Problem 31 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

An ionic crystalline solid, \(\mathrm{MX}_{3}\), has a cubic unit cell. Which of the following arrangement of the ions is consistent with the stoichiometry of the compound? (a) \(\mathrm{M}^{3+}\) ions at the corners and \(\mathrm{X}^{-}\) ions at the face centres (b) \(\mathrm{M}^{3+}\) ions at the corners and \(\mathrm{X}^{-}\) ions at the body centres. (c) \(\mathrm{X}^{-}\) ions at the corners and \(\mathrm{M}^{3+}\) ions at the face centres. (d) \(\mathrm{X}^{-}\) ions at the corners and \(\mathrm{M}^{3+}\) ions at the body centres.

Step-by-Step Solution

Answer

The only arrangement consistent with the stoichiometry of the compound \( \mathrm{MX}_{3} \) is option (a), where the \( \mathrm{M}^{3+} \) ions are at the corners of the cubic unit cell and \( \mathrm{X}^{-} \) ions are at the face centers.

Step 1: Understanding the composition

An ionic compound with formula \( \mathrm{MX}_{3} \) means that for every metal ion \( \mathrm{M}^{3+} \) there are three \( \mathrm{X}^{-} \) ions. This stoichiometry must be reflected in the arrangement of the ions within the cubic unit cell.

Step 2: Analyzing option (a)

An \( \mathrm{M}^{3+} \) ion at each of the eight corners contributes \( \frac{1}{8} \) of its ion per corner to the unit cell. Since there are 8 corners, this results in one \( \mathrm{M}^{3+} \) ion. Six \( \mathrm{X}^{-} \) ions at the face centers contribute \( \frac{1}{2} \) of an ion each to the unit cell, totaling three \( \mathrm{X}^{-} \) ions. This matches the stoichiometry of \( \mathrm{MX}_{3} \) and indicates this arrangement is consistent.

Step 3: Analyzing options (b), (c), and (d)

Option (b) results in one \( \mathrm{M}^{3+} \) ion and one \( \mathrm{X}^{-} \) ion, which does not match the stoichiometry. Option (c) would have one \( \mathrm{X}^{-} \) ion (from the corners) and three \( \mathrm{M}^{3+} \) ions (from the face centers), which also does not match the desired stoichiometry. Lastly, option (d) provides one \( \mathrm{X}^{-} \) ion and a single \( \mathrm{M}^{3+} \) ion in the center, not matching the 1:3 ratio. Therefore, (b), (c), and (d) are not consistent with the stoichiometry.

Key Concepts

Cubic Unit Cell

A cubic unit cell is the simplest 3D repeating unit that makes up the crystal lattice of a crystalline solid. Imagine it as a tiny box with ions, molecules, or atoms located at specific points within the unit. In ionic compounds, the arrangement of these ions is crucial to determine the crystal structure and properties of the material.

The corners of the cubic unit cell are shared by adjacent cells, so an ion placed at a corner contributes only a fraction of its presence to any one particular cell. For example, an ion at a corner is shared by eight unit cells, so it contributes only \(\frac{1}{8}\) of an ion to each cell. Similarly, an ion on a face is shared by two cells, contributing \(\frac{1}{2}\) of an ion to each cell's composition.

When analyzing the structure of an ionic crystalline solid, it's essential to take into account how the position of each ion within the cubic unit cell contributes to the overall structure and stoichiometry of the compound.

Ionic Compound Composition

The composition of an ionic compound is critical in determining the arrangement of ions within a crystalline solid. Stoichiometry, which is the calculated relationship between the amounts of reactants and products in a chemical reaction, extends to solid state chemistry as well.

For an ionic compound represented by the general formula \(\text{MX}_{3}\), the stoichiometry demands a fixed ratio of one metal ion (\(\text{M}^{3+}\)) to three halide ions (\(\text{X}^{-}\)). This ratio must be maintained within the structure of a cubic unit cell to ensure the correct properties and stability of the ionic compound. Deviations from this stoichiometric ratio can lead to an imbalance of charge and potentially alter the physical and chemical characteristics of the resulting material.

Stoichiometry in Solid State Chemistry

Stoichiometry in solid state chemistry entails understanding the precise ratios and arrangements of ions within a solid structure. This not only involves counting the numbers of each type of ion but also considering their spatial arrangement within the unit cell.

As seen in the exercise, only option (a) provides the correct ratio of \(\text{M}^{3+}\) to \(\text{X}^{-}\) ions, which is integral to the stoichiometry of the compound \(\text{MX}_{3}\). This stoichiometric balance is a fundamental aspect of creating stable ionic solids with predictable properties. Any deviation, as illustrated in options (b), (c), and (d), would result in an incorrect stoichiometry, leading to a crystal structure that does not reflect the intended chemical composition and could exhibit different properties than those expected or required for specific applications.