Problem 80 - A Complete Resource Book in Chemistry for JEE Main

Question

If three elements A, B and C crystallized in cubic solid lattice with A atoms at corners, \(\mathrm{B}\) atoms at cube centres and \(\mathrm{C}\) atoms at the edges, the formula of the compound is (a) \(\mathrm{ABC}\) (b) \(\mathrm{ABC}_{3}\) (c) \(\mathrm{AB}_{3} \mathrm{C}\) (d) \(\mathrm{A}_{3} \mathrm{~B}_{2} \mathrm{C}_{3}\)

Step-by-Step Solution

Answer

The formula of the compound is \(\mathrm{AB}_{3} \mathrm{C}\).

Step 1: Determine Atomic Contribution of A Atoms

In a cubic lattice, A atoms are located at the corners of the cube. Each corner atom is shared by 8 adjacent cubes, so each A atom contributes \(\frac{1}{8}\) to the unit cell. Since a cube has 8 corners, the number of A atoms per unit cell is \(8 \times \frac{1}{8} = 1\).

Step 2: Determine Atomic Contribution of B Atoms

B atoms are located at the centers of the cubes. Each central atom belongs entirely to one unit cell, so the contribution of B atoms is 1 per unit cell.

Step 3: Determine Atomic Contribution of C Atoms

C atoms are located along the edges of the cube. Each edge-shared C atom is shared by 4 cubes. Since there are 12 edges in a cube, the contribution of C atoms per unit cell is \(12 \times \frac{1}{4} = 3\) atoms.

Step 4: Derive the Empirical Formula

Add the contributions of each element in the unit cell: A has 1, B has 1, and C has 3 atoms. Therefore, the formula of the compound is \(\mathrm{AB}_{1} \mathrm{C}_{3}\), simplifying to \(\mathrm{AB}_{3} \mathrm{C}\).

Key Concepts

Unit Cell Calculations

Understanding the unit cell calculations is crucial for determining the formula of a compound in a crystal lattice. A unit cell is the smallest repeating structural unit of a crystal. It defines the entire crystal's structure by repeating in three-dimensional space.
A cubic unit cell is particularly easy to visualize since it resembles a box shape with equal edges.When performing unit cell calculations, we determine the contribution of atoms from different lattice positions to a single cell.
These positions could be corners, edges, or centers of the unit cell. Each position contributes a fractional amount to the unit cell based on its sharing among adjacent cells.

• Atoms at corners are shared by eight unit cells, hence each contributes \( \frac{1}{8} \).
• Atoms at the centers contribute wholly, with a value of 1.
• Atoms on edges are shared by four cells, contributing \( \frac{1}{4} \) of an atom each.

By calculating these contributions, we can sum them up to find the total number of each type of atom in the unit cell.

Atomic Contribution in Lattice

In a crystal lattice, the atomic contribution refers to how much of an atom is part of the unit cell. Different atoms in the lattice can occupy different sites, and their contributions can differ based on these sites.

Determining Atomic Contributions

Calculating atomic contributions helps in establishing which atoms are actually present in a unit cell. For example, in a cubic crystal structure:

• Corner atoms each contribute only \( \frac{1}{8} \), as they are shared by 8 different unit cells.
• Center atoms are exclusive to one unit cell, counting as a whole (1 atom per cell).
• Edge atoms are shared by 4 adjacent cells, thus contributing \( \frac{1}{4} \) each.

By performing these calculations, we identify how many complete atoms exist in the unit cell. This is a vital step in determining the empirical formula, as it provides the ratio of each type of atom.

Empirical Formula Derivation

Once we know the contribution of each type of atom in the unit cell, we can deduce the empirical formula of the compound. An empirical formula represents the simplest whole-number ratio of atoms in a compound.
It provides a straightforward representation of how many of each type of atom exist relative to one another within the crystal.

Steps to Derive the Empirical Formula

• Calculate the total number of each atom type in the unit cell using their contributions. For instance, if the contributions are: 1 for atom A, 1 for atom B, and 3 for atom C.
• Align these figures to form the simplest whole-number ratio.
• Write the empirical formula using these ratios, such as \( \mathrm{AB}_{3} \mathrm{C} \) for the example given, where A and B each appear once, while C appears three times.

Deriving the empirical formula effectively summarizes the composition of the compound represented by the unit cell.