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 the nearest neighbors for each ion in \( \mathrm{ZnS} \) is 8.

Step 1: Identify the lattice structure

The given structure is a cubic zinc blende structure, which is the crystal arrangement of \( \mathrm{ZnS} \). This structure consists of a face-centered cubic (fcc) lattice of \( \mathrm{S}^{2-} \) ions with \( \mathrm{Zn}^{2+} \) ions occupying half of the tetrahedral holes.

Step 2: Determine the Positions of Ions in the Unit Cell

In the zinc blende structure, \( \mathrm{S}^{2-} \) ions form an fcc lattice, while \( \mathrm{Zn}^{2+} \) ions occupy some of the tetrahedral voids within this lattice. Specifically, only every other tetrahedral site is occupied by \( \mathrm{Zn}^{2+} \), resulting in each \( \mathrm{Zn}^{2+} \) being surrounded by \( \mathrm{S}^{2-} \) ions and vice versa.

Step 3: Calculate Coordination Number for \( \mathrm{Zn}^{2+} \)

Each \( \mathrm{Zn}^{2+} \) ion in a zinc blende structure is tetrahedrally coordinated, meaning it is surrounded by 4 nearest \( \mathrm{S}^{2-} \) ions. This can be visualized as each \( \mathrm{Zn}^{2+} \) sitting at the center of a tetrahedron formed by the \( \mathrm{S}^{2-} \) ions.

Step 4: Calculate Coordination Number for \( \mathrm{S}^{2-} \)

Similarly, each \( \mathrm{S}^{2-} \) ion is surrounded by 4 \( \mathrm{Zn}^{2+} \) ions, also forming a tetrahedron. In this structure, \( \mathrm{Zn}^{2+} \) and \( \mathrm{S}^{2-} \) have a mutual tetrahedral coordination with each other.

Step 5: Sum of Nearest Neighbors

Since both \( \mathrm{Zn}^{2+} \) and \( \mathrm{S}^{2-} \) have a coordination number of 4, the sum of the coordination numbers in the unit cell will be \(4 + 4\). This results in a sum of 8 nearest neighbors for each ion pair within the \( \mathrm{ZnS} \) unit cell.

Key Concepts

Cubic Crystal System

In the world of crystal structures, the cubic crystal system is one of the most common and simplest forms. It features lattice points that are arranged in a cube form, giving it significant symmetry and simplicity.

This system includes several variations, including simple cubic, body-centered cubic, and face-centered cubic lattices. The \( \mathrm{ZnS} \) zinc blende structure, specifically, falls under the face-centered cubic (fcc) category.

This means that the atoms are arranged at all the corners and the centers of each face of the cube. The cubic crystal system provides a robust framework that eases the study of more complex structures, allowing scientists and students to visualize and predict the behavior of different solid materials.

Tetrahedral Coordination

Tetrahedral coordination is a vital concept when discussing structures like \( \mathrm{ZnS} \). In the zinc blende structure, each zinc ion is surrounded by four sulfur ions, creating a geometric shape known as a tetrahedron.

This configuration results in each \( \mathrm{Zn}^{2+} \) ion occupying a tetrahedral site within the sulfur ion lattice. Since every zinc ion is surrounded by four sulfur ions and vice versa, this mutual arrangement is called tetrahedral coordination.

• The tetrahedron shape provides stability and maximizes the distance between the surrounding ions.
• This arrangement helps minimize repulsion and contributes to the overall energy efficiency of the structure.

Tetrahedral coordination is essential for understanding the spatial distribution within the crystal and predicting its properties.

Face-Centered Cubic Lattice

The face-centered cubic (fcc) lattice, as seen in the zinc blende structure, is a highly symmetrical structure where each cell of the lattice contains lattice points at all eight corners and the centers of each cube face.

In the context of \( \mathrm{ZnS} \), the sulfur ions form an fcc lattice. Each face-centered position serves as a regular anchor point that helps maintain structural integrity.

• The fcc arrangement is not only prevalent in \( \mathrm{ZnS} \) but is also seen in metals such as aluminum and copper.
• It is characterized by high packing density and efficient use of space.

The face-centered cubic lattice underpins the zinc blende's capacity to maintain a stable, repeating three-dimensional pattern.

Coordination Number

Coordination number is a critical concept, referring to the number of nearest neighbors surrounding an ion in a crystal lattice.

In the zinc blende \( \mathrm{ZnS} \) structure, each \( \mathrm{Zn}^{2+} \) ion is tetrahedrally coordinated by four \( \mathrm{S}^{2-} \) ions, and each sulfur ion is similarly surrounded by four zinc ions.

• Thus, both \( \mathrm{Zn}^{2+} \) and \( \mathrm{S}^{2-} \) ions have a coordination number of 4.
• This count is derived directly from the tetrahedral coordination nature of the structure.

This commonality in coordination number is key to maintaining the structural consistency and chemical properties of the \( \mathrm{ZnS} \) zinc blende.