Problem 112 - A Complete Resource Book in Chemistry for JEE Main

Question

An alloy of copper, silver and gold is found to have copper constituting the cop lattice. If silver atoms occupy the edge centres and gold is present at body centre, the alloy will have the formula (a) \(\mathrm{Cu}_{4} \mathrm{Ag}_{4} \mathrm{Au}\) (b) \(\mathrm{Cu} \mathrm{Ag} \mathrm{Au}\) (c) \(\mathrm{Cu}_{4} \mathrm{Ag}_{2} \mathrm{Au}\) (d) \(\mathrm{Cu}_{4} \mathrm{Ag}_{3} \mathrm{Au}\)

Step-by-Step Solution

Answer

(d) \( \mathrm{Cu}_4 \mathrm{Ag}_3 \mathrm{Au} \)

Step 1: Understanding the Crystal Structure

We need to understand how atoms are positioned in the crystal. In a cubic lattice, copper is at the corners, silver occupies the edge centers, and gold is at the body center. A cubic structure typically features 8 corner atoms, 12 edge-center atoms, and 1 body-center atom.

Step 2: Determine Contribution of Copper

Since copper constitutes the corner lattice points, and each corner atom contributes \( \frac{1}{8} \) of an atom per unit cell (as there are 8 corners), the total contribution of copper atoms is \( 8 \times \frac{1}{8} = 1 \) copper atom per unit cell.

Step 3: Calculate Contribution of Silver

Silver atoms are located at edge centers. Each edge-centered atom contributes \( \frac{1}{4} \) of an atom per unit cell because one edge is shared between four unit cells. With 12 edges, silver's total contribution is \( 12 \times \frac{1}{4} = 3 \) silver atoms per unit cell.

Step 4: Determine Contribution of Gold

Gold is located at the body center. In a cubic structure, the atom at the body center is wholly contained within a single unit cell, contributing \( 1 \) atom of gold per unit cell.

Step 5: Derive the Chemical Formula

Now, we combine the contributions of all atoms to form the chemical formula of the alloy: \( \text{Cu}_{1} \text{Ag}_{3} \text{Au}_{1} \). However, in chemical formulations, it's usual to represent the smallest whole numbers if possible, so it becomes \( \text{Cu}_{4} \text{Ag}_{3} \text{Au} \).

Step 6: Selecting the Correct Answer

Comparing the derived formula \( \text{Cu}_{4} \text{Ag}_{3} \text{Au} \) with the given options, the correct choice is option (d).

Key Concepts

Understanding the Unit Cell

A unit cell is the smallest structural unit or building block of a crystal lattice that, when repeated in all directions, creates the entire lattice of a solid material. The unit cell contains information about the arrangement and position of atoms. It acts like a miniature version of the whole structure and helps us understand the properties of the material.
In our scenario, where we have an alloy of copper, silver, and gold, the unit cell helps us to see how these different types of atoms fit together. For example, copper atoms are situated at the corners of the unit cell, influencing how they interact and connect with the rest of the crystal. Understanding the unit cell in such systems allows us to decipher the mystery of material composition and formulation.

Exploring the Cubic Lattice

The cubic lattice is a popular type of crystal structure and features a highly symmetrical arrangement of atoms. There are different kinds of cubic lattices, such as simple cubic, body-centered cubic (BCC), and face-centered cubic (FCC). Each one is characterized by how the atoms are positioned within the lattice.
In our particular problem, copper atoms are positioned at the corners of the cubic lattice, gold at the center of the unit cell, and silver at the edges. This configuration is a great example of how different types of atoms can inhabit various sites in a cubic lattice to form complex structures. Understanding cubic lattices is crucial for identifying the properties and behaviors of materials, as it influences factors like density and stability.

Deciphering the Chemical Formula

A chemical formula represents the proportion of elements within a compound. It offers a concise way to convey complex information about the composition and structure of a substance. In the context of our alloy, the chemical formula is derived from understanding the roles of copper, silver, and gold atoms in the unit cell.
Here's a quick breakdown of how we concluded the formula:

• Copper occupies corner positions, contributing effectively 1 complete atom per unit cell.
• Silver, found at the edge centers, together contributes 3 atoms per unit cell.
• Gold, central to the unit cell, contributes 1 full atom.

With these contributions, we arrive at the formula \( \text{Cu}_{4} \text{Ag}_{3} \text{Au}\) which expresses the composition of the alloy in the simplest ratio. This formula is essential for chemists to understand and manipulate material properties.