Question

In sodium chloride crystal, the number of next nearest neighbours of each \(\mathrm{Na}^{+}\) ions is: (a) \(8 \mathrm{Cl}^{-}\) ions (b) \(12 \mathrm{Na}^{+}\) ions (c) \(12 \mathrm{Cl}^{-}\) ions (d) \(24 \mathrm{Cl}^{-}\) ions

Short answer

The number of next nearest neighbours of each \text{Na}^{+} ion in sodium chloride crystal is: (c) 12 \text{Cl}^{-} ions.

Step-by-step solution

- Understanding the Crystal Structure of Sodium Chloride

First, it is important to understand the crystal structure of sodium chloride (NaCl). Sodium chloride forms a cubic crystal lattice structure, which is also known as a face-centered cubic (fcc) lattice for Cl- ions with Na+ ions in the octahedral holes. In this arrangement, each ion is surrounded by others in a specific pattern.

- Identifying Nearest and Next Nearest Neighbours

In an fcc lattice, each atom (or ion) has its nearest neighbours and next nearest neighbours. The nearest neighbours for Na+ in NaCl are Cl- ions that are touching it directly. The next nearest neighbours are those that are a step further away from the nearest neighbours.

- Counting the Next Nearest Neighbours of a Na+ Ion

Each Na+ ion is surrounded by 6 Cl- ions at the faces of the octahedral void which are at an equal distance (nearest neighbours). The next nearest neighbours are the 12 Cl- ions located at the edges of the cube that formulates the fcc lattice. These 12 Cl- ions are equidistant from the Na+ ion and are the next nearest neighbours.

Key concepts

Face-Centered Cubic Lattice

The face-centered cubic (fcc) lattice is one of the most prevalent lattice structures found in metal crystals, such as the chloride ions (Cl-) in a sodium chloride crystal. Imagine a cube that not only has an atom at each corner but also one atom positioned in the center of each of the cube's faces. This results in each unit cell having a total of 14 atoms: 8 contributed by the corners (each corner atom shared by eight adjacent unit cells) and 6 from the faces (each face atom shared by two unit cells).

The significance of this structure is vivid when considering how tightly the atoms are packed together, which in turn affects the physical properties of the material. The fcc lattice is known for its high density and closely packed structure, which is particularly efficient for ionic compounds like NaCl where each ion's electric field influences the positioning of neighboring ions, creating a highly orderly arrangement.

In the context of sodium chloride, Cl- ions form this fcc lattice while Na+ ions fit snugly in the spaces, or 'holes', between them, specifically in the octahedral holes, which we'll talk about next.

Octahedral Holes

When delving into the study of crystal structures, one discovers the concept of 'holes' within the lattice—spaces where other atoms or ions can reside. In a face-centered cubic lattice, two types of holes are significant: octahedral and tetrahedral. Focusing on octahedral holes, these are voids surrounded by six atoms or ions, arranged at the vertices of an octahedron.

In sodium chloride, these holes are occupied by Na+ ions. For each Cl- ion, there is an equal-sized space in between that snugly fits a Na+ ion. These octahedral holes are located at the center of each edge and in the very center of the cube itself. The spatial efficiency of the fcc lattice allows each Na+ ion to be surrounded and stabilized by an electric field emanating from the six nearest Cl- ions at the vertices of the octahedron. This positioning is key to understanding the stability and formula of the NaCl crystal since it dictates the 1:1 ratio of sodium to chloride ions present in the compound.

Nearest and Next Nearest Neighbours

An ion's neighbours in a crystal lattice are differentiated by their proximity—nearest and next nearest neighbours. For a Na+ ion in sodium chloride's fcc structure, the immediate, touching ions are its nearest neighbours, which are 6 Cl- ions located at each face of the octahedron which comprises the octahedral hole the Na+ resides in.

What about the next nearest neighbours? Continuing our exploration further afield from the Na+, we find an additional layer of neighbouring ions. Each Na+ ion is surrounded by 12 Cl- ions, which are located at the edges of the cube forming the larger crystal lattice surrounding the Na+. These Cl- ions are the next nearest neighbours as they are a step further away in the spatial arrangement but still significantly affect the overall stability and electrostatic balance of the ionic crystal. Since the nearest Cl- ions provide an immediate stabilizing effect while the next nearest play a secondary role, both levels of proximity are crucial for the crystal's integrity and chemical properties.