is an effective average volume per molecule in the liquid. It represents the total space associated with each molecule—including the small gaps between the molecules and the constraints imposed by intermolecular forces. One may think of V_a as the size of a “parking space” required for a single molecule. Just as each car in a parking lot requires an allocated space larger than its physical dimensions, each molecule in a liquid is associated with an effective average volume depending on its thermal motion and intermolecular interactions. From a quantum mechanics standpoint, even the notion of a molecule’s sharply defined “classical” volume is itself an idealization.

 

“We shall suppose that each molecule at the surface of the liquid occupies a certain cross-sectional area A. Then the number of molecules per unit area of liquid surface will be 1/A. And now, how long does it take a molecule to escape? If the molecules have a certain average speed v, and have to move, say, one molecular diameter D, the thickness of the first layer, then the time it takes to get across that thickness is the time needed to escape, if the molecule has enough energy (Feynman et al., 1963, p. 42-3).

Feynman's idealization of a well-behaved discrete monolayer at the liquid’s surface transforms a complex phenomenon into a solvable problem. First, he defines each surface molecule as having a fixed “cross-sectional area” A, so that the number of molecules per unit area of liquid surface becomes 1/A; this ignores the fact that real molecules—especially non-spherical ones—rotate, vibrate, and present fluctuating effective area. Second, he treats the liquid-vapor interface as a sharp boundary (thickness D) as if only the outermost molecules in liquid are waiting their turn to depart; in reality, the interface is a fuzzy and dynamic region where molecules continually moving between liquid and vapor. Third, he represents molecular motion by a single average speed v, ignoring the Maxwell distribution of velocities—only a fraction of molecules have the right direction and sufficient energy to escape. Together, these idealizations provide a simple picture of molecules moving upward like orderly particles, allowing Feynman to develop a toy model of evaporation.

 

2. Approximations

“So formulas such as (42.1) are interesting only when W is very much bigger than kT… Thus the number evaporating should be approximately N_e = (1/A)(v/D)e^−W/kT (42.3) (Feynman et al., 1963, p. 42-2,3).”

 

Feynman’s approximate equation N_e = (1/A)(v/D)e^-W/kT is built on deliberate idealizations that isolate the essential physics while temporarily setting aside molecular complexities. The surface density 1/A means each molecule occupies a fixed cross-sectional area, ignoring rotation motion and thermal fluctuation, but it gives the number of molecules per unit area at liquid surface. The factor D/v represents the escape time—the time required for a molecule moving outward at speed v to cover one molecular diameter D, neglecting collisions, angular spread, and possible barrier recrossing of the surface, but captures the correct dimensional link between speed and distance. Together, the three variables form (1/A)(v/D), a rough geometric “attempt rate” estimating how frequently surface molecules try to leave the liquid. Multiplying this rate by the Boltzmann factor, it accounts for the fraction of the molecules with sufficient energy to escape, providing a calculable evaporation flux.

 

Feynman’s approximation can also be explained by the Boltzmann factor, expressed in terms of W, kT and exponential e^-W/kT.  First, he models the excess (binding) energy needed as a single, well-defined energy “hill” W that must be overcome for a molecule to escape. Second, the quantity kT sets the characteristic thermal energy scale, even though there is always temperature fluctuation at the surface region of a liquid. Implicitly, he assumes that evaporation occurs when a molecule acquires an excess energy W above its typical thermal energy kT, treating the liquid approximating as a classical system with weak correlations among molecules. When Feynman uses the exponential factor e^{-W/kT}—the Boltzmann Factor—he is applying a statistical shortcut: rather than tracking detailed molecular motion, he estimates the fraction of molecules capable of overcoming the energy “hill” (W).

 

3. Limitations

“Even though we have used only a rough analysis so far as the evaporation part of it is concerned, the number of vapor molecules arriving was not done so badly, aside from the unknown factor of reflection coefficient. So therefore we may use the fact that the number that are leaving, at equilibrium, is the same as the number that arrive. True, the vapor is being swept away and so the molecules are only coming out, but if the vapor were left alone, it would attain the equilibrium density at which the number that come back would equal the number that are evaporating. Therefore, we can easily see that the number that are coming off the surface per second is equal to the unknown reflection coefficient R times the number that would come down to the surface per second were the vapor still there, because that is how many would balance the evaporation at equilibrium (Feynman et al., 1963, p. 42-4).”

 

Feynman acknowledges the presence of an unknown reflection coefficient to account for vapor molecules that return to the liquid rather than escape permanently. However, he does not state the limitations of his equation—for example, the temperature range over which the simple Boltzmann factor remains accurate, or how increasing vapor density (and thus back-collisions) would modify the net flux. His model is intentionally pedagogical: it isolates the essential statistical idea—attempt frequency multiplied by Boltzmann factor—without attempting a full kinetic theory treatment and systematic  experimental validation across regimes. By  contrast, the Hertz-Knudsen equation (e.g., F = aP/Ö[2pmkT]) is the standard framework for estimating evaporation and condensation fluxes in applications ranging from metallurgy to fusion engineering. In this equation, the evaporation coefficient (a is effectively equivalent to 1-R) quantifies the probability that a molecule with sufficient energy undergoes phase change, thereby addressing Feynman's acknowledged uncertainty about the process.

 

In the literature, there are many different versions of the Hertz–Knudsen equation. This is because the equation evolved from an idealized theory of evaporation to the complicated reality of industrial manufacturing. In 1882, Hertz derived the equation through experiments on mercury evaporation in vacuum, assuming ideal conditions in which vapor molecules do not return to the surface—that is, no condensation occurs. In 1915, Knudsen refined it by introducing the evaporation coefficient to explain the partial reflection of molecules at the interface. There are many other versions, for example, Schrage (1953) incorporated corrections for macroscopic drift velocity (net movement of vapor molecules). Interestingly, in physical vapor deposition for thin film deposition, the equation could be used to forecast evaporation rates from heated sources for achieving desired coating thicknesses across substrates of IC chips (see below).

Source: [Learn Display] 43. PVD (Physical Vapor Deposition)

(Source: Sze, 1983)

 

Note: Chemists may prefer the term Langmuir’s Equation for Evaporation. Irving Langmuir was an American chemist, physicist, and engineer, who was awarded the Nobel Prize in Chemistry in 1932 for his discoveries in surface chemistry. For a derivation of Langmuir’s equation, please visit: Langmuir’s Equation for Evaporation | Jun's Notes

 

Key Takeaways:

Feynman's section on evaporation teaches that the Boltzmann factor is the universal key to understanding thermally activated processes, and learning to recognize its dominance is more important than memorizing amplitudes or prefactors (in this case, attempt rate).

This is why he says his analysis is "highly inaccurate but essentially right"—because he has identified and elevated the one feature that truly matters.

Feynman’s structure:

Evaporation rate = (attempt rate) ´ (Boltzmann factor)

This same idea appears in the remaining four sections of Chapter 42:

• Thermionic emission
• Thermal Ionization
• Chemical kinetics
• Einstein’s law of radiation

In a sense, Feynman’s Chapter 42 acts as a hidden blueprint for an AI chip fab:  (1) Evaporation: Physical Vapor Deposition is a process where metallic atoms are evaporated to coat wafers in high-purity metal interconnects. (2) Thermionic emission:  The emission of electrons in Scanning Electron Microscope is used to inspect nano-scale defects. (3) Thermal ionization: In an Ion Implanter, atoms like Boron or Phosphorus are ionized and accelerated at high speeds into the silicon lattice to form P-type or N-type regions. (4) Chemical kinetics: Atomic Layer Deposition (ALD) relies on self-limiting surface chemical reactions to build the ultra-thin insulating layers. (5) Einstein’s law of radiation: In Extreme Ultraviolet (EUV) Lithography, laser-produced plasmas generate the 13.5 nm light needed to "print" billions of 2 nm features that give AI chips their massive processing power.

       Instead of “Applications of Kinetic Theory,” Chapter 42 could be slightly revised to include the manufacturing process of Modern AI Chips, and titled “From Jiggling Atoms to Artificial Intelligence: The Boltzmann Factor Behind Modern AI Chips.”

 

The Moral of the Lesson: Humidity, Evaporation, and Survival

In Israel, summer feels like “a tale of two climates.” Along the coast in Tel Aviv, the humidity often reaches 70–80%, producing the familiar "sticky" sensation. As you move toward Eilat and the Negev, the humidity can fall below 20%. The humidity dramatically changes both the physics of evaporation and the way your body regulates temperature:

 

1. The Physics: Net Evaporation

Using Feynman’s logic, evaporation is the difference between molecules leaving your skin and molecules returning from the air.

• High Humidity (Coastal regions, e.g., Carmel Coast): The air contains a high density of water vapor. While sweat molecules escape from your skin, some vapor molecules from the air hit the skin and re-condense. The net evaporation rate is slow.
• Low Humidity (Desert regions, e.g., Eilat): The air contains very few vapor molecules. Sweat molecules escape from your skin at roughly the same rate, but almost none return. This imbalance creates a strong net evaporation flux, so sweat evaporates rapidly.

2. Perspiration vs. Evaporation: The Physiological Feedback Loop

The relationship between humidity and sweating is governed by a feedback loop designed to maintain a stable core body temperature. Humidity may disrupt this loop by decoupling the act of sweating from the effect of cooling.

• Low Humidity: Evaporative cooling is efficient. As sweat evaporates, it removes heat from the skin, keeping the body temperature stable. Your body is unlikely to detect a rise in core temperature, and it does not signal the sweat glands to overproduce. However, the air is a "hungry" vacuum for moisture in the desert. You may lose fluids rapidly, but without the feedback of being “sweaty,” you can easily underestimate the rate of loss.
• High Humidity: Cooling is inefficient. Sweat accumulates and drips rather than evaporating. As your temperature rises, the body increases perspiration in an attempt to cool itself, but without much evaporation, that effort provides limited relief. In July 15, 2023, Netanyahu was apparently dehydrated after spending several hours in the sun at the Sea of Galilee on Friday amid an intense heatwave across the country. 

 

Practical Health Implications

• In dry climates (Hydrate Proactively, Not Reactively): Do not wait for thirst—it is a late indicator. Sip water consistently throughout the day. Consider using a humidifier indoors and moisturize skin to prevent excessive dryness.
• In humid climates: Drink water regularly even if you don't feel sweaty. Seek shade or air-conditioned spaces and be aware of the signs of heat-related illness.

 

A Broader Water Reality

Beyond comfort and thermoregulation, humidity and evaporation connect to a much larger issue: access to drinkable water. In Gaza Strip, water scarcity has long been severe. Even before the recent conflict, many of the local aquifers were contaminated and overdrawn, and desalination capacity was insufficient to meet demand. As a result, a very high percentage of available water has been considered unsafe for human consumption.

This contrast reveals a profound truth: while the physics of evaporation is universal, access to clean water—and protection from heat stress—is not. It remains contingent on infrastructure, geography, and the fragile stability of the societies we build.

 

Atmospheric Water Harvesting (AWH)

Israel’s AWH technology operates as a high-tech reversal of evaporation, extracting water from air by enhancing condensation. These systems cool intake air below its dew point, initiating water vapor to transition from gaseous state to liquid state. Crucially, energy consumption depends on local humidity conditions. In coastal regions, high moisture content results in elevated dew points. This allows condensation to be triggered with only modest cooling, enabling high water output with relatively low energy expenditure. Conversely, in desert regions, low moisture content leads to lower dew points, forcing systems to achieve extreme temperature differentials for condensation. This drastically increases power consumption while producing lesser water—a dual penalty of high energy input for low output that defines the challenge of desert-based atmospheric water harvesting.

 

Review Questions

1. Idealizations: Feynman presents an idealized toy model where each molecule in the liquid has a definite volume and a constant binding energy. What are the key simplifications or idealizations hidden in this picture of the liquid state?

2. Approximations: Feynman derives the formula but states that the "factors in front are not really interesting to us." How would you explain the approximation being made and why is it considered valid to focus on the exponential term (or Boltzmann factor)?

3. Limitations: Feynman explicitly states his analysis is “highly inaccurate but essentially right.” What are the limitations of his toy model with respect to the unknown reflection coefficient or range of applicability? (A better model is Hertz–Knudsen equation or Langmuir’s Equation for Evaporation?)

 

References:

Beigtan, M., Gonçalves, M., & Weon, B. M. (2024). Heat transfer by sweat droplet evaporation. Environmental science & technology, 58(15), 6532-6539.

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.

Hertz, H. (1882). On the evaporation of liquids, especially mercury, in vacuo. Ann. Phys, 17, 178-193.

Knudsen, M. (1915). Maximum rate of vaporization of mercury. Ann. Phys, 47, 697-705.

Schrage, R. W. (1953). A Theoretical Study of Interphase Mass Transfer. New York: Columbia University Press.

Sze, S.M. (1983). VLSI Technology. New York: McGraw-Hill.