In the Classroom
Secondary School Chemistry
Crystal Models Made from Clear Plastic Boxes and Their Use in Determining Avogadro’s Number Thomas H. Bindel Pomona High School, 8101 West Pomona Drive, Arvada, CO 80005;
[email protected] The crystal models presented here are fairly inexpensive and easy to construct, and they clearly show the relationship between the unit cell and the crystal lattice. They help students visualize the basic formula unit within the unit cell and thus facilitate the determination of “Z ”, where Z represents the number of formula units within the cell. These models are also used to support a calculation of Avogadro’s number. The concept of Avogadro’s number is important in the chemistry curriculum. A recent article reports use of X-ray crystallographic data, density, and gram-atomic masses for various chemical elements to calculate Avogadro’s number (1). The theoretical basis for the calculation is fairly involved for the general chemistry/introductory level. The use of the crystallographic models described here, however, allows students to visualize some of the key concepts involved in the calculation, such as unit cell, Z, and crystal lattice. The models represent significant portions of crystal lattices. The lattices are composed of thin-walled transparent plastic cubes (except for the one composed of hexagonal unit cells, which is hand-constructed from acrylic plastic). The unit cells are stacked to demonstrate the relationship between unit cells and crystal lattices. The models have several desirable features. They are colorful, rigid, durable, stackable, relatively easy to construct,1 inexpensive,2 and transparent. A number of mineral structural types, including sodium chloride and graphite, are represented, as are a variety of unit cells, including face-centered cubic, body-centered cubic, and hexagonal. The unit cells have partial atoms3 (polystyrene snowballs) attached to the
interior perimeter of the cells (along corners, edges, and faces, as dictated by the crystal structure). The unit cells are easily stacked in three dimensions to produce a significant portion of the crystal lattice. All of the atoms are easily seen in the stacked arrangement. A survey of the literature finds many creative unit cells: ones made from Styrofoam balls and rods (2), pom pons (3), stacked square tissue culture Petri dishes containing marbles (4), balsa wood (5), tinned copper wire and galvanized iron (stereographic projection model) (6 ), stacked Plexiglas shelves containing marbles (7 ), polystyrene foam spheres (8), “toy playing marbles” (9), paper and cellophane (10), sponge rubber balls and oak doweling (11), light-emitting diodes (12), and cork balls glued to Plexiglas (13). All but one of these cells contain whole atoms on the exterior and consequently cannot be stacked to make an extended lattice. Olsen’s cell (13) is the only one that could be reproduced many times and made to stack into an extended lattice. The unit cells or models presented in this paper stack easily to make an extended lattice. Since the unit cells contain the correct fraction of particles, represented by slices of spheres, at the appropriate cell locations, they are easily stacked to show readily and visibly how whole particles (spheres) will come together at the corners, edges, and faces.
Figure 1. The clear plastic box model of the sodium chloride structure with uniform spheres.
Figure 2. The clear plastic box model of a crystal system composed of body-centered cubic unit cells.
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Classroom Presentation A familiarity with some basic concepts is needed by the instructor: crystal (14), unit cell (16, 17), lattice (14–16 ),
Journal of Chemical Education • Vol. 79 No. 4 April 2002 • JChemEd.chem.wisc.edu
In the Classroom
crystal systems (16 ) and “net lattice ions” (15). Other concepts, such as crystal geometry and basis vectors, are for those more advanced (17, 18). JCE Software programs (19) and books (20, 21) are available. The following is an example of how the demonstration models are used. Of course, this may be modified to meet the particular needs of the instructor or chemistry curriculum. Included are the calculations of Avogadro’s number for cubic and hexagonal systems and directions for the construction and assembly of the demonstration models.
Unit Cell/ Crystal Lattice The class is shown a commercial crystal lattice model of sodium chloride4 and asked what is the smallest repeating unit within the model that will, when it is repeated, generate the entire structure. This is a parallelepiped. Most students identify a simple cubic unit cell. This unit cell is referred to as the “incorrect unit cell” of the sodium chloride structure. Attempts to stack together the appropriate clear plastic unit cells that represent the incorrect unit cells of the sodium chloride structure, of course, fail. The boxes do not stack properly for there is always a mismatch of colors. Each unit cell needs to be translated exactly the same each time. That is, the partial atom in the front left corner of the cube must be the same color each time the unit cell is placed into the crystal structure. A cubic cell of the sodium chloride structure is shown to the class (edge-centered, face-centered, and body-centered atoms). Using the clear plastic unit cells of the sodium chloride structure (Fig. 1), the class is shown that the cells match up when stacked together. These cells have a high degree of symmetry, which is a general characteristic. Next are shown the clear plastic crystal lattices for body-centered cubic (Fig. 2) and the acrylic model of the graphite structure (Fig. 3), which is composed of hexagonal unit cells. This illustrates the relationship between a unit cell and a crystal lattice. The hexagonal unit cells are hand-constructed from acrylic plastic. Even though this is technically difficult, the hexagonal crystal
Figure 3. The acrylic model of the graphite structure composed of hexagonal unit cells.
system is a nice one to have because the unit cell is composed of angles other than 90°. Z (The Number of Formula Units within Unit Cell) First, the body-centered cubic cells are stacked two across and two high. Note that the partial atom in the center of the front face of this structure is made up of four corner atoms and is half an atom in total. Next, additional cubes are placed one at a time in front, so as to make another stack two across and two high. It is then apparent that the center atom is made up of a total of eight corner atoms. Therefore, corner atoms represent 1Ú8 of an entire atom. The body-centered atoms are not shared between adjacent cells and thus contribute one entire atom to the cell. The Z value for the cell is 2. Z = 8(1Ú8) + 1 = 1 + 1 = 2 A similar analysis is done for the face-centered crystal lattice (Fig. 4). Z = 8(1Ú8) + 6(1Ú2) = 1 + 3 = 4 The hexagonal system is especially good, for it demonstrates that unit cells do not have to be cubic or have right angles. The hexagonal parallelepipeds are stacked together in a fashion similar to the other lattices. Next, three unit cells are placed together in such a way as to make a hexagon. This demonstrates why the unit cell is referred to as hexagonal. The analysis of the Z value is a little more involved. For a unit cell, there are two face atoms (1Ú2 atom each), four corner atoms in the 60° angles (1Ú12 atom each), four corner atoms in the 120° angles (1Ú6 atom each), two edge-centered atoms along the long edge formed by a 60° angle (1Ú6 atom each), two edge-centered atoms along the long edge formed by a 120° angle (1Ú3 atom each), and 1 atom in the body. This totals to a Z of 4. A more convenient method of counting uses the entire stack of eight parallelepipeds and looks at the average contribution for a corner atom and edge-centered atoms. This gives Z = 1 + 2(1Ú2)+ 4(1Ú4) + 8(1Ú8) = 1 + 1 + 1 + 1 = 4
Figure 4. The clear plastic box model of a crystal system composed of face-centered cubic unit cells.
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Calculation of Avogadro’s Number for Crystal Systems Composed of Hexagonal Unit Cells Avogadro’s number is calculated easily for the various crystal systems (1) (see Table 1). The calculation is extended for crystal systems composed of hexagonal unit cells (Table 2). Materials Each crystal system consists of 8 parallelepipeds. All cubic systems are constructed from empty AMAC boxes5 with lids (cubes 10.2 cm on edge). The boxes are the 700 series (popular flat-top boxes), model number 774, and the color is crystal. Craft, hobby, candy, and housewares stores carry these boxes. Polystyrene snowballs (11Ú2 in.) are cut with X-acto knives6 into the appropriate slices and then painted with water-based fresco tempera paint (green, red, black, and blue). The painted polystyrene slices are glued into the cubes using a low-temperature glue gun and low-temperature glue sticks. Wooden toothpicks (6.6 cm) are used to attach body-centered atoms. Acrylic plastic (1Ú8-in. thick) was used for the hexagonal system. Acrylic plastic pieces are glued using a clear, water-thin, moderately fast-curing solvent cement for joining acrylic, WELD-ON 4.7,8 The following materials are used for the designated models. 1. Body-centered cubic 8 polystyrene boxes with lids 16 Styrofoam snowballs green tempera paint
Table 1. Calculation of Avogadro’s Number for Cr ystal Systems Composed of Cubic Unit Cells Element
Crystal System
d/ (g/cm3) (22a)
ma / (g /mol) (22a)
a/Å (22b)
NA a
22.990
4.2906
6.0 × 1023
3.1652
6.01 × 1023
Na
BCC; Z = 2
0.97
W
BCC; Z = 2
19.30
Fe
BCC; Z = 2
7.87
55.845
2.8665
6.03 × 1023
Ni
FCC; Z = 4
8.90
58.693
3.5240
6.03 × 1023
Cu
FCC; Z = 4
8.96
63.546
3.6146
6.01 × 1023
Ag
FCC; Z = 4
107.868
4.0857
6.03 × 1023
10.5
183.84
aThe volume of the unit cell is a3. N = [m Z (1 × 1024 Å3 cm᎑3)]/dV, A a where ma is the atomic mass, d is density, and V is the volume of the 3 unit cell in Å .
Table 2. Calculation of Avogadro’s Number for Cr ystal Systems Composed of Hexagonal Unit Cells Crystal System
d/ (g/cm3) (22b)
ma (g/mol) (22b)
a/Å (22c)
c/Å (22c)
NAa
Cb
Hex; Z = 4
2.26
12.011
2.4612
6.7079
6.04 × 1023
Zn
Hex: Z = 2
7.14
65.39
2.665
4.947
6.02 × 1023
Te
Hex; Z = 3
6.24
127.60
4.4570
5.9290
6.01 × 1023
Se
Hex: Z = 3
4.81
78.96
4.3642
4.9588
6.02 × 1023
Element
aThe
volume of the unit cell is 0.8660 a2c. NA is calculated as in Table 1.
bGraphite.
2. Face-centered cubic 8 polystyrene boxes with lids 32 Styrofoam snowballs blue tempera paint 3a. Sodium chloride (space filling) 1 polystyrene box with lid 4 Styrofoam snowballs (21Ú2-in. diameter) 4 Styrofoam snowballs (11Ú2-in. diameter) 3b. Sodium chloride (uniform sphere) 8 polystyrene boxes with lids 64 Styrofoam snowballs red tempera paint 4. Hexagonal crystal system (graphite structure) 8 acrylic hexagonal cells 32 Styrofoam snowballs black tempera paint 5. The incorrect unit cell of the sodium chloride structure 4 polystyrene boxes with lids 4 Styrofoam snowballs
Construction 1. Body-Centered Cubic. Eight Styrofoam snowballs are left whole and painted green; 8 are cut in 8ths (64 pieces) and painted green. The 8th pieces are glued into the 8 corners of the polystyrene cube. A whole piece is placed in the middle of the cube and fastened to two opposite corners along the cube diagonal with two toothpicks. This is repeated for a total of eight polystyrene cubes. Coordinates are (0,0,0), (1Ú2,1Ú2,1Ú2).9 470
Figure 5. The space-filling unit cell of the sodium chloride structure.
2. Face-Centered Cubic. Twenty-four Styrofoam snowballs are cut into halves (48 pieces) and painted blue; 8 are cut into 8ths (64 pieces) and painted blue. The 8th pieces are glued into the 8 corners of the polystyrene cube. One half piece is glued into the middle of each face of the cube. This is repeated for a total of eight polystyrene cubes. Coordinates are (0,0,0), (0,1Ú2,1Ú2). 3. Sodium Chloride Structure. One space-filling unit cell (Fig. 5) and eight unit cells with uniform spheres are constructed. 3a. SPACE-FILLING MODEL. In the case of ionic solids, it may be desirable to have a model that also shows the relative sizes of the ions, so as not to mislead students about the microscopic state. A single unit cell will accomplish this. The ratio of the edge length of the unit cube to the diameter of Na+ to the diameter of Cl᎑ is 5.6:2.04:3.6 (22d ). If these are
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In the Classroom
scaled to the actual clear plastic box edge length of 4 in., then Styrofoam balls of 11Ú2- and 21Ú2-in. diameter are required. One Styrofoam snowball (11Ú2 in.) is left whole and painted red. One (21Ú2 in.) is cut into 8ths and left unpainted. Three (21Ú2 in.) are cut into halves and left unpainted. Three (11Ú2 in.) are cut into quarters and painted red. One layer at a time is glued, starting at the bottom. The 8th pieces are glued into the corners, the halves into the middle of the faces, and the 4ths into the middle of the edges. 3b. UNIFORM-SPHERE MODEL. Eight unit cells composed of uniform spheres are constructed. This allows for a greater visibility of the individual atoms and layers. Eight Styrofoam snowballs are left whole and painted red. Eight are cut into 8ths (64 pieces) and left unpainted. Twenty-four are cut into halves (48 pieces) and left unpainted. Twenty-four are cut into quarters (96 pieces) and painted red. Each unpainted eighth piece is glued into a different corner of the polystyrene cube. One unpainted half piece is glued into the middle of each face of the polystyrene cube. One red quarter piece is glued into the middle of each edge. This is repeated for a total of eight cubes. A whole red piece is secured to the middle of the cube by attaching it to a face atom with a toothpick. Coordinates are (0,0,0), (0,1Ú2,1Ú2), (0,1Ú2,0), (1Ú2,1Ú2,1Ú2). 4. Hexagonal Crystal System (Graphite Structure [13]). The construction of this model is technically difficult. Some assistance may be needed from individuals experienced in woodworking. Sixteen rhombi (angles 120° and 60°, 10.2 cm on edge) are cut from acrylic plastic (top and bottom plates), using a table saw. Thirty-two rectangles (10.2 × 12.7 cm, beveled at 30° and 60°) are cut from acrylic plastic (sides). A cell, without top and bottom plates, is constructed by gluing the four sides together. The 12.7-cm edge is the long edge (c axis). A template is used to cut the Styrofoam snowballs into 6ths and 12ths. In the middle of a blank sheet of paper are drawn three intersecting lines 120° apart and about 7 cm long. A circle 11Ú2 in. in diameter is drawn, using a compass, with the intersection of the three lines as the center. The snowballs are cut in half and one half is placed on top of the circle. Using a marker, the outer edges of the half are marked where each line touches. Using an X-acto knife with a 6-cm blade and starting on the top center of the half, incisions are made to the marks, cutting all the way through. This gives 6th pieces. The sixth pieces are cut in half to make the 12ths. Altogether, 8 snowballs are left whole; 8 are cut into halves (16 pieces); 102Ú3 are cut into 6ths (64 pieces); 51Ú3 are cut into 12ths (64 pieces). After cutting, the pieces are painted black. Figure 6 depicts the assembly of the hexagonal structure. One-sixth pieces are glued into the 120° corners and 1Ú12 pieces are glued into the 60° corners. Two half-pieces are centered and glued onto the top and bottom plates (5.9 cm in from a vertex formed by a 120° angle along the long diagonal). Two 1Ú6 pieces are glued together (1Ú3) into the 120° angles in the middle of the long edge (c axis) of the cell. Two 1Ú12 pieces are glued together (1Ú6) into the 60° angles in the middle of the long edge (c axis) of the cell. Three toothpicks (6 cm) are pushed into a whole snowball to a distance of about 2 cm and the snowball is positioned in the middle of the cell according to Figure 6. The toothpicks are then pulled out with hemostats or tweezers until they are firmly embedded into the two
5.9 cm 1/ 2
top plate
long diagonal
1
/12
1/ 6
top layer
1/ 6
1/ 12
1/ 6
middle layer
1/ 3
1/ 3
1 1/ 6
1
/12
bottom layer
1/ 6
1/ 6
1
/12
5.9 cm 11 /2 /2
bottom plate
long diagonal
Figure 6. The hexagonal unit cell construction of the graphite structure.
neighboring 1Ú6 pieces and the neighboring 1Ú3 piece. The whole piece should be centered between these three pieces. The top and bottom plates then are hot-glued onto the cell. Coordinates are (0,0,0), (0,0,1Ú2), (1Ú3,2Ú3,0), (2Ú3,1Ú3,1Ú2).10 5. The Incorrect Unit Cell of the Sodium Chloride Structure. Eight Styrofoam snowballs are cut into 8ths (64 pieces); half are painted red and the other half are left unpainted. The red and unpainted pieces are glued into the corners of the 8 cubes in such a way as to have alternating colors at the corners. In other words, no two adjacent corners have the same colors.
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Acknowledgments I am indebted to the following students who contributed their time and effort in helping to construct the unit cells: Kerri Hahn, Nicole Perrine, Annemarie DellaGuardia, Leah Phillips, Jaime Villemonte, and Crystal Cantor. I am indebted to Lee Omohundro for assistance in the construction of the acrylic hexagonal parallelepipeds. I would like to thank Sheila E. B. Gould, Director, Beevers Miniature Models University of Edinburgh, for assistance with the hexagonal unit cell of the graphite structure. Lastly, I would like to thank the reviewers for their time and effort. Notes 1. A crystal model takes approximately 5 hours to construct. 2. The cost for a crystal model is under $40. 3. The reference to atoms can also include ions and molecules, whichever is appropriate for the material being discussed. 4. A very nice one containing 64 wooden spheres was obtained for less than $130: Sargent-Welch, P.O. Box 5229, Buffalo Grove, IL 60080-5229 (http://www.sargentwelch.com). Catalog no. WLS61917-10H. 5. AMAC Plastic Products, P.O. Box 750249, Petaluma, CA 94975-0249 (http://www.amacplastics.com/ ). (AMAC is not a retailer.) 6. An X-acto tool set is used. Styrofoam balls are easily halved using the X-acto aluminum miter box (12.5-cm saw blade). 7. IPS Corporation, Gardena, CA 90248. 8. Dichloromethane may be substituted. 9. The coordinates designate the locations of atoms within the unit cell (23). They are found by superimposing three mutually orthogonal coordinate axes onto one of the corners of the Bravais cell. The axes are labeled a, b, c and are parallel to the cell edges. The coordinates of any point are expressed as (a,b,c). The a axis is scaled in units of a0 (length of side “a” of the unit cube) and the other axes are scaled accordingly. For a cubic cell, α = β = γ = 90 and a0 = b0 = c0. As an example, in the body-centered cubic arrangement there are two atoms located within the cubic unit cell. One is at the origin (0,0,0) and the other (the body-centered atom) lies at the intersection of the cube diagonals with coordinates (1Ú2,1Ú2,1Ú2). When this unit cell is translated, the entire crystal structure is generated. 10. The coordinates are found by superimposing three coordinate
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axes onto one of the corners having a 120° angle. The coordinate axes are parallel to the edges of the cell. The a and b axes form a 120° angle (γ) and the c axis is mutually orthogonal to the a and b axes. (Note: for a hexagonal bravais cell a0 = b0.)
Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
21.
22.
23.
Bindel, T. H. J. Chem. Educ. 1992, 69, 305. Mattson, B. J. Chem. Educ. 2000, 77, 622. Cady, S. G. J. Chem. Educ. 1997, 74, 794. Laing, M. J. Chem. Educ. 1997, 74, 795. Li, T.; Worrell, J. H. J. Chem. Educ. 1989, 66, 73. Brady, K. T. J. Chem. Educ. 1983, 60, 36. Kildahl, N. K.; Bersa, L. H.; Bodner, G. M. J. Chem. Educ. 1986, 63, 62. Mann, A. W. J. Chem. Educ. 1973, 50, 652. Scattergood, A. J. Chem. Educ. 1937, 14, 140. Komuro, Y.; Sone, K. J. Chem. Educ. 1961, 38, 580. Seymour, K. M. J. Chem. Educ. 1938, 15, 192. Kennard, C. H. L. J. Chem. Educ. 1979, 56, 238. Olsen, R. C. J. Chem. Educ. 1967, 44, 728. Brock, C. P.; Lingafelter, E. C. J. Chem. Educ. 1980, 57, 552. Kotz, J. C.; Purcell, K. F. Chemistry and Chemical Reactivity; Saunders: Philadelphia, 1987; pp 401–408. Noggle, J. H. Physical Chemistry, 3rd ed.; HarperCollins: New York, 1996; pp 958–971. McKie, D.; McKie, C. Crystalline Solids; Wiley: New York, 1974; p 6. Livingston, R. L. J. Chem. Educ. 1967, 44, 377. Lisensky, George; Blackwell, Joey M. Solid State Resources, 2nd ed. [CD-ROM]; JCE Software 1995, SP-12. Ellis, A. B.; Geselbracht, M. J.; Johnson, B. J.; Lisensky, G. C.; Robinson, W. R. Teaching General Chemistry: A Materials Science Companion; Oxford University Press: New York, 1993. Lisensky, G. C.; Covert, J. C.; Mayer, L. A. Solid-State Model Kit Instruction Manual, 2nd ed.; ICE Publ. 94-007; Institute for Chemical Education: Madison, WI, 1994. CRC Handbook of Chemistry and Physics, 81st ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 2000; (a) p 12-19; (b) p 4-50; (c) p 4-149; (d) p 12-14. Pauling, L. General Chemistry, 3rd ed.; Freeman: San Francisco, 1970; pp 24–25.
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