Structural and Mental Models

Abstract

In this example both the original and the model can be observed: the photo of the castle as well as the model in Disneyland can be compared to the Bavarian original in detail. In scientific models for chemical structures the spheres of the model cannot be compared to the submicroscopic original atoms, ions or molecules, because it is impossible to see them – neither with a magnifying glass nor with the best microscope.

Appendices

Appendix A. Problems and Exercises

• P6.1 You will find models in your chemistry collection at school: NaCl packing of spheres, NaCl crystal structure and NaCl unit cell (see Fig. 6.3). Give the features of depiction of the three models. Discuss and compare the irrelevant components of these models.

• P6.2 The NaCl-structure can be described by the cubic face-centered lattice of chloride ions where the octahedral holes are filled with sodium ions. Describe the Li₂O-structure and the ZnS-structure in a similar way and draw the corresponding models (see Fig. 6.13).

• P6.3 Planning of chemistry lessons applies two levels: level 1 deals with phenomena and lab experiences, level 2 with structural and mental models (see Fig. 6.11). Describe and draw your mental model of the diffusion of hydrogen sulfide (H₂S) in air on the basis of (a) the particle model of matter, (b) Dalton’s atomic model.

• P6.4 Usually mental models are introduced in chemistry lessons from the particle model, via Dalton’s atomic model to the nucleus–shell model. Chose (a) a substance and (b) a chemical reaction and make model drawings on the basis of these three models. Discuss the differences in interpretation based on those models.

• P6.5 The chemical equilibrium can be illustrated with model experiments or with every-day experiences. Give one example for each and establish connections to an example of a real chemical equilibrium.

Appendix B. Modeling Course: Structures of Metals and Salts

Material: 100 white cellulose balls d = 30 mm [26], 50 red cellulose balls d = 12 mm [26], triangular wooden frame (a = 17.5 cm), square wooden frame (a = 15 cm), modeling clay, glue, two equilateral ball-triangles (six balls with d = 30 mm each).

Structure the plural: Close-packing of spheres (1 metal atom ≡ 1 sphere) are structural models to describe the structure of many metal crystals (find drawing models at the end of this sheet).

• M 6.1: Fill up the triangular wooden frame with a closest layer of balls in the triangular pattern. Put as many layers of balls on top as possible. Draw the layers of balls.

• M 6.2: The coordination number stands for the number of balls that touch one ball in the middle of the packing. Find out the coordination number of one sphere in the close-packing of spheres. Draw three layers of balls to visualize this number.

• M 6.3: Two different close-packings of spheres with the coordination number 12 are possible:

  (a) With the layer sequence ABCABC…, (b) with the layer sequence ABAB…

  Build both packings! Draw the layers of balls with triangular pattern so that (a) and (b) become apparent.

  Definition: A layer sequence ABCA… exists, when the fourth layer of balls is congruent with the first layer – seen from above. The layer sequence ABA… exists, when the third layer of balls is already congruent with the first layer (the layers in triangular pattern are meant).

  Information: An elementary cube (see picture) can be found in the ABCA-packing of spheres. Therefore this packing is also called cubic close-packing of spheres.

• M 6.4: Draw the crystal structure model next to the shown packing. Draw a perspective cube: instead of the balls only give the central points of the balls and connect these points.

  

• M 6.5: Put together the shown model with the help of the two equilateral 6-spheres triangles and two additional spheres.

  Try to build this cube into the closest ABC-packing (M 6.3).

  Draw two possible ways for building the elementary cube:

  (a) Connect layers in a triangle pattern (1 + 6 + 6 + 1), (b) in a square pattern (5 + 4 + 5).

• M 6.6: Take the square wooden frame. Build the cubic close-packing of spheres beginning with the square pattern. Try to get the elementary cube into the packing. Determine the coordination number. Draw the sphere layers so that the coordination number can be determined.

  Information: The following models show the structure of metal crystals:

  • 1. The hexagonal close-packing of spheres with the layer sequence ABA: it shows how crystals of magnesium and zinc are built up of atoms. One can say that there are crystals with hexagonal symmetry or crystals of the Mg-type.

  • 2. The cubic close-packing of spheres with the layer sequence ABCA in triangular pattern: it shows in which way crystals of copper, silver or gold are built up of atoms. One can say that there are crystals with cubic symmetry or crystals of the Cu-type. The elementary cube has one sphere in every face center – therefore this structure is also called face-centered cubic.

  • 3. The name “face-centered cubic” is possible to point out the difference to the body-centered cubic packing of spheres. It is not a close-packing anymore, the coordination number is 8 (see picture). Metal crystals of tungsten and alkaline metals have this structure, the W-type.

• M 6.7: The nine-spheres packing shows the elementary cube of body-centered cubic metal structures. Draw the crystal structure next to the picture: draw a perspective cube, instead of the balls only give the central points of the balls and connect these points. Compare it with M6.4.

  

Structures of salt crystals: The structure of many metal crystals can be illustrated by packings of spheres of one kind – the structure of salt crystals needs two kinds of spheres. Models for the sodium chloride structure and three other salt crystals will be built in the following (drawing models of the crystal structures are shown at the end of this section).

• M 6.8: Na⁺ ions in sodium chloride are to be represented with red balls (d = 12 mm), Cl⁻ ions with white balls (d = 30 mm). With the help of the triangle frame build a close-packing of spheres with both kinds of balls. Draw the layers of balls.

• M 6.9: Determine the coordination number for both kinds of balls. Draw the layers of balls to visualize the coordination number of both kinds of balls.

• M 6.10: In the close-packing of spheres there are two different-sized kinds of holes or gaps. Determine the number of balls that form those two different gaps. Draw the gap-producing balls for both kinds of holes in the form of the layers of balls: (a) for big gaps, (b) for small gaps.

  Information: Two different types of holes can be found in the close-packing of spheres. Convince yourself with the help of the model M 6.8:

  • 1. Big holes are formed by six balls with octahedral geometry: octahedral gaps (OG)

  • 2. Small holes are formed by four balls with tetrahedral geometry: tetrahedral gaps (TG)

  • 3. Spheres, OG and TG exist in a ratio of 1:1:2 in the close-packing of spheres.

  The structure of the sodium chloride crystal can therefore be explained like this: Cl⁻ ions form a cubic close-packing of spheres, all octahedral gaps are occupied by smaller Na⁺ ions. The coordination numbers are 6 and 6, the ion ratio is 1:1, the ratio symbol should be (Na⁺)₁(Cl⁻)₁

  Explain the cubic shape of the sodium chloride crystal.

• M 6.11: Take the elementary cube from M6.5, look for octahedral gaps and fill them with small red balls.

  • (a) Complete the model drawing by filling the gaps with small spheres (see picture).

  • (b) Draw the crystal structure next to the picture. Draw a perspective cube: instead of the balls only give the central points of the balls and connect these points (see M6.4).

• M 6.12: Build the cubic close-packing of spheres with the help of the square frame and with both kinds of spheres. Draw the sequence of the 2-spheres layers.

• M 6.13: Convince yourself that the elementary cube can be build into the packing of spheres beginning with the triangle pattern (M 6.8) as well as the one beginning with the square pattern (M 6.12). Which position does it take in both models?

  Draw the elementary cube with the help of (a) layers of spheres in the triangular pattern, (b) layers of spheres in the square pattern.

• M 6.14: A model for the aluminium oxide structure (corundum structure) may be built:

  • (a) Stick together three layers with 15 white balls each (see picture).

  • (b) Add ten small balls in the shown pattern (see picture).

  • (c) Stack up three layers so that the sequence is ABA. The coordination number of the small balls has to be six, of the big balls has to be four. What is the ratio of the balls?

  Information: The O²⁻ ions form a hexagonal close-packing in aluminium oxide crystals, only 2/3 of octahedral gaps are occupied by Al³⁺ ions. The coordination is 6/4, the ratio of ions is 2:3. Therefore the models have to be abbreviated to formulae like {(Al³⁺)₂(O²⁻)₃} or Al₂O₃.

• M 6.15: Form a few small balls with modeling clay that fit into the tetrahedral gaps of big spheres. With the small and the big spheres build an elementary cube

  • (a) For the zinc sulfide structure (see picture).

  • (b) For the lithium oxide structure (see picture).

  Information: zinc sulfide can be described as a cubic close-packing of S²⁻ ions, half of the tetrahedral gaps are occupied by Zn²⁺ ions. The coordination is 4/4, the formula for the unit cell is {(Zn²⁺)₄(S²)₄}, the empirical formula ZnS.

  Lithium oxide can be shown as a cubic close-packing of O²⁻ ions, all of the tetrahedron gaps are occupied by Li⁺ ions. The coordination is 4/8, the formula for the unit cell is {(Li⁺)₈(O²⁻)₄}, the empirical formula Li₂O.

On the following two pages you will find the expected model drawings for the models M6.13 to M6.15. Compare your own drawings with these.