Problem 29 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The simplest formula of a solid having CCP arrangement for 'A' atoms in which alternate face-centres are occupied by 'B' atoms and alternate edge centres are occupied by 'C' atoms, is (a) \(\mathrm{ABC}\) (b) \(\mathrm{A}_{4} \mathrm{BC}\) (c) \(\mathrm{A}_{2} \mathrm{BC}\) (d) \(\mathrm{A}_{4} \mathrm{~B}_{2} \mathrm{C}\)

Step-by-Step Solution

Answer

\(\text{A}_4 \text{BC}\)

Step 1: Understanding CCP Structure

In a cubic close-packed (CCP) structure, the 'A' atoms occupy the corners and the centers of the faces of a cube. There are a total of 8 corners with 1/8th of each corner atom within the unit cell and 6 faces with half of each atom on a face within the unit cell. Thus, this accounts for 8 * 1/8 + 6 * 1/2 = 4 'A' atoms.

Step 2: Calculating 'B' Atoms

Since 'B' atoms occupy alternate face-centers in a CCP arrangement and there are 6 faces, only half of these faces will have 'B' atoms. This gives us 6 * 1/2 = 3 faces, but each B atom is shared by two unit cells, so 3 * 1/2 = 1.5, which we count as one 'B' atom per unit cell because atoms cannot be divided.

Step 3: Calculating 'C' Atoms

The 'C' atoms are located at alternate edge centers. There are 12 edges, half of which will be occupied by 'C' atoms, giving us 12 * 1/2 = 6 edges. Each edge is shared by 4 unit cells, so 6 * 1/4 = 1.5, which, like for 'B' atoms, is counted as one 'C' atom per unit cell.

Step 4: Determining the Simplest Formula

From the above calculations, the unit cell of the solid contains 4 'A' atoms, 1 'B' atom, and 1 'C' atom. Therefore, the simplest formula of the solid is \( \mathrm{A}_{4}\mathrm{BC} \).

Key Concepts

Cubic Close-Packed Structure

The cubic close-packed (CCP) arrangement is one of the most dense ways atoms can be arranged in solid matter. Students need to visualize this geometrically appealing structure as spheres (atoms) packed together, leaving the least amount of space between them. In the CCP structure, each layer of atoms is placed over the gaps of the previous layer, creating a pattern that repeats in three dimensions. A good way to imagine this is by stacking layers of oranges at a grocery store.

When looking at a CCP structure, the layers follow a specific sequence often denoted as ABCABC... meaning that the third layer is a repeat of the first. This creates a three-dimensional structure that is both highly organized and efficient in its use of space. Due to this repetitive stacking, students can be confident that understanding one unit cell—a repeating unit that can regenerate the entire crystal lattice through spatial translations—provides the key to grasping the whole crystal structure.

Unit Cell Atom Counting

Counting atoms within a unit cell is a foundational skill in crystallography. The challenge comes from the fact that atoms at various points within the unit cell are shared with neighboring cells. For instance, atoms at the corners of the cube are shared with eight unit cells, so only 1/8th of each corner atom belongs to one particular cell. Similarly, atoms on the face centers are shared between two unit cells, translating to half of each atom belonging to a unit cell.

It's useful to employ simple fractions to keep track of these shared portions. After adding up these fractions for all atom positions—corners, face centers, edge centers, and internal positions—we arrive at the total number of atoms within the unit cell. For many students, drawing the cell and visually partitioning the atoms can be a substantial aid in understanding this concept.

Composition Determination in Solid-State Chemistry

Determining the composition of a crystalline solid requires the ability to count the numbers of different types of atoms within the unit cell and then scale this knowledge to understand the macroscopic properties of the material. This is a crucial step in material science, as it allows one to deduce the formula of a substance based on its microscopic arrangements.

In our exercise, for example, the unit cell contains 'A', 'B', and 'C' type atoms in specific locations with a defined pattern. By knowing the structural layout and counting the relevant portions of 'A', 'B', and 'C' atoms, we can derive the empirical formula of the substance. This approach merges geometrical understanding with chemical composition to reveal critical insights into the material's characteristics and potential applications.

Coordination Number in Crystal Structures

The coordination number concept is a measure of how many nearest neighbors an atom has in a crystal structure. In CCP structures, each atom typically has a coordination number of 12, signifying that each atom touches 12 other atoms. For students, this can be a key insight into the strong bonding and high stability of CCP crystals.

The coordination number also informs us about the space each atom occupies and the resultant density of the material. When resolving complex structures, understanding the coordination number helps in predicting various properties of the material, such as melting point, hardness, and electronic properties. Thus, unraveling the coordination number is a gateway to the fundamental understanding of material behavior at the atomic level.