Problem 84 - General Chemistry

Question

How many atoms are there in a body-centered cubic unit cell of an atomic crystal in which all atoms are at lattice points?

Step-by-Step Solution

Answer

There are 2 atoms in a body-centered cubic unit cell.

Step 1: Understand the Structure

In a body-centered cubic (BCC) unit cell, the structure consists of atoms located at each of the eight corners and a single atom at the center of the cube.

Step 2: Calculate the Contribution from Corner Atoms

Corner atoms in a cube are shared by 8 adjacent cubes, so each corner atom contributes 1/8 of an atom to the unit cell. Since there are 8 corners, the total contribution from corner atoms is \( 8 \times \frac{1}{8} = 1 \text{ atom} \).

Step 3: Calculate the Contribution from Center Atom

The atom at the center of the BCC unit cell is not shared with any other unit cells. Hence, it fully belongs to the unit cell, contributing 1 whole atom.

Step 4: Total Atoms in the Unit Cell

Add the contributions from corner atoms and the center atom. The total number of atoms per BCC unit cell is \( 1 \text{ atom (from corners)} + 1 \text{ atom (from center)} = 2 \text{ atoms} \).

Key Concepts

Atomic Crystal

An atomic crystal is a solid material where the atoms are arranged in a highly ordered, repeating pattern. This ordered structure is uniform throughout the crystal. Atoms in such crystals form a lattice, which is a 3D array of points signaling where atoms are positioned.

Crystals are composed of a unit cell that repeats in every direction to form the entire lattice. Understanding the atomic crystal concept helps in comprehending how elements like metals are structured at a microscopic level. This concept is vital for studying various properties of materials, such as their strength, brittleness, and ability to conduct electricity.

• An example of an atomic crystal is a metal like iron, which forms a specific crystal structure.
• Body-centered cubic (BCC) unit cells are a type of atomic crystal structure, containing at least one atom at the center of the unit cell.

In summary, the understanding of atomic crystals forms a foundation for exploring material properties and applications.

Lattice Points

Lattice points are specific locations within an atomic crystal that define the positions of atoms in a repeated pattern. In a body-centered cubic (BCC) unit cell, lattice points are located at the corners and the center of the unit cell.

Each lattice point in a BCC structure is indicative of an atom's position, helping to create the crystal's ordered grid-like formation. This formation utilizes symmetry, which can significantly affect the crystal's physical traits.

Lattice points are integral for:

• Calculating distances between atoms, which impacts the crystal's density.
• Defining how atoms pack in a crystal and contribute to its stability.
• Determining coordination numbers, which describe how many atoms surround a lattice point.

Therefore, lattice points are essential in visualizing how atoms organize themselves in solid structures, aiding in material design and analysis.

Corner Atoms

In a body-centered cubic (BCC) unit cell, the corner atoms are fundamental as they make up the basic framework of the crystal structure. There are eight corner atoms in a unit cell, one on each corner.

However, each corner atom is shared equally among eight unit cells. This sharing mechanism means that each corner atom contributes only 1/8th toward a single unit cell. The calculation for all the corner atoms is therefore:
When multiplied by the eight corners, the total contribution to the unit cell sums up to one whole atom ( 8 imes rac{1}{8} = 1 ext{ atom} ). Understanding this sharing process is crucial as it ensures accuracy in counting the total number of atoms per unit cell.

Additionally, corner atoms:

• Help in creating a delicate balance of forces within the crystal structure.
• Serve as connection points that hold the entire crystal structure together.

This framework is necessary to maintain the stability and integrity of the crystal.

Center Atom

The center atom in a body-centered cubic (BCC) unit cell is positioned at the very center of the cube. It plays a pivotal role as it is unique to its own unit cell. Unlike corner atoms, which are shared, the center atom belongs solely to the BCC unit cell it occupies.

For the body-centered cubic structure:

• The center atom contributes entirely (1 whole atom) to the unit cell's atom count, as it does not extend into any neighboring cells.
• The presence of the center atom increases the density of the packing, enhancing the material's properties like strength and hardness.

This center atom is crucial because it completes the two atoms that a BCC unit cell comprises. Its unique position effectively balances out the contributions made by corner atoms, resulting in a cohesive and stable crystal structure.