Section 42–2 Thermionic emission

Idealizations / Approximations / Limitations

 

This section is about thermionic emission, historically termed the Edison effect, which occurs when a material is heated sufficiently to enable electrons the kinetic energy required to overcome the surface’s potential barrier. While Owen Willans Richardson was awarded the 1928 Nobel Prize in Physics for formalizing the relationship, Feynman simplified the complex equation into a toy model involving the Boltzmann factor. The principle can be used in the modern AI chip manufacturing, e.g., it governs the emission of electrons from tungsten tips enclosed within the Scanning Electron Microscope.

 

1. Idealizations:

“Then there would be a certain density of electrons at equilibrium which would, of course, be given by exactly the same formula as (42.1), where Va is the volume per electron in the metal, roughly, and W is equal to qeϕ, where ϕ is the so-called work function, or the voltage needed to pull an electron off the surface….. In other words, the answer is that the current of electricity that comes in per unit area is equal to the charge on each times the number that arrive per second per unit area, which is the number per unit volume times the velocity, as we have seen many times: I = q_env = (q_ev/Va)e^−q_eϕ/kT (Feynman et al., 1963, p. 42-4).”

 

Feynman’s equation I = q_env = (q_ev/V_a)e^−qeϕ/kT contains the essential Boltzmann factor in the exponential, but the prefactor q_ev/V_a embodies several idealizations. First, replacing the electron number density n with 1/V_a treats conduction electrons as if each occupies a fixed average volume in the metal, but thermionic emission is essentially a surface phenomenon. Second, the use of a single average speed v ignores the Fermi–Dirac velocity distribution and directional effects; in reality, only electrons with sufficiently large outward velocity components normal to the surface can escape. Third, the expression assumes that every electron reaching the surface with enough energy overcomes the barrier, neglecting reflection, surface scattering, and impurities. Thus, the prefactor functions as a simplified “attempt rate” — charge × available carriers × average speed — providing a toy model while omitting the quantum statistics that is incorporated in the Richardson–Dushman law.

 

“We may give another example of a very practical situation that is similar to the evaporation of a liquid—so similar that it is not worth making a separate analysis. It is essentially the same problem (Feynman et al., 1963, p. 42-4).”

 

Strictly speaking, evaporation and thermionic emission are not identical physical problem, even though both involve particles escaping from a surface. The Richardson-Dustman law describes electron emission from a metal and is characterized by a T² prefactor multiplied by a Boltzmann factor, reflecting the presence of a work function barrier. In contrast, Langmuir’s law (often associated with Hertz-Knudsen equation) describes the evaporation of neutral atoms and follows classical kinetic theory, yielding a 1/ÖT dependence in the flux expression, which is commonly written in pressure form without a Boltzmann factor. Although the two laws differ in detail—electrons versus neutral atoms, Fermi–Dirac statistics versus Maxwell–Boltzmann statistics—they share a similar foundation of surface-related processes.

 

It is also worth mentioning that Irving Langmuir had an exceptionally broad research program centered on surface phenomena. His investigations into surface adsorption, thin films, and tungsten-filament bulb naturally led him to study both evaporation and thermionic emission. Awarded the 1932 Nobel Prize in Chemistry in 1932 for his work in surface chemistry, Langmuir also contributed to the development of Child-Langmuir Law (for thermionic emission), which describes the space-charge limited current in vacuum tubes and was foundational for early electronics technology.

 

2. Approximations

“The filament of the tube may be operating at a temperature of, say, 1100 degrees, so the exponential factor is something like e⁻¹⁰; when we change the temperature a little bit, the exponential factor changes a lot. Thus, again, the central feature of the formula is the e^−qeϕ/kT (Feynman et al., 1963, p. 42-4).”

 

In Feynman’s equation, the central approximation lies in the use of the Boltzmann factor e^−q_eϕ/kT, where q_eϕ (or W) is the work function—the energy barrier electrons must overcome to escape the metal. At a filament temperature around 1100 K, the ratio W/kT may be about 10, making the emission proportional to e^−10, a very small number; because this exponential contains 1/T, even a slight increase in temperature significantly reduces the exponent and therefore produces a large increase in current. Interestingly, Richardson noted in his Nobel Lecture, it is experimentally difficult to distinguish between emission laws proportional to T^½e^-W/kT and T²e^-W/kT: the algebraic power of T changes slowly compared with the exponential term, and small adjustments in the constants can mask the difference. However, the essential physics of thermionic emission still lies in the exponential Boltzmann factor, which governs the fraction of electrons energetic enough to overcome the work-function barrier.

 

In his Nobel Lecture, Richardson (1929) mentions: “In 1901, I was able to show that each unit area of a platinum surface emitted a limited number of electrons. This number increased very rapidly with the temperature, so that the maximum current i at any absolute temperature T was governed by the law i=AT^½e^-W/kT …Eq.(1)……In 1911 as a result of pursuing some difficulties in connection with the thermodynamic theory of electron emission I came to the conclusion that i=AT²e^-W/kT … Eq.(2) was a theoretically preferable form of the temperature emission equation to Eq.(1) with, of course, different values of the constants A and w from those used with (1). It is impossible to distinguish between these two equations by experimenting. The effect of the T² or T^½ term is so small compared with the exponential factor that a small change in A and w will entirely conceal it. In fact, at my instigation K. K. Smith in 1915 measured the emission from tungsten over such a wide range of temperature that the current changed by a factor of nearly 10¹², yet the results seemed to be equally well covered by either (1) or (2).”

 

3. Limitations

As a matter of fact, the factor in front is quite wrong—it turns out that the behavior of electrons in a metal is not correctly described by the classical theory, but by quantum mechanics, but this only changes the factor in front a little. Actually, no one has ever been able to get the thing straightened out very well, even though many people have used the high-class quantum-mechanical theory for their calculations (Feynman et al., 1963, p. 42-5).

 

The Paradox of Feynman's Pessimism: Why Agreement and Disagreement Both Clarify the Truth

Feynman's seemingly pessimistic statements about the development of thermionic emission present a productive paradox: by simultaneously agreeing and disagreeing with him, we gain a richer understanding of how physics progresses.

 

1. The Humility of Agreeing: Surface Physics is Messy

Agreeing with Feynman is an exercise in intellectual humility. His prefactor is indeed “quite wrong” because it ignores Fermi–Dirac statistics. Even after the quantum mechanical refinement to the Richardson–Dushman law—where the Richardson constant (A = 4pm_ek²/h³) replaces the earlier empirical prefactor—experimental values often deviate from the theoretical prediction. Surface contamination, surface roughness, and space-charge effects all complicate ideal experimental conditions. In practice, while the Boltzmann factor remains robust and reliable, the prefactor is sensitive to the “messy” material-specific surface conditions that are difficult to control. Feynman’s pessimism is not cynicism but a methodological caution: theoretical elegance does not ensure experimental exactness.

 

2. The Optimism of Disagreeing: A Theoretical Achievement

Disagreeing with Feynman allows us to recognize what was genuinely “straightened out.” By the mid-1920s, Saul Dushman and others had used quantum statistics to transform the empirical prefactor into one that is derived from fundamental constants (m_e, q_e, k, and h). This was not yet a revolution but a revelation: thermionic emission is a manifestation of connections between Fermi–Dirac statistics, phase space, and electron behavior. When experimental values deviate from the Richardson constant, the discrepancy typically reflects imperfect surfaces rather than a breakdown of quantum theory. In this sense, much was “straightened out”, that is, there are corrections in the foundational physics, even if real materials introduce unavoidable complications.

 

3. Synthesis: Where Theory meets Reality

Holding both views simultaneously provides a mature scientific perspective. When designing thermionic energy converters or optimizing electron sources for AI chip fabrication, engineers may refine the Richardson–Dushman law*. Yet Feynman's skepticism may function like a craftsman's caliper, continually emphasizing the gap between theoretical predictions and real surfaces. The tension between theoretical completeness and experimental complexity is not evidence of failure; it is the engine of refinement in surface science. Thus, the “optimist” may still use the Richardson’s constant as a guide, while the “pessimist” accounts for the surface contamination and imperfections—and progress emerges from the dialogue between the two.

 

*In VLSI Technology, the formula related to thermionic emission is modified as shown below:

Source: VLSI Technology (Sze, 1983)

Note: In his 1928 Nobel Lecture, Richardson explicitly acknowledged the importance of Sommerfeld’s quantum-theoretical treatment of the electron gas in metals: “This great problem was solved by Sommerfeld in 1927. Following up the work of Pauli on the paramagnetism of the alkali metals, which had just appeared, he. showed that the electron gas in metals should not obey the classical statistics as in the older theories, such as that of Lorentz for example, but should obey the new statistics of Fermi and Dirac… The only clear exceptions which emerged were the magnitude of the work function in relation to temperature as deduced from the cooling effect and the calculation of the actual magnitude of the absolute constant A which enters into the AT²e-w/kT formula. As this contains Planck’s constant h its elucidation necessarily involved some form of quantum theory.”

 

Key Takeaways: Why Feynman Embraces the “Wrong” Formula

Feynman’s goal is not to provide a handbook for industrial engineering, but to illuminate the conceptual core of Statistical Mechanics.

• The Scaffolding: His central aim is to show that the Boltzmann Factor is the universal engine behind virtually all “escape” processes. Whether the subject is evaporation, thermionic emission, or chemical reaction rates, the exponential suppression associated with an energy barrier is the main physical principle.
• The Essence of Physics: From this perspective, the precise temperature dependence of the prefactor—whether it is ÖT or T² in the Richardson-Dushman law—is secondary. The exponential term governs the scale of the effect; the prefactor refines it.

Thus, the deeper lesson is this: “once you understand the 'thermal jiggle' and the 'energy hill,' you are 99% of the way to the truth. The last 1% is just coefficient-hunting.” The remaining refinements—coefficients, quantum statistics, and material-specific corrections—are crucial for quantitative precision, but they do not change the underlying physics. Feynman teaches us to see the forest first; the trees, however beautiful and necessary, can be examined later.

The Moral of the Lesson:

Placing aluminum foil inside a microwave oven (Magnetron) is effectively introducing a highly reflective conductor into an electromagnetic cavity. While the magnetron generates microwaves through thermionic emission, the oven chamber is designed to bounce those waves until they are largely absorbed by your food. When foil is added, it does more than “shield”: it alters the boundary conditions of the cavity. Used carefully, foil is a tool for selective shielding to prevent portions of food from overcooking; used carelessly, it transforms the oven into an uncontrolled discharge chamber.

Why Foil Sparks: Load Geometry, Not Heat

There is a fundamental difference between the thermionic emission inside the microwave oven and electric field concentration on the foil’s surface.

• Inside the Magnetron (The Source): Electrons are “boiled” off a cathode via thermionic emission. The frequency of microwaves depends on the geometry (size and shape) of the microwave oven and the strength of the magnetic field. This “Source Geometry” determines how fast the electrons move and oscillate.
• On the Foil (The Load): Conversely, arcing on the foil is not directly related to the thermal “boiling” of electrons; it is driven by electric field concentration. Here, the shape of the foil (Load Geometry) determines the electric field strength and the possibility of sparks or arcing.

 

The Physics of Field Enhancement

When microwaves (oscillating electromagnetic fields) strike a conductor like aluminum, they induce surface currents. The resulting electric field is governed by the geometry of the conductor.

• Smooth Surfaces: On a flat sheet of aluminum foil, the electric charge spreads evenly, and electric fields remain moderate.
• Sharp Edges/Points: According to the Gauss’ law, the surface charge density — and thus the local electric field — is inversely proportional to the radius of curvature. In short, sharper points or edges ® stronger electric field.

This is the same principle that makes lightning rods work. If the electric field at a sharp edge becomes strong enough to ionize air, it can create a visible spark or arc.

 

Practical Guidelines for Safe Foil Use

To use foil safely, you should manage both geometry and energy absorption:

1. The Anti-Crumple Rule: Never use crumpled foil. Every wrinkle creates numerous microscopic “points” that intensify the local electric field. Keep foil as flat and smooth as possible.

2. Maintain Clearance: Keep foil well away from the oven walls. If they are too close, the potential difference can produce a direct arc between them, damaging the interior.

3. “Round off” the corners: If you are shielding a turkey wing or delicate pastry, tuck or fold the edges of foil into curves. Curved shapes distribute charge more evenly, significantly reducing the risk of arcing compared to jagged edges.

4. Manage the Load: Absorption Matters

Microwaves need a "load" (something to absorb energy, like water or fat in food).

• If you wrap your food entirely in foil, you effectively create a “No-Load” condition. Thus, it reflects most of the energy rather than absorbing it. Excessive reflection may increase standing waves and stress internal components.
• Use only small patches of foil, allowing most of the food to absorb microwaves.  

 

iPhone Prank: The "Apple Wave" Catastrophe

In 2014, a notorious hoax originated on the internet claiming that a feature called “Apple Wave” allow users to charge an iPhone by “microwaving” it. This incident provides a good review on the contributions of geometry:

The Geometry Trap: A smartphone is a dense thicket of complex metal structures. The internal circuitry and components have thousands of sharp features that may trigger arcing or fire.

Thermal Runaway: The most dangerous component is the Lithium-Ion battery. Microwaves induce massive currents in the battery's conductive layers. This leads to thermal runaway—a state where the battery's internal temperature rises so fast it causes a self-sustaining fire, releasing flammable gases and potentially exploding.

Source: How Not To Charge Your iPhone: Users Fall For 'Apple Wave' Microwave Prank | IBTimes

Youtube: Microwaving iPhone Battery!! Don't Try this at Home!

Summary: Geometry is Destiny

In a microwave oven, aluminum foil is neither inherently safe nor inherently destructive, but it is a “passive” element whose safety is determined entirely by its shape. It only becomes potentially dangerous when its geometry—sharp points and narrow gaps—allows the electric field to break down the insulating properties of the air. Feynman would likely appreciate this contrast—while the exponential factor governs the birth of electrons in thermionic emission, but in the realm of microwave safety, geometry is destiny.

 

Review Questions:

1. Does Feynman suggest that thermionic emission is fundamentally similar to the evaporation of a liquid that a separate, specialized analysis is redundant?

2.Why is the Boltzmann factor regarded as the “central feature” of thermionic emission, while the pre-exponential factor is often treated as secondary? Does this distinction reflect deeper theoretical stability in the exponential term versus material sensitivity in the prefactor?

3. How would you evaluate Feynman’s claim that no one has ever been able to straighten out" thermionic emission despite the use of “quantum mechanics. In light of Saul Dushman’s derivation of the universal constant in Richardson-Dushman law, should this be interpreted as a failure of quantum mechanics, or as an acknowledgment of the complexity of real-world surface physics?

 

References:

Crowell, C. R. (1965). "The Richardson constant for thermionic emission in Schottky barrier diodes". Solid-State Electronics. 8(4), 395–399.

Dushman, S. (1930). Thermionic emission. Reviews of Modern Physics, 2(4), 381.

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.

Richardson, O. W. (1929). Thermionic phenomena and the laws which govern them. Nobel Lecture, December, 12, 1929.

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

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 is applied in the five sections of Chapter 42:

• Evaporation → molecules overcome a cohesive energy barrier
• Thermionic emission → electrons overcome a work-function barrier
• Thermal Ionization → electrons overcome atomic binding energy
• Chemical kinetics → atoms overcome activation-energy barriers
• Einstein’s law of radiation → photons emission (Laser)

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. On 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.