Problem 130 - General Chemistry: Principles and Modern Applications

Question

A metal that crystallizes in the body-centered cubic (bcc) structure has a crystal coordination number of (a) \(6 ;\) (b) \(8 ;\) (c) \(12 ;\) (d) any even number between 4 and 12

Step-by-Step Solution

Answer

The crystal coordination number of a metal that crystallizes in the body-centered cubic (bcc) structure is (b) 8.

Step 1: Understanding Body-Centered Cubic (bcc) Structure

In a bcc unit cell, there is one atom at each of the eight corners and one atom at the center of the cube. The atoms at the corners are shared by eight adjacent cubes, while the atom at the center belongs only to the unit cell. Therefore, there are in total 1 + 1/8 * 8 = 2 atoms in the unit cell.

Step 2: Calculating Coordination Number

The coordination number of a central atom in a molecule is the number of atoms, ions, or molecules directly bonded to it. In a body-centered cubic (bcc) structure, the atom in the center of the cube is in contact with the atoms at the eight corners. Therefore, the coordination number of a metal that crystallizes in the bcc structure is 8.

Key Concepts

Crystal Coordination Number

The 'crystal coordination number' refers to the number of nearest neighbor atoms directly surrounding a particular atom within a crystal lattice. This number is significant because it gives insight into how atoms are packed in a solid material and consequently influences the properties of that material, such as its strength, density, and melting point.

Imagine a playground with children holding hands: the number of hands each child holds is akin to the coordination number in a crystal. In the body-centered cubic (bcc) structure, each atom is like a child surrounded by eight others, much like being in the middle of a small crowd, all reaching out and touching hands. This is unlike other structures, such as the face-centered cubic or hexagonal close-packed structures, where the coordination numbers vary. Thus, the bcc crystal coordination number provides a fundamental understanding of the geometric arrangement in metallic crystals.

Unit Cell

A 'unit cell' is akin to the basic building block of a crystal, representing the smallest portion of the crystal lattice that repeats in three-dimensional space to form the entire crystal. Think of it as a 3D pattern used to create a larger textile design; just as you would repeat this pattern to create a beautiful quilt, the unit cell repeats to form the entire crystal structure.

For body-centered cubic (bcc) lattices, the unit cell appears as a cube with one atom at each of its corners and a single atom at its center. Each corner atom is shared among eight adjacent unit cells, and thus, contributes one-eighth of its volume to the unit cell in question. The architecture of the bcc unit cell is simple yet unique, and understanding it is essential for visualizing how atoms are organized within a metallic crystal. Moreover, the unit cell concept aids in calculations that pertain to the properties of materials, such as density and atomic packing factor.

Coordination Number

It is important to not confuse the 'coordination number' with the crystal coordination number, as they can be contextually different. Generally, the coordination number specifies the number of immediate neighboring atoms, ions, or molecules surrounding a central atom in any molecular or crystalline structure. This concept is not limited to solids; it also applies to molecules in chemistry.

Identifying the coordination number is like determining how many close friends you can have in a seating arrangement at a round table — it defines the immediate interaction level. In the context of a body-centered cubic (bcc) structure, as in our original exercise, the central atom can be thought of sitting at the table with eight close friends at each corner of the cube. These eight directly adjacent atoms determine its coordination number, which is crucial for understanding the local atomic environment and properties of the metal. For a bcc structure, the coordination number is uniformly eight, which reflects the high level of symmetry and the specific touch points between the atoms within the framework.