Problem 70 - Chemistry Matter and Change

Question

Compare the number of particles in one unit cell for each of the following types of unit cells. a. simple cubic b. body-centered cubic

Step-by-Step Solution

Answer

A simple cubic unit cell contains 1 particle, as the 8 corners each contribute 1/8 of a particle. In contrast, a body-centered cubic unit cell contains 2 particles, with 8 corners contributing 1/8 of a particle and an additional whole particle at the center. Therefore, a body-centered cubic unit cell contains 1 more particle than a simple cubic unit cell.

Step 1: a. Simple Cubic Unit Cell

In a simple cubic unit cell, the lattice points are only at the corners of the cube. There are 8 corners in a cube, and each corner contributes 1/8 of a particle to the unit cell, as it is shared among 8 adjacent unit cells. So, the total number of particles in a simple cubic unit cell can be found as: Number of particles in a simple cubic unit cell = 8 corners × (1/8) = 1

Step 2: b. Body-centered Cubic Unit Cell

In a body-centered cubic (BCC) unit cell, there are lattice points at the corners of the cube as well as at its center. The 8 corner lattice points contribute 1/8 of a particle, just like in a simple cubic unit cell. But the lattice point in the center is not shared with any other unit cell, so it contributes a complete particle to the unit cell. The total number of particles in a BCC unit cell can be calculated as: Number of particles in a BCC unit cell = 8 corners × (1/8) + 1 center = 1 + 1 = 2

Step 3: Comparison

Now that we have calculated the number of particles for each type of unit cell, we can compare them. The simple cubic unit cell contains 1 particle, while the body-centered cubic unit cell contains 2 particles. So a body-centered cubic unit cell contains 1 more particle than a simple cubic unit cell.

Key Concepts

Simple Cubic Unit Cell

A simple cubic unit cell is the most straightforward type of crystal lattice structure. Imagine a cube where particles are located at each of its eight corners. What makes a simple cubic lattice unique is its high symmetry and simplicity.

• Each corner particle is shared by eight neighboring unit cells, meaning that only \( \frac{1}{8} \) of each particle actually "belongs" to one unit cell.
• Therefore, when you calculate the total amount of particles in one unit cell, it comes out to 1: \( 8 \times \frac{1}{8} = 1 \).

The packing efficiency in this arrangement is relatively low, approximately 52%, meaning there is a lot of empty space within the structure. This type of structure is not commonly found in nature but provides fundamental understanding of the packing arrangement in solids.

Body-centered Cubic Unit Cell

The body-centered cubic (BCC) unit cell has a more complex arrangement compared to the simple cubic structure. In addition to particles at the eight cube corners, there is also one particle at the center of the cube. This central particle isn't shared with any other unit cell, making BCC more dense.

• The corners contribute the same as in the simple cubic structure: \( 8 \times \frac{1}{8} = 1 \) particle.
• The center particle contributes fully, adding one whole particle to the body-centered unit cell.
• Thus, the total comes to 2 particles per unit cell: \( 1 + 1 = 2 \).

This results in a slightly higher packing efficiency of about 68%, which means less empty space compared to the simple cubic structure. Many metals, like iron and chromium, crystallize in the BCC arrangement, illustrating its significance in the material world.

Crystal Lattice

The term "crystal lattice" refers to the ordered, repeating three-dimensional arrangement of atoms, ions, or molecules in a crystal. This intricate structure is fundamental in characterizing materials and determining their physical properties. Think of each lattice point as a repetitive unit that collectively forms the complete crystal.

• A crystal lattice can be visualized as a network of intersecting lines or points, which represent the positions of particles in the structure.
• These arrangements are categorized into different types of unit cells, such as simple cubic and body-centered cubic.
• The properties of the crystal, like its symmetry and compactness, define both chemical behavior and material characteristics.

Understanding crystal lattices helps in comprehending the microscopic origins of macroscopic material properties, such as hardness, electrical conductivity, and optical characteristics. They form the basis of disciplines like crystallography, which explores the atomic structure of crystalline materials.