In the Classroom
W
Density Visualization Richard L. Keiter and Whitney L. Puzey Department of Chemistry, Eastern Illinois University, Charleston, IL 61920-3099 Erin A. Blitz* Department of Art, Eastern Illinois University, Charleston, IL 61920-3099
Our interest in densities of metals began with the publication of Oliver Sacks’s Uncle Tungsten (1). Research in our group has long centered on complexes of tungsten and so naturally this book stimulated our curiosity. The enthusiasm of Sacks’s uncle for the properties of tungsten led us to order a 5-cm slug so that we too could heft it and sense its enormous density. The next step in the evolution of this project came in recalling a lecture given by a blacksmith who crafted jewelry made of iron. In the question and answer period a student asked whether anyone would be willing to wear something as heavy as iron in place of lighter metals such as silver and gold. Clearly the general public associates iron with massive heavy equipment while silver and gold are viewed as delicate metals. Producers of movies who show actors running with suitcases of gold bars fail to realize (or do not care) that their stars are often carrying quantities that would weigh hundreds of pounds if the bars were genuine.1 Even our students of chemistry often have a blurred view of density. It is best not to place on an exam the give-away question, “if the density of one marble is 2.5 g cm᎑3, what is the density of four marbles?” Our experience is that one out of three students will choose 10 g cm᎑3, not grasping the difference between an intensive and extensive property. In talking about relative densities of metals in the classroom setting it is not uncommon to pass around samples of metals such as aluminum and lead so that students can experience their relative heaviness. It has been pointed out in this Journal that if a student is asked to choose the heavier of two similarly-sized objects when one weighs more but is less dense, the student will choose the one which weighs less, but is more dense (2). In other words the hand senses pressure (thus density) rather than force (weight). Thus hefting two objects of similar volume allows determination of relative density. It seemed to us that a more visual evaluation of metal densities would greatly augment heft comparisons. With that in mind we sought commercial metal rods of constant diameter, cut to lengths such that all would have the same mass. These rods were placed into round holes drilled to accommodate one-quarter of their lengths to ensure stability. The lengths of the metal rods exposed thus are inversely proportional to density; the taller the rod, the smaller its density. To see the magnesium rod standing eleven times higher than the tungsten rod leaves a strong impression on students. The rods are removable so that they can be passed around the classroom. Here the students express surprise when they heft the equal weight samples of tungsten and magnesium because the much shorter tungsten sample seems much heavier. To assist us in the construction of our display we collaborated with a colleague in the art department so that our densometer would be both artistic and functional. The display is shown in Figures 1 and 2.2 Nine metal samples of www.JCE.DivCHED.org
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Figure 1. The densometer—a display of relative volumes of nine metals (W, Pb, Ag, Cu, Fe, V, Ti, Al, and Mg) having the same mass. Photo by Erin Blitz.
Figure 2. Sophomore Whitney Puzey discussing relative densities of metals. Photo by Bev Kruse.
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equal mass were chosen for the display on the basis of density variation, availability, and cost. While it would be ideal to display the least dense metal (Li) and the most dense metal (Ir or Os), the reactivity of the former and the cost of the latter led to their elimination from consideration.3,4 A rod of gold, of course, would be quite attractive but hardly within the budgets of most departments.5
Table 1. Metallic Radii and Densities for Selected Metals Metalsa
Density/(g cm᎑3)b
Radius/Åc
0.534
1.53
bc
K
0.862
2.29
bc
oNa
0.971
1.86
bc
oRb
1.532
2.43
bc
oCa
1.55
1.97
ohcp
oMg
1.738
1.60
ohcp
Be
1.848
1.12
ohcp
Cs
1.873
2.65
bc ohcp
Sr
2.54
2.15
Al
2.6989
1.43
occp
Sc
2.989
1.64
ohcp
3.5
2.18
bc
Y
4.469
1.82
ohcp
Ti
4.54
1.47
ohcp
6.11
1.31
bc
oZr
V
6.506
1.60
ohcp
oZn
7.133
1.37
ohcp
oCr
7.19
1.26
bc
oFe
7.874
1.23
bc
oNb
8.57
1.43
bc
oCo
8.9
1.25
ohcp
oNi
8.902
1.25
occp occp
oCu
8.96
1.28
oMo
10.22
1.36
bc
oAg
10.5
1.44
occp
oPb
11.35
1.75
occp
Tl
11.85
1.71
ohcp
oPd
12.02
1.37
occp
oRu
12.41
1.34
ohcp
oRh
12.41
1.34
occp
oHf
13.31
1.59
ohcp
oTa
16.654
1.43
bc
oAu
19.3
1.44
occp
oW
19.3
1.37
bc
oRe
21.02
1.37
ohcp
oPt
21.45
1.39
occp
oIr
22.42
1.36
occp
oOs
22.57
1.35
ohcp
Metals listed in order of increasing density. Metals with unit cells that deviate significantly from idealized hcp, ccp, and bc structures (e.g., Mn) were not included in our plots. The lanthanides, actinides, and other radioactive elements were also omitted. bData from ref 10. cData from ref 11 (see note 5). a
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4M
Density ( ccp ) =
3
1
(1)
r3
NA 8 2
Structure
Li
oBa
The density of a metal is proportional to its atomic mass and inversely proportional to its atomic radius cubed (3–7). It also depends on the arrangement of the atoms in the unit cell as well as the temperature and pressure. Metals may crystallize in cubic close-packed (ccp), hexagonal close-packed (hcp), or body-centered cubic (bc) arrangements. For perfect cubic close-packed and hexagonal close-packed structures the theoretical densities, based on their unit cells, are the same and can be expressed mathematically as
2M
Density ( hcp ) =
1
( ) ( 2r )
NA sin 60
8 3
2
o
2
2r
(2)
where M is the atomic mass, NA is Avogadro’s number, and r is atomic radius (8, 9). The density of the body-centered cubic structure is given by 2M
Density ( bc ) =
3
NA
4 1
3
r
(3)
2
While the metals chosen for the display were not selected primarily to show periodic trends, it can be seen that the density of the first-row transition series increases in going from Ti (4.54) to V (6.11) to Fe (7.87) to Cu (8.96). As the above equations make clear a small radius and a large mass lead to a greater density. For this series radii (atomic masses) are 1.47 (47.9), 1.31 (50.9), 1.23 (55.8), and 1.28 (63.6) Å, respectively. The slight increase in radius for Cu compared to Fe does not offset the increase in mass. Furthermore, Cu crystallizes as a cubic close-packed structure, while Fe is a bodycentered cubic, which in itself leads to an 8.8% increase in density.6 Both magnesium and aluminum have close-packed structures but the latter has a greater density because it has a smaller radius and a larger mass. We also see to a limited extent that density increases as one descends a group (Cu, Ag) and that the elements in the sixth period (W, Pb) are the most dense of those in the display. While the second- and third-row elements have larger atomic radii than those of the first row, which would lead to a smaller density, the large increase in mass on descending a group dominates, leading to a significant increase in density. A useful out-of-class exercise for students is to have them plot the experimental densities of metals versus (atomic mass)兾(radius)3 as found in the literature and from the slopes of these plots determine Avogadro’s number (10, 11). Densities and radii for the metals used for this exercise are found in Table 1.7 For perfect close-packed and body-centered structures, theoretical slopes of 0.293 and 0.269 are expected. Many structures are distorted from ideal geometries, how-
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In the Classroom
ccp & hcp bc linear (ccp & hcp) linear (bc)
25
20
15
10
5
0
Summary
0
Metal rods of high purity for many elements are now commercially available and may be used to construct a display of relative densities. An attractive feature of such a display for teaching purposes is that the rods may be removed from it for student examination. Such examination leaves a more lasting impression of relative density than looking at tabulated densities or bar graphs. The number and types of metals chosen for the display can be varied according to the available budget and periodic trends of interest. Density understanding may be further reinforced by having students prepare plots of experimental densities of metals versus (atomic mass)兾(radius)3 for close-packed and body-centered cubic structures. These plots can be used to obtain Avogadro’s number and to show that, for a wide range of metals, structural deviations from perfect geometries are small. W
30
Density / (g/cm3)
ever, to give small deviations from a perfect straight line (Figure 3).8 In addition, the uncertainties associated with the densities in ref 10 are not given and some may be less precisely determined than others. Nevertheless, slopes obtained from these plots, 0.2930(11) and 0.2698(12) for hcp兾ccp and bc, respectively, are in excellent agreement with the theoretical expectations. Avogadro’s number, calculated from these slopes, was found to be 6.0216(12) × 1023 and 6.033(11) × 1023, comparing well with the literature value, 6.0221335(30) × 1023 (12).9 While an individual crystal structure and density may be used to calculate Avogadro’s number, this exercise shows how broadly applicable the model is to a large collection of metals.
Supplemental Material
Detailed instructions for the construction of the densometer are available in this issue of JCE Online. Acknowledgments We acknowledge with gratitude the advice and guidance given by Ellen A. Keiter and Douglas E. Brandt. We are grateful to Scott M. Tremain and Wen Zhang for technical assistance. Notes 1. For example, in The Italian Job (2003) boxes and satchels of gold bars are carried with ease. About twelve thousand pounds (208 bars of 12 in. × 4 in. × 2 in.) of gold is loaded into a couple of Mini Coopers that weigh about twenty-five hundred pounds each; the cars show no noticeable sag or loss of performance. Michael Caine can sword fight in Secondhand Lions (2003) while multiple bags of gold are hanging from his body. In the Three Kings (1999) George Clooney and his buddies carry containers (weighing hundreds of pounds) filled with gold bars. In The Mask of Zorro (1998) a gold bar is lifted as easily as balsa wood, and in Raiders of the Lost Ark (1981) Harrison Ford has no trouble moving a gold idol. 2. To complement the set of rods of constant mass and varying volume it is also helpful to have a set of rods of constant volume and varying mass to be used for the traditional heft test. A
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10
20
30
40
Mass Radius3
50
60
70
80
90
100
g mol Å3
Figure 3. Plots of density versus mass/radius3 for cubic close-packed (ccp), hexagonal close-packed (hcp), and body-centered cubic (bc) structures.
variety of density samples are available from Educational Innovations Inc. For example, a density cubes kit is available that features six equally sized cubes made of six different metals. In addition, racks of aluminum or brass rods of different lengths are available for density determinations in the laboratory. 3. Lithium has a density of 0.534 g cm᎑3. It can be compared to other metals in the display by representing it with a 0.5 in. wooden rod (181 cm in length). 4. The densities of iridium and osmium are so similar that there is some uncertainty over which is the more dense. Reference (10) reports the values to be 22.42 (Ir) and 22.57 (Os) g cm᎑3, but suggests that crystallographic values of 22.65 (Ir) and 22.61 (Os) g cm᎑3 may be more accurate. Crystallographic values of 22.560 (Ir) and 22.590 (Os) g cm᎑3 have also been reported (9) as well as values of 22.57 (Ir) and 22.59 (Os) g cm᎑3 (13). 5. The densities of gold and tungsten are the same (19.3 g cm᎑3) to three significant figures. 6. The ccp structure of Cu has a 0.74 packing efficiency and the bc structure of Fe has a 0.68 packing efficiency. The ratio of 0.74兾0.68 corresponds to an 8.8% increase. 7. A note of caution here. Metallic radii for body-centered cubic structures in most tables are expressed as 12-coordination values. To obtain 8-coordination values the former must be multiplied by 0.973. Otherwise the same straight line will be obtained for all the metals regardless of whether they have hcp兾ccp or bc structures (11). 8. For example, if the density of osmium is calculated with eq 2 from its mass and radius, assuming a perfect hexagonal structure, c = (8兾3)1/2a where c is the unit cell height and a is the length of the unit cell base, a value of 21.85 g cm᎑3 is obtained. The hcp lattice is distorted, however, as shown in ref 9. The experimental value of a is 2.743 Å, which would require c to be 4.4651 for a perfect hcp lattice. The value of c is actually 4.3200 Å leading to a calculated density of 22.59 g cm᎑3. 9. The value of Avogadro’s number has been determined very precisely with extraordinarily pure silicon-28 crystals and X-ray inteferometry (12).
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Literature Cited 1. 2. 3. 4. 5. 6. 7.
Sacks, O. Uncle Tungsten; Alfred A. Knopf: New York, 2001. Blanck, H. F. J. Chem. Educ. 1977, 54, 628. Singman, C. N. J. Chem. Educ. 1984, 61, 137–142. Pauling, L.; Herman, Z. S. J. Chem. Educ. 1885, 62, 1086–1088. DeMeo, S. J. Chem. Educ. 2001, 78, 201–203. Laing, M. J. Chem. Educ. 2001, 78, 1054–1058. Hawkes, S. J. J. Chem. Educ. 2004, 81, 14–15.
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8. Rodgers, G. E. Descriptive Inorganic, Coordination, and SolidState Chemistry, 2nd ed.; Brooks/Cole: Toronto, 2002. 9. Arblaster, J. W. Platinum Metals Rev. 1989, 33, 14–16. 10. Handbook of Chemistry and Physics, 84th ed.; Lide, D., Ed.; CRC Press: New York, 2003–2004. 11. Wells, A. F. Structural Inorganic Chemistry, 5th ed.; Clarendon Press: Oxford, 1991. 12. Becker, P. Rep. Prog. Phys. 2001, 64, 1945–2008. 13. Crabtree, R. H. J. Less–Common Metals 1979, 64, 7–9.
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