Question

\(\mathrm{Na}_{2} \mathrm{O}\) has antifluorite structure. In \(\mathrm{Na}_{2} \mathrm{O}\), the coordination number of \(\mathrm{N}^{\mathrm{a}^{+}}\)and \(\mathrm{O}^{2}\) are respectively (a) 4,4 (b) 6,6 (c) 4,8 (d) 8,8

Short answer

(c) 4,8

Step-by-step solution

Understanding Antifluorite Structure

The antifluorite structure is the opposite of the fluorite (CaF₂) structure. In the antifluorite structure, the anions (here, O²⁻) form a face-centered cubic (fcc) lattice, and the smaller cations (here, Na⁺) occupy all the tetrahedral holes within that cubic structure.

Analyze the Anion Coordination Number

In the antifluorite structure, each O²⁻ is surrounded by 8 cations (Na⁺) because it occupies a position in the fcc lattice, and the tetrahedral holes are filled with the smaller cations. Hence, the coordination number for O²⁻ is 8.

Analyze the Cation Coordination Number

Each Na⁺ occupies a tetrahedral hole, which means it is surrounded by 4 anions (O²⁻) as it can only touch four oxygen ions in the tetrahedral arrangement. Therefore, the coordination number of Na⁺ is 4.

Selection of the Correct Option

Based on the coordination numbers derived, Na⁺ has a coordination number of 4 and O²⁻ has a coordination number of 8. We compare these numbers to the given options.

Key concepts

Coordination Number

In a crystalline structure, the coordination number is a fundamental concept that refers to the number of ions surrounding a central ion in a crystal lattice. The antifluorite structure provides an excellent example of this concept. Here, each cation and anion has a distinct coordination number. For Na\(^+\) ions in Na\(_2\)O, the coordination number is determined by the number of surrounding O\(^{2-}\) ions. In this arrangement, each Na\(^+\) ion is surrounded by four O\(^{2-}\) ions, leading to a coordination number of 4. Similarly, each O\(^{2-}\) ion is surrounded by eight Na\(^+\) ions, giving it a coordination number of 8. These numbers highlight the geometric arrangement and balance within the crystal structure.

Face-centered Cubic Lattice

A face-centered cubic (fcc) lattice is a specific and highly efficient way atoms are packed in a crystal structure. In the antifluorite structure of Na\(_2\)O, the O\(^{2-}\) ions form a face-centered cubic lattice. This means that the larger O\(^{2-}\) ions are located at each corner and the center of each face of the cube. This arrangement maximizes the space occupied by the ions and allows the Na\(^+\) ions to fit into the voids of the lattice effectively. The fcc structure is advantageous because it offers high density packing, which is crucial for maintaining the stability and durability of the structure.

Tetrahedral Holes

Tetrahedral holes are small spaces in a crystal lattice where smaller ions or atoms can fit in. In the antifluorite structure, these holes are critical as they house the Na\(^+\) ions. Situated amid larger O\(^{2-}\) ions in the face-centered cubic lattice, these tetrahedral holes form when four atoms arrange themselves in the shape of a tetrahedron, creating a small void. The presence of these holes ensures that the cations are effectively positioned to optimize the interaction with the anions, consolidating the strength of the crystal lattice.

Anion and Cation Arrangement

In the antifluorite structure, the arrangement of anions and cations plays a crucial role in defining the material's properties. Specifically, the oxygen anions (O\(^{2-}\)\()\) are arranged in a face-centered cubic lattice, acting as the building blocks of the entire structure. Meanwhile, the sodium cations (Na\(^{+}\)) find their places within the interstitial sites—namely the tetrahedral holes—of the O\(^{2-}\) lattice. This careful positioning is vital in ensuring the overall neutrality and stability of the compound. Understanding these arrangements helps explain why certain materials have unique physical and chemical properties. By precisely filling these positions, Na\(_2\)O achieves an efficient packing and maintains its crystalline integrity.