Question

In a cubic unit cell, seven of the eight corners are occupied by atoms \(A\) and centres of faces are occupied by atoms \(B\). The general formula of the compound is : (a) \(A_{7} B_{6}\) (b) \(A_{7} B_{12}\) (c) \(A_{7} B_{24}\) (d) \(A_{24} B_{7}\)

Short answer

The general formula of the compound is: \(A_7B_{24}\).

Step-by-step solution

Understand the Unit Cell Structure

In a cubic unit cell, there are 8 corners and each corner atom is shared among 8 adjoining unit cells. Similarly, each face-centered atom is shared between 2 unit cells. First, count the number of atoms of type A and B that are actually within one unit cell, accounting for the shared atoms.

Calculate Contribution of Corner Atoms (A)

Since 7 corners are occupied by atom A, and each corner atom contributes to 1/8th of an atom to the unit cell, total contribution of corner atoms A to the unit cell is: 7 atoms * 1/8 per unit cell = 7/8 of an atom A.

Calculate Contribution of Face-Centered Atoms (B)

There are 6 faces in a cubic cell, each occupied by atom B. Since face-centered atoms are shared between 2 unit cells, each contributes 1/2 of an atom to the cell. Thus, total contribution of face-centered atoms B is: 6 atoms * 1/2 per unit cell = 3 atoms B.

Formulate the General Formula

Combine contribution of A and B to get the formula of the compound within a unit cell. Since we have 7/8 A per unit cell and 3 B per unit cell, we multiply these numbers by the least common multiple to get whole numbers. The LCM of 8 and 1 is 8, so: (7/8 * 8)A + (3 * 8)B = 7A + 24B.

Key concepts

Understanding Solid State Chemistry

Solid state chemistry is a fascinating area of study that delves into the properties, structure, and bonding of solid materials. At the heart of this field is the analysis of how the arrangement of atoms in a material's structure affects its overall characteristics. A basic yet vital concept in this area is the 'unit cell,' which is the smallest repeating unit that shows the full symmetry of the crystal structure. Think of it as a building block from which the entire crystal is constructed, much like a Lego piece used to build larger structures.

In the case of the given exercise, we're considering a cubic unit cell — the simplest form of crystal structure, with atoms located at the corners and centers of the faces. When examining such structures, it's important to note that the atoms at the corners and faces do not entirely belong to one unit cell. This is because they are shared with neighboring unit cells. As a result, each of these positions contributes only a fraction of an atom to the individual unit cell's chemical composition. By understanding how these contributions piece together, we can unlock the mystery behind the stoichiometry of the solid material in question.

Deciphering the Crystal Lattice Structure

The crystal lattice structure is a systematic, geometric arrangement of atoms in a crystalline solid. Envisage it as a three-dimensional pattern that extends in all directions. A cubic lattice, which is the focus of our textbook problem, has atoms positioned at each corner of the cube and at the centers of all the faces.

Within this highly ordered structure, each positioning of an atom holds significance. For instance, corner atoms are shared among eight different unit cells, whereas face-centered atoms straddle just two. This means that when calculating the composition of a single unit cell, we must account for this sharing by dividing the contributions accordingly. In essence, the crystal lattice structure determines how atoms are packed together and how they interact with each other, which in turn influences the physical and chemical properties of the material.

Chemical Stoichiometry in the Solid State

Chemical stoichiometry is fundamental to comprehending the fixed ratios of atoms that comprise a compound. It is the quantitative relationship that governs the conservation of mass in chemical reactions and formulations.

In the context of the solid state, stoichiometry allows us to describe the exact chemical makeup of a unit cell. Considerations of stoichiometry come into play when we calculate the contribution of atoms in a crystal lattice to determine the chemical formula of the unit cell. For our cubic unit cell, this calculation involves recognizing that the atoms are not fully contained within one unit cell but are shared with adjacent unit cells. By calculating the proportional contribution of each atom, and then scaling these up to whole numbers, we determine the stoichiometry of the compound.

In the exercise provided, atom A occupies seven of the eight corners, each contributing 1/8 of an atom, and atom B occupies the centers of the faces, each contributing 1/2 of an atom. Understanding these contributions, we can deduce the correct formula of the compound, which provides insight into the substance's stoichiometry within the crystalline solid.