Arrangements of Particles in Crystals

Description

Styrofoam models are built to illustrate cubic closest packing, hexagonal closest packing, body-centered cubic, and simple cubic crystal structures. The sodium chloride lattice is also demonstrated.

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• The crystalline state of matter consists of a regular arrangement of atoms, molecules, or ions. In this experiment, using styrofoam spheres as the building blocks, some of the packing arrangements will be studied.
• The regular geometric shapes for crystalline solids are evidence for the highly ordered arrangement of the atoms which comprise them. The persistence of this characteristic noticed when a large crystal is broken into succeedingly smaller fragments strongly suggests that the organization is present at the atomic level.
• Proof of the existence of microscopic order comes from the diffraction of X rays by crystals.
• This experiment examines three types of packing in metals: hexagonal closest packing; face-centered cubic packing; and body-centered cubic packing. The number of nearest neighbors (the coordination number) of the particles in each of the structures will be observed.
• In addition, an ionic crystal will be simulated by packing spheres of different radius.

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Coordination Number in a Plane

1. Pack spheres together in a plane, and label one central sphere. Count the number of spheres that this labeled sphere touches.

   

   I. Does the number of spheres touching a central sphere depend on the size of the spheres when all of the spheres are identical? Check your prediction.

   

2. Place spheres above the labeled sphere touching it. Imagine placing a like number of spheres below the labeled sphere.

   II. From this arrangement, decide the maximum number of spheres that can touch the labeled sphere.

3. This is called the coordination number, and the arrangement is called closest packing.

Hexagonal Closest Packing

1. Connect the groups of 7 spheres you used above with short lengths of pipe cleaner pieces to obtain the layers shown. Make 2 3-sphere layers.
2. Place one 3-sphere layer on the bench top. Place the layer of seven on top. Place another triangle, with its orientation identical to the first layer, on top as a third layer. Every other plane is identical. This is an ABABABAB... pattern.

Face-Centered Cubic Packing (also called Cubic Closest Packing)

1. Construct the layers shown using 2-inch spheres and pipe cleaner pieces as above. Place the first layer flat on the desk. Place the second layer on it so that the spheres rest in the spaces between the corner spheres of the first layer. Place the third layer directly over the first layer. Lift the top layer, and rotate is 60°. In this way, the top-most layer is not exactly the same as the bottom-most layer. Every third layer is identical. this is an ABCABCABC... pattern.
2. Set aside the closest packing model just prepared from planes of spheres.

Comparison of Hexagonal Closest Packing with Cubic Closest Packing

1. Construct a model for cubic closest packing (face-centered cubic) from square layers.
2. Place this face-centered packing model alongside the hexagonal closest packing. Rotate the top layer by 60° to make the ABCABC...pattern.
3. Remove the top layer of the face-centered packing model. Tilt the remainder toward you, and rotate it by 60°.
4. Note that these models, built by different schemes, are really identical in terms of the shown internal arrangement of spheres.

Body-Centered Cubic Packing

1. Construct the layers shown leaving a space of about 1 cm between the spheres as indicated. Place one square layer on the surface. Place the unconnected sphere over the center of the first layer, and then place the second square layer on top so that it lies over the first square layer.

III. Why is this called a "body centered" arrangement?

The Sodium Chloride Lattice

1. Ionic crystals are formed when positive and negative ions pack in a regular array or lattice. Sodium and chloride ions have diameters of 190 pm and 362 pm, respectively. We will use 1-inch spheres for Na⁺ and 2-inch spheres for Cl^- in our model. Build the layers shown. Note that the NaCl lattice is an "interpenetrating set" of face-centered cubes-one involving Na⁺ ions, and the other Cl^- ions.

   IV. What type of ion surrounds each Na⁺? Each Cl^-? Find the coordination number of the spheres representing Na⁺. Find the coordination number of the spheres representing Cl^-.

2. In order to achieve this type of lattice, limits are placed upon the sizes of the spheres. If you imagined the sodium ions to somehow increase in size, they could take on relatively more negative ions. Likewise, were they to shrink, then the six negative ions might find themselves too crowded, and try to reduce their coordination number. In fact, different ionic crystal packing arrangements are known, and these are related to the radius ratios of the ions.

   V. Find the radius ratio for Na⁺/Cl^- in this model.

3. Tables of ionic radii are included with the Handout for you to answer the questions.

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Name _____________________________ Class _______

Teacher______________________________

DoChem 042 Arrangements of Particles in Crystals

Makeup student should watch the movies and answer the questions after each step in the procedure section.

Answer the 5 Roman-numbered questions in the procedure section.

Answer the following questions after you watch the movie.

Questions:

1. Write a brief description of each type of packing of metallic crystals that you studied.
2. In one of the types of cubic packing, the spheres occupy about two-thirds of the space and in the other they fill about three-fourths of the space available. Identify which type is which.

   Other factors being equal, which is more dense?

   In which does each atom form the larger number of bonds?

3. Suppose you have a crystal XY with the sodium chloride structure. The ions are doubly charged (X²⁺ and Y^2-) but have the same size as Na⁺ and Cl^-. Predict the melting temperature of XY compared to that of NaCl. Suggest a real pair of ions which meets the criteria above, and look up their melting temperatures to check your prediction.
4. Suppose you have a crystal MX with the sodium chloride packing in which each of the ions has the same charge M⁺ and X^- as Na⁺ and Cl^-, but the radii of M and X are proportionately larger. Predict the melting temperature of MX compared to that of NaCl. Suggest a pair of ions that meets this criterion, and look up the melting temperature of the solid to check your prediction.

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Purpose

To construct and examine models of ions and metallic solids.

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(per 10 students)

• 180 2-inch styrofoam spheres
• 65 1-inch styrofoam spheres
• 190 pieces of pipe cleaner or toothpicks

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• Due to the large number of spheres required you may wish the students to work in pairs or larger groups.
• If a large number of spheres is available, other models may be constructed.
• Show and discuss a display of crystals which can be made from samples found around the laboratory.

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Model building: 2-3 hours

Presentation: 20 minutes

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No unusual hazards are presented during this experiment. (Hazards during assembly include cuts from sharp blades, and sensitivity to glues.)

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Follow normal precautions for laboratory work.

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Store materials for future use.

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Presentation Question:

• What is the effect of using a closed square -- in which four spheres touch -- in place of the open square for this model?
  • By forcing the spheres in the square to touch, a distortion from cubic symmetry would be introduced into the structure.

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The students should write answers to all questions raised in the procedure section. Have them label the questions by steps.

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• Except for the NaCl lattice, the models represent unit cells of perfect metallic crystals. In these crystals the lattice points are imagined to be occupied by identical, positive, spherically-shaped, metallic ions. In general, the atoms of most metallic elements tend to pack so as to achieve a maximum coordination number.
• It can be shown that crystals which pack in closest-packed structures have 26% empty space; 74% of the volume is filled by atoms. Less efficient packing of atoms in the body-centered structure gives rise to softer metals with lower densities. The density of a crystal is related to the mass of the atoms and the way in which they are packed together.

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Closure Questions:

1. Write a brief description of each type of packing of metallic crystals that you studied.
2. In one of the types of cubic packing, the spheres occupy about two-thirds of the space and in the other they fill about three-fourths of the space available. Identify which type is which. Other factors being equal, which is more dense? In which does each atom form the larger number of bonds?
3. Suppose you have a crystal XY with the sodium chloride structure. The ions are doubly charged (X²⁺ and Y^2-) but have the same size as Na⁺ and Cl^-. Predict the melting temperature of XY compared to that of NaCl. Suggest a real pair of ions which meets the criteria above, and look up their melting temperatures to check your prediction.
4. Suppose you have a crystal MX with the sodium chloride packing in which each of the ions has the same charge M⁺ and X^- as Na⁺ and Cl^-, but the radii of M and X are proportionately larger. Predict the melting temperature of MX compared to that of NaCl. Suggest a pair of ions that meets this criterion, and look up the melting temperature of the solid to check your prediction.

Answers to Closure Questions:

1. Body-centered cubic has coordination number 8 with two types of interleaved planes. The closest packed structures have coordination number 12. Hexagonal has two alternating planes -- the first and third and fifth planes are identical, as are the second and fourth and sixth. Cubic closest packing also has coordination number 12. It has 3 arrangements of planes: the first, fourth, seventh; the second, fifth, eighth; and the third, sixth and ninth would be identical. (Simple cubic structures, not studied in this experiment, have coordination number 6, and considerable empty space, with alternating planes being identical.) The closest packed structures have less empty space than do the body-centered cubic structures.
2. Closest packing has the least space and implies the most dense packing. Body-centered cubic is implied to be less dense than either of the closest packed structures -- which are predicted to be equally dense. (Simple cubic has the most space and implies the least dense materials. There is only one example among the elements of this structure, however.) The largest number of bonds is 12 (coordination number) which is found in both closest packed structures.
3. The larger charge implies stronger forces and tighter bonds. Barium oxide might be such a compound. A larger cation is selected to keep about the same radius ratio. The melting temperatures are: NaCl, 801 °C; and BaO, 1918 °C.
4. Rubidium bromide or cesium iodide would be appropriate compounds. The larger distances of separation for the same charge imply weaker forces which, in turn, imply lower melting temperatures. The experimental melting temperatures are: NaCl, 801 °C; RbBr, 693 °C; and CsI, 626 °C.

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• crystal lattice
• ions
• ionic solids
• metallic solids
• cubic closest packing
• body-centered cubic
• simple cubic
• face-centered cubic
• hexagonal closest packing

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