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Chapter 3: Problem 2

If the atomic radius of lead is \(0.175 \mathrm{~nm}\), calculate the volume of its unit cell in cubic meters.

Short Answer

Expert verified
Answer: The approximate volume of a unit cell of lead is \(1.20 \times 10^{-28} \mathrm{m}^3\).

Step by step solution

01

Determine the geometry of the unit cell

Lead has a face-centered cubic (FCC) unit cell structure.
02

Calculate the edge length of the unit cell using the atomic radius

In a face-centered cubic unit cell, the edge length (a) can be calculated using the atomic radius (r) with the following formula: $$a = 2\sqrt{2} \cdot r$$ Given the atomic radius of lead \((r = 0.175 \mathrm{nm})\), we can find the edge length by plugging the value into the formula: $$a = 2\sqrt{2} \cdot 0.175 \mathrm{nm} \approx 0.494 \mathrm{nm}$$
03

Convert the edge length to meters

The edge length is given in nanometers, so we need to convert it to meters: $$a = 0.494 \mathrm{nm} \cdot \frac{1 \mathrm{m}}{10^9 \mathrm{nm}} \approx 4.94 \times 10^{-10} \mathrm{m}$$
04

Calculate the volume of the unit cell

Now that we have the edge length in meters, we can calculate the volume of the unit cell using the formula for the volume of a cube: $$V = a^3$$ Plugging in the edge length: $$V = (4.94 \times 10^{-10} \mathrm{m})^3 \approx 1.20 \times 10^{-28} \mathrm{m}^3$$ So the volume of the unit cell of lead is approximately \(1.20 \times 10^{-28} \mathrm{m}^3\).

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Most popular questions from this chapter

Magnesium (Mg) has an HCP crystal structure and a density of \(1.74 \mathrm{~g} / \mathrm{cm}^{3}\). (a) What is the volume of its unit cell in cubic centimeters? (b) If the \(c / a\) ratio is \(1.624\), compute the values of \(c\) and \(a\).

Sketch a unit cell for the face-centered orthorhombic crystal structure.

Magnesium (Mg) has an HCP crystal structure, a \(c / a\) ratio of \(1.624\), and a density of \(1.74 \mathrm{~g} / \mathrm{cm}^{3}\). Compute the atomic radius for \(\mathrm{Mg}\).

For which set of crystallographic planes will a first-order diffraction peak occur at a diffraction angle of \(44.53^{\circ}\) for BCC tantalum (Ta) when monochromatic radiation having a wavelength of \(0.1937 \mathrm{~nm}\) is used?

Within a cubic unit cell, sketch the following directions: (a) [101] (e) \([\overline{1} 1 \overline{1}]\) (b) [211] (f) \([\overline{2} 12]\) (c) \([10 \overline{2}]\) (g) [3\overline{12} ] (d) \([3 \overline{13}]\) (h) [301]
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