Problem 25 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Consider an ionic solid \(M X\) with \(\mathrm{NaCl}\) structure. Construct a new structure \((Z)\) whose unit cell is constructed from the unit cell of \(M X\) following the sequential instructions given below. Neglect the charge balance. (i) Remove all the anions \((X)\) except the central one (ii) Replace all the face centered cations ( \(M\) ) by anions \((X)\) (iii) Remove all the corner cations \((M)\) (iv) Replace the central anion \((X)\) with cation \((M)\) The value of \(\left(\frac{\text { number of anions }}{\text { number of cations }}\right)\) in \(Z\) is __ .

Step-by-Step Solution

Answer

6

Step 1: Understand the Structure of NaCl

The NaCl structure is a face-centered cubic (FCC) lattice. In this structure, cations (\(M\)) occupy the corner and face-centered positions, and anions (\(X\)) occupy the octahedral sites, which includes the center of the unit cell.

Step 2: Initial Composition in NaCl Unit Cell

The NaCl unit cell has 4 formula units of \(MX\). This consists of 4 cations \(M\) and 4 anions \(X\) distributed as follows:- Corner cations: 8 corners each contributing \(\frac{1}{8}\) of \(M\), totaling 1 \(M\) - Face-centered cations: 6 faces each contributing \(\frac{1}{2}\) of \(M\), totaling 3 \(M\) - Central anion: 1 \(X\) at the center.

Step 3: Remove All Anions Except Central X

Following instruction (i), remove all \(X\) except the central one. You will be left with:- Central \(X\)- All \(M\) atoms still in their positions.

Step 4: Replace Face-Centered Cations with Anions

According to instruction (ii), replace the face-centered \(M\) with \(X\). Now, you have:- 6 \(X\) at face-centered positions- 1 central \(X\)- Corner \(M\) atoms still remain.

Step 5: Remove Corner Cations

Instruction (iii) asks to remove corner cations \(M\). Therefore, all cations at the corner positions are removed.

Step 6: Replace Central Anion with Cation

Following instruction (iv), replace the central remaining anion \(X\) with cation \(M\). Now, we have:- 6 \(X\) at face-centered positions- 1 \(M\) at the center.

Step 7: Calculate Anion to Cation Ratio

Finally, determine the ratio of anions to cations:- Number of anions = 6 (from face-centered \(X\))- Number of cations = 1 (central \(M\))Thus, the ratio \(\frac{\text{number of anions}}{\text{number of cations}} = \frac{6}{1} = 6\).

Key Concepts

NaCl structure

The NaCl structure is one of the most common types of ionic crystal lattices. It is also known as the face-centered cubic (FCC) lattice. In this structure, each sodium ion (\( \text{Na}^+ \)) is surrounded by six chloride ions (\( \text{Cl}^- \)), and each chloride ion is surrounded by six sodium ions. This arrangement results in a highly symmetrical and stable structure.

• Sodium ions (\( \text{Na}^+ \)) are positioned at the corners and face-centered positions of the cube.
• Chloride ions (\( \text{Cl}^- \)) fill the octahedral positions within the cube, including the center.

This geometric alignment places cations and anions in close proximity, minimizing the combination of attractive forces. Each unit cell of NaCl contains a net total of four sodium ions and four chloride ions.

Face-centered cubic lattice

A face-centered cubic lattice is a type of crystal structure that is common in many metallic and ionic compounds. In the FCC lattice:

• Atoms or ions are arranged in a cube, with one particle at each of the eight corners and one at the center of each of the six faces of the cube.
• Each corner atom is shared among eight different unit cells, and face-center atoms are shared among two unit cells.

In ionic compounds, specifically those with the NaCl structure, this arrangement allows ions to maximize attractions and minimize repulsions. The result is a robust, repeating pattern of anions and cations. Understanding the face-centered cubic lattice is crucial in visualizing and designing new materials via unit cell transformations.

Anion-cation ratio

The anion-cation ratio in ionic structures is a crucial aspect that influences the properties of the compound. This ratio is derived from the balancing of charges and sizes of the ions.
In the context of NaCl and similar structures:

• The basic NaCl unit cell has an equal ratio of 1:1, meaning there are equal numbers of anions and cations.
• This ratio maintains electrical neutrality, ensuring that the overall structure is stable.

In the transformed \( Z \) structure, after following the complex instructions to remove and replace various ions, the anion to cation ratio changes significantly from \(1:1\) to an uneven \(6:1\). Such transformations can lead to the creation of structures with unusual properties, offering new functionalities in materials science.

Unit cell transformation

Unit cell transformations involve modifying a basic crystal structure to create a new material with unique properties. This can be achieved by systematically altering the positions and types of ions in the structure.
In the given NaCl structure transformation:

• Original positions of ions in the NaCl unit cell are modified based on a sequence of steps.
• Each step of removing and replacing ions—such as replacing face-centered cations with anions—contributes to the creation of a totally distinct structure.

Understanding these transformations allows scientists and engineers to discover new compounds and anticipate their properties. Such knowledge is pivotal in fields like solid-state chemistry and materials engineering, where designing innovative materials can lead to significant technological advancements.