VISUALIZING VAPOR PRESSURE A Mechanical Demonstration of Liquid–Vapor Phase Equilibrium

by Dennis Lamb and Raymond A. Shaw

The exponential dependence of the equilibrium vapor pressure of water is illustrated through a simple mechanical analogy involving bouncing balls on a vibrating plate.

he atmosphere would be a much simpler system if balance of Earth (Durran and Frierson 2013; Stevens it did not contain water vapor—but it would also and Bony 2013). The simplest radiative equilibrium T be far less interesting to atmospheric scientists models do a surprisingly poor job of capturing Earth’s and meteorologists, not to mention less hospitable to mean temperature when water vapor and clouds life. The presence of a trace substance—water—that are not included at least in some simplified way. Of exists as a gas, a liquid, and a solid under the range of course the hydrological cycle depends crucially on typical atmospheric conditions, changes everything: the evaporation of water from the warm oceans of water vapor is a primary greenhouse (infrared active) Earth and the eventual condensation of water vapor gas, the latent heat release in cloud convection fun- to generate precipitation; and one of the key pathways damentally changes the temperature structure of the for precipitation formation depends on the differential atmosphere, and of course the presence of clouds and equilibrium vapor pressure of liquid and solid phases precipitation themselves profoundly alter the radiative of water. It is therefore central to the teaching of at- mospheric physics to correctly convey the concept of phase equilibrium. Indeed, the Clausius–Clapeyron AFFILIATIONS: Lamb—Department of Meteorology, The Penn- equation that expresses the temperature dependence sylvania State University, University Park, Pennsylvania; Shaw— of equilibrium water vapor pressure on temperature Atmospheric Sciences Program, and Department of Physics, could be considered one of a handful of equations that Michigan Technological University, Houghton, Michigan an undergraduate meteorology student should really CORRESPONDING AUTHOR: Raymond A. Shaw, Dept. of Physics, Michigan Technological University, 1400 Townsend Drive, internalize, right up there with the hydrostatic equa- Houghton, MI 49931 tion and equations for geostrophic balance. It is our E-mail: [email protected] experience, though, that the concept of equilibrium

The abstract for this article can be found in this issue, following the vapor pressure and its temperature dependence is table of contents. deceptively subtle and challenges even the brightest DOI:10.1175/BAMS-D-15-00173.1 of students [and sometimes their teachers; see Bohren

A supplement to this article is available online (10.1175/BAMS-D-15-00173.2) and Albrecht (1998) for a discussion of some of the more infamous pitfalls]. In this essay we introduce In final form 4 November 2015 ©2016 American Meteorological Society a demonstration experiment that provides a vivid, mechanical analogy for the concepts of evaporation,

AMERICAN METEOROLOGICAL SOCIETY AUGUST 2016 | 1355 Unauthenticated | Downloaded 09/30/21 02:06 PM UTC condensation, and the temperature dependence of because they capture essential features of the physics equilibrium vapor pressure. The demonstration is and offer opportunities for detailed quantitative analy- useful for conveying a conceptual understanding of sis. Maxwell’s kinetic theory of gases, for instance, while these concepts but also can illustrate the essential still serving as a cornerstone of theoretical physics, gen- concepts underlying the functional form of the most tly introduces new students to the molecular world and common expression for equilibrium vapor pressure. to the rigors of . Physical analogies, At a given temperature, the rates of molecular ex- while not capable of the precision provided by mathe- change across the liquid–vapor interface must differ matical models, serve to bring the invisible microscopic for a net transfer of matter to occur between the phases. world directly to students through demonstrations We speak therefore of net evaporation (when the flux and experimentation. Physical analogies have been of molecules leaving the liquid exceeds the rate mol- used, for instance, by Prentis (2000) to illustrate how ecules reenter the liquid) and net condensation (when mechanical systems with “motorized molecules” can be the condensation flux exceeds the evaporation flux). It described well by Boltzmann statistics when using the is important to distinguish between the practical usage ergodic hypothesis of statistical mechanics. Their two- of the terms (evaporation and condensation) and the level “Boltzmann machine” utilizing multiple balls is of physical processes of molecular exchange across the particular relevance to our study of vaporization, which liquid–vapor interface, which occur simultaneously. is modeled with a vibrating plate containing a depres- When the rate that molecules escape from a liquid sion and a countable number of metal balls. However, exactly equals the rate of return, no net transfer of mat- whereas Prentis varied the potential energy between ter occurs, and the system is said to be in equilibrium, the levels while keeping the mechanical equivalent a dynamic steady state. Knowing the precise point of of temperature fixed, we suggest that evaporation is equilibrium is of course key to distinguishing the con- best modeled as a system in which the potential (i.e., ditions for net evaporation and net condensation. The “bond”) energy is fixed while the degree of mechanical equilibrium concentration of vapor in contact with the agitation (“temperature”) is varied. In both studies, the liquid surface at a particular temperature is unique to important thermodynamic and statistical–mechanical the substance in question. Here, we focus on water and condition of temperature uniformity throughout the use the term “vapor pressure” as the measure of water system has been maintained. vapor concentration (through the ideal gas equation) This paper describes a mechanical system devel- that maintains phase equilibrium. We wish to explore oped for use in the teaching of atmospheric physics. the nature of this equilibrium state and how the vapor We use it as a way of introducing students to the pressure varies with temperature. growth and evaporation of cloud drops, for which The equilibrium state is intimately linked to the a clear understanding of equilibria between phases process by which molecules leave the liquid state. is imperative. We attempt here to draw parallels Molecules in the liquid surface escape only if they ac- between the classical Clausius–Clapeyron equation quire kinetic energies large enough to break the bonds (which describes the dependence of vapor pres- holding them to neighbors. The observed vapor densi- sure on temperature) and the escape of balls from a ties therefore tell us something about the magnitudes gravitational potential energy well. We first review of the cohesive bonding energies. Whereas a weakly essential theoretical principles before describing the bonded condensate (e.g., a light alcohol) evaporates physical system and the data derived from it. Water readily under normal conditions and exhibits a large is the substance of most atmospheric relevance and vapor pressure, a strongly bonded liquid (e.g., mer- the focus of the discussion here, but the principles are cury) evaporates only slowly and yields relatively low general and apply to any volatile liquid or solid. This concentrations of molecules in the gas phase even at demonstration helps answer such questions as “What elevated . Water is a substance whose causes vapor pressure to increase with temperature?” molecules are bound to each other with intermediate and “What physical principles determine the form strength through hydrogen bonding. of the curve describing equilibrium vapor pressure Detailed explanations of molecular phenomena are versus temperature?” important in the sciences and engineering, particularly at the university level. Analogies can be valuable aids to THE BOLTZMANN FACTOR: VAPOR PRES- instruction, in part because they highlight key features SURE AND A MECHANICAL ANALOG. of the underlying physics, and in part because they A molecular interpretation via the Boltzmann equa- help students visualize scales not sensed by humans. tion (see the sidebar for an overview) provides a com- Mathematical analogies abound in technical fields pelling perspective for interpreting the temperature

1356 | AUGUST 2016 Unauthenticated | Downloaded 09/30/21 02:06 PM UTC dependence of water vapor pressure. Conceptually, versus the reciprocal of temperature. The slope of the it is a struggle between water molecules in the liquid line allows one to calculate the latent heat of evapora- state effectively trapped in a depression of potential tion (approximately the energy needed to break two energy (“potential well”) of magnitude lυ and the hydrogen bonds in the liquid). Such macroscopic random thermal energy kBT that occasionally al- measurements thus tell us how strongly the molecules lows molecules to escape to the vapor. To first ap- must be joined to neighbors. So, at its most basic level, proximation, lυ is the binding energy per molecule. evaporation is simply a result of random thermal agi- [The correction due to expansion that is needed to tation of molecules in liquid water resulting in a lucky convert binding energy to , e.g., shown by few near-surface molecules getting jostled in just the Baierlein (1999), section 12.3, is neglected here for right way to gain enough kinetic energy to exceed reasons of pedagogical simplicity.] Binding energy is the binding energy and so escape from the liquid and essentially the potential energy, due to Coulomb or enter the vapor phase. Simply because molecules in a other intermolecular forces, that must be overcome liquid are embedded in a potential well (roughly equal to extract a water molecule from the condensed in magnitude to the latent heat of evaporation lυ), the phase. Indeed, the latent heat of evaporation of water vapor pressure must increase with increasing tem- −1 (Lυ ≈ 45 kJ mol ) when converted to molecular units perature, again in accord with the Boltzmann factor. is lυ = 0.47 eV. This value is strikingly close to twice This recognition that molecules in a liquid ef- the energy of a hydrogen bond (εH = 0.24 eV) between fectively reside in a potential energy well imposed by two water molecules in the liquid phase. The ratio the electrical forces holding them to neighbors serves lυ/εH ≈ 1.9 provides a rough estimate of the average as the inspiration for our mechanical demonstration. number of hydrogen bonds per molecule in liquid Evaporation is here made visible by replacing the mol- water that must be broken for a water molecule to es- ecules with small metal balls and the electric potential cape the bound, liquid state and enter the unbound, well with a gravitational potential well. The mechani- gas phase. How does this binding energy compare cal demonstration consists of a horizontally oriented to the thermal energy in the Boltzmann factor? At metal plate with a central depression (the “well”) that ~ room temperature kBT = 0.025 eV, so we note that serves to represent the liquid state as a “mean field” of lυ >> kBT, which shows via the Boltzmann factor that the electrical binding energy. Forced vibrations of the the fraction of molecules escaping the liquid (i.e., plate serve as the analog of thermal energy and cause a the fraction of molecules having energy equal to or few of the balls in the well to jump up to the flat upper greater than lυ) is small. level (see Fig. 2). The number of particles in the “gas” This same molecular interpretation of evaporation phase (level 2) compared with that in the “liquid” phase also explains the tendency for the rate of evaporation (level 1) can then be written as N2/N1 = exp(–ε0/kBT), and hence vapor pressure to increase with tempera- where ε0 is the gravitational potential energy differ- ture. Whereas classical thermodynamics shows this ence. Now we also need to recognize that temperature tendency to stem from the relative changes of per se has little meaning in a mechanical analogy. We and volume across the phase boundary, it is equally therefore replace thermal energy kBT with a mean me- valid to see the increase as arising from the need for chanical kinetic energy Ekin. Normalization of the num- the molecules to overcome an energy barrier: the ber densities to a reference state allows us to develop enthalpy of phase change lυ. Higher temperatures an equation that is similar in form to the integrated lead to larger molecular kinetic energies and higher Clausius–Clapeyron equation. When we designate probabilities of molecules in the liquid surface being the reference condition as Ekin,0 (analogous to kBT0 in able to break the bonds holding them to liquid-phase a thermal system), the normalized ratio of densities in neighbors. This larger evaporation flux must be bal- the “vapor” state (level 2) is expressed as anced, in equilibrium, by a larger impingement flux,  ε  which arises in turn from a larger concentration of exp − 0  NE2 ()kin  Ekin  vapor over the liquid surface. Such a kinetic viewpoint = . (2) NE20()kin, mechanical  ε  is consistent with chemical kinetics and Boltzmann exp − 0  model  E  statistics, as seen by taking the logarithm of Eq. (SB5):  kiin,0  Equation (2) is the relationship used to analyze the  p  l ln   =−ln A υ . (1) data from our mechanical analogy. This ratio of two  p0  kTB Boltzmann factors is strikingly similar to the nor- Figure 1 shows the same data for water as from malized concentration ratio one obtains from the Fig. SB1, now plotted as the ratio of vapor pressures thermodynamic treatment [Eq. (SB5)]:

AMERICAN METEOROLOGICAL SOCIETY AUGUST 2016 | 1357 Unauthenticated | Downloaded 09/30/21 02:06 PM UTC CLASSICAL THERMODYNAMICS AND MOLECULAR VIEWS OF THE TEMPERATURE DEPENDENCE OF VAPOR PRESSURE he term “vapor pressure” as used (e.g., Wallace and Hobbs 2006, p. 98; the work done in moving the molecules There is a partial pressure of the Lamb and Verlinde 2011, p. 134). A through distance dx against the force, gaseous component of interest in molecular perspective interprets the integration yields the result equilibrium with its pure liquid. Vapor exponential expression as a Boltzmann n  U  escapes from a unit surface of liquid factor, a concept already familiar to =−exp  . (SB4) n kT with a rate that increases with increas- atmospheric scientists in the guise of 0  B  ing temperature, a fact reflected in the hydrostatic balance equation for an This equation, one form of Boltzmann’s the higher vapor pressures and larger isothermal atmosphere: law, states that at constant vapor densities at higher temperatures.  mgz  temperature the concentration n of The Clausius–Clapeyron equation, pptots=−fc exp  , (SB3) molecules at elevated potential energy kT derived from the principles of classical  B  U is reduced over the concentration in thermodynamics, relates an increase in where ptot is the total pressure (sub- the base state by the Boltzmann factor, vapor pressure p with temperature to script “total” to differentiate from exp(–U/kBT). This expression lies at the difference in entropy relative to the vapor pressure), psfc is the surface pres- the foundation of the atomistic view of difference in specific volume between sure (at height z = 0), m is the mass of physics (statistical mechanics). vapor and liquid. Ultimately, this can be a gas molecule, g is the gravitational We thus gain a compelling perspec- written approximately as acceleration, and kB is the Boltzmann tive for interpreting vapor pressure constant (the molecular gas constant). resulting from the integrated Clau- dpln L υ , (SB1) The hydrostatic balance equation is sius–Clapeyron equation [Eq. (SB2)]. = 2 dT RT typically derived by equating the pres- Expressing Eq. (SB2) in molecular where L is the molar latent heat and R sure difference across a horizontal slab terms, we see that the pressure ratio υ is the gas constant. Integration of Eq. of air of thickness dz to the weight of becomes (SB1) over a modest range such that the slab per unit area. One readily sees variations in L can be ignored gives a that the numerator of the exponential p  l  υ =−Aexp u , dependence on temperature that can be term (mgz) is nothing more than the   (SB5) p0  kTB  written as an Arrhenius relationship: gravitational potential energy of the gas where is the latent heat per mol- molecules above the surface. lυ pT()  Lυ  =−Aexp  . (SB2) This result can be generalized to ecule. We can interpret the latent heat p0  RT  l as the potential energy of molecules any potential field, not just that due to υ in the vapor phase relative to those Here p0 (=611 Pa for water) is the gravity (Feynman et al. 1963, section vapor pressure at the ice point 40-2). For example, we can envision in the liquid. We have emphasized the T = 273.15 K, and A = exp(L /RT ). some generic of magnitude common roots of Eqs. (SB3) and (SB5) 0 υ 0 As seen in Fig. SB1, the vapor F acting on each of the molecules in but should also recognize that we pressure of water increases exponen- population n and directed along coordi- typically consider variations in z (in the tially with temperature, in accord with nate x. We thus find a balance between numerator) for hydrostatic balance and Eq. (SB2), which suggests that water the difference in pressure across a slab variations in T (in the denominator) for molecules escape with increasing fre- oriented perpendicular to x and the vapor pressure; thus, vapor pressure quency as the temperature is raised. force per unit area acting on the slab. p must vary in the opposite sense that Equation (SB2) is obtained from Assuming the at constant molecular concentration n does in the classical thermodynamics, as commonly T, and noting that –Fdx = dU is just the atmosphere, as the independent vari-

developed in atmospheric physics texts change in potential energy by virtue of able changes.

0

p /

s p

Fig. SB1. The dependence of the equilibrium vapor pressure ratio of water on tempera- ture, as calculated from the Magnus equa- tion (Pruppacher and Klett 1997, p. 854).

1358 | AUGUST 2016 Unauthenticated | Downloaded 09/30/21 02:06 PM UTC  l  exp − υ  nT() T kT = 0  B  , (3) nT()0 T  l  real exp − υ 

 kTB 0 

)

when the ideal gas law is used to convert pressure 0

p /

s s p

into concentration. (The factor T0/T, arising from the ( expansion of the real vapor with temperature, can be LN ignored here.)

MECHANICAL SYSTEM, DATA COLLEC- TION, AND ANALYSIS. The fundamental con- cept behind this demonstration, replacement of the mean electrostatic binding potential in a real liquid by a gravitational potential energy well, is realized by Fig. 1. An alternative way of showing the dependence milling a depression into the center of a flat alumi- of the equilibrium vapor pressure ratio of water on temperature: Arrhenius plot of the vapor pressure num disk (refer again to Fig. 2). Small spherical balls normalized to the ice point T = 273.15 K (at which of copper or aluminum, the analogs of molecules, 0 p0 = 611 Pa). were able to bounce and migrate anywhere within the confines of the outer wall in response to the plate vibrations. Particles (balls in the mechanical system was determined optically. As illustrated in Fig. 2 (top), or molecules in a real liquid) naturally seek the lowest a (green) light from a laser pointer was focused and gravitational potential energy in the absence of other made to reflect first off a plane mirror attached to forces, but they may be ejected from the potential well the bottom of the vibrating plate, then off a spheri- if they happen to acquire sufficient kinetic energy cal mirror (focal length ~20 cm) that was mounted (from the plate or from neighboring particles). The near the opposite side of the plate. This technique circular symmetry of the mechanical system (Fig. 2, allowed the variations in the light path arising from bottom) offers the perspective, when viewed from the small (<1 mm) vertical excursions of the plate to above, similar to that of a drop of water in air. be greatly amplified (~240 times) when projected onto The mechanical system was devised with a few con- the room wall about 6 m from the apparatus. A set siderations in mind. The plate (15 cm in diameter) was of calibration experiments was performed in which constructed of aluminum (6 mm thick) to minimize its the peak-to-peak range of the projected light was mass and provide a hard, robust surface upon which measured for each voltage setting of the function gen- the metal balls could bounce readily. Also, the rigidity erator. The resulting regression equation was used, in of the plate served to prevent unwanted mechanical conjunction with the measured magnification factor, resonances. It was found that machining the surface to calculate the amplitude A of the plate oscillations with a very gentle slope (~1/4°) toward the center from the measured voltages with a precision of a few helped compensate for the friction that naturally percent for each experiment. arises in a mechanical system. The central depression Data of particle “evaporation” were gathered from (2.5 cm in diameter) was cut 3 mm deep and placed in video records of the vibrating plate with bouncing the center of the disk, mainly for symmetry. The outer balls. The digital video camera that generated these wall was constructed of parchment paper and attached records was mounted above the plate and yielded to the outer edge of the plate to keep the balls confined many complete images (as in Fig. 2, bottom) for a to the plate. The entire plate was spray-painted black to period of 3 min at each amplitude. The stored images give visual contrast to the shiny balls. The bottom of were later played back and stopped every 5 s, yielding the aluminum disk was outfitted with an adapter that 37 frames for data collection. This time interval was permitted attachment to a mechanical wave generator, chosen as a compromise between the needs for the the frequency and amplitude of which were driven by separate realizations of the particle distributions to a function generator. Further details on constructing be independent and for the number of samples (deter- the apparatus are provided in the online supplement minations) to be statistically meaningful. A relatively (http://dx.doi.org/10.1175/BAMS-D-15-00173.2). few (~60) large (2.4-mm diameter) copper balls work The amplitude of plate oscillations, needed to well for demonstration purposes, but the data here estimate the kinetic energy imposed on the particles, were gathered using a greater number (300) of small

AMERICAN METEOROLOGICAL SOCIETY AUGUST 2016 | 1359 Unauthenticated | Downloaded 09/30/21 02:06 PM UTC (0.79-mm diameter) aluminum balls (McMaster-Carr) uncertainty in rk by standard methods (e.g., Taylor to improve statistics. The data from each voltage set- 1997), as outlined in the online supplement.

ting (yielding amplitude category k) of the mechanical The mean kinetic energy Ekin imparted to the balls

vibrator are expressed as the average number N2 of balls on the plate by the mechanical vibrator served as

observed to be out of the well (i.e., “evaporated” and on the analog of the thermal energy kBT that drives the level 2), from which a normalized ratio was calculated: motions of molecules in a real liquid–vapor system. Determining the mechanical kinetic energy is chal-

rk = (N2/N1)k/(N2/N1)0. (4) lenging, in part because no convenient “thermom- eter” is available and because some of the mechanical

An “uncertainty” δN2 about the mean, taken as the energy is quickly dissipated by friction, something standard deviation, was propagated to the overall that does not exist as such in the molecular world. However, our purposes here are served adequately by assuming that the energy used for analysis of the data are linearly related to the hy- pothetical “true” energy imparted to the balls. We therefore use the square of maximum plate speed for the 2 ∝ “relative energy” Erel = (Aω) Ekin, where A is the amplitude of plate motion (as described above) and ω = 2πf is the angular frequency of the applied voltage.

RESULTS AND DISCUSSION. The results presented here were derived from a series of nine experi-

ments using the Ntot = 300 aluminum balls. The conditions and experi- mental results are summarized in Table ES1 in the online supplement.

The number ratio N2/N1 (where

N1 = Ntot − N2) for each amplitude category k is shown in Fig. 3 (top) plotted against the relative energy

Erel of the plate. One clearly sees the qualitative trend that the balls “evaporate” and jump from the lower level (the potential energy well) to the upper level more readily as the imposed energy (the “temperature”) increases. Low energies permit only a tiny fraction of the balls in the well to gain the requisite energy needed for promotion to the higher level; large energies, by contrast, let many of the balls escape the well. To first Fig. 2. The mechanical system used to demonstrate phase equilib- approximation, the balls mimic the rium. (top) Schematic diagram of the plate and optical arrangement quasi-exponential pattern from the for determining the amplitude of imposed oscillations. (bottom) Photograph of the plate from above, showing a set of copper balls and evaporation of water (Fig. 1) as the various ancillary components: the leveling base, electrical connection temperature increases. to the driving speaker, a “wall” to hold in the balls, and a mirror for The data from balls jumping out measurement of amplitude. of a potential well in this mechanical

1360 | AUGUST 2016 Unauthenticated | Downloaded 09/30/21 02:06 PM UTC system can be interpreted further with the help of Boltzmann statistics. An underlying feature of clas- sical statistical mechanics is the fact that all energy states are equally likely, so a closed system overall tends toward the most probable distribution. If the balls in our mechanical analogy interact with the plate and with each other in random and indepen- dent ways, then the form of the dependence of the relative number densities in the upper and lower states should depend on the mean kinetic energy in the same way it does for the evaporation of true liq- uids [Eq. (3)]. We have put our data into an analogous form [Eq. (2)] by normalizing the number densities to a particular value (N20, from amplitude category k = 9) and then plotting the normalized ratio [Eq. (4)] against the reciprocal of the relative energy. We see (by comparing the bottom panel of Fig. 3 with Fig. 1) that the overall pattern exhibited by the data from the mechanical analogy is, to good approximation, linear in semilogarithmic coordinates, just as it is with true molecular evaporation. It surprised us to find that such a few short segments of data from a system containing a mere 300 particles can be so well described by Boltzmann statistics. The scatter in the data, too, provides insight into the statistics of balls bouncing on a plate. The “error” bars shown in Fig. 3 (bottom) are everywhere large compared with the standard errors of the means (white boxes), which shows the distinction that must Fig. 3. Data from the vibrating plate using a total of be made between the fluctuations inherent in any Ntot = 300 aluminum balls. (top) Count ratio vs rela- statistical system and the ability to approach physi- tive energy applied to the vibrating plate. Here N2 is the number in the upper level and N = N − N is the cally meaningful expectation values (the means) by 1 tot 2 repetitive sampling. The individual determinations number in the well. (bottom) Arrhenius plot of the concentration ratios normalized to the maximum (of r from the counting of balls on the upper level of k degree of “vaporization” from the vibrating-plate ex- the plate) will always vary greatly when the numbers periment. The filled circles represent the mean values are small. Nevertheless, we see from Fig. 3 (bottom) of the normalized ratio from 37 trials at each energy that the standard deviations (indicated by the error level, while the white boxes represent the standard bars) follow the boundaries (shaded region) described error of the means. The solid line is the best fit to the by Poisson statistics quite well. Increasing the number mean values and accounts for 99.5% of the variance. of determinations will improve the estimation of the The error bars indicate the plus and minus standard deviations about the means. The shaded region identi- mean but not the statistical fluctuations. The only fies the theoretical range of uncertainty arising from way to beat down that statistical uncertainty is to a system obeying Poisson statistics. increase the number of particles in the system. Using more balls in the vibrating-plate experiment would help here; real liquid–vapor systems do this naturally all similar to the random way in which thermal because of the huge numbers of molecules involved. energy is transferred to or from a temperature bath. (Note that uncertainties in vapor pressure data arise Moreover, most of the particle motions arise from from imprecision in the instruments making the interactions with the plate itself and not with other measurements and not from the type of statistical particles. The fact that this mechanical system obeys uncertainties discussed here.) Boltzmann statistics so well may arise from the cha- Several caveats need to be stated about our physi- otic nature of particle–particle collisions and even cal system. The regular sinusoidal motion imposed of particle–plate interactions when many impacts on the plate by the mechanical actuator is not at occur per second.

AMERICAN METEOROLOGICAL SOCIETY AUGUST 2016 | 1361 Unauthenticated | Downloaded 09/30/21 02:06 PM UTC SUMMARY. The mechanical analogy of balls on error analysis. It is our hope that this demonstration a vibrating plate containing a depression serves a serves to enlighten all of us students about the inner couple of useful purposes. The mechanical system workings of liquid–vapor interactions. has a behavior predicted by Boltzmann statistics and therefore also illustrates the essential physics of the ACKNOWLEDGMENTS. This project was funded in Clausius–Clapeyron equation. The mere existence part by the National Science Foundation. Discussions with of a potential energy well in a system of agitated Craig Bohren, Will Cantrell, Jerry Harrington, Alex Kos- particles is sufficient to guarantee that the equilib- tinski, and Brian Suits are appreciated, and feedback about rium “vapor” density increase with “temperature,” the demonstrations from students of a thermodynamics class a measure of the energy imposed on the particles at Michigan Technological University was helpful. We are comprising the system. Escape of molecules from the grateful for the technical assistance from the staff of the MTU “liquid” phase is simply a matter of probability; those Physics Department, particularly Marvin Manninen and lucky few with kinetic energy exceeding the energy Jesse Nordeng, as well as from Sue Hill in the development depth of the potential well are likely to escape; oth- of the animation. One of the authors (DL) appreciates the erwise, they remain trapped in the well. Finally, the opportunities afforded by his association with MTU during coupling of theory (thermodynamics and statistical a sabbatical leave from The Pennsylvania State University. mechanics) with data from a macroscopic system yields valuable information about the particle-scale and thermodynamic properties of the system. Vapor REFERENCES pressure above a liquid increases with temperature Baierlein, R., 1999: Thermal Physics. Cambridge Uni- simply by virtue of the cohesive forces holding mol- versity Press, 442 pp. ecules in the liquid. It is one of the beauties of science Bohren, C. F., and B. A. Albrecht, 1998: Atmospheric that a molecular phenomenon like the probability of Thermodynamics. Oxford University Press, 402 pp. escape from a binding energy in a liquid can have Durran, D. R., and D. M. W. Frierson, 2013: Conden- global consequences: the physics of balls bouncing sation, atmospheric motion, and cold beer. Phys. out of a depression in a vibrating plate or of molecules Today, 66, 74–75, doi:10.1063/PT.3.1958. escaping from a liquid underlies the “water vapor Feynman, R. P., R. B. Leighton, and M. Sands, 1963: The feedback,” the increase of water vapor concentra- Feynman Lectures on Physics. Vol. 1. Addison-Wes- tion in the atmosphere as global-mean temperature ley, 560 pp. [Available online at www.feynmanlectures increases, and the concomitant enhancement of the .caltech.edu/.] greenhouse effect. Lamb, D., and H. Verlinde, 2011: Physics of Chem- For teachers, we envision that the system described istry and Clouds. Cambridge University Press, here can be useful as a classroom demonstration in an 584 pp. atmospheric thermodynamics course or similar. To Prentis, J. J., 2000: Experiments in . further that end, we have made an animation of the Amer. J. Phys., 68, 1073–1083, doi:10.1119/1.1315604. mechanical demonstration, which is available online Pruppacher, H. R., and J. D. Klett, 1997: Microphysics of (http://phy.mtu.edu/vpt/). Moving the slider along the Clouds and Precipitation. Kluwer Academic, 954 pp. temperature scale in the animation shows how adding Stevens, B., and S. Bony, 2013: Water in the atmosphere. energy to the system increases the likelihood of par- Phys. Today, 66, 29–34, doi:10.1063/PT.3.2009. ticles escaping the central potential well. Or the system Taylor, J. R., 1997: An Introduction to Error Analysis. can be reproduced without great expense or effort to University Science Books, 327 pp. serve as part of an undergraduate atmospheric phys- Wallace, J. M., and P. V. Hobbs, 2006: Atmospheric ics laboratory. We find the quantitative illustration of Science: An Introductory Survey. 2nd ed. Academic Boltzmann statistics to be highly instructive, and the Press, 504 pp. analysis also serves to teach several useful concepts in

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