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Chapter 12: Problem 38

Calculate the volume in \(\hat{A}^{3}\) of each of the following types of cubic unit cells if it is composed of atoms with an atomic radius of 1.82 A. (a) primitive (b) face-centered cubic.

Short Answer

Expert verified
The volume of a primitive cubic unit cell with an atomic radius of 1.82 Šis approximately 48.14 ų. The volume of a face-centered cubic unit cell with the same atomic radius is approximately 136 ų.

Step by step solution

01

Calculate the lattice parameter of a primitive cubic unit cell

In a primitive cubic unit cell, atoms are present only at the corner of the cube. To find the lattice parameter (length of one edge of the cube), we use the relation: Lattice parameter (a) = 2 * Atomic radius We substitute the given value of the atomic radius (1.82 Å) and calculate the lattice parameter: a = 2 × 1.82 Å a = 3.64 Å
02

Calculate the volume of a primitive cubic unit cell

Now that we have the lattice parameter, we can calculate the volume of a primitive cubic unit cell using the formula: Volume = a³ Volume = (3.64 Å)³ Volume = 48.14 ų (b) Face-Centered Cubic Unit Cell
03

Calculate the lattice parameter of a face-centered cubic unit cell

In a face-centered cubic unit cell, atoms are present at the corners and center of each face. To find the lattice parameter, we need to use the relation: Lattice parameter (a) = \(2√2\) * Atomic radius We substitute the given value of the atomic radius (1.82 Å) and calculate the lattice parameter: a = \(2√2\) × 1.82 Å a ≈ 5.14 Å
04

Calculate the volume of a face-centered cubic unit cell

Now that we have the lattice parameter, we can calculate the volume of a face-centered cubic unit cell using the formula: Volume = a³ Volume = (5.14 Å)³ Volume ≈ 136 ų

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Most popular questions from this chapter

Teflon is a polymer formed by the polymerization of \(\mathrm{F}_{2} \mathrm{C}=\mathrm{CF}_{2}\) . (a) Draw the structure of a section of this polymer. (b) What type of polymerization reaction is required to form Teflon?

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