Chemistry Everyday for Everyone Applications and Analogies
Using Balls from Different Sports To Model the Variation of Atomic Sizes Gabriel Pinto Departamento de Ingeniería Química Industrial, E.T.S.I. Industriales, Universidad Politécnica de Madrid, José Gutiérrez Abascal 2, 28006-Madrid, Spain
Teachers have found that analogies that appeal to reallife examples and students’ daily experience in the classroom usually help students to understand certain science concepts (1, 2). Such analogies help students to develop nonobservable pictures by comparing them to something observable with which they are familiar (3). Moreover, as pointed out by Fortman (4), analogical demonstrations are suitable in chemistry because the teaching of this science involves a great deal of abstract reasoning and the majority of students up through high school and beginning college still are making the transition from the concrete–operational to the formal–operational mode of thinking. Periodic variation of atomic sizes, normally evaluated by atomic radii, is an important topic in introductory chemistry courses, as is shown in most chemistry textbooks (5 ); but the order of magnitude of this variation, involving submicroscopic scales, is difficult for students to imagine. A model of this variation is to draw circles (or to cut them from cardboard) with radii proportional to the radii of different elements. But in this case, the students see circles, whereas atoms are spherical. When we say, for example, that the radius of one atom is twice that of another, students usually forget that according to the formula for volume (four-thirds × pi × radius cubed), the volume of one is eight times larger than that of the other! It is this massive increase in volume that students rarely realize and that is illustrated by the use of three-dimensional balls. In this article, I describe an analogy that can be used for discussion of this matter in introductory chemistry classes. Most students are very familiar with the world of sports. In any case, the teacher can make use of the wide informative coverage given to the Olympic Games or similar events, where different sports are televised in a few days. The radii of official balls of seven well-known sports are given in Figure 1. These sports are ping-pong (table tennis), tennis, baseball, handball (team handball), volleyball, soccer (association football), and basketball. For several sports, the given value is an average between allowed values. This can serve to show students that the tabulated covalent radius for an atom is an average value based on interatomic distances in different molecules. (In contrast, to illustrate the difference between the covalent radius and the van der Waals radius, an excellent analogy was proposed recently by Rayner-Canham [6 ], comparing the radii to close dancing and brushing past individuals.) Students are involved in the activity of finding atoms that correspond to the various sports balls because, for homework or for a classroom activity, they must assign an atom for each ball, by using tabulated single-bond covalent radii (7) and by considering that the smallest ball (the ping-pong ball) is assigned to the smallest atom (the hydrogen). They
must find the most similar atom in proportion to the ball of each radius, accepting a deviation of, for example, 5%. Excepting the tennis ball, which has no correspondence with any atom, the remainder of the balls correspond with the following atoms (radii in parentheses): H (0.037 nm), F (0.072 nm), Ca (0.174 nm), Sr (0.191 nm), K (0.203 nm), and Cs (0.235 nm), as it is shown in Figure 1. With this analogy the following effects can be debated in the classroom. •
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Comparison of the order of magnitude of variation between the extreme tabulated covalent radii—those of hydrogen and cesium. This variation, for example, is smaller than the difference between a marble and a large beach ball, but greater than the difference between a cherry and an orange. Typical trend for atomic size within a family in the periodic table: IA (H, K, and Cs) or IIA (Ca and Sr). Typical trend for atomic size for a row in the periodic table: fourth row (K and Ca).
An analogy and application is even more useful if it can be reused. Using the ionic radii values from the same data book by Stark and Wallace (7), the balls can be used to show how the radii change upon ionization. For example, cesium decreases from 0.235 nm (basketball) to 0.169 (handball) in its monopositive ion. During anion formation, nitrogen increases from 0.074 nm (baseball) to 0.171 nm (handball) in its trinegative ion. Most amazingly, hydrogen increases from 0.037 nm (ping-pong) to 0.208 nm (soccer) in its mononegative ion. Finally, there are two balls that can be used as typical cation–anion size comparisons: the tennis ball is about
Figure 1. Balls of several sports and corresponding atoms, by analogy in terms of relative size. The radius of typical official balls, in centimeters, is given in parentheses.
JChemEd.chem.wisc.edu • Vol. 75 No. 6 June 1998 • Journal of Chemical Education
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Chemistry Everyday for Everyone
the size of an Al3+ ion (0.050 nm), while the volleyball is about the size of a Br{ ion (0.195 nm). It then becomes very apparent why crystal packing of ionic compounds is often considered in terms of the anion packing in arrays with the cations fitting in the interstices. The most obvious limitation of the analogy is that, unlike balls, atoms do not have sharply defined boundaries: the judgment for deciding arbitrarily where an atom ends is based primarily on the way the atom interacts with what is around it. But, as pointed out by Webb (8), the identification of limitations by students gives the teacher an appreciation of whether the analogy has been understood. Acknowledgment I am grateful to the reviewers for their valuable suggestions.
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Literature Cited 1. Goh, N. K.; Chia, L. S.; Tan, D. J. Chem. Educ. 1994, 71, 733–734. 2. Thiele, R. B.; Treagust, D. F. Aust. Sci. Teach. J. 1991, 37 (2), 10–12. 3. Last, A. M. J. Chem. Educ. 1985, 62, 1015–1016. 4. Fortman, J. J. J. Chem. Educ. 1992, 69, 323–324. 5. See for example: Boikess, R. S.; Edelson, E. Chemical Principles; Harper and Row: New York, 1978; p 172. Zumdahl, S. S. Chemistry; Heath: Lexington, MA, 1986; p 279. Chang, R. Chemistry, 4th ed.; McGraw-Hill: New York, 1991; p 317. Pinto, G. Fundamentos de Química; Sección de Publicaciones ETSII-UPM: Madrid, 1996; p 187. 6. Rayner-Canham, G. J. Chem. Educ. 1994, 71, 943–944. 7. Stark, J. G.; Wallace, H. G. Chemistry Data Book. SI Edition; John Murray: London, 1980; p 27. 8. Webb, M. J. School Sci. Math. 1985, 85, 645–650.
Journal of Chemical Education • Vol. 75 No. 6 June 1998 • JChemEd.chem.wisc.edu
