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Chapter 3: Problem 19

Cobalt has an HCP crystal structure, an atomic radius of \(0.1253 \mathrm{nm},\) and a \(c / a\) ratio of 1.623 Compute the volume of the unit cell for Co.

Short Answer

Expert verified
Answer: The volume of the unit cell for cobalt is approximately 0.0661 nm³.

Step by step solution

01

Find the lattice parameter a from the atomic radius

In an HCP structure, the relationship between the lattice parameter a and the atomic radius r can be expressed as: \(a = 2\sqrt{2} \cdot r\) Given, r = 0.1253 nm Now substituting the value of r in the equation, \(a = 2\sqrt{2} \cdot 0.1253 = 0.3544 \mathrm{nm}\)
02

Find the lattice parameter c from the c/a ratio

Given, c/a ratio = 1.623 We found the value of a in step 1, so now we can compute the value of c as: \(c = (c/a) \cdot a\) \(c = 1.623 \cdot 0.3544 = 0.5753 \mathrm{nm}\)
03

Calculate the volume of the unit cell

Using the formula \(V_{cell} = (3\sqrt{3}/2) \cdot a^2 \cdot c\), and substituting the values of a and c found in steps 1 and 2, we get: \(V_{cell} = (3\sqrt{3}/2) \cdot (0.3544)^2 \cdot 0.5753\) \(V_{cell} \approx 0.0661 \mathrm{nm^3}\) The volume of the unit cell for cobalt is approximately 0.0661 nm³.

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Most popular questions from this chapter

Convert the [110] and \([00 \overline{1}]\) directions into the four-index Miller-Bravais scheme for hexagonal unit cells.

Cite the indices of the direction that results from the intersection of each of the following pair of planes within a cubic crystal: (a) (110) and (111) planes, (b) (110) and \((1 \overline{1} 0)\) planes and (c) \((11 \overline{1})\) and (001) planes.

Calculate the radius of a palladium atom, given that \(\mathrm{Pd}\) has an \(\mathrm{FCC}\) crystal structure, a density of \(12.0 \mathrm{g} / \mathrm{cm}^{3},\) and an atomic weight of \(106.4 \mathrm{g} / \mathrm{mol}.\)

(a) Derive planar density expressions for FCC (100) and (111) planes in terms of the atomic radius \(R.\) (b) Compute and compare planar density values for these same two planes for aluminum.

Sketch within a cubic unit cell the following planes: (a) \((10 \overline{1})\) (b) \((2 \overline{1} 1)\) (c) (012) (d) \((3 \overline{1} 3)\) (e) \((\overline{1} 1 \overline{1})\) (f) \((\overline{2} 12)\) (g) \((3 \overline{1} 2)\) (h) (301)
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