Problem 72 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

Sapphire is aluminium oxide. Aluminium oxide crystallizes with aluminium ions in two-third of the octahedral voids in the closest packed array of oxide ions. What is the formula of aluminium oxide? (a) \(\mathrm{Al}_{2} \mathrm{O}_{3}\) (b) \(\mathrm{AlO}_{2}\) (c) \(\mathrm{Al}_{3} \mathrm{O}_{4}\) (d) \(\mathrm{Al}_{3} \mathrm{O}_{2}\)

Step-by-Step Solution

Answer

The formula of aluminium oxide is \(\mathrm{Al}_{2} \mathrm{O}_{3}\), which is option (a).

Step 1: Understanding the Packing Structure

Aluminium oxide crystallizes in a closest packed array of oxide ions. For each oxide ion, there is a possibility to occupy either a tetrahedral or an octahedral void by another ion. The question states that aluminium ions occupy two-thirds of the octahedral voids.

Step 2: Determining the Number of Voids

In a closest packed structure, the number of octahedral voids is equal to the number of oxide ions. As aluminium ions occupy two-thirds of these octahedral voids, the ratio of the number of aluminium ions (Al) to the number of oxide ions (O) can be calculated as 2:3.

Step 3: Calculating the Ratio of Aluminium to Oxide Ions

Given that two-thirds of the octahedral voids are occupied by aluminium ions, we can represent the ratio as follows: number of Al ions = 2/3 * number of O ions. Since the number of octahedral voids is equal to the number of O ions, the ratio can be simplified to give us 2 Al ions : 3 O ions.

Step 4: Determining the Simplest Whole Number Ratio

The simplest whole number ratio of Al ions to O ions is 2:3. This leads us to the empirical formula where the aluminium oxide formula is \(\mathrm{Al}_{2} \mathrm{O}_{3}\).

Key Concepts

Stoichiometry

When we look at the stoichiometry of a compound, we are essentially observing the ratio of the elements contained within it. This plays a crucial role in understanding the chemical composition of substances like aluminium oxide. In the given problem, stoichiometry serves as the mathematical way of expressing the proportion between aluminium ions and oxygen ions within the crystal structure of aluminium oxide.

By understanding that aluminium ions fill two-thirds of the octahedral voids in a packed array of oxide ions, we can deduce that for every three oxide ions, there are two aluminium ions. This ratio must be expressed in whole numbers, as fractional elements would not make sense in practical chemistry. The exercise determined that the simplest whole number ratio is 2 aluminium ions to 3 oxide ions, leading to the empirical formula \( \mathrm{Al}_2\mathrm{O}_3 \) which agrees with choice (a).

It's critical to note that stoichiometry is not just a numerical exercise; it reflects real physical quantities and their ratios in a chemical substance. Hence, the stoichiometric analysis in our problem connects the abstract concept of ratios with tangible chemical substances.

Crystal Lattice Structure

The crystal lattice structure of a solid defines how its particles are arranged in three-dimensional space. In the case of aluminium oxide, it takes on a close-packed structure, which is one of the most efficient ways to pack spherical particles. This arrangement allows for a high level of structural stability and is commonly seen in metals and ionic compounds.

In sapphire, which is essentially aluminium oxide, the lattice structure is composed of oxide ions (\( \mathrm{O}^{2-} \) ions) arranged in a closely packed system. Into the gaps, or voids, within this arrangement, aluminium ions (\( \mathrm{Al}^{3+} \) ions) are slotted. The specific locations where the aluminium ions go are known as octahedral voids, which we'll look at more closely in the next section.

Understanding the crystal lattice structure not only provides a visual understanding of how the materials are constructed at the atomic level but also helps explain properties such as melting points, hardness, and density. The arrangement of ions in a crystal lattice is a fundamental factor that can influence these physical properties.

Octahedral Voids

We've mentioned that in the crystal lattice of aluminium oxide, aluminium ions fill the so-called octahedral voids. But what is an octahedral void? In simple terms, it's a space in the lattice that is surrounded by six particles, forming an octahedron shape. These voids are found in the spaces between the packed spheres (or ions) in a close-packed array.

An octahedral void is larger than a tetrahedral void and, in the case of aluminium oxide, just the right size for an aluminium ion to fit in. In our exercise, not all octahedral voids are filled—only two-thirds of them are. The fact that aluminium ions occupy these specific voids and in such proportion provides a direct link to the stoichiometry of aluminium oxide, as it dictates the ratio of 2 aluminium ions to every 3 oxygen ions.

Understanding the concept of octahedral voids aids in visualizing how different ions arrange themselves in a structured and predictable pattern, which is critical for the formation of solid structures with specific chemical and physical properties. It's an excellent example of how geometry and chemistry intertwine within the realm of material science.