Problem 18 - A Complete Resource Book in Chemistry for JEE Main

Question

What is the contribution of the atom present at the edge centre to the unit cell? (a) \(\frac{1}{8}\) (b) \(\frac{1}{2}\) (c) \(\frac{1}{4}\) (d) \(\frac{1}{3}\)

Step-by-Step Solution

Answer

The contribution of the edge center atom to the unit cell is (c) \( \frac{1}{4} \).

Step 1: Understanding the Edge Atom Position

In a cubic unit cell, an edge center atom is located at the midpoint of any edge. This implies it is shared by four adjacent unit cells since a cube is composed of 12 edges and each edge atom contributes to four different cubes.

Step 2: Calculating the Contribution

Since the edge center atom is shared by four unit cells, its contribution to a single unit cell is calculated by dividing the whole atom by the number of unit cells it is part of, which is 4. Thus, the contribution is \ \( \frac{1}{4} \), representing a quarter of each edge atom.

Key Concepts

Understanding Edge Atoms in Unit Cells

In crystalline structures, atoms are arranged in a highly ordered pattern. These patterns often form a cubic unit cell, which is a small, repeating unit that makes up the larger structure. An edge atom refers to an atom that is positioned at the center of any one of the 12 edges of a cubic unit cell. Each of these edge atoms is shared by four neighboring unit cells. This placement is crucial as it determines how much of the atom's volume or contribution is considered part of a specific unit cell.

Unlike corner or face atoms which may be shared by more or fewer unit cells, the edge atom always spans across four unit cells. This sharing is important for calculating its exact contribution to the composition of each unit cell.

The Role of the Cubic Unit Cell

A cubic unit cell is the basic building block of many crystal structures. Its cubic shape makes it symmetrical and straightforward to model in three dimensions. Each unit cell includes corner atoms, face center atoms, and, as mentioned, edge center atoms.

Understanding the cubic unit cell is essential for grasping how substances like metals and ionic crystals are structured. All of these positions within the unit cell - corners, edges, and faces - define how atoms are packed and shared among multiple cells. This configuration dictates the properties and strength of the material as a whole. The symmetry and repetitive nature of the cubic structure are what make it such a prevalent and essential arrangement in crystallography.

Calculating Atomic Contribution

When dealing with atoms in a unit cell, especially those that are not entirely within one cell, such as edge atoms, calculating their contribution can initially seem complex. But it follows a straightforward division principle.

• The contribution of an edge atom to a cubic unit cell is determined by how much it is shared among unit cells.
• Since an edge atom is shared by four unit cells, only a quarter of that atom belongs to each individual unit cell. This is mathematically expressed as a c contribution for each edge atom for any given cubic unit cell.
• Thus, when determining density or atomic composition of the larger structure, this fractional accounting is crucial.

Understanding these contributions is essential for calculations in material science and helps explain properties like density, weight, and the geometry of crystal formation.