Problem 138 - A Complete Resource Book in Chemistry for JEE Main

Question

An ionic compound has a unit cell consisting of A ions at the corners of a cube and \(\mathrm{B}\) ions on the centres of the faces of the cube. The empirical formula for this compound would be \([\mathbf{2 0 0 5}]\) (a) \(\mathrm{AB}\) (b) \(\mathrm{A}_{2} \mathrm{~B}\) (c) \(\mathrm{AB}_{3}\) (d) \(\mathrm{A}_{3} \mathrm{~B}\)

Step-by-Step Solution

Answer

The empirical formula for the compound is \(\mathrm{AB}_3\).

Step 1: Determine the number of A ions in the unit cell

In a cubic unit cell with ions at the corners, each corner ion is shared by 8 adjacent unit cells. Therefore, an ion positioned at a corner contributes \(\frac{1}{8}\) of an ion to the unit cell. Since there are 8 corners in a cubic unit cell, the total contribution of A ions is \(8 \times \left(\frac{1}{8}\right) = 1\). Thus, there is a total of 1 A ion.

Step 2: Determine the number of B ions in the unit cell

In a cubic unit cell with ions at the centers of the faces, each face ion is shared by 2 adjacent unit cells. Therefore, an ion positioned at a face center contributes \(\frac{1}{2}\) of an ion to the unit cell. Since there are 6 faces in a cubic unit cell, the total contribution of B ions is \(6 \times \left(\frac{1}{2}\right) = 3\). Thus, there are 3 B ions.

Step 3: Derive the empirical formula

Based on the contributions from Step 1 and Step 2, there are 1 A ion and 3 B ions in the unit cell. This leads to an empirical formula of \(\mathrm{AB}_3\).

Key Concepts

Unit Cell

A unit cell is the smallest repeating structure of a crystal lattice that shows the entire symmetry of the arrangement within the crystal. Think of it as the building block of a crystal, like one marble tile in a floor covered entirely with such tiles.
The unit cell contains all the lattice points where ions, atoms, or molecules are located, and its repetition in three-dimensional space forms the whole crystal lattice.

• Corner ions: Located at each of the eight corners of the cube. Each corner ion actually belongs to eight separate unit cells because the corner is shared.
• Face-centered ions: These ions sit on the faces of the cube and contribute to two unit cells as each face is shared between two cells.

The correct counting of ions within a unit cell involves understanding how much of each ion actually contributes to that cell.
For instance, corner ions contribute one-eighth each, as indicated in the original exercise.

Empirical Formula

The empirical formula of a compound represents the simplest whole number ratio of the atoms or ions present in it. This doesn't necessarily reflect the actual number of atoms but rather the simplest ratio between them.
Knowing the contributions of corner and face ions helps deduce the empirical formula from a unit cell, as done in the exercise.

• Corner contributions: When you have ions like A at the corners, you calculate their total contributions to the formula using the fraction each shares with the cell, as we saw with A ions.
• Face contributions: B ions at the faces also have their contributions calculated by considering how many unit cells they share a face with.

In the provided exercise solution, adding these contributions gave us a formula based on one A ion to three B ions, leading to the empirical formula of \( \text{AB}_3 \).

Cubic Lattice Structure

The cubic lattice structure is a very symmetrical and common arrangement in which atoms or ions are arranged at the corners of a cube. In this structure:

• Each unit cell is cube-shaped, making calculations straightforward, as symmetry simplifies many aspects of geometry.
• It often leads to high coordination numbers due to its simplicity. The ions are tightly packed, minimizing empty space.

In the exercise, the ionic compound takes advantage of this structure to arrange A and B ions efficiently, optimizing space and stabilizing interactions.
By placing B ions at the centers of each face, you assure they touch adjacent unit cells, showing the crisp "tiling" effect typical of cubic structures. This arrangement results in the stable empirical formula \( \text{AB}_3 \).