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Crookes radiometer
Crookes radiometer
Crookes radiometer
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A Crookes radiometer

The Crookes radiometer (also known as a light mill) consists of an airtight glass bulb containing a partial vacuum, with a set of vanes which are mounted on a spindle inside. The vanes rotate when exposed to light, with faster rotation for more intense light, providing a quantitative measurement of electromagnetic radiation intensity.

The reason for the rotation was a cause of much scientific debate in the ten years following the invention of the device,[1][2] but in 1879 the currently accepted explanation for the rotation was published.[3][4] Today the device is mainly used in physics education as a demonstration of a heat engine run by light energy.

It was invented in 1873 by the chemist Sir William Crookes as the by-product of some chemical research. In the course of very accurate quantitative chemical work, he was weighing samples in a partially evacuated chamber to reduce the effect of air currents, and noticed the weighings were disturbed when sunlight shone on the balance. Investigating this effect, he created the device named after him.

It is still manufactured and sold as an educational aid or for curiosity.

General description

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A Crookes radiometer in action

The radiometer is made from a glass bulb from which much of the air has been removed to form a partial vacuum. Inside the bulb, on a low-friction spindle, is a rotor with several (usually four) vertical lightweight vanes spaced equally around the axis. The vanes are polished or white on one side and black on the other.

When exposed to sunlight, artificial light, or infrared radiation (even the heat of a hand nearby can be enough), the vanes turn with no apparent motive power, the dark sides retreating from the radiation source and the light sides advancing.

Cooling the outside of the radiometer rapidly causes rotation in the opposite direction.[5]

Effect observations

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The effect begins to be observed at partial vacuum pressures of several hundred pascals (or several torrs), reaches a peak at around 1 pascal (0.0075 torrs) and has disappeared by the time the vacuum reaches 1×10−4 pascals (7.5×10−7 torrs) (see explanations note 1). At these very high vacuums the effect of photon radiation pressure on the vanes can be observed in very sensitive apparatus (see Nichols radiometer), but this is insufficient to cause rotation.

Origin of the name

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The prefix "radio-" in the title originates from the combining form of Latin radius, a ray: here it refers to electromagnetic radiation. A Crookes radiometer, consistent with the suffix "-meter" in its title, can provide a quantitative measurement of electromagnetic radiation intensity. This can be done, for example, by visual means (e.g., a spinning slotted disk, which functions as a simple stroboscope) without interfering with the measurement itself.

Thermodynamic explanation

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A Crookes radiometer in action with the light switched on and off. (Note that the explanation given in the caption to the clip does not agree with the modern explanation.)

Movement with absorption

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When a radiant energy source is directed at a Crookes radiometer, the radiometer becomes a heat engine.[6] The operation of a heat engine is based on a difference in temperature that is converted to a mechanical output. In this case, the black side of the vane becomes hotter than the other side, as radiant energy from a light source warms the black side by absorption faster than the silver or white side. The internal air molecules are heated up when they touch the black side of the vane. The warmer side of the vane is subjected to a force which moves it forward.

The internal temperature rises as the black vanes impart heat to the air molecules, but the molecules are cooled again when they touch the bulb's glass surface, which is at ambient temperature. This heat loss through the glass keeps the internal bulb temperature steady with the result that the two sides of the vanes develop a temperature difference. The white or silver side of the vanes are slightly warmer than the internal air temperature but cooler than the black side, as some heat conducts through the vane from the black side. The two sides of each vane must be thermally insulated to some degree so that the polished or white side does not immediately reach the temperature of the black side. If the vanes are made of metal, then the black or white paint can be the insulation. The glass stays much closer to ambient temperature than the temperature reached by the black side of the vanes. The external air helps conduct heat away from the glass.[6]

The air pressure inside the bulb needs to strike a balance between too low and too high. A strong vacuum inside the bulb does not permit motion, because there are not enough air molecules to cause the air currents that propel the vanes and transfer heat to the outside before both sides of each vane reach thermal equilibrium by heat conduction through the vane material. High inside pressure inhibits motion because the temperature differences are not enough to push the vanes through the higher concentration of air: there is too much air resistance for "eddy currents" to occur, and any slight air movement caused by the temperature difference is damped by the higher pressure before the currents can "wrap around" to the other side.[6]

Movement with radiation

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When the radiometer is heated in the absence of a light source, it turns in the forward direction (i.e. black sides trailing). If a person's hands are placed around the glass without touching it, the vanes will turn slowly or not at all, but if the glass is touched to warm it quickly, they will turn more noticeably. Directly heated glass gives off enough infrared radiation to turn the vanes, but glass blocks much of the far-infrared radiation from a source of warmth not in contact with it. However, near-infrared and visible light more easily penetrate the glass.

If the glass is cooled quickly in the absence of a strong light source by putting ice on the glass or placing it in the freezer with the door almost closed, it turns backwards (i.e. the silver sides trail). This demonstrates radiation from the black sides of the vanes rather than absorption. The wheel turns backwards because the net exchange of heat between the black sides and the environment initially cools the black sides faster than the white sides. Upon reaching equilibrium, typically after a minute or two, reverse rotation ceases. This contrasts with sunlight, with which forward rotation can be maintained all day.

Explanations for the force on the vanes

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Over the years, there have been many attempts to explain how a Crookes radiometer works:

Incorrect theories

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Crookes incorrectly suggested that the force was due to the pressure of light.[7] This theory was originally supported by James Clerk Maxwell, who had predicted this force. This explanation is still often seen in leaflets packaged with the device. The first experiment to test this theory was done by Arthur Schuster in 1876, who observed that there was a force on the glass bulb of the Crookes radiometer that was in the opposite direction to the rotation of the vanes. This showed that the force turning the vanes was generated inside the radiometer. If light pressure were the cause of the rotation, then the better the vacuum in the bulb, the less air resistance to movement, and the faster the vanes should spin. In 1901, with a better vacuum pump, Pyotr Lebedev showed that in fact, the radiometer only works when there is low-pressure gas in the bulb, and the vanes stay motionless in a hard vacuum.[8] Finally, if light pressure were the motive force, the radiometer would spin in the opposite direction, as the photons on the shiny side being reflected would deposit more momentum than on the black side, where the photons are absorbed. This results from conservation of momentum – the momentum of the reflected photon exiting on the light side must be matched by a reaction on the vane that reflected it. The actual pressure exerted by light is far too small to move these vanes, but can be measured with devices such as the Nichols radiometer. It is in fact possible to make the radiometer spin in the opposite direction by either heating it or putting it in a cold environment (like a freezer) in absence of light, when black sides become cooler than the white ones due to the thermal radiation.

Another incorrect theory was that the heat on the dark side was causing the material to outgas, which pushed the radiometer around. This was later effectively disproved by both Schuster's experiments[9] (1876) and Lebedev's (1901)[8]

Partially correct theory

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A partial explanation is that gas molecules hitting the warmer side of the vane will pick up some of the heat, bouncing off the vane with increased speed. Giving the molecule this extra boost effectively means that a minute pressure is exerted on the vane. The imbalance of this effect between the warmer black side and the cooler silver side means the net pressure on the vane is equivalent to a push on the black side and as a result the vanes spin round with the black side trailing. The problem with this idea is that while the faster moving molecules produce more force, they also do a better job of stopping other molecules from reaching the vane, so the net force on the vane should be the same. The greater temperature causes a decrease in local density which results in the same force on both sides. Years after this explanation was dismissed, Albert Einstein showed that the two pressures do not cancel out exactly at the edges of the vanes because of the temperature difference there. The force predicted by Einstein would be enough to move the vanes, but not fast enough.[10]

Currently accepted theory

[edit]

The currently accepted theory was formulated by Osborne Reynolds, who theorized that thermal transpiration was the cause of the motion.[11] Reynolds found that if a porous plate is kept hotter on one side than the other, the interactions between gas molecules and the plates are such that gas will flow through from the cooler to the hotter side. The vanes of a typical Crookes radiometer are not porous, but the space past their edges behaves like the pores in Reynolds's plate. As gas moves from the cooler to the hotter side, the pressure on the hotter side increases. When the plate is fixed, the pressure on the hotter side increases until the ratio of pressures between the sides equals the square root of the ratio of absolute temperatures. Because the plates in a radiometer are not fixed, the pressure difference from cooler to hotter side causes the vane to move. The cooler (white) side moves forward, pushed by the higher pressure behind it. From a molecular point of view, the vane moves due to the tangential force of the rarefied gas colliding differently with the edges of the vane between the hot and cold sides.[3]

The Reynolds paper went unpublished for a while because it was refereed by Maxwell, who then published a paper of his own, which contained a critique of the mathematics in Reynolds's unpublished paper.[12] Maxwell died that year and the Royal Society refused to publish Reynolds's critique of Maxwell's rebuttal to Reynolds's unpublished paper, as it was felt that this would be an inappropriate argument when one of the people involved had already died.[3]

All-black light mill

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To rotate, a light mill does not have to be coated with different colors across each vane. In 2009, researchers at the University of Texas, Austin created a monocolored light mill which has four curved vanes; each vane forms a convex and a concave surface. The light mill is uniformly coated by gold nanocrystals, which are a strong light absorber. Upon exposure, due to geometric effect, the convex side of the vane receives more photon energy than the concave side does, and subsequently the gas molecules receive more heat from the convex side than from the concave side. At rough vacuum, this asymmetric heating effect generates a net gas movement across each vane, from the concave side to the convex side, as shown by the researchers' direct simulation Monte Carlo modeling. The gas movement causes the light mill to rotate with the concave side moving forward, due to Newton's third law. This monocolored design promotes the fabrication of micrometer- or nanometer-scaled light mills, as it is difficult to pattern materials of distinct optical properties within a very narrow, three-dimensional space.[13][14]

Horizontal vane light mill

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The thermal creep from the hot side of a vane to the cold side has been demonstrated in a mill with horizontal vanes that have a two-tone surface with a black half and a white half. This design is called a Hettner radiometer. This radiometer's angular speed was found to be limited by the behavior of the drag force due to the gas in the vessel more than by the behavior of the thermal creep force. This design does not experience the Einstein effect because the faces are parallel to the temperature gradient.[15]

Nanoscale light mill

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In 2010 researchers at the University of California, Berkeley, succeeded in building a nanoscale light mill that works on an entirely different principle to the Crookes radiometer. A gold light mill, only 100 nanometers in diameter, was built and illuminated by laser light that had been tuned. The possibility of doing this had been suggested by the Princeton physicist Richard Beth in 1936. The torque was greatly enhanced by the resonant coupling of the incident light to plasmonic waves in the gold structure.[16]

Practical applications

[edit]

The radiometric effect has not been often used for practical applications. Marcel Bétrisey made in 2001 two different clocks (Le Chronolithe and Conti) powered by the light. Their pendulums had bulb lamps located outside the glass dôme and pointing against 4 mica vanes. One meter pendulum gives one second, two lamps placed in either side light up alternately, thus "pushing" the 4 kilos pendulum each time. As there was vacuum inside, its accuracy was of the order of 2 seconds per month.

Radiometers are now commonly sold worldwide as a novelty ornament; needing no batteries, but only light to get the vanes to turn. They come in various forms, such as the one pictured, and are often used in science museums to illustrate "radiation pressure" – a scientific principle that they do not in fact demonstrate.

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Crookes radiometer

View on Grokipedia
from Grokipedia
The Crookes radiometer, also known as a light mill, is a scientific device consisting of an airtight glass bulb partially evacuated to a low of approximately 1 Pa, containing a low-friction spindle supporting a lightweight rotor with four to eight thin vanes, each coated black on one side (typically with ) and reflective or white on the other. When exposed to , the rotor spins continuously, with the black sides trailing and moving away from the light source, converting into mechanical rotation at speeds proportional to light intensity. Invented by British chemist and physicist in 1873 during his investigations into the effects of light on precise chemical measurements involving , the was first publicly described in his 1874 "On Attraction and Repulsion Resulting from ," published in the Philosophical Transactions of the Royal Society. Crookes initially attributed the motion to from , a that sparked intense scientific debate in the late , as it seemed to challenge prevailing views on the nature of light and heat. However, subsequent analyses, including those by Osborne Reynolds in 1879, established that the phenomenon arises not from direct photon momentum but from thermal effects in the residual gas: the black surface absorbs light and heats up more than the reflective side, creating a that drives gas molecules to rebound with higher velocity from the hotter surface, imparting greater momentum and causing the vane to move. This radiometric force, or thermal transpiration, is most pronounced at low pressures where the of gas molecules approaches the vane dimensions, and it diminishes in high vacuum or . The device's historical significance lies in its role as an early tool for studying and phenomena, influencing developments in vacuum technology and inspiring later inventions like the for experiments. Patented by Crookes in , it became a popular demonstration apparatus in and museums, exemplifying principles of and gas kinetics while highlighting the interplay between light, heat, and molecular motion. Modern variants continue to illustrate these concepts, though the core design remains unchanged since the 19th century.

Description

Components and Assembly

The Crookes radiometer is constructed with an airtight glass bulb that houses a partial , typically maintained at a of approximately 1 Pa (about 0.01 or 0.0075 mmHg), achieved through evacuation using a pump such as the Sprengel pump in historical designs. The bulb, often spherical or pear-shaped with a diameter of around 70-100 mm and height up to 210 mm, is sealed after assembly to preserve the low-pressure environment essential for the device's function. In original constructions by , the glass was selected from types like soft German glass or for optical clarity and vacuum integrity. At the center of the is a low-friction pivot or spindle, typically a thin or wire supporting a rotating assembly, with the pivot point formed by a needle resting in a hard steel cup to minimize friction. Mounted radially on this spindle are 4 to 8 lightweight vanes, usually square or diamond-shaped and measuring about 1-2 cm on each side, made from thin sheets of mica or aluminum leaf for their low mass and rigidity. One surface of each vane is blackened with an absorbent material such as lampblack, soot, or India ink to maximize light absorption, while the opposite surface is left polished, silvered, or naturally reflective to enhance light reflection. Assembly involves attaching the vanes to radial arms—often of aluminum—forming a winged or that is balanced on the central pivot and connected via sealed wires ( or aluminum) to external terminals if needed for experimentation. The entire rotating system is then placed inside the glass bulb, which is evacuated to the desired pressure and hermetically sealed, often using or fusion techniques to ensure airtightness and allow free rotation with minimal resistance. Historical versions by Crookes incorporated materials like or for supportive elements, emphasizing durability under conditions.

Observed Rotation Effect

When illuminated by , the vanes of a Crookes radiometer rotate such that the black, absorbing surfaces trail behind the reflective surfaces, moving away from the light source. This directional motion is consistent across typical setups with asymmetrically coated vanes, one side blackened for absorption and the other polished for reflection. The activates under exposure to in the visible or spectrum, requiring a minimum light intensity on the order of 1 mW/cm² for observable motion, though brighter sources accelerate the effect. The device responds proportionally to light intensity, with ceasing below threshold levels or in darkness, but resuming promptly upon re-illumination. Under standard sunlight conditions, the radiometer achieves rotational speeds of 1-10 revolutions per second, varying with vane dimensions, ambient , and incident strength—for instance, a focused yields about 2 revolutions per second, while intense sources can exceed 20 revolutions per second. Optimal performance occurs in a partial at pressures around 1-10 Pa, where rotation is brisk; at higher atmospheric pressures, motion slows due to viscous drag, and in complete high below 10^{-3} Pa, no rotation occurs as residual gas is insufficient. Uniformly colored vanes produce no net rotation, emphasizing the role of surface asymmetry. The effect is readily demonstrated qualitatively using or a strong incandescent lamp, where the contactless spinning surprised early observers like , who noted the vanes' indefinite motion under illumination without mechanical input.

History

Invention by William Crookes

(1832–1919), an English chemist and physicist renowned for his spectroscopic discoveries, invented the Crookes radiometer in 1873 as a byproduct of precise quantitative experiments on the element , which he had identified in 1861. While determining the atomic weight of thallium using a highly sensitive balance in his laboratory, Crookes observed that radiant heat from an incandescent lamp caused the balance arm to deflect upward, making the heated body appear lighter than its cold counterpart even in a near-perfect vacuum. This unexpected repulsion effect, noted initially in his June 1872 paper to the Royal Society on thallium's atomic weight, motivated him to develop a device capable of detecting and quantifying such interactions between and matter in low-pressure environments. In his private laboratory at his home on Mornington Road in , Crookes constructed initial prototypes by suspending lightweight vanes—typically made of with one side blackened and the other polished—on a fine pivot within a partially evacuated glass bulb, refining the design through iterative testing to amplify the rotational response to . These experiments built directly on his 1873 investigations, culminating in a detailed description of the device's construction and behavior in his paper "On Attraction and Repulsion Resulting from Radiation," received by the Royal Society on August 12, 1873, and read on December 11, 1873. Crookes first publicly demonstrated the radiometer at the Royal Society's soirée on April 22, 1874, where the device's vanes rotated visibly under lamplight, captivating attendees and marking a pivotal moment in his research trajectory. The invention occurred amid Crookes' pioneering work on high-vacuum technology and "radiant matter," where he adapted and improved upon Geissler tubes—sealed glass discharge tubes developed in the that enabled visualization of electrical phenomena in rarefied gases—to achieve pressures as low as 10^{-6} atmospheres for studying and molecular trajectories. Inspired by these tools, Crookes aimed to create an instrument sensitive enough to measure subtle forces from , bridging his chemical analyses with emerging insights into gaseous dynamics and light propagation in vacua. Crookes' personal interests further shaped his pursuits in light-matter interactions; his early studies on , beginning in the 1850s with compounds, had already drawn him to and energy absorption, while his growing fascination with spiritualism—sparked in the late 1860s after his brother's death and intensified through investigations of mediums like Florence Cook from 1871 to 1874—fueled a broader curiosity about invisible forces and ethereal phenomena that paralleled his radiometer research.

Early Recognition and Naming

Following its invention, the Crookes radiometer garnered significant scientific and public interest through demonstrations by . He first publicly demonstrated the device at the Royal Society's soirée on April 22, 1874. A further presentation occurred at a meeting of the Royal Society on April 7, 1875, where its novel rotation under light exposure astonished attendees and prompted widespread discussion. The instrument was subsequently exhibited at the Special Loan Collection of Scientific Apparatus in in 1876, further amplifying its visibility to both experts and the general public. The device was patented by Crookes in 1876. Crookes coined the term "radiometer" for the device in his 1874 paper, deriving it from the Latin radius to denote its perceived function in detecting and quantifying radiant energy, particularly from light and heat sources. This nomenclature reflected his initial belief that the motion resulted from , a concept inspired by Maxwell's electromagnetic theory. Over time, the full name "Crookes radiometer" became standard in , while the colloquial term "light mill" emerged to describe its spinning vanes. The radiometer received early acclaim from leading figures in physics, including , who hailed it as a "grand discovery" in his popular lectures, emphasizing its potential to illuminate fundamental interactions between and . It featured prominently in contemporary journals, such as , with articles in 1874 and 1875 detailing Crookes' experiments and their implications for radiant phenomena. However, the device also ignited controversies, as scientists debated whether its rotation truly demonstrated or involved other mechanisms, such as residual gas effects in the partial vacuum—disputes that persisted through the late 1870s and challenged Crookes' interpretation. By the 1880s, commercial production of radiometers proliferated, with versions manufactured in and for educational use in schools and laboratories to illustrate principles of and dynamics. Crookes' meticulous work on achieving high vacuums for the device—employing improved mercury pumps and exhaustion techniques—additionally advanced 19th-century vacuum technology, influencing applications in and early electric lighting.

Theoretical Developments

Initial Incorrect Theories

In 1874, William Crookes proposed that the rotation of the radiometer resulted from direct exerted by photons, with the reflective surfaces of the vanes experiencing greater repulsion than the light-absorbing black surfaces due to the doubled transfer upon reflection. This hypothesis drew support from observations of natural phenomena, such as the orientation of tails pointing away from the Sun and the apparent repulsion in stellar atmospheres, both attributed to solar . Furthermore, Crookes' idea aligned with James Clerk Maxwell's electromagnetic theory of from 1873, which theoretically predicted the existence of such pressure from 's . Early evidence against pure radiation pressure emerged from the observed direction of rotation, which was opposite to expectations: the vanes spin with the black sides trailing and moving away from the light source, rather than the black sides leading the motion as predicted by greater push on the reflective side. Additional disproof came from the device's performance in partial vacuums around 1 Pa, where rotation is maximal but the mean free path of residual gas molecules exceeds the vane spacing, rendering direct photon momentum transfer insufficient to overcome mechanical friction—yet the device fails entirely in higher vacuums where gas effects are minimized. Other initial suggestions, including electrostatic attractions between charged vanes or magnetic influences from , were quickly ruled out through experiments substituting various non-conductive and non-magnetic materials, which produced no change in rotational behavior. These erroneous theories lingered into the , largely because contemporary vacuum technology could not achieve the high needed to isolate light pressure effects from residual gas interactions.

Partially Correct Explanations

In the late 19th century, intermediate theories began to recognize the thermal nature of the radiometer's motion while still falling short of a complete description. James Clerk Maxwell proposed in 1879 that the absorption of light by the black surfaces of the vanes generates heat, leading to convection currents in the residual gas within the that propel the vanes. This idea marked a shift from purely mechanical explanations to ones involving gas dynamics driven by temperature gradients. Building on this, Osborne Reynolds introduced the concept of "radiometric streaming" in 1879, positing that gas molecules in contact with the hotter side of a vane acquire higher thermal velocities upon rebounding, creating a net streaming force away from the heated surface and causing . Reynolds' model emphasized the role of molecular interactions in rarefied gases under differences. These explanations were partially accurate in identifying the essential disparities across the vanes, where the side becomes approximately 10–20 °C hotter than the reflective side due to differential absorption. They also correctly highlighted the necessity of residual gas, as experiments varying the demonstrated optimal speeds around 1 Pa, with negligible motion at higher pressures (where viscous drag dominates) or in high vacuum (where insufficient molecules are present). Supporting evidence included tests with uniform heating applied to both sides of the vanes, which significantly reduced or halted , confirming that localized temperature gradients are crucial for the effect. Nevertheless, these models had key limitations: they inadequately accounted for along the vane boundaries, where much of the force actually originates, and struggled to explain sustained motion in pressure regimes too low for significant bulk . Such gaps prompted refinements in 20th-century investigations that integrated molecular kinetic theory more fully.

Currently Accepted Theory

The currently accepted theory explains the rotation of the Crookes radiometer through the radiometric force generated by thermal transpiration, or thermal creep, resulting from the across the vanes within the rarefied gas. This effect occurs because the partial vacuum inside the device creates conditions where gas molecules interact asymmetrically with the heated black surfaces and cooler reflective surfaces of the vanes. In this environment, the of gas molecules is approximately 5 mm, comparable to the vane dimensions, enabling molecules rebounding from the hot black side to carry higher average and velocity than those from the cool side. The net force arises primarily at the vane edges, where the thermal gradient drives a tangential flow of gas from the cold to the hot side, imparting momentum that propels the black (hot) side backward, consistent with the observed rotation direction. This edge-dominated mechanism was rigorously formalized by in 1924, who derived the force in terms of molecular agitation analogous to , emphasizing the role of pressure differences over narrow boundary layers near the edges. Radiation pressure, involving photon momentum transfer, contributes negligibly to the motion, as the momentum per photon is on the order of 10−2710^{-27} N s and the total radiation force is orders of magnitude smaller than the thermal radiometric forces, which can reach micro-Newton scales under typical operating conditions. Experimental validations, including pressure-dependent torque measurements and simulations, confirm that thermal effects dominate, with the approximate force scaling as F≈P2(ΔTT)AF \approx \frac{P}{2} \left( \frac{\Delta T}{T} \right) A, where PP is gas pressure, ΔT\Delta T the temperature difference, TT the ambient temperature, and AA the effective vane area.

Thermodynamic Principles

Temperature Gradients and Gas Interactions

The black side of each vane in a Crookes radiometer absorbs nearly all of the incident due to its dark , while the reflective side rejects most of it, establishing a pronounced across the vane. Under illumination from a bright source such as , the black side heats to approximately 30°C, whereas the reflective side remains close to ambient conditions around 20°C, resulting in a differential of about 9–10 . This asymmetry arises from the radiative heating process, where absorbed photons convert to primarily on the blackened surface. The partial vacuum inside the bulb, maintained at pressures around 1–10 Pa, places the system in the Knudsen regime of rarefied gas dynamics. In this regime, the of air molecules λ is approximately 0.7–7 mm, exceeding the vane thickness or inter-vane gap of about 0.01–0.1 mm, such that molecules undergo ballistic trajectories with few collisions among themselves and primarily interact with the vane surfaces. The velocity distribution of these molecules follows the Maxwell-Boltzmann distribution, given by f(v)∝v2exp⁡(−mv22kT)f(v) \propto v^2 \exp\left(-\frac{m v^2}{2 k T}\right), where mm is the molecular mass, kk is Boltzmann's constant, and TT is the local temperature. Molecules desorbing from the hot black side acquire higher thermal velocities, with the most probable speed vhot≈2kThotmv_\text{hot} \approx \sqrt{\frac{2 k T_\text{hot}}{m}}
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