Problem 85 - A Complete Resource Book in Chemistry for JEE Main

Question

In a cubic closed packed structure of mixed oxides, the lattice is made up of oxide ions, \(20 \%\) of tetrahedral Voids are occupied by divalent \(\mathrm{A}^{2+}\) ions and \(50 \%\) of the octahedral voids by trivalent \(\mathrm{B}^{3+}\). The formula of the oxide is? (a) \(\mathrm{A}_{4} \mathrm{~B}_{5} \mathrm{O}_{10}\) (b) \(\mathrm{A}_{2} \mathrm{~B}_{5} \mathrm{O}_{5}\) (c) \(\mathrm{A}_{2} \mathrm{BO}\) (d) \(\mathrm{A}_{4} \mathrm{~B}_{5} \mathrm{O}_{8}\)

Step-by-Step Solution

Answer

The formula of the oxide is \( \text{A}_4 \text{B}_5 \text{O}_{10} \).

Step 1: Understand the Cubic Closed Packed Structure

In a cubic closed packed (ccp) structure or face-centered cubic (fcc) arrangement, the oxide ions form the lattice. For any face-centered cubic unit cell, there are 4 oxide ions.

Step 2: Calculate Tetrahedral Voids

In the ccp structure, there are 2 tetrahedral voids for each oxide ion. Since there are 4 oxide ions, the total number of tetrahedral voids is 4 * 2 = 8.

Step 3: Calculate Octahedral Voids

In the ccp structure, the number of octahedral voids equals the number of oxide ions. Therefore, there are 4 octahedral voids since we have 4 oxide ions.

Step 4: Determine A ions in Tetrahedral Voids

20% of the tetrahedral voids are occupied by divalent \( A^{2+} \) ions. This means 20% of 8 tetrahedral voids are filled, which equals 0.2 * 8 = 1.6 \( A^{2+} \) ions.

Step 5: Determine B ions in Octahedral Voids

50% of the octahedral voids are occupied by trivalent \( B^{3+} \) ions. This means 50% of 4 octahedral voids are filled, which equals 0.5 * 4 = 2 \( B^{3+} \) ions.

Step 6: Formulate the Formula Based on the Ratio

Given 1.6 \( A^{2+} \) ions, 2 \( B^{3+} \) ions, and 4 oxide ions, the simplest ratio is determined by rescaling. Each ion count should be divided by the smallest number among them, which are 1.6 \( A^{2+} \), giving approximately \( A_2 \), \( B_{2.5} \), and \( O_5 \) after scaling, which simplifies to \( A_4 B_5 O_{10} \).

Key Concepts

Tetrahedral Voids

In a cubic closed packed structure (also known as a face-centered cubic arrangement), we come across fascinating spaces called tetrahedral voids. These voids are formed in such a way that four atoms, located at the corners of a tetrahedron, create a small, empty space within their collective volume.
When analyzing the cubic closed packed structure, each oxide ion contributes to creating such voids. Specifically, for each oxide ion in this structure, there are 2 tetrahedral voids formed.

• For 4 oxide ions, there are a total of 4 x 2 = 8 tetrahedral voids available.
• In this exercise, we see that only 20% of these tetrahedral voids are occupied by divalent \( A^{2+} \) ions.

This percentage translates into 0.2 x 8 = 1.6 \( A^{2+} \) ions predominantly filling these voids, indicating a balanced yet specific distribution of divalent ions.

Octahedral Voids

Octahedral voids are another type of interstitial space found in cubic closed packed structures. These forms are unique, created by a cluster of six atoms—three above and three below the plane—encompassing a central void.
In a face-centered cubic structure, such octahedral voids are equal in number to the oxide ions present.

• For each unit cell containing 4 oxide ions, there are 4 octahedral voids.
• In the given problem, 50% of these octahedral voids are occupied by trivalent \( B^{3+} \) ions.

This occupancy translates into 0.5 x 4 = 2 \( B^{3+} \) ions, showing the affinity of these ions for these specific structural spaces.

Coordination Number

The coordination number in crystalline structures is a crucial concept representing the number of closest neighboring atoms or ions surrounding a particular atom or ion in the lattice. In a cubic closed packed structure, coordination numbers provide insight into stability and bonding.

• For oxide ions in a ccp structure, the coordination number is 12, meaning each oxide ion is surrounded by 12 closest atoms.
• For tetrahedral voids, the coordination number is 4, as they are surrounded by four oxygen ions forming the corners of a tetrahedron.
• Meanwhile, octahedral voids have a coordination number of 6, reflecting the six neighboring ions that frame the void.

Understanding these figures helps visualize how ions like \( A^{2+} \) and \( B^{3+} \) fit and stabilize in the structure, emphasizing the nature of ionic occupancies and potential interactions within the lattice.