Problem 95 - A Complete Resource Book in Chemistry for JEE Main

Question

A compound contains two types of atoms: \(\mathrm{X}\) and \(\mathrm{Y}\). It crystallizes in a cubic lattice with atoms \(\mathrm{X}\) at the corners of the unit cell and atoms \(\mathrm{Y}\) at the body centres. The simplest possible formula of this compound is: (a) XY (b) \(\mathrm{X}_{2} \mathrm{Y}_{2}\) (c) \(\mathrm{XY}_{6}\) (d) \(\mathrm{X}_{8} \mathrm{Y}\)

Step-by-Step Solution

Answer

The simplest possible formula of the compound is XY.

Step 1: Determine Atoms at Lattice Points

First, we consider the positions of the atoms in a cubic unit cell. Atoms of element \( \mathrm{X} \) are located at the corners of the cube, while atoms of element \( \mathrm{Y} \) are located at the body center.

Step 2: Calculate Number of X Atoms Per Unit Cell

Each corner atom \( \mathrm{X} \) is shared by 8 adjacent unit cells. Since there are 8 corners in a cubic cell, the total contribution of corner atoms to a single unit cell is: \[ \frac{1}{8} \times 8 = 1 \text{ atom of } \mathrm{X} \text{ per unit cell}. \]

Step 3: Calculate Number of Y Atoms Per Unit Cell

The body center atom \( \mathrm{Y} \) is entirely inside the unit cell and is not shared with any other cells. Thus, it contributes 1 whole atom of \( \mathrm{Y} \) to the unit cell.

Step 4: Determine Empirical Formula

Since there is 1 atom of \( \mathrm{X} \) and 1 atom of \( \mathrm{Y} \) per unit cell, the simplest empirical formula of the compound based on this ratio is: XY.

Key Concepts

Cubic Lattice

A cubic lattice is a three-dimensional structure consisting of a grid formed by a repeating pattern of points in space. It's a foundational concept in solid-state physics and chemistry, as it describes how atoms or molecules organize themselves in crystalline materials. Here are some key features:

• All angles in a cubic lattice are 90 degrees.
• All edges are of equal length, making the lattice symmetrical from all sides.
• The cubic lattice can have various configurations such as simple cubic, body-centered cubic (BCC), or face-centered cubic (FCC), each defined by the placement of atoms within the unit cell.

In our exercise, the compound is described to crystallize in a cubic lattice, with specific positioning of atoms. The choice of lattice influences how atoms interact and bond with each other, affecting physical properties like density and melting point. This helps explain why two elements like X and Y have particular roles in creating a stable structure.

Unit Cell

A unit cell is the simplest repeating unit in a crystalline structure that reflects the entire crystal's symmetrical properties. It acts as a building block for the crystal lattice. By repeating the unit cell in all directions, you generate the full crystalline form. Key points of a unit cell include:

• It comprises atoms arranged in a specific three-dimensional pattern.
• The dimensions of the unit cell correspond to the lengths of edges and internal angles.
• In a cubic system, unit cells contain cubic shapes, such as simple, body-centered, or face-centered cubes.

In the provided exercise, the unit cell consists of atoms in distinct positions: X atoms at the corners and Y atoms at the body center. This arrangement is crucial because it dictates the number of each type of atom within a single unit cell, directly affecting the empirical formula calculation for the compound.

Crystalline Structure

Crystalline structures are characterized by an orderly and repeating pattern of atoms, ions, or molecules. This highly structured arrangement results in the macroscopic properties we observe in crystals, such as their shape, optical properties, and strength.

• Crystals can be classified by their symmetry and axial systems, of which the cubic system is just one example.
• Within a crystal, the atoms’ orderly arrangement allows for specific cleavage planes and predictable behavior under external conditions like pressure and temperature.
• Understanding crystalline structures is key in fields like material science, where the arrangement influences material properties such as hardness, solubility, and conductivity.

In our example, the atoms X and Y form a crystalline structure through their positions in a cubic lattice. This structured arrangement not only determines the physical properties of material composed of these atoms but also the stoichiometry captured in the empirical formula XY. By studying these structures, scientists can design and predict new materials with desired properties.