In the Classroom edited by
JCE DigiDemos: Tested Demonstrations
Ed Vitz Kutztown University Kutztown, PA 19530
Demonstrating Void Space in Solids: A Simple Demonstration To Challenge a Powerful Misconception submitted by:
Mary Whitfield Department of Chemistry, Edmonds Community College, Lynnwood, WA 98036-5999;
[email protected] checked by:
Ed Vitz Department of Chemistry, Kutztown University, Kutztown, PA 19530
The atomic-level structure of crystalline solids can be difficult to comprehend. This simple demonstration is used to illustrate the substantial quantity of empty space that remains when solid spheres of any size are packed together. The demonstration is straightforward. A beaker or large (250-mL) graduated cylinder is filled to an arbitrary mark (100-mL mark is simplest) with small glass beads or metal shot of uniform diameter (2–5 mm). A second container is filled with beads of much smaller diameter, perhaps 500 µm or less. Sand can also be used. For the greatest effect, it is best if the particles in the second beaker are so small as to make the sample appear nearly uniform. Students are then queried as to which sample contains the greatest percentage of empty space. In a class of beginning students nearly all will choose the sample with the larger particles. To demonstrate how much empty space remains, water is poured into each container until it is level with the surface of the solids. To measure the quantity of water added it is easiest to start with a filled graduated cylinder or buret and to then note the quantity dispensed. (In large lecture halls, colored water may help with visibility.) While crude, this makes for a quick and effective demonstration. In both cases, approximately 30–35 mL of water are dispensed, which is reasonably consistent with calculated values for the packing efficiency of spherical objects.1 For students, the most surprising result is that the same quantity of water is dispensed in each case, demonstrating that the percentage of void space in a solid is independent of particle size. With coaching from the instructor, students can be led to the conclusion that the same must hold true for atomic-sized particles; that is, that there must indeed be roughly 30% void space between the atoms of a seemingly uniform metallic solid. There are of course many possible variations to the demonstration presented here. It can be structured as an inquiry activity for students. More than two samples can be used, providing additional experimental data and a greater sense of the continuum from macroscopic to microscopic. For greater precision, the quantity of water added can be determined from its mass and density. However there are many sources of error, including bubbles of air that adhere to the solids and the “edge effect” that becomes more pronounced when larger particles are used. Thus, time spent improving measurement precision does not necessarily lead to more ac-
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curate results and may in fact distract from the “big idea”; namely, void space is independent of particle size. Materials Any small spherical objects can be used, although the diameter should not exceed 6 mm or the results become less reliable. If larger spheres are desired, the container volume should be scaled up accordingly, remembering that sphere volume increases as the radius cubed. Good results are obtained with small borosilicate glass beads (0.2–6 mm) of the type used for packing chromatography columns.2 Less expensive options include metal bbs, shotgun shot, or beads from craft stores. (Note however that bbs tend to rust.) As previously noted, sand can also be used and gives reasonably good results, though of course the particles are neither perfectly spherical nor uniform. If you are going to purchase sand rather than just visiting the local playground, 20–40 mesh sea sand seems to work best. Discussion
Void Space: A Brief History Pick up any freshman chemistry text, turn to the section on the solid state, and there you will find a description of the various ways that atoms, modeled as solid spheres, can pack together. (You will probably also find a picture of stacked oranges). Read further and you will discover that the most efficient way to pack atoms is in the face-centered cubic (FCC) arrangement, which yields a 74% packing efficiency (or, conversely, 26% void space). What you will not find, and may not know, is that mathematicians have only recently been able to prove that the FCC is indeed the densest possible arrangement of spherical objects. The so-called Kepler conjecture, first put forth by Johannes Kepler in the 1600s, troubled mathematicians for several hundred years until a satisfactory proof was offered in 1998 (1). Until its solution the Kepler conjecture was considered one of the last great unsolved problems in mathematics, comparable in significance to, for example, Fermat’s last theorem. It was so significant that an entire book has been devoted to the problem and its solution (2). Though all of this may be of greater interest to mathematicians than to chemists, it does show
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that problems of packing efficiency are not as simple as they might at first appear. The reader may wonder why the results of the demonstration presented here do not come closer to the 26% void space predicted for cubic closest packing. While some of the discrepancy may be attributed to experimental error and edge effects, it is also likely that the beads do not pack in a perfectly ordered arrangement. Physicists, as it turns out, have a name for this disordered packing, which they call random close packing or RCP (3). It has a 64% packing efficiency (36% void space), which agrees well with values obtained in this experiment. This depends of course on how the beads are added to the container. As physicist David Weitz (4) notes, a jar of packed marbles may be transitioned from RCP to FCC by shaking the jar so that the marbles “jump up slightly and rearrange themselves”. Instructors who are interested in a materials science application for this demonstration may find it interesting to explore with their students how the quantity of void space changes when the container is shaken to order the beads. Researchers who are interested in packing problems do not limit their attention to spherical objects. One recent study (5), widely reported in the popular science media (6, 7) concerned the packing of ellipsoids, in particular M&M candies. It turns out that ellipsoids have a much higher RCP efficiency than spheres (about 72%), which may be useful information if you are ever trying to win a contest by guessing the number of M&Ms in a candy jar. All jokes aside, the point is that questions about how solid particles pack together is actually an active and complex area of research with applications in a variety of disciplines.
The Conceptual Challenge for Students For students, the very concept of void space in a solid can be difficult to grasp. Most students, when asked, would never guess that there can be as much as 48% empty space between the atoms in a crystalline solid. And who can blame them? Nothing about the visual appearance of a solid gives any clues to its atomic-level structure. There are a number of well-documented misconceptions in this area. Johnson (8) found that “The idea of ‘nothing’ between the particles … appears to cause considerable difficulties for students. Many seem to prefer to think of ‘something’, usually referred to as ‘air’, as being between the particles.” Nussbaum (9), in a review of existing literature on student conceptions of the gaseous state, found that even students who recognized that gases were particulate in nature still resisted the notion that there was empty space between the particles. Most research, however, has centered on student conceptions of the liquid or gaseous state, with little or no attention given to the solid state. Perhaps this is because there are numerous demonstrations and activities to challenge misconceptions about gases and liquids. The compressibility of gases provides some indirect evidence for the existence of empty space between the molecules or atoms. The fact that salt dissolves in water without noticeably increasing its volume suggests that there is empty space between individual water molecules. There are, unfortunately, no compelling demonstrations or activities that provide convincing evidence for the void space in a solid.
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Since this void space cannot be observed directly or indirectly, instructional approaches usually rely on molecular models or graphics in which atoms are analogized with macroscopic spherical objects. Unfortunately, existing research has shown that students do not always make the desired connection between the target concept and the analog (10, 11). Students may believe that a model of a unit cell constructed from model kits or oranges contains 30% void space, but will not necessarily extend this result to the atomic scale.
Bridging Analogies The idea of bridging analogies has been used in the physics education community to help students make connections between an easily observed “anchoring” phenomenon and an abstract “target” phenomenon or concept. This has proved to be an especially useful approach in situations, such as the one discussed here, in which the similarities between the target concept and the analog are not immediately apparent to the novice learner. In one example from a widely cited article by Clement (12), it is reported that students will accept the idea that a spring pushes up on a book resting on top of it, but do not agree that a book resting on a table experiences the same upward force. The target concept, in this case, is Newton’s third law, or the principle of equal and opposite forces. To help students more clearly make the connection between the target concept and the analog, Clement proposes the use of bridging analogies, a series of analogies that successively approximate the target concept. In Clement’s example one intermediate analogy is of a bent, “springy” board. The springy board shares properties of both the bouncy spring and the rigid table, helping students to ultimately see the similarity between the table and the spring. Research conducted with beginning students found that the experimental group (students exposed to the bridging analogies) had larger posttest gains than the control group. In the example presented here, the beaker filled with the small glass beads or sand is the bridge that can help students grasp the idea that even substances that appear uniformly solid can still contain large quantities of void space. Instructional Applications There are many places in the introductory curriculum where this demonstration might be used. In introductory or high school classes it provides a way to address deep-seated misconceptions about the structure of matter at the atomic scale. In a majors-level course it can be used in conjunction with a unit on the properties of solids. Students in these classes often calculate packing efficiency using the atomic radius, crystal structure, and dimensions of a unit cell; this demonstration provides a visual way to comprehend what they otherwise simply calculate. The demonstration can also be applied to discussions about density. When students in introductory classes are asked to devise a method for determining the density of a crystalline solid (e.g. NaCl) many choose to pour the solid into a graduated cylinder and incorrectly assume that they are measuring the volume of their sample. Even when they do recognize that some volume is unaccounted for, they usually assume that the error is a small one. The above demonstration is an excellent way to dispel that
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notion. In an interdisciplinary science class with a geology or earth science focus, the demonstration might be relevant to a discussion about soil percolation and drainage. Lastly, there is growing interest in the inclusion of materials science concepts in the general chemistry curriculum. The ACS (13) has published a companion text on materials science that can be used as part of a general chemistry curriculum and numerous articles in this Journal (14–16) have offered additional resources for instructors who wish to incorporate materials science concepts into their curricula. This demonstration would fit well within any of these instructional programs. Notes 1. The MathWorld Web site, http://mathworld.wolfram.com/ SpherePacking.html (accessed Feb 2006), summarizes the packing efficiency calculation for a number of different packing arrangements. 2. Fisher Scientific sells soda lime glass beads in 3–6 mm diameter (#11-312A for the 3 mm). Borosilicate beads (#NC9768938 for 3 mm) are considerably more expensive. Supelco has 250 µm beads (#59202).
Literature Cited 1. Sloane, N. J. A. Nature 1998, 395, 436–436. 2. Szpiro, G. Kepler’s Conjecture: How Some of the Greatest Minds in History Helped Solve One of the Oldest Math Problems in the World; John Wiley: Hoboken, NY, 2003.
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3. Torquato, S.; Truskett, T. M.; DeBenedetti, P. G. Phys. Rev. Lett. 2000, 84, 2064–2067. 4. Weitz, D. Science 2004, 303, 968–969. 5. Donev, A.; Cisse, I.; Sachs, D.; Variano, E. A.; Stillinger, F. H.; Connelly, R.; Torquato, S.; Chaiken, P. M. Science 2004, 303, 990–993. 6. Weiss, P. Sci. News 2004, 165 (7), 102. 7. Chang, Kenneth. A Treat That Fills the Void. The New York Times, Jul 20, 2004, p F2. 8. Johnson, P. Int. J. Sci. Ed. 1998, 20, 393–412. 9. Nussbaum, J. The Particulate Nature of Matter in the Gaseous Phase. In Children’s Ideas in Science; Driver, R., Guesne, E., Tiberghien, A., Eds.; Open University Press: Philadelphia, 1985; pp 124–144. 10. Taber, K. S. Phys. Ed. 2001, 36, 222–226. 11. Harrison, A. G.; Treagust, D. F. Sci. Ed. 1996, 80 (5), 509– 534. 12. Clement, J. J. Res. Sci. Teach. 1993, 30 (10), 1241–1257. 13. Ellis, A. B.; Geselbracht, M. J.; Johnson, B. J.; Lisensky, G. C.; Robinson, W. R. Teaching General Chemistry: A Materials Science Companion; American Chemical Society Books: Washington, DC, 1993. 14. Widstrand, C. G.; Nordell, K. J.; Ellis, A. B. J. Chem. Educ. 2001, 78, 1044. 15. Ellis, A. B. J. Chem. Educ. 1997, 74, 1033. 16. Gulden, T. D.; Norton, K. P.; Streckert, H. H.; Woolf, L. D.; Baron, J. A.; Brammer, S. C.; Ezell, D. L.; Wynn, R. D. J. Chem. Educ. 1997, 74, 785.
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