# 10.6 Lattice structures in crystalline solids  (Page 10/29)

 Page 10 / 29

## Key concepts and summary

The structures of crystalline metals and simple ionic compounds can be described in terms of packing of spheres. Metal atoms can pack in hexagonal closest-packed structures, cubic closest-packed structures, body-centered structures, and simple cubic structures. The anions in simple ionic structures commonly adopt one of these structures, and the cations occupy the spaces remaining between the anions. Small cations usually occupy tetrahedral holes in a closest-packed array of anions. Larger cations usually occupy octahedral holes. Still larger cations can occupy cubic holes in a simple cubic array of anions. The structure of a solid can be described by indicating the size and shape of a unit cell and the contents of the cell. The type of structure and dimensions of the unit cell can be determined by X-ray diffraction measurements.

## Key equations

• $n\lambda =2d\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta$

## Chemistry end of chapter exercises

Describe the crystal structure of iron, which crystallizes with two equivalent metal atoms in a cubic unit cell.

The structure of this low-temperature form of iron (below 910 °C) is body-centered cubic. There is one-eighth atom at each of the eight corners of the cube and one atom in the center of the cube.

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?

eight

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?

12

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 3.165 Å.

(a) What is the atomic radius of tungsten in this structure?

(b) Calculate the density of tungsten.

(a) 1.370 Å; (b) 19.26 g/cm

Platinum (atomic radius = 1.38 Å) crystallizes in a cubic closely packed structure. 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 5.025 Å

(a) What is the atomic radius of barium in this structure?

(b) Calculate the density of barium.

(a) 2.176 Å; (b) 3.595 g/cm 3

Aluminum (atomic radius = 1.43 Å) crystallizes in a cubic closely packed structure. Calculate the edge length of the face-centered cubic unit cell and the density of aluminum.

The density of aluminum is 2.7 g/cm 3 ; that of silicon is 2.3 g/cm 3 . Explain why Si has the lower density even though it has heavier atoms.

The crystal structure of Si shows that it is less tightly packed (coordination number 4) in the solid than Al (coordination number 12).

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?

how many isotopes does hydrogen have
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how many isotopes does hydrogen have
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