Question

Which of the following has the smallest packing efficiency for atoms of a single type? (a) Body centered cubic (b) Face centered cubic (c) Simple cubic (d) None of these

Short answer

The simple cubic (c) structure has the smallest packing efficiency for atoms of a single type.

Step-by-step solution

Understanding Packing Efficiency

Packing efficiency is the fraction of volume in a crystal structure that is occupied by the atoms themselves. It can be calculated by the ratio of the volume of atoms in a unit cell to the total volume of the unit cell.

Comparing Packing Efficiencies of Different Structures

Let's compare the packing efficiencies for each given structure. For single type atoms, we can use the known packing efficiencies: simple cubic (sc) has a packing efficiency of approximately 52.4%, body centered cubic (bcc) has a packing efficiency of approximately 68.0%, and face centered cubic (fcc) has a packing efficiency of approximately 74.0%.

Identifying the Structure with the Smallest Packing Efficiency

By comparing the values, we can see that the simple cubic structure has the lowest packing efficiency out of the given options.

Key concepts

Crystal Structure

To appreciate the diversity of how atoms are organized in solids, one must first understand the term 'crystal structure'. In essence, crystal structure refers to the orderly and repeating pattern in which atoms are arranged in a three-dimensional space. Think of it as stacking oranges in a market stand – the way you arrange the oranges can vastly differ. Similarly, atoms can be arranged in different patterns to form various crystal structures, each with unique physical properties. These structures are crucial in determining the characteristics of materials, such as strength, ductility, conductivity, and more.

Understanding crystal structures is fundamental to material science and serves as a gateway to exploring other related concepts like packing efficiency, which is essential in predicting the properties of materials. It sets the foundation for how we can expect atoms to behave when they are packed together in different configurations within a solid.

Simple Cubic

Imagine stacking spheres in the simplest way possible, with each sphere directly on top of another - this is the essence of the simple cubic (sc) structure. In more technical terms, it has atoms located at each of the eight corners of a cube, and no other atoms inside the box. While this may sound efficient, in reality, the simple cubic structure isn't tightly packed.

The atoms only touch along the cube's edge, and as a result, they utilize just about 52.4% of the available space – this is its packing efficiency. This structure isn't common among pure elements, but understanding it provides a basis for comparison against more complex and tightly packed structures.

Body Centered Cubic

Taking a step beyond the simple cubic, let’s consider the body centered cubic (bcc) structure. Picture the same cube from the simple cubic, but now there is an extra atom nestled right in the center of the cube, in addition to the eight corner atoms. This central atom touches each corner atom through the body diagonal of the cube.

This slight adjustment leads to a better utilization of space, and thus, a higher packing efficiency of approximately 68.0%. Despite still having voids within the structure, the bcc crystals are denser than simple cubic crystals. Bcc structures are commonly found in metals and play a pivotal role in explaining the properties of these materials, including their density and how they interact with light and other forms of energy.

Face Centered Cubic

Face centered cubic (fcc) crystals represent yet another level of sophistication in atomic arrangement. This structure takes the cube and places an atom at each of the corners, and one at the center of each face of the cube, leading to a total of 14 atoms considering the sharing with adjacent unit cells.

What sets the fcc structure apart is its packing efficiency, which is a whopping 74.0%, the highest of the three types discussed. This highly efficient packing results in each atom being touched by twelve others – more than any other discussed structure. This is why materials with fcc structures, such as many metals, are known for their ductility and ability to conduct electricity and heat effectively. From building constructions to electrical wiring, fcc crystals serve as the backbone of numerous critical applications.