Answer: TiO 2. Answer: The radius of the potassium ion is 1. Answer: Figure This illustration shows an X-ray diffraction image similar to the one Franklin found in her research. Chemistry End of Chapter Exercises Describe the crystal structure of iron, which crystallizes with two equivalent metal atoms in a cubic unit cell.
Describe the crystal structure of Pt, which crystallizes with four equivalent metal atoms in a cubic unit cell. What is the coordination number of a chromium atom in the body-centered cubic structure of chromium?
What is the coordination number of an aluminum atom in the face-centered cubic structure of aluminum? Cobalt metal crystallizes in a hexagonal closest packed structure.
What is the coordination number of a cobalt atom? Nickel metal crystallizes in a cubic closest packed structure. What is the coordination number of a nickel atom? Tungsten crystallizes in a body-centered cubic unit cell with an edge length of 3.
Calculate the edge length of the face-centered cubic unit cell and the density of platinum. Barium crystallizes in a body-centered cubic unit cell with an edge length of 5. Calculate the edge length of the face-centered cubic unit cell and the density of aluminum. The density of aluminum is 2. Explain why Si has the lower density even though it has heavier atoms. The free space in a metal may be found by subtracting the volume of the atoms in a unit cell from the volume of the cell.
Calculate the percentage of free space in each of the three cubic lattices if all atoms in each are of equal size and touch their nearest neighbors. Which of these structures represents the most efficient packing? That is, which packs with the least amount of unused space? Cadmium sulfide, sometimes used as a yellow pigment by artists, crystallizes with cadmium, occupying one-half of the tetrahedral holes in a closest packed array of sulfide ions.
What is the formula of cadmium sulfide? Explain your answer. A compound of cadmium, tin, and phosphorus is used in the fabrication of some semiconductors. It crystallizes with cadmium occupying one-fourth of the tetrahedral holes and tin occupying one-fourth of the tetrahedral holes in a closest packed array of phosphide ions.
What is the formula of the compound? What is the formula of the magnetic oxide of cobalt, used in recording tapes, that crystallizes with cobalt atoms occupying one-eighth of the tetrahedral holes and one-half of the octahedral holes in a closely packed array of oxide ions? A compound containing zinc, aluminum, and sulfur crystallizes with a closest-packed array of sulfide ions.
Zinc ions are found in one-eighth of the tetrahedral holes and aluminum ions in one-half of the octahedral holes. What is the empirical formula of the compound? A compound of thallium and iodine crystallizes in a simple cubic array of iodide ions with thallium ions in all of the cubic holes. What is the formula of this iodide? Which of the following elements reacts with sulfur to form a solid in which the sulfur atoms form a closest-packed array with all of the octahedral holes occupied: Li, Na, Be, Ca, or Al?
What is the percent by mass of titanium in rutile, a mineral that contains titanium and oxygen, if structure can be described as a closest packed array of oxide ions with titanium ions in one-half of the octahedral holes? What is the oxidation number of titanium? Explain why the chemically similar alkali metal chlorides NaCl and CsCl have different structures, whereas the chemically different NaCl and MnS have the same structure. As minerals were formed from the molten magma, different ions occupied the same cites in the crystals.
Lithium often occurs along with magnesium in minerals despite the difference in the charge on their ions. Suggest an explanation. Rubidium iodide crystallizes with a cubic unit cell that contains iodide ions at the corners and a rubidium ion in the center. One of the various manganese oxides crystallizes with a cubic unit cell that contains manganese ions at the corners and in the center. Oxide ions are located at the center of each edge of the unit cell. NaH crystallizes with the same crystal structure as NaCl.
The edge length of the cubic unit cell of NaH is 4. Thallium I iodide crystallizes with the same structure as CsCl. The edge length of the unit cell of TlI is 4. A cubic unit cell contains manganese ions at the corners and fluoride ions at the center of each edge. What is the spacing between crystal planes that diffract X-rays with a wavelength of 1. A diffractometer using X-rays with a wavelength of 0.
Determine the spacing between the diffracting planes in this crystal. A metal with spacing between planes equal to 0. What is the diffraction angle for the first order diffraction peak? Gold crystallizes in a face-centered cubic unit cell. The wavelength of the X-rays is 1. What is the density of metallic gold? When an electron in an excited molybdenum atom falls from the L to the K shell, an X-ray is emitted.
These X-rays are diffracted at an angle of 7. Simple Cubic 2. Face-Centered Cubic 2a. Diamond Cubic 3. Body-Centered Cubic 4. Simple Hexagonal 4a. Hexagonal Close-Packed 4b. Double Hexagonal Close-Packed La-type 5. Rhombohedral 5a. Rhombohedral Close-Packed Sm-type 6. Simple Tetragonal 7. Body-Centered Tetragonal 7a. Diamond Tetragonal White Tin 8. Simple Orthorhombic 9. Base-Centered Orthorhombic Face-Centered Orthorhombic Body-Centered Orthorhombic Simple Monoclinic Because the ccp structure contains hexagonally packed layers, it does not look particularly cubic.
The hcp and ccp structures differ only in the way their layers are stacked. The illustrations in a show an exploded view, a side view, and a top view of the hcp structure. The simple hexagonal unit cell is outlined in the side and top views. Note the similarity to the hexagonal unit cell shown in Figure The ccp structure in b is shown in an exploded view, a side view, and a rotated view.
The rotated view emphasizes the fcc nature of the unit cell outlined. The line that connects the atoms in the first and fourth layers of the ccp structure is the body diagonal of the cube. Table Most metals have hcp, ccp, or bcc structures, although several metals exhibit both hcp and ccp structures, depending on temperature and pressure.
The smallest repeating unit of a crystal lattice is the unit cell. The simple cubic unit cell contains only eight atoms, molecules, or ions at the corners of a cube. A body-centered cubic bcc unit cell contains one additional component in the center of the cube. A face-centered cubic fcc unit cell contains a component in the center of each face in addition to those at the corners of the cube.
The simple cubic and bcc lattices have coordination numbers of 6 and 8, respectively. A crystalline solid can be represented by its unit cell, which is the smallest identical unit that when stacked together produces the characteristic three-dimensional structure.
Why is it valid to represent the structure of a crystalline solid by the structure of its unit cell? What are the most important constraints in selecting a unit cell?
All unit cell structures have six sides. Can crystals of a solid have more than six sides? Explain your answer.
Explain how the intensive properties of a material are reflected in the unit cell. Are all the properties of a bulk material the same as those of its unit cell? The experimentally measured density of a bulk material is slightly higher than expected based on the structure of the pure material.
Propose two explanations for this observation. The experimentally determined density of a material is lower than expected based on the arrangement of the atoms in the unit cell, the formula mass, and the size of the atoms. What conclusion s can you draw about the material? Only one element polonium crystallizes with a simple cubic unit cell. Why is polonium the only example of an element with this structure?
What is meant by the term coordination number in the structure of a solid? How does the coordination number depend on the structure of the metal? Arrange the three types of cubic unit cells in order of increasing packing efficiency. What is the difference in packing efficiency between the hcp structure and the ccp structure?
The structures of many metals depend on pressure and temperature. Which structure—bcc or hcp—would be more likely in a given metal at very high pressures?
Explain your reasoning. A metal has two crystalline phases. Sketch a phase diagram for this substance. The metal is known to have either a ccp structure or a simple cubic structure. Label the regions in your diagram appropriately and justify your selection for the structure of each phase.
However, when cooled below their melting point, metals rearrange to form ordered, crystalline structures. Figure 2: Arrangement of atoms in a liquid and a solid. Crystals To form the strongest metallic bonds, metals are packed together as closely as possible. Several packing arrangements are possible. Instead of atoms, imagine marbles that need to be packed in a box. The marbles would be placed on the bottom of the box in neat orderly rows and then a second layer begun.
The second layer of marbles cannot be placed directly on top of the other marbles and so the rows of marbles in this layer move into the spaces between marbles in the first layer. The first layer of marbles can be designated as A and the second layer as B giving the two layers a designation of AB.
Notice that layer B spheres fit in the holes in the A layer. Packing marbles in the third layer requires a decision. Again rows of atoms will nest in the hollows between atoms in the second layer but two possibilities exist. If the rows of marbles are packed so they are directly over the first layer A then the arrangement could be described as ABA. If the rows of atoms are packed in this third layer so that they do not lie over atoms in either the A or B layer, then the third layer is called C.
Both arrangements give the closest possible packing of spheres leaving only about a fourth of the available space empty.
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