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

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VISUALIZING VAPOR PRESSURE a Mechanical Demonstration of Liquid–Vapor Phase Equilibrium VISUALIZING VAPOR PRESSURE A Mechanical Demonstration of Liquid–Vapor Phase Equilibrium BY DENNIS LAMB AND RAYMOND A. SHAW The exponential temperature 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 statistical mechanics. 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 temperatures. 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 enthalpy, e.g., shown by few near-surface molecules
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