Ch.12 - Solids and Modern Material

Problem 86

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Chapter 12, Problem 86

Calculate the fraction of empty space in cubic closest packing to five significant figures.

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Understand that cubic closest packing (CCP) is also known as face-centered cubic (FCC) packing.

In FCC, each unit cell contains 4 atoms. The atoms are located at each corner and the centers of each face of the cube.

Calculate the volume of the unit cell. If 'a' is the edge length of the cube, the volume is given by \( V_{cell} = a^3 \).

Determine the volume occupied by the atoms in the unit cell. The effective radius of each atom is \( r \), and the relationship between 'a' and 'r' in FCC is \( a = 2\sqrt{2}r \). The volume occupied by the atoms is \( 4 \times \frac{4}{3}\pi r^3 \).

Calculate the fraction of empty space by subtracting the volume occupied by the atoms from the total volume of the unit cell, and then divide by the total volume of the unit cell: \( \text{Fraction of empty space} = 1 - \frac{\text{Volume occupied by atoms}}{V_{cell}} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cubic Closest Packing (CCP)

Cubic Closest Packing (CCP) is a method of packing spheres in three-dimensional space to maximize density. In this arrangement, spheres are stacked in layers, with each layer fitting into the gaps of the layer below. This structure achieves a packing efficiency of approximately 74%, meaning that 74% of the volume is occupied by spheres, while the remaining 26% is empty space.

Packing Efficiency

Packing efficiency is a measure of how effectively a given volume is filled with particles, expressed as a percentage. In the case of cubic closest packing, the packing efficiency is calculated by dividing the volume occupied by the spheres by the total volume of the packing arrangement. Understanding packing efficiency is crucial for determining the fraction of empty space in a given packing structure.

Fraction of Empty Space

The fraction of empty space in a packing arrangement is calculated by subtracting the packing efficiency from 1 (or 100% if expressed as a percentage). For cubic closest packing, with a packing efficiency of 74%, the fraction of empty space is 1 - 0.74 = 0.26. This concept is essential for understanding how much of the volume in a packed structure is not occupied by the particles.

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Textbook Question

An unknown metal is found to have a density of 7.8748 g/cm³ and to crystallize in a body-centered cubic lattice. The edge of the unit cell is 0.28664 nm. Calculate the atomic mass of the metal.

When spheres of radius r are packed in a body-centered cubic arrangement, they occupy 68.0% of the available volume. Use the fraction of occupied volume to calculate the value of a, the length of the edge of the cube, in terms of r.

Open Question

Potassium chloride crystallizes in the rock salt structure. Estimate the density of potassium chloride using the ionic radii provided in Chapter 8.

Open Question

What is a tetrahedral site in a closest-packed lattice formed by?

Textbook Question

X-ray diffractometers often use metals that have had their core electrons excited as a source of X-rays. Consider the 2p → 1s transition for copper, which is called the K⍺ transition. Calculate the wavelength of X-rays (in Å) given off by the K⍺ transition if the energy given off by a mole of copper atoms is 7.77⨉10⁵ kJ.(1Å = 10^-10 m)

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Open Question

Why is it necessary to use the Kα transition (2p → 1s) in copper (see Problem 88) to generate X-rays? Why not use, for example, the 4s → 3p transition?