LABORATORY EXPERIMENT pubs.acs.org/jchemeduc
A Unit Cell Laboratory Experiment: Marbles, Magnets, and Stacking Arrangements David C. Collins* Chemistry Department, Brigham Young University—Idaho, Rexburg, Idaho 83460, United States
bS Supporting Information ABSTRACT: An undergraduate first-semester general chemistry laboratory experiment introducing face-centered, body-centered, and simple cubic unit cells is presented. Emphasis is placed on the stacking arrangement of solid spheres used to produce a particular unit cell. Marbles and spherical magnets are employed to prepare each stacking arrangement. Packing efficiency is calculated using simple measurements of marble or magnet diameters and the dimensions of the stacking arrangement. Edge effects are introduced. Packing efficiency is subsequently calculated for various metal samples employing density measurements and literature atomic radii values. The stacking arrangement for each metal and its corresponding unit cell are determined by comparing packing efficiency values. Estimated percent unoccupied space values for water and air are also determined. KEYWORDS: First-Year Undergraduate/General, High School/Introductory Chemistry, Laboratory Instruction, Physical Chemistry, Hands-On Learning/Manipulatives, Crystals/Crystallography, Metals, Physical Properties, Solid State Chemistry, Solids
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arious classroom demonstrations using crystal-lattice models have been described in this Journal and others presenting face-centered, body-centered, and simple cubic unit cells for firstsemester general chemistry courses. Models presented include software programs,1,2 BBs and watch glasses,3 paper and glue,46 Styrofoam balls,7,8 and others.912 Commercial display models are also available through companies such as Carolina Biological Supply Company (Burlington, NC) and Indigo Instruments (Waterloo, ON, Canada). A very popular commercial crystallattice model set is the Solid-State Model Kit distributed through the Institute for Chemical Education. This kit is used to construct various unit-cell arrangements using hard spheres and strategically placed metal rods. Although models for in-class demonstrations appear commonplace, there are few published laboratory exercises covering this topic. A current review of 11 published general chemistry laboratory manuals1323 resulted in only one experiment on the subject. The single experiment17 uses traditional commercial model sets. A novel laboratory exercise is presented that emphasizes the stacking arrangement and the associated packing efficiency of marbles and spherical magnets resulting in a particular unit cell. The use of marbles and magnets reduces cost, offers versatility, and allows for emphasis on stacking arrangements. Most textbooks when discussing simple crystal-lattice structures stress the properties of the unit cell without offering an adequate discussion (if presented at all) of the stacking arrangement. Often a brief discussion of stacking for the face-centered cubic unit cell is presented simply to emphasize its difference from the hexagonal closest-packed arrangement. The process of having students first physically stack marbles or magnets in a particular arrangement and then connect the unit cell to the stacking arrangement through Copyright r 2011 American Chemical Society and Division of Chemical Education, Inc.
packing efficiency offers a unique perspective to aid understanding. Many more marbles or magnets (e.g., 148 or 216) can be stacked using the described technique than what is possible with the mentioned commercial model kits. Layers of marbles or magnets can be added and removed to further emphasize the stacking arrangement. In addition, when using magnets, deformation of layers can be modeled in an attempt to show disruption to the crystalline structure. Although the use of marbles was suggested over 70 years ago for the construction of classroom models,24 its use is not ordinary and there are no known published laboratory exercises on their use. Additionally, there are no reports of using spherical magnets for classroom models or laboratory exercises. This exercise can be completed in one, 3-h laboratory period with 2030 students.
’ EXPERIMENTAL PROCEDURES Stacking Arrangements
Spherical magnets (5 mm diameter) purchased from Zen Magnets LLC (Boulder, CO) or Maxfield & Oberton Holdings LLC (New York, NY) are arranged into a simple cubic stacking arrangement (216 magnets) as shown in Figure 1. The simplecubic stacking arrangement is easily prepared by making one long chain of 216 spherical magnets and then folding the chain every six magnets to form a band of 36 rows, 6 magnets wide. The band can then be folded every six rows to form a cube 6 6 6. Calipers are used to measure magnet diameters and the dimensions of the stacking arrangement. The volume of the simple Published: June 30, 2011 1318
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LABORATORY EXPERIMENT
Figure 1. Picture of a simple cubic stacking arrangement using 216 spherical magnets.
Figure 3. Top view of four layers of a face-centered cubic stacking arrangement.
layer 180° may be required to allow the layers to form the desired structure. To find the volume of the face-centered cubic stacking arrangement, the area of each hexagonal layer is first determined using the traditional equation for the area of a hexagon pffiffiffi 3 3 2 D ð2Þ area ¼ 8 where D is the distance of a line drawn from one corner of the hexagon, through the center, to another corner of the hexagon as seen in Figure 2. This dimension is easily measured using calipers. Because of the rhombus-like shape for the stacked hexagonal layers, Va is determined by simply multiplying the area of a hexagonal layer by the height (see Figure 4) of the facecentered cubic stacking arrangement. pffiffiffi 3 3 2 D ð3Þ Va ¼ height 8
Figure 2. Top view of the first layer of a face-centered cubic stacking arrangement.
cubic stacking arrangement (i.e., Va) is simply its length width height. The volume of space occupied by the magnets (i.e., Vm) within the stacking arrangement is calculated by multiplying the volume of a single magnet by the total number of whole magnet equivalents stacked within the arrangement. Because of the space between magnets, a correctly calculated Vm is less than Va. Packing efficiency (i.e., PE), which represents the percentage of space within the arrangement occupied by the spheres, is calculated using the following equation: PE ¼
Vm 100 Va
ð1Þ
The percentage of unoccupied space is simply 100% minus PE. Magnets are then arranged into a face-centered cubic (148 magnets) stacking arrangement as shown in Figures 25. The face-centered cubic stacking arrangement is easily prepared by making four hexagonal-shaped layers with 37 magnets each (Figure 2) and then stacking the layers one at a time to form the rhombus-like structure with slanting sides as seen in Figures 35. Owing to the polarity of the magnets, flipping each
To emphasize that the face-centered cubic stacking arrangement shown in Figures 35 truly is an ABCA stacking arrangement as presented in most general chemistry texts, each layer is identified in Figure 4. Although challenging to see owing to the slanting nature of the stacking arrangement, one magnet in the fourth layer, identified with an arrow in Figure 5, is directly over another magnet in the first layer, also identified with an arrow. Students (and instructors alike) are encouraged to examine the stacking arrangement to verify that it is truly an ABCA arrangement with the magnets in the fourth layer directly over the magnets in the first layer. For the face-centered cubic stacking arrangement, the number of whole magnet equivalents that can occupy Va (as calculated using eq 3) is more than simply the number of stacked magnets within the arrangement. This is due to portions of magnets able to occupy regions between magnets on the faces of the arrangement. This is collectively called edge effects. For example, if we were to add a fifth layer to the face-centered cubic stacking arrangement in Figure 5, portions of the magnets in the fifth layer would occupy the dimensions of Va as they sit within the dimples (see Figure 2) of the fourth layer of magnets. Approximately 8.5 additional magnets occupy the dimensions of Va for the facecentered cubic stacking arrangement if the stacking arrangement were to continue in all three dimensions (i.e., 148 + 8.5 = 156.5 total magnets). This correction, determined empirically, is an artifact of how Va is defined and must be incorporated to accurately 1319
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Figure 4. Side view of four layers of a face-centered cubic stacking arrangement: A, B, and C identify the layers. Figure 6. Top view of the first layer of a body-centered cubic stacking arrangement: r is the radius of a single marble.
Figure 7. Side view of three layers of a body-centered cubic stacking arrangement: r is the radius of a single marble. Figure 5. Picture of a face-centered cubic stacking arrangement using 148 spherical magnets. An example of two magnets directly over each other in the fourth and first layers are labeled with arrows.
find the packing efficiency. Packing efficiency is then calculated using eq 1.25 Marbles are stacked together in a body-centered cubic stacking arrangement (16 mm diameter, 27 marbles) using adhesive putty obtained from Scotch 3M (St. Paul, MN), or a similar product, as shown in Figures 68.26 It is easiest to first prepare three 9-marble layers (each layer appearing as in Figure 6) and then stack each layer as shown in Figures 7 and 8. The volume of the body-centered cubic stacking arrangement (i.e., Va) is simply length width height; however, measurements are made as shown in Figures 6 and 7. The width and height of the arrangement are both approximately 5.3r, and the length is approximately 7.8r, with r being the radius of a single marble. (These values are theoretical and determined from the actual geometry for a body-centered cubic unit cell.) When calculating the volume of the stacking arrangement occupied by the marbles (i.e., Vm), edge effects must be considered as successive layers occupy the dimples of layers found within the dimensions of Va. As before, approximately 8.5 additional marbles, determined empirically, can occupy the dimensions of the prepared bodycentered cubic stacking arrangement (i.e., 27 + 8.5 = 35.5 total marbles). Packing efficiency is then calculated using eq 1. Unit Cells
A connection is made between a particular stacking arrangement and a unit cell when packing efficiency values are similar. Packing efficiency for the unit cells is calculated using the same ideas as presented above, that is, (i) calculate the volume of the unit cell occupied by the atoms (i.e., Vm) and (ii) determine the total volume of the unit cell (i.e., Va). Face-centered, body-centered, and simple cubic unit cells are shown in Figure 9. A unit cell is typically defined as the smallest
repeating unit of a particular arrangement, and although practically impossible, portions of atoms rather than whole atoms are commonly represented. Because face-centered, body-centered, and simple cubic unit cells are all cubes, Va is easily calculated by cubing the edge length of the unit cell. In postlab questions, students are asked to use geometry and validate the expressions for the edge length of each unit cell as a function of atomic radius as seen in Figure 9 (e.g., 2r, for the simple cubic unit cell). Students then calculate unit-cell volume as a function of r3 by cubing the edge length, that is, Va = (2r)3 = 8r3 for the simple cubic unit cell. Recognizing there is one whole atom equivalent in the simple cubic unit cell (i.e., 1/8-th of an atom at each of the eight corners) and that the total volume of the unit cell occupied by these atoms (i.e., Vm = 4/3πr3 1) is also a function of r3, a theoretical packing efficiency can be calculated for each unit cell with r3 dropping out of the PE equation. An example calculation for the packing efficiency of a simple cubic unit cell is shown below. PE ¼
Vm 4=3πr 3 1 100% ¼ 100% ¼ 52:36% ___________ ð4Þ 8r 3 Va
Students perform these calculations for all three unit cells recognizing there are 2 and 4 whole atom equivalents for the body-centered and face-centered cubic unit cells, respectively. Because the unit cells already contain portions of atoms at their faces, there are no edge effects to consider. This exercise emphasizes that packing efficiency is independent of atomic radii. Students then compare these theoretical packing efficiency values to their experimental values obtained from the stacking arrangements to make a connection between a particular stacking arrangement and unit cell. Density
Packing efficiency is also calculated for actual metal samples using density. In this experiment, the densities of metal samples are conventionally measured employing a balance and a graduated cylinder. Density is described as the mass of atoms present 1320
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Figure 8. Picture of a body-centered cubic stacking arrangement using 27 marbles and adhesive putty.
Figure 9. (Left) Face-centered cubic unit cell; (center) body-centered cubic unit cell; (right) simple cubic unit cell. All cells modified from ref 27.
in 1 mL of a substance; 1 mL being analogous with Va. Vm is calculated by first converting the mass of atoms found in 1 mL of each metal sample to its corresponding number of atoms (i.e., using molar mass and Avogadro’s number), and then multiplying this value by the volume of one metal atom. Literature atomic radii values are given to calculate the volume of one metal atom. Packing efficiency is subsequently calculated as above and compared to the packing efficiency values obtained for both (i) the stacking arrangements and (ii) the unit-cell calculations in an attempt to identify the unit cell for three metal samples (i.e., iron, aluminum, and lead). The above technique is additionally used to calculate percent unoccupied space (i.e., 100% minus PE) for water and air. Percent unoccupied space is chosen over packing efficiency for this part of the experiment because no “packing” occurs in water and air. Va is treated as 1 mL, and Vm is calculated multiplying the total number of molecules found in 1 mL of each sample (determined using density, molar mass, and Avogadro’s number) by the volume of each molecule. To simplify calculations, air is treated as pure nitrogen and the volume of one molecule (i.e., H2O or N2) is estimated as the summation of the volumes of their atoms. For example, the volume of one water molecule is treated as the volume of two hydrogen atoms plus the volume of one oxygen atom. As above, literature atomic radii values are given to calculate the estimated volume of each atom. The mass of air used to determine density is measured by subtracting the mass of a large 150 mL plastic syringe with a 120 mL vacuum from the mass of the same syringe with 120 mL of air. The vacuum can be created by capping the end of the syringe, pulling the plunger to the 120 mL position, and arresting the plunger with a nail placed through a hole in the exposed plunger (or similar procedure). The mass of the syringe with air must, of course, also include the mass of the cap and nail. This simple laboratory technique works very well to measure the mass of gases.
LABORATORY EXPERIMENT
Figure 10. Picture of a marble-and-epoxy-glue model of a bodycentered cubic stacking arrangement with the unit cell accentuated with darker-colored marbles. The unit cell within the stacking arrangement is a cube sitting on edge with one slightly hidden darker marble in the center.
’ HAZARDS Water, air, aluminum, iron, and lead are the only chemicals needed; there are no major chemical hazards. Lead may be replaced with other inexpensive face-centered metals such as copper or nickel. Ingestion of more than one magnet may cause intestinal damage as the magnets become attracted to each other potentially pinching tissue. Marbles or magnets falling on the floor may also be a concern. There is no waste for this experiment; adhesive putty can be reused. ’ DISCUSSION Students preparing their own stacking arrangements find it helpful to have available larger premade models prepared by the instructor using 25 mm marbles and epoxy glue (Figure 10). If these larger models are prepared with clear marbles, colored marbles can be appropriately placed to accentuate the unit cell within the stacking arrangement. Students seem to greatly appreciate how a stacking arrangement produces a unit cell; this point is often only weakly emphasized in most general chemistry textbooks. Theoretical packing efficiency (and percent unoccupied space) values for face-centered, body-centered, and simple cubic unit cells are 74.05% (25.95%), 68.02% (31.98%), and 52.36% (47.64%), respectively. Students obtain experimental packing efficiencies (from their stacking arrangements) within a few percentage points of these true values. The simple and face-centered cubic stacking arrangements constructed with the magnets typically produce more accurate values than the body-centered cubic stacking arrangement constructed with marbles. Students often struggle producing the body-centered cubic stacking arrangement. They have a tendency to stack the marbles touching within each row. However, as shown in Figure 6, there must be space between each marble within a row. Many students are surprised to calculate percent unoccupied space values greater than 95% for water and air! Magnets cost about 12 cents each and are typically sold in sets of 216 ($24.74) or 1728 ($172.80) at Zen Magnets LLC. The larger set contains eight smaller sets of 216 magnets. Each student/group (24 students) should have 216 magnets to create the stacking arrangements in this experiment. It is suggested that when the students have completed the experiment they return the magnets to the instructor in a chosen geometric shape (e.g., cube or hexagon) to ensure no magnets have been lost or stolen. If magnets do become lost, spares can be ordered 1321
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Journal of Chemical Education for 15 cents each. The magnets unfortunately cannot be used for the body-centered cubic stacking arrangement.26 The primary advantage of using marbles or magnets over traditional model kits is their use to emphasize stacking arrangements. Stacking arrangements can be made of any desirable size highlighting the rows and layers of the arrangement. If desired, the hexagonal closest-packed stacking arrangement can also be created using the magnets by simply stacking the third layer of magnets directly above the first layer creating the traditional ABA arrangement. Similar calculations can be performed to verify that this arrangement has the same packing efficiency as the facecentered cubic arrangement. The unit cell for iron is body-centered cubic; aluminum and lead are face-centered cubic. Because of the simple nature of this laboratory experiment, it can easily be performed as a classroom activity if density values are given instead of measured.
’ ASSOCIATED CONTENT
bS
Supporting Information Editable graphics; complete laboratory experiment with directions for students and instructor notes including a description of hazards. This material is available via the Internet at http://pubs. acs.org.
LABORATORY EXPERIMENT
(16) Postma, J. M.; Robert, J. L.; Hollenberg, J. L. Chemistry in the Laboratory, 7th ed.; W. H. Freeman: New York, NY, 2011. (17) Slowinski, E.; Wolsey, W. C. Chemical Principles in the Laboratory, 10th ed.; Brooks/Cole: Belmont, CA, 2011. (18) Bauer, R.; Birk, J.; Sawyer, D. Laboratory Inquiry in Chemistry, 3rd ed.; Brooks/Cole: Belmont, CA, 2009. (19) Murov, W. L.; Experiments in General Chemistry, 5th ed.; Brooks/Cole: Belmont, CA, 2007. (20) Block, T. F.; McKelvy, G. M. Lab Experiments for General Chemistry, 5th ed.; Brooks/Cole: Belmont, CA, 2006. (21) Williamson, K. L.; Little, J. Microscale Experiments for General Chemistry, 1st ed.; Brooks/Cole: Belmont, CA, 1997. (22) Murov, S.; Stedjee, B. Experiments and Exercises in Basic Chemistry, 7th ed.; Wiley: Hoboken, NJ, 2009. (23) Beran, J. Laboratory Manual for Principles of General Chemistry, 8th ed.; Wiley: Hoboken, NJ, 2009. (24) Scattergood, A. J. Chem. Educ. 1937, 14, 140. (25) There are no edge effects for the simple cubic stacking arrangement because magnets of successive layers do not occupy dimples and are placed outside of the determined volume for Va. (26) Marbles are used for the body-centered cubic stacking arrangement because spaces not possible to produce using spherical magnets are needed between the marbles in the rows of each layer. (27) University of WisconsinMadison Materials Research Science and Engineering Center (UW MRSEC) Interdisciplinary Education Group. http://mrsec.wisc.edu/Edetc/SlideShow/ (accessed Jun 2011).
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT The author would like to graciously thank the Department of Chemistry at Brigham Young University—Idaho for their kind support and advice during the development of the laboratory and for the purchase of the marbles and magnets. The author additionally thanks the students of Brigham Young University— Idaho for piloting the experiment and providing suggestions. Most importantly, the author thanks his wife and family for their unceasing support. ’ REFERENCES (1) Robinson, W. R.; Tejchma, J. F. J. Chem. Educ. 1997, 74, 1143. (2) Robinson, W. R. J. Chem. Educ. 1994, 71, 300. (3) Foote, J. D.; Blanck, H. F. J. Chem. Educ. 1991, 68, 777. (4) Yamana, S. J. Chem. Educ. 1987, 64, 1040. (5) Yamana, S. J. Chem. Educ. 1987, 64, 1033. (6) Birk, J. P.; Ellen, J. Y. J. Chem. Educ. 2003, 80, 157. (7) Birk, J. P.; Coffman, P. R. J. Chem. Educ. 1992, 69, 953. (8) Mattson, B. J. Chem. Educ. 2000, 77, 622. (9) Brown, W. J. Chem. Educ. 1941, 18, 182. (10) Bindel, T. H. J. Chem. Educ. 2008, 85, 822. (11) Hawkings, J. A.; Rittenhouse, J. L.; Soper, L. M.; Rittenhouse, R. C. J. Chem. Educ. 2008, 85, 90. (12) Kelly, B. S.; Splittgerber, A. G. J. Chem. Educ. 2005, 82, 756. (13) Vincent, J. J.; Livingston, E. J. Laboratory Manual for Chemistry: A Molecular Approach, 2/E; Prentice Hall: Upper Saddle River, NJ, 2010. (14) Brown, T. E.; LeMay, H. E.; Bursten, B. E.; Murphy, C.; Woodward, P.; Nelson, J. H.; Kemp, K. C. Laboratory Experiments for Chemistry: The Central Science, 11th ed.; Prentice Hall: Upper Saddle River, NJ, 2008. (15) Weiss, G. S; Rickard, L. H. Experiments in General Chemistry, 9th ed.; Prentice Hall: Upper Saddle River, NJ, 2007. 1322
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