Problem 24 - Chemistry in 30 days

Question

A compound alloy of gold and copper crystallizes in a cubic lattice in which the gold atoms occupy the lattice points at the corners of a cube and a copper atom occupies the center of each cube face. The formula of this compound is (a) \(\mathrm{Au}_{3} \mathrm{Cu}\) (b) \(\mathrm{AuCu}_{3}\) (c) \(\mathrm{Au}_{4} \mathrm{Cu}\) (d) \(\mathrm{AuCu}_{2}\)

Step-by-Step Solution

Answer

The formula of the compound is \( \text{AuCu}_3 \).

Step 1: Understand the Lattice Structure

The cubic lattice has gold atoms at the corners of the cube and copper atoms at the center of each face. In a unit cell of a cubic lattice, there are 8 corners and 6 faces.

Step 2: Calculate Contribution of Gold Atoms

Each corner atom is shared by 8 adjacent unit cells in a cubic lattice. Therefore, the contribution of gold atoms per unit cell is \( \frac{8}{8} = 1 \) gold atom.

Step 3: Calculate Contribution of Copper Atoms

Each face-centered atom is shared by 2 unit cells. Therefore, the contribution of copper atoms per unit cell is \( \frac{6}{2} = 3 \) copper atoms.

Step 4: Determine the Empirical Formula

Combine the number of gold and copper atoms calculated to determine the empirical formula of the compound: \( \text{1 Au atom} + \text{3 Cu atoms} = \text{AuCu}_3 \).

Key Concepts

Cubic Lattice

A cubic lattice is one of the simplest and most common ways that atoms are arranged in a crystal structure. Imagine a cube where atoms are positioned at the corners. These corner atoms are shared among multiple cubes. Specifically, each atom at a corner connects with eight neighboring cubes. This sharing is efficient in packing the atoms into a repeating pattern. Cubic lattices are characterized by their symmetrical geometry, which means the sides and angles are all the same. They can be found in different variations such as simple cubic, body-centered cubic (BCC), and face-centered cubic (FCC). Understanding the cubic lattice gives a good foundation for further insights, including those involving more complex structures.

Empirical Formula Calculation

Finding the empirical formula of a compound involves determining the simplest whole-number ratio of atoms present. In the context of a crystal lattice, this means counting the contribution of each type of atom in a unit cell. When calculating the empirical formula for compounds in a crystal structure:

• Identify the different sites that each type of atom occupies.
• Calculate the number of each type of atom in a unit cell by considering shared and unshared positions.
• For corner atoms, because each corner atom is shared by eight cubes, its contribution is \( \frac{1}{8} \) per cube.
  
• For face-centered atoms, shared by two cubes, the contribution is \( \frac{1}{2} \) per cube face.
  

By adding up these contributions, you can derive the total count of each atom type and simplify them to get the empirical formula, such as \( \text{AuCu}_3 \) in the provided exercise.

Face-Centered Cubic Structure

The face-centered cubic (FCC) structure is a specific type of cubic lattice. In this configuration, atoms are located at each corner and the center of each face of the cube. The FCC structure is one of the most closely packed arrangements possible, second only to the hexagonal close-packed structure. Because of this dense packing:

• FCC structures are highly efficient in how they use space.
• They typically have high coordination numbers, meaning each atom in the lattice is in close contact with many others.

In our example, the FCC structure allows copper atoms to occupy the center of each face, contributing significantly to the lattice without actually increasing the volume of the crystal. This efficient packing is why so many metals adopt this structure, offering strength and stability to the overall material. Understanding FCC can provide insights into material properties like ductility and conductivity.