Problem 134 - A Complete Resource Book in Chemistry for JEE Main

Question

The \(\mathrm{ZnS}\) structure is cubic. The unit cell may be described as a face centered sulphide ion sub-lattice with zinc ions in the centers of alternating mini cubes made by partitioning the main cube into eight equal parts. The sum of the nearest neighbors around each \(\mathrm{Zn}^{2+}\) and \(\mathrm{S}^{2-}\) ion in the unit cell with be equal to

Step-by-Step Solution

Answer

The sum of nearest neighbors is 4 for each ion.

Step 1: Understanding the Unit Cell Structure

The ZnS structure is a cubic crystal system. It is specifically known as the zinc blende structure, where sulphide ions (S²⁻) form a face-centered cubic (fcc) lattice, and zinc ions (Zn²⁺) fill in half of the tetrahedral sites. This means that if you partition the cube into eight mini-cubes, each corner of a mini-cube has a sulphide ion, and the centers of these mini-cubes are occupied by zinc ions, but only in alternate mini-cubes.

Step 2: Counting Nearest Neighbors for Zn²⁺

In this structure, a Zn²⁺ ion is surrounded by sulphide ions in a tetrahedral arrangement. This means that each Zn²⁺ is bonded to 4 nearest neighbor S²⁻ ions.

Step 3: Counting Nearest Neighbors for S²⁻

Similarly, each sulphide ion (S²⁻) is also tetrahedrally coordinated. It is surrounded by 4 nearest neighbor Zn²⁺ ions. This is due to the arrangement of Zn²⁺ ions in alternating mini-cubes, giving the same coordination number for S²⁻ as for Zn²⁺.

Step 4: Determining Total Coordination Number

The coordination number of the structure is the sum of the nearest neighbors. Since each Zn²⁺ ion is surrounded by 4 S²⁻ ions and each S²⁻ ion is surrounded by 4 Zn²⁺ ions, the total number of nearest neighbors for each ion is 4 for Zn²⁺ and 4 for S²⁻.

Key Concepts

Cubic Crystal System

The cubic crystal system is one of the simplest and most symmetric crystal categories in the field of crystallography. Crystalline materials are grouped into lattice systems based on their symmetry and unit cell geometry, with the cubic system being particularly noted for its three equal axes forming right angles. This means that if you imagine the three-dimensional space defined by the unit cell, all edges are of equal length, and all angles between edges are 90 degrees.

In this system, there are three primary types:

• Simple Cubic (SC)
• Body-Centered Cubic (BCC)
• Face-Centered Cubic (FCC)

These types differ in terms of how points (representative of atom positions) are arranged within the cell. The simple cubic has atoms only at the corners, BCC has an additional atom at the center, and FCC, as seen in the ZnS structure, has additional atoms at the center of each face of the cube.

The arrangement of atoms within a cubic system affects the material's properties, such as density and packing efficiency. Cubic systems are widely studied because of their ease of analysis and their presence in many naturally occurring compounds and synthetic materials.

Zinc Blende Structure

The zinc blende structure is a specific type of cubic crystal system observed in compounds like zinc sulfide (ZnS). In this arrangement, the lattice is based on the face-centered cubic (FCC) design, but it gets its unique structural characteristics from the specific positioning of Zn²⁺ and S²⁻ ions within this framework.

In the zinc blende structure:

• S²⁻ ions form the face-centered cubic sub-lattice.
• Zn²⁺ ions occupy half of the available tetrahedral sites, ensuring that each Zn²⁺ ion is surrounded by four S²⁻ ions, and vice versa.

This results in a coordination number of 4 for both ions, which means that each type of ion is in a tetrahedral coordination environment. This coordination leads to a stable structure with significant implications for the properties of the material, including its electronic, optical, and thermal properties.

Additionally, because the zinc blende structure is a perfect example of an FCC lattice interpenetrated by a second atomic configuration, it is an important model for understanding semiconductor properties, making it foundational for materials science studies.

Face-Centered Cubic (FCC) Lattice

A face-centered cubic (FCC) lattice is a highly efficient arrangement of atoms where each cube within the lattice contains atoms at each of the corners and in the center of each face. This kind of packing is quite dense compared to other types of cubic lattices. For example, it's more packed than the simple cubic or body-centered cubic structures.

The FCC lattice is noted for having:

• A coordination number of 12, meaning each atom is adjacent to 12 others.
• A packing efficiency of about 74%, which is the highest for any type of straightforward lattice arrangement.

In the context of the ZnS crystal structure, the S²⁻ ions are arranged in an FCC configuration. This allows Zn²⁺ ions to occupy the tetrahedral sites provided by the FCC lattice. These sites are voids or gaps in the lattice structure that create an ideal environment for coordination of atoms as seen in elements and compounds like copper, aluminum, and the ZnS itself.

The FCC lattice's superior packing efficiency and symmetry allow for exceptional stability and predictability of properties across the entire crystal, which is why it is frequently utilized in the study and application of various crystalline materials.