Question

In a cubic unit cell, seven of the eight corners are occupied by atom \(\mathrm{A}\) and having of faces are occupied of \(\mathrm{B}\). The general formula of the substance having this type structure would be (a) \(\mathrm{A}_{7} \mathrm{~B}_{14}\) (b) \(\mathrm{A}_{14} \mathrm{~B}_{7}\) (c) \(\mathrm{A}_{7} \mathrm{~B}_{24}\) (d) \(\mathrm{A}_{9} \mathrm{~B}_{24}\)

Short answer

The general formula is \(\mathrm{A}_7\mathrm{B}_{24}\). Choice (c).

Step-by-step solution

Understand the Arrangement of Atoms

In a cubic unit cell, atoms at the corners and faces respectively contribute partially to the unit cell due to sharing with adjacent cells. Each corner atom is shared among 8 unit cells, and each face atom is shared between 2 unit cells. The structure described has atoms of type A at 7 out of 8 corners and faces are occupied by atoms of type B.

Calculate Contribution of Atom A

Since each corner atom contributes 1/8 to the unit cell and 7 corners are occupied by A atoms, the contribution of A is calculated as \(7 \times \frac{1}{8} = \frac{7}{8}\).

Calculate Contribution of Atom B

Face atoms are shared between two unit cells. Since it is a cubic cell, there are 6 faces, and all are occupied by B atoms. Each face atom contributes \(\frac{1}{2}\), giving a total contribution from B as \(6 \times \frac{1}{2} = 3\).

Determine Empirical Formula

The effective contribution from the atoms per unit cell is A: \(\frac{7}{8}\) and B: 3. To write the chemical formula, we need whole numbers. This involves finding a common multiplier, which in this case is 8. Therefore, we multiply the contributions by 8: A: \(\frac{7}{8} \times 8 = 7\), B: \(3 \times 8 = 24\).

Select the Correct Option

With the empirical formula components calculated, we determine the formula of the compound as \(\mathrm{A}_7\mathrm{B}_{24}\). Thus, the correct option among the given choices is (c).

Key concepts

Atom Contribution in Cubic Unit Cells

In the world of solid-state chemistry, the concept of atom contribution is essential to understanding the structure of cubic unit cells. A unit cell is the smallest repeating unit that defines the crystal structure of a solid. In a cubic unit cell, atoms can occupy different positions such as corners, edges, or faces. But critically, these atoms often belong to more than one unit cell.

• Corner Atoms: Atoms located at the corners of a cubic cell are shared by 8 adjacent unit cells. Therefore, each corner atom contributes only about \( \frac{1}{8} \) of itself to a single unit cell.
• Face Atoms: Atoms residing on the face of the cubic unit cell are shared between just 2 unit cells. Thus, the contribution here is \( \frac{1}{2} \) per face.

In the given structure, 7 corner atoms are occupied by atom \( \mathrm{A} \), leading to a total contribution of \( 7 \times \frac{1}{8} = \frac{7}{8} \). Similarly, with 6 face atoms of \( \mathrm{B} \), the total contribution is \( 6 \times \frac{1}{2} = 3 \). Understanding these contributions is crucial in determining the overall composition of the structure.

Determining the Empirical Formula in Solid-State Structures

An empirical formula represents the simplest whole-number ratio of the atoms present in a compound. In solid-state chemistry, finding the empirical formula involves considering the contribution of each type of atom in the unit cell.

The problem describes a structure where the contribution of atom \( \mathrm{A} \) is \( \frac{7}{8} \) and that of atom \( \mathrm{B} \) is 3. Since empirical formulas must have whole-number subscripts, we need a strategy to convert these fractional contributions into whole numbers.

• We find a common factor to multiply both contributions. Here, 8 is a suitable choice.
• Multiplying \( \frac{7}{8} \) of \( \mathrm{A} \) by 8 gives 7, and multiplying 3 of \( \mathrm{B} \) by 8 gives 24.

This yields an empirical formula of \( \mathrm{A}_7\mathrm{B}_{24} \). Such calculations help describe the chemical composition of complex crystal structures accurately.

Fundamentals of Solid State Chemistry

Solid state chemistry is a branch of chemistry focused on the study of the synthesis, structure, and properties of solid materials. It is an essential field because so much of the world is composed of solids—from fundamental elements to complex compounds.

In solid-state chemistry, understanding the arrangement and contribution of atoms within unit cells like cubic ones is key. Here are some core aspects:

• Structure Determination: Analyzing how atoms are distributed in the crystal lattice, using models like the cubic unit cell.
• Properties Correlation: How these distributions affect the physical and chemical properties of the material.
• Synthesis Methods: Techniques to create materials with desirable properties by manipulating their crystal structures.

Throughout, solid state chemistry evolves our understanding of materials, leading to innovations in technology, medicine, and industry. The cubic unit cell, often discussed in foundational chemistry courses, serves as a starting point for exploring more complex materials and their properties.