Problem 21 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

A substance \(A_{x} B_{y}\) crystallizes in a face centred cubic (FCC) lattice in which atoms 'A' occupy each corner of the cube and atoms 'B' occupy the centres of each face of the cube. Identify the correct composition of the substance \(A_{x} B_{y}\) (a) \(A B_{3}\) (b) \(A_{4} B_{3}\) (c) \(A_{3} B\) (d) Compostion cannot be specified

Step-by-Step Solution

Answer

The composition is \(AB_{3}\), option (a).

Step 1: Understand the FCC Lattice Structure

In an FCC lattice, there are atoms at each corner of the cube and in the center of each face. A cube has 8 corners and 6 faces.

Step 2: Calculate the number of 'A' atoms

Atoms 'A' are present at each of the 8 corners of the cube. Each corner atom belongs to 8 different cubes, so in one cube, each corner contributes \( \frac{1}{8} \) of an atom. Therefore, the total number of 'A' atoms in one unit cell is: \[ 8 \times \frac{1}{8} = 1 \] atom of 'A'.

Step 3: Calculate the number of 'B' atoms

Atoms 'B' occupy the centers of the 6 faces of the cube. Each face-centered atom is shared between 2 adjacent unit cells. Thus, each face atom contributes \( \frac{1}{2} \) of an atom to one unit cell. Therefore, the total number of 'B' atoms in one unit cell is: \[ 6 \times \frac{1}{2} = 3 \] atoms of 'B'.

Step 4: Formulate the Compound

From the calculations, we obtained 1 atom of 'A' and 3 atoms of 'B'. Hence, the empirical formula of the compound is \(A_{1}B_{3}\) or simply \(AB_{3}\).

Step 5: Identify the Correct Composition

Refer back to the given options and match the empirical formula obtained: \(AB_{3}\) is option (a).

Key Concepts

Face Centred Cubic Lattice

In crystallography, the face centred cubic (FCC) lattice is one of the most common three-dimensional structures that atoms can form. Imagine a cube where you place an atom at each of the eight corners and additionally on each of the six face centers of the cube. This results in a highly efficient and densely packed structure.

Each corner atom in an FCC lattice is shared with seven other cubes. This sharing means that only one-eighth of each corner atom belongs to one unit cell. For the face-centered atoms, each face atom is shared with an adjacent cube, therefore, only half of it belongs to a single unit cell.

Due to this efficient arrangement:

• The entire structure can include many atoms in a compact space.
• This structure is utilized by numerous metals such as copper, aluminum, and gold.
• It leads to unique properties like malleability and high density.

Understanding the FCC lattice will aid in grasping how atoms are configured in crystalline solids.

Empirical Formula

The empirical formula is a way of expressing the simplest ratio of the elements in a compound. It provides the most reduced form of the chemical composition, indicating the relative number of atoms of each type in the smallest possible integers.

For example, in our exercise, we determined the empirical formula for the face centred cubic lattice where atoms 'A' and 'B' arrange in a specific pattern. After the calculation, we had one 'A' atom and three 'B' atoms within the unit cell. This led us to the empirical formula of the compound being expressed as \(AB_{3}\). It's the simplest representation that conveys useful information about the composition of a substance.

Key points about empirical formulas:

• They only show the ratio, not the actual number of atoms.
• The simplest version is not necessarily the actual molecular formula.
• Understanding empirical formulas is essential for identifying compound compositions.

Recognizing and working with empirical formulas is crucial in chemistry for analyzing and conveying information about substances.

Atoms in Unit Cell

The concept of atoms in a unit cell deals with counting how many of each type of atom actually reside in the smallest repeating unit of a crystal lattice. This calculation is essential to ascertain the empirical formula of a compound.

In the case of an FCC lattice like the one described in our exercise, we see that each unit cell effectively contains different fractions of atoms depending on their positions within the lattice.

• 'A' atoms are on corners: Each contributes \(\frac{1}{8}\) of an atom per corner.
• Total for 'A' atoms: \(8 \times \frac{1}{8} = 1\).
• 'B' atoms are in the face centers: Each contributes \(\frac{1}{2}\) of an atom.
• Total for 'B' atoms: \(6 \times \frac{1}{2} = 3\).

This total calculation shows exactly how many atoms reside within a single unit cell. Accurately accounting for these helps predict and confirm the empirical formula of the substance.

Coordination Number

Coordination number is a term that describes the number of atoms directly surrounding a given atom within a crystal lattice. It's a fundamental value used to understand the geometry and bonding within crystalline structures.

In an FCC lattice, the coordination number plays a crucial role because it illustrates the close-packing efficiency of the structure. In an FCC lattice:

• Each corner or face-centered atom is surrounded by 12 equidistant nearest neighbors.
• This high coordination number signifies a strong atomic bonding environment, leading to high stability in many metals like copper and silver.
• The high density resulting from such close packing makes these materials ductile and capable of withstanding deformation.

Understanding coordination numbers is critical for interpreting the structural properties seen in crystalline materials and predicting their possible uses and behaviors in practical applications.