Question

A solid contains \(\mathrm{A}^{\mathrm{n}}\) and \(\mathrm{B}^{\mathrm{m}-}\) ions. The structure of solid is \(\mathrm{FCC}\) for \(\mathrm{B}^{\mathrm{m}-}\) ions and \(\mathrm{A}^{\mathrm{n}+}\) ions are present in one-fourth of tetrahedral voids as well as in one fourth of octahedral voids. What is the simplest formula of solid? (a) \(\mathrm{A}_{3} \mathrm{~B}_{4}\) (b) \(\mathrm{A}_{4} \mathrm{~B}_{3}\) (c) \(\mathrm{AB}_{2}\) (d) \(\mathrm{A}_{2} \mathrm{~B}\)

Short answer

The simplest formula of the solid is \(\mathrm{A}_{3}\mathrm{B}_{4}\).

Step-by-step solution

Identify Number of FCC Ions

FCC structure has 4 atoms/unit cell. Since B ions occupy the FCC positions, number of B ions per unit cell is 4.

Calculate Number of Tetrahedral Voids

In an FCC structure, the number of tetrahedral voids is twice the number of FCC atoms. Therefore, there are 4 * 2 = 8 tetrahedral voids per unit cell.

Calculate Occupied Tetrahedral Voids

Given that A atoms occupy one-fourth of the tetrahedral voids, the number of A ions in tetrahedral voids is 1/4 * 8 = 2.

Calculate Number of Octahedral Voids

In an FCC structure, the number of octahedral voids is equal to the number of FCC atoms, which is 4.

Calculate Occupied Octahedral Voids

Given that A atoms also occupy one-fourth of the octahedral voids, the number of A ions in octahedral voids is 1/4 * 4 = 1.

Determine Total Number of A Ions

Total number of A ions is the sum from tetrahedral and octahedral voids: 2 (from tetrahedral voids) + 1 (from octahedral voids) = 3 A ions.

Write the Simplest Formula

With 3 A ions and 4 B ions per unit cell, the simplest formula of the compound is A3B4.

Key concepts

FCC structure

The Face-Centered Cubic (FCC) structure is a highly compact way atoms, ions, or molecules can arrange themselves in a crystalline lattice. In an FCC lattice, each unit cell has atoms at all the corners and at the center of each face of the cube, which translates to a coordination number of 12. This coordination number is a measure indicating how each particle (atom or ion) is surrounded by other particles.

An important characteristic of an FCC structure is its packing efficiency, which is about 74%. This means in an FCC crystal, 74% of the space is occupied by the particles themselves, while the remaining 26% is void space that can be occupied by smaller atoms or ions, giving rise to different types of voids. The determination of void locations is crucial for understanding the properties of materials and how different ions might incorporate into the lattice.

Tetrahedral voids

Tetrahedral voids are the spaces left between atoms when they are packed in a crystal lattice, specifically in an FCC arrangement. Imagine placing one sphere atop three others that are closely packed together; the space in the center of the four spheres forms a tetrahedron, hence the name 'tetrahedral void'.

Within the FCC structure, the number of tetrahedral voids is always twice the number of atoms in the lattice. For example, if an FCC unit cell contains four atoms, it will have eight tetrahedral voids. These voids are smaller than the octahedral voids and are often occupied by smaller ions in ionic compounds. Importantly, when ions occupy these voids, they influence the electrical properties, stability, and density of the resulting solid-state material.

Octahedral voids

Octahedral voids are another type of gap found in the FCC structure. These voids are shaped like an octahedron and are larger than tetrahedral voids. An octahedral void is surrounded by six atoms, forming a shape that resembles two square pyramids with their bases joined together.

In an FCC lattice, the number of octahedral voids is equal to the number of atoms: hence, a unit cell with four atoms possesses four octahedral voids. These voids can be occupied by ions in ionic solids, affecting the material's structural and electronic properties. In the context of the original problem, knowing that each octahedral void could be partially filled provides a method to calculate the precise stoichiometry of the compound.