Question

In a body-centred cubic arrangement, A ions occupy the centre while \(\mathrm{B}\) ions occupy the corners of the cube. The formula of the solid is (a) \(\mathrm{AB}\) (b) \(\mathrm{AB}_{2}\) (c) \(\mathrm{A}_{2} \mathrm{~B}\) (d) \(\mathrm{AB}_{3}\)

Short answer

The formula of the solid is \( \mathrm{AB} \).

Step-by-step solution

Understanding the Structure

In a body-centred cubic (BCC) arrangement, each unit cell consists of one atom at the center of the cube and atoms at each corner of the cube. In this problem, the A ion is at the center, and B ions are at the corners.

Calculating the Effective Atoms

Each corner atom in a cubic structure is shared by eight adjacent unit cells, so each B ion contributes \(\frac{1}{8}\) to the cell. Since there are eight corners, the effective number of B ions per cell is: \[ \text{Effective B ions} = 8 \times \frac{1}{8} = 1 \] The A ion is entirely within the unit cell, contributing 1 fully.

Determining the Formula

With these calculations, we have 1 A ion and 1 B ion per unit cell. This means the ratio of A to B in the unit cell is 1:1. Thus, the chemical formula of the solid is \( AB \).

Key concepts

Effective Number of Atoms

Understanding the concept of the effective number of atoms in a unit cell is central to grasping body-centred cubic structures. In such structures, atoms are arranged in a specific geometric pattern. The placement and sharing of atoms define how many atoms are "effectively" present in a unit cell.

Let’s break it down with the different positions in the cell:

• Corner Atoms: Each B ion in this problem is located at the corner of the unit cell. Corner atoms are shared among eight neighboring unit cells. Therefore, each corner atom contributes only \(\frac{1}{8}\) of an atom to the cell. There are eight corners, hence the total contribution of B ions per unit cell is \ 8 \times \frac{1}{8} = 1. \
• Center Atom: The A ion is at the center of the unit cell, and unlike the corner atoms, it is not shared with any other unit cell. Hence, this A ion contributes fully as 1 atom per unit cell.

The effective number of atoms takes into account these positions and sharing rules, which is a crucial stepping stone toward determining the correct chemical formula.

Chemical Formula Determination

Determining the chemical formula from a crystalline structure like a body-centered cubic arrangement involves understanding the ratio of effectively present atoms.

In this exercise, with one A ion at the center and eight B ions at the corners, you need to consider the effective number of atoms calculated.

• We found that there is effectively 1 B ion per unit cell due to sharing at the corners.
• Since the A ion at the center contributes fully, there is 1 A ion per unit cell.

The ratio of A ions to B ions in the cell is 1:1. It shows that, from a chemical perspective, for every A ion, there is exactly one B ion. This 1:1 ratio reflects in the simplest empirical formula for the solid, which is simply expressed as \(AB\). This clear and pure ratio is essential for defining the solid’s composition.

Ionic Arrangement in Solids

Ionic arrangement in solids is the spatial distribution of ions within a crystal lattice, which determines the solid's physical and chemical properties. Understanding this concept involves recognizing how ions are organized in a particular structure, such as the body-centred cubic (BCC) configuration discussed here.

In a BCC lattice:

• A ions: These are located at the center of the cube, offering stability and holding the structure together, not being shared with any other unit cells.
• B ions: These corner ions contribute to the unit cell through partial sharing across neighboring cubes, emphasizing the interconnected nature of solid structures.

This arrangement shows how ionic solids can be efficiently packed while maintaining specific stoichiometric ratios, like the 1:1 A to B ratio here. Such precise arrangements dictate the ionic solid's material properties, including its reactivity, melting point, and hardness, demonstrating the importance of understanding the ionic arrangement in diverse chemical applications.