Heterogeneous Catalysis: On Bathroom Mirrors and Boiling Stones

In the Classroom

Heterogeneous Catalysis: On Bathroom Mirrors and Boiling Stones Albert P. Philipse Van't Hoff Laboratory for Physical and Colloid Chemistry, Utrecht University, Debye Institute for Nanomaterials Science, Padualaan 8, 3584 CH Utrecht, The Netherlands [email protected]

A catalyst is defined as a substance that accelerates a process without undergoing a net change due to that process. Most chemistry students learn about catalysts in the context of chemical reactions, such as the enzymes in biochemistry or the heterogeneous metal catalysts in inorganic chemistry (1, 2). However, it is both important and instructive to show students that the concept of catalysis is broader than molecular chemistry as it also includes physical processes such as crystallization, freezing, boiling, and liquid condensation. Students are familiar with measures to accelerate these processes but probably without realizing that they apply a catalyst, according to the definition given above. In basic laboratory work, for example, students learn about the precautions to prevent “bumping” when heating solutions or distilling liquids. This bumping is the sudden boiling that results in hot liquid and vapor shooting uncontrollably out of the heating vessel. Addition of boiling stones to the liquid before heating will prevent this uncontrolled boiling; alternative antibumping measures are stirring of the liquid with a mechanical stirrer or magnetic flea or the use of an inert gas (nitrogen) capillary bleed in the case of vacuum distillation (3). The effect of these precautions is that they facilitate the nucleation, that is, the formation of vapor bubbles in the heated liquid, such that the liquid boils at its normal boiling point (for either ambient or reduced pressure). Without antibumping measures the liquid temperature may rise above the liquid's boiling point. Any bubble nucleation in this so-called superheated liquid will lead to the uncontrollable boiling mentioned above. Erne (4) describes an instructive classroom experiment, using a microwave oven, that demonstrates this superheating of clean liquids and the effect of antibumping agents. The homogeneous nucleation of vapor bubbles in a pure, very clean liquid is a slow process with a high activation energy; boiling stones provide sites for the much faster heterogeneous nucleation of vapor bubbles. The boiling stones can be filtered off to verify that they were not affected by the whole process, with the conclusion that, by definition, the stones are heterogeneous catalysts for the process of bubble nucleation. This heterogeneous catalysis is also nicely illustrated by adding boiling stones to carbonated drinks or beer. Investigation of the effect of other substances (glass, sand, a student's finger, etc.) shows that almost anything acts as catalyst for the nucleation of carbon dioxide bubbles; see the illuminating discussion by Bohren (5). This example suggests that the catalytic effect has a very general, physical cause; students may be asked to speculate on this cause before they are asked to read this article. In this article, we consider another educational example of catalyzed nucleation, the formation of water droplets on a smooth surface in contact with a supersaturated water vapor. This example is perhaps easier to visualize than bubble nucleation on antibumping

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agents in stirred liquids; however, the underlying thermodynamics is the same. In addition, most students probably witness the catalysis of droplet nucleation almost everyday in the bathroom, according to Bohren (5), a home laboratory second in importance only to the kitchen. It is also instructive to realize the close analogy between the fogging of bathroom mirrors and the action of boiling stones in distillation. Catalyzed Droplet Nucleation As a result of a hot bath or shower, the moist air in a bathroom is much warmer than the dry mirror. Thus, the environment of the mirror is a supersaturated water vapor from which droplets condense onto the mirror surface. The resulting foggy appearance is due to the light scattering of the droplets; if the water would condense as a continuous film, the mirror would remain transparent. Such film formation actually occurs when the mirror has been treated with soap or liquid detergent (5). In catalysis language, we can say that the mirror is a heterogeneous catalyst for droplet nucleation and that soap makes it an even better catalyst. This can all be understood on the basis of classical nucleation theory (CNT), which we will briefly recapitulate, referring for more details to various extensive monographs (6-12). The key assumption in CNT is that internal and surface properties of microscopic droplets are the same as for a macroscopic bulk liquid. In other words, there is no difference between droplets and bulk other than droplets having a much larger specific surface area. Then, the change in the Gibbs energy function, ΔG, for the formation of a droplet at constant temperature T and pressure can be written as ΔG ¼ Aγlv þ nΔμ

ð1Þ

where γlv is the surface tension of the liquid-vapor interface, Δμ is the chemical potential difference for transfer of one molecule from the vapor to a liquid bulk, and n is the number of molecules in the droplet with area A. Δμ is a function of the concentration ratio (6-12) c Δμ ¼ - kT ln S; S ¼ ð2Þ cs in which k is the Boltzmann constant and S is the supersaturation defined as the concentration c of molecules in the vapor phase relative to the saturation concentration cs. From eqs 1 and 2, it follows that for the homogeneous formation of a spherical liquid droplet with radius a, the change in the Gibbs energy function can be expressed as kT ð3Þ ln S ΔGhom ¼ Asph γlv - Vsph Vm

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r 2010 American Chemical Society and Division of Chemical Education, Inc. pubs.acs.org/jchemeduc Vol. 88 No. 1 January 2011 10.1021/ed100364y Published on Web 10/14/2010

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Journal of Chemical Education

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In the Classroom

Figure 1. Sketch of the Gibbs energy change, ΔG, in eq 3 for the formation of a sphere with radius a, homogeneously (hom) in a vapor or solution and heterogeneously (het) on a surface. A smooth, inert surface catalyzes the sphere nucleation as it only lowers the activation Gibbsenergy barrier, see also eqs 9 and 11. 2

3

where Asph = 4πa and Vsph = (4/3)πa are, respectively, surface area and volume of the spherical droplet and Vm is the liquid volume per molecule. The formation Gibbs energy, eq 3, has a maximum at the critical droplet radius a* (Figure 1). This maximum corresponds to an unstable equilibrium because the critical droplet either spontaneously dissolves or spontaneously grows. In the latter case, the droplet manages to pass the Gibbs energy barrier, which is referred to as a nucleation event. The magnitude of the critical radius follows from the derivative of eq 3 with respect to the radius a and putting dΔGhom/da = 0, ahom

2Vm γlv ¼ kT ln S

ð4Þ

which on substitution into eq 3 yields the height of the Gibbs energy barrier (Figure 1):

ΔGhom ¼

4 1 πðahom Þ2 γlv ¼ Asph γlv 3 3

ð5Þ

This, incidentally, is an insightful formulation of the nucleation barrier: the Gibbs energy penalty to form a critical droplet is the surface energy Asph*γ1v but the barrier in eq 5 is just 1/3 of this penalty owing to the spontaneous formation of bulk phase in the interior of the critical droplet. In addition to the critical radius a*, another characteristic radius is the radius ahom0, at which ΔGhom in eq 3 equals zero, 3Vm γ1v 3 ¼ ahom ahom 0 ¼ ð6Þ 2 kT 1n S We now consider heterogeneous nucleation in which liquid water nucleates as sphere caps (Figure 2) on a flat mirror, the substrate s. The formation Gibbs energy change for a liquid cap equals kT ð7Þ ln S ΔGhet ¼ Acap γlv þ Abase ðγls - γsv Þ - Vcap Vm Here Acap is the surface area of the liquid sphere cap that is in contact with the vapor phase; Abase is the circular contact area between liquid and solid, and γls is the surface tension of the corresponding liquid-solid interface, subtracted in eq 7 by the surface tension γsv of the solid-vapor interface. This subtraction is necessary because the liquid cap occupies an area of the mirror that previously was in contact with vapor. 60


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Vol. 88 No. 1 January 2011

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Figure 2. Heterogeneous nucleation of a water droplet (l), in the form of a spherical cap of radius a and height h, having a contact angle θ to a flat surface (s), all within a supersaturated vapor (v).

To eliminate the interfacial tensions γls and γsv from eq 7, we assume that the liquid sphere caps on the bathroom mirror are equilibrium shapes. In other words, the sum of forces on the circumference of Acap (the three-phase contact line) is zero, according to the well-known Laplace-Young equation (13) γls - γsv ¼ - γlv cos θ

ð8Þ

where θ is the contact angle as indicated in Figure 2. Using this equation, we can simplify eq 7 to ΔGhet ¼ ðAcap - Abase cos θÞγlv - Vcap

kT ln S Vm

ð9Þ

In the supporting information, we derive the following identity for the ratio of the sphere cap volume Vcap to the sphere volume Vsph Vcap Acap - Abase cos θ ¼ ð10Þ Vsph Asph Substitution of this identity in eq 9 yields ΔGhet

Vcap Vcap kT Asph γlv Vsph ln S Vm Vsph Vsph Vcap ¼ ΔGhom Vsph

¼

ð11Þ

where ΔGhom is the Gibbs energy for homogeneous nucleation in eq 3. Thus, the Gibbs energy curve for heterogeneous nucleation is a constant fraction of the homogeneous energy curve (Figure 1); the volume fraction Vcap/Vsph only depends on the wetting angle θ (see the supporting information). Multiplying ΔGhom in eq 3 by a constant volume fraction cannot change the location of its maximum, so the critical radii of curvature for homogeneous and heterogeneous droplet nucleation must be the same

ahet ¼ ahom ¼

2Vm γlv kT ln S

ð12Þ

Multiplying ΔGhom in eq 1 by the volume ratio Vcap/Vsph, of course, does not changes its zero point

pubs.acs.org/jchemeduc

ahet 0 ¼ ahom 0 ¼

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3 a 2

ð13Þ

r 2010 American Chemical Society and Division of Chemical Education, Inc.

In the Classroom

What does change in heterogeneous nucleation is the height of the nucleation barrier; the homogeneous nucleation barrier is multiplied with the volume fraction Vcap/Vsph from eq 10:

ΔGhet ¼

Vcap ΔGhom Vsph

ð14Þ

This reduction of the barrier may strongly increase the nucleation flux I, that is, the number of critical nuclei that per second hop over the barrier in Figure 1. It is customary in nucleation theory (6-12) to take this flux proportional to the number density F* of critical droplets (i.e., spheres with critical radius a*), which in turn is determined by the Boltzmann exponent of the activation energy. Thus, for the homogeneous nucleation flux, we have in order of magnitude ! - ΔGhom ð15Þ Ihom µ Fhom ¼ C exp kT where C is the number density of single molecules in the supersaturated vapor. So using eq 14, we can write for the ratio of heterogeneous to homogeneous nucleation " !# Vcap Ihet ΔGhom 1 exp ð16Þ Ihom kT Vsph For a nucleating substance that wets the substrate well, the volume ratio Vcap/Vsph is small such that ! Ihet ΔGhom exp ð17Þ ; Vcap , Vsph Ihom kT In other words, if the contact angle θ in Figure 2 is very small, the nucleation barrier is almost completely removed, which accelerates the nucleation with a substantial factor of exp[ΔGhom*/(kT)]. A numerical example illustrates that the acceleration in eq 17 may actually be quite staggering. The surface tension of the water-air interface at room temperature is γlv ≈ 72 10-3 N/m. The molar volume of water is 18 cm3/ mol and taking T = 298 K, we then find from eqs 4 and 5:

ΔGhom 80 kT ðln SÞ2

ð18Þ

For a modest super saturation, for example, S = 5, the Gibbs energy barrier in eq 18 is about ΔGhom* ≈ 31 kT. Then, homogeneous nucleation will be very slow and the catalytic effect of a well-wetted substrate according to eq 17 is an acceleration of the order exp 31 ≈ 2.9 1013. Only for a much larger super saturation the effect of the substrate is not significant; for example, ΔGhom* ≈ 3.8 kT for S = 100. Discussion What the preceding section shows is an example of catalysis of a physical process, namely, liquid condensation, which is valid within the approximations of classical nucleation theory (CNT). Here we can estimate the increase in reaction (nucleation) rate due to a catalyst; such a simple calculation is not feasible for catalysis of chemical reactions. The calculation not only is instructive to better understand the concept of catalysis, but also


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requires the student to comprehend wetting phenomena and Young's equation (eq 8). In addition, the proof in the supporting information is a geometrical exercise with a clear practical application. The presentation here is somewhat different from CNT textbooks (6-11), which are generally too elaborate for use in a basic physical chemistry course. Moreover, these textbooks do not make any link between heterogeneous nucleation and catalysts; the word catalysis is nowhere mentioned in refs 6-11. The CNT model for heterogeneous nucleation should, as any model, not be used unreservedly. A notorious problem in CNT is the quantitative prediction of nucleation rates, which is a challenge even with respect to order of magnitude (6-12). Nevertheless, for processes such as bubble nucleation by boiling stones and fogging of bathroom mirrors, the correct inference from CNT is that the wetting angle θ (Figure 2) is the essential parameter for the energetics of heterogeneous nucleation. This, however, will not be generally true when it comes to nucleation or precipitation of solid particles in a liquid or on a substrate. Here crystallinity and nonspherical particle shape will complicate the issue, apart from the fact that interfacial tensions of solids are often unknown. In addition, the precipitation of, for example, catalytic nanoparticles on a substrate is also affected by specific chemical interactions that are beyond CNT, as further discussed in refs 1 and 14. It is a pity, but the synthesis of a heterogeneous solid catalyst is not a clear example of heterogeneous catalysis itself. Acknowledgment Marina Uit de Bulten-Weerensteyn and Laura Rossi are thanked for their help in the preparation of the manuscript. Literature Cited 1. Synthesis of Solid Catalysts; de Jong, K. P., Ed.; Wiley-VCH: New York, 2009. 2. Claesson, E. M.; Mehendale, N. C.; Klein Gebbink, R. J. M.; Koten, G.; van; Philipse, A. P. J. Magn. Magn. Mater. 2007, 311, 41–45. 3. Dean, J. R.; Jones, A. M.; Holmes, D.; Reed, R.; Weyers, J.; Jones, A. Practical Skills in Chemistry; Pearson Education: Harlow, U.K., 2002; pp 31-32. 4. Erne, B. H. J. Chem. Educ. 2000, 77, 1309. 5. Bohren, C. F. Clouds in a Glass of Beer; Simple Experiments in Atmospheric Physics; Wiley: New York, 1987; pp 1-7, 44-52. 6. Abraham, F. F. Homogeneous Nucleation Theory; Academic Press: New York, 1974; pp 1-30. 7. Debenedetti, P. G. Metastable Liquids; Concepts and Principles; Princeton University Press: Princeton, NJ, 1996; pp 147-233. 8. Nielsen, A. E. Kinetics of Precipitation; Pergamon Press: New York, 1964; pp 1-29. 9. Nucleation; Zettlemoyer, A. C., Ed.; Marcel Dekker: New York, 1969; pp 1-224, 10. Nucleation Phenomena. Gushee, D. E., Ed.; American Chemical Society:Washington, DC, 1966; pp 1-24. 11. Walton, A. G. The Formation and Properties of Precipitates; Interscience Publishers: New York, 1967; pp 1-43. 12. Philipse, A. P. Particulate Colloids; Aspects of preparation and Characterization. In Fundamentals of Colloids and Interface Science, Vol. IV; Lyklema, J., Ed.; Elsevier: Amsterdam, 2005; pp 2.27-2.30.


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13. Atkins, P.; de Paula, J. Physical Chemistry; Oxford UP: Oxford, 2010; pp 647-648. 14. van der Lee, M. K.; van Dillen, J.; Bitter, J. H.; de Jong, K. P. J. Am. Chem. Soc. 2005, 127, 13573–13582.

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Supporting Information Available Derivation for the ratio of the sphere cap volume Vcap to the sphere volume Vsph. This material is available via the Internet at http://pubs. acs.org.


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Heterogeneous Catalysis: On Bathroom Mirrors and Boiling Stones

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