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 ccp 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

Option (d) \( \mathrm{Cu}_{4} \mathrm{Ag}_{3} \mathrm{Au} \) is the correct formula.

Step 1: Understand the Structure

First, realize that the copper atoms form the ccp (cubic close-packed) lattice, which is the same as FCC (face-centered cubic) structure. In an FCC lattice, the primitive cell contains 4 atoms of the constituent element (in this case, copper).

Step 2: Locate Silver Atoms

Silver atoms occupy the edge centers of the cube. Each edge of a cube in a cubic crystal system has 4 neighbors and since there are 12 edges, the contribution of silver atoms can be calculated as \( \frac{1}{4} \times 12 = 3 \) silver atoms per unit cell.

Step 3: Locate Gold Atom

Gold atom is located at the body center. The contribution of a single atom at the body center to the CCP unit cell is 1. Thus, there is 1 gold atom in the formula.

Step 4: Formulate Chemical Formula

Now, combining all the information, we have Copper: 4 atoms (from FCC structure), Silver: 3 atoms (from edge centers), Gold: 1 atom (from body center). Therefore, the chemical formula for the alloy is \( \mathrm{Cu}_{4} \mathrm{Ag}_{3} \mathrm{Au} \).

Key Concepts

ccp lattice

When we talk about the ccp lattice, we're discussing a kind of atomic arrangement. Ccp stands for "cubic close-packed," which is one of the most efficient ways of packing spheres (or atoms) in three dimensions. Imagine a three-dimensional grid where each point can be considered an atom.
In a ccp lattice, atoms stack in such a way that each layer of atoms is nestled into the gaps left by the layer below it. This results in a highly organized and dense formation.
The ccp structure is essential because it ensures maximum contact and minimal space between atoms, meaning it's highly stable.

• Every layered pattern is repeated every third layer, giving it a unique ABCABC pattern.
• This structure is very similar to the fcc (face-centered cubic) structure, so similar that they are often referred to interchangeably.

Understanding how atoms are arranged in a lattice helps in figuring out the properties of a material, including strength and conductivity.

fcc structure

The fcc structure, or face-centered cubic structure, is a particular type of crystal lattice.
Visualize a cube with an atom at every corner and one in the middle of each face. That's the fcc structure! This arrangement is not just theoretical; it's how many metals and compounds naturally organize at the atomic level.
Each atom at the corner of the cube is shared by eight other cubes, and each face-centered atom is shared by two cubes. Ultimately, this leads to:

• Each cubic cell containing an equivalent of 4 complete atoms for calculation purposes.
• An optimal use of space, with approximately 74% of the space being occupied by atoms.

This efficient packing means substances with fcc structures are often very compact and often have superior malleability and ductility, making them easy to bend and shape.

crystal systems

Crystal systems form the backbone of understanding mineral structures and behaviors. In chemistry, these systems describe the geometrical arrangement of atoms in space, forming different shapes and structures. Each system has unique symmetry and shape characteristics.
There are seven basic crystal systems: cubic, tetragonal, orthorhombic, hexagonal, trigonal, monoclinic, and triclinic. For simplicity, we'll focus on the cubic system:

• The cubic system includes structures like simple cubic, body-centered cubic (bcc), and face-centered cubic (fcc).
• It features all three axes of equal length and at right angles to each other.
• The versatility of the cubic system allows for varied atomic packing and plays a crucial role in determining the material's physical properties.

Knowing the crystal system of a material either helps predict or explain its durability, melting point, and even how it conducts electricity. This knowledge is more than just academic; it's critically applied in materials science to develop new materials and enhance existing ones.