Problem 37 - Chemistry

Question

What is the coordination number of each sphere in (a) a simple cubic cell, (b) a body-centered cubic cell, and (c) a face- centered cubic cell? Assume the spheres are all the same.

Step-by-Step Solution

Answer

The coordination number for a simple cubic cell is 6, for a body-centered cubic cell it's 8, and for a face-centered cubic cell it's 12.

Step 1: Coordination Number for Simple Cubic Cell

In a simple cubic cell, each sphere (i.e., atom or ion) is surrounded by 6 other spheres. Hence, the coordination number is 6. These spheres are located top, bottom, left, right, front and back respectively.

Step 2: Coordination Number for Body-centered Cubic Cell

In a body-centered cubic cell, one sphere in the center is surrounded by 8 other spheres located at the corners of the cube. Hence, the coordination number is 8.

Step 3: Coordination Number for Face-centered Cubic Cell

In a face-centered cubic cell, each sphere is surrounded by 12 other spheres. These spheres are located at the centers of each of the faces and at the corners of the cube. Hence, the coordination number is 12.

Key Concepts

Simple Cubic Cell

In the world of crystal structures, the simple cubic cell is one of the most basic forms you will encounter. Imagine a box where each corner is occupied by a sphere or atom, forming a cube. In this crystalline arrangement, each sphere is touched by six adjacent spheres. This specific type of cell is called "simple" because each atom is only in contact with these six neighbors. The geometric distribution forms a basic cube, with spheres positioned at the top, bottom, left, right, front, and back.
Understanding this can be even easier with a mental visualization:

• Think of a single sphere sitting in a stack of marbles—touching one directly next to it in left, right, and above and below directions.
• Coordination number for this setup is 6, representing these direct contacts.

Even though the simple cubic structure is easy to envisage, it is rather rare in natural materials due to its inefficient usage of space.

Body-Centered Cubic Cell

The body-centered cubic cell (BCC) spices up our crystal structure exploration a bit. This structure features not only spheres at the corners of the cube but also an additional sphere precisely in the center of the cube. This simple addition significantly changes the packing.
Here's how the coordination works:

• The central sphere touches eight spheres located at the cube's corners.
• This results in a coordination number of 8 for the body-centered sphere.

BCC is known for its stability in metals, such as iron at certain temperatures. It's a slightly more efficient arrangement than the simple cubic cell, providing stronger bonding abilities among the spheres.

Face-Centered Cubic Cell

Welcome to the highly efficient and densely packed face-centered cubic cell (FCC) structure! In this layout, imagine each face of the cube adorned with a sphere right at the center in addition to those at each corner.
This gives each sphere numerous neighbors:

• Each sphere in this arrangement is in contact with 12 other spheres.
• The spheres are located at the face centers and the corners of the cube.

This coordination number of 12 means that each sphere is surrounded and supported by 12 neighboring spheres. The face-centered cubic structure is often found in metals like aluminum and copper. Its efficient space usage allows for tightly packed structures, making it a preferred pattern for many elements in the periodic table.