Problem 10 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

A metal crystallises in a face centred cubic structure. If the edge length of its unit cell is 'a', the closest approach between two atoms in metallic crystal will be: (a) \(2 a\) (b) \(2 \sqrt{2} a\) (c) \(\sqrt{2} a\) (d) \(\frac{a}{\sqrt{2}}\)

Step-by-Step Solution

Answer

The closest approach between two atoms is (d) \(\frac{a}{\sqrt{2}}\).

Step 1: Understanding the FCC structure

A face-centered cubic (FCC) structure has atoms at each corner and at the centers of each face of the cube. This structure is important when considering the distance between atoms.

Step 2: Visualize closest distance in FCC

In an FCC unit cell, the closest distance between atoms is along the face diagonal. A face diagonal connects two face-centered atoms, passing through the center of a corner atom.

Step 3: Calculate the face diagonal length

The face diagonal in a cube is given by the formula \(d = a\sqrt{2}\), where \(d\) is the diagonal length and \(a\) is the edge length of the cube.

Step 4: Determine distance between closest atoms

Since there are two face-centered atoms along the face diagonal, the shortest distance between the centers of two adjacent atoms is half the face diagonal: \(\frac{a\sqrt{2}}{2}\).

Step 5: Simplify and find the formula for shortest distance

Simplified, the shortest distance between two atoms in an FCC structure is \(\frac{a\sqrt{2}}{2} = \sqrt{2} \frac{a}{2}\). However, the question asks for the closest approach, which is the radius to radius contact, i.e., \(\frac{a}{\sqrt{2}}\) for the closest approach between two atoms.

Key Concepts

Closest Distance Between Atoms

In a face-centered cubic (FCC) structure, understanding how atoms are positioned is crucial for figuring out the closest distance between atoms. Atoms are located at each corner and in the center of each face of the cube. The path that provides the shortest distance is the face diagonal, which is the line that runs diagonally across a face of the cube.

On this diagonal, there are three atoms in a row: two face-centered atoms with a corner atom in-between. Since the face diagonal is the key to determining the distance, calculating its length is the first step. The formula to find the length of the face diagonal in a cube is given by \(d = a\sqrt{2}\), where \(d\) stands for face diagonal length and \(a\) is the cube's edge length. The closest distance between two atoms is actually half of this diagonal, which simplifies to \(\frac{a}{\sqrt{2}}\). Thus, this distance represents the closest approach between neighboring atoms in the FCC structure.

FCC Unit Cell

An FCC unit cell, or face-centered cubic cell, is one particular arrangement that atoms can be organized into within a crystal lattice. In this type of crystal structure, each atom is surrounded by others in a densely packed manner, leading to the high density commonly seen in metals like aluminum and copper.

The structure of an FCC unit cell includes:

• Atoms at each of the eight corners of the cube. Each corner atom is shared among eight separate cells.
• Atoms in the center of each face of the cube, shared between two cells.

This unique arrangement allows each FCC unit cell to contain effectively four atoms. Understanding the FCC structure helps in comprehending how metals conduct electricity and the strength of such materials since the atoms are tightly packed and have minimal space between them.

Crystallography

Crystallography is essential in understanding the arrangement of atoms within solids. Specifically, it focuses on the geometric arrangement of atoms in crystalline solids. The face-centered cubic (FCC) structure is a primary example studied in crystallography, as it is highly symmetric and densely packed.

When studying crystallography, it's important to note:

• The symmetry of the structure, which affects how materials break under stress and how they conduct electricity.
• The arrangement of atoms, understanding their closest distance, and how it brings about the unique properties of metals with FCC structures.
• The repeating pattern of unit cells forming the larger lattice structure of the material.

By using techniques such as X-ray diffraction, scientists can determine the exact arrangement of atoms within a crystal. This knowledge helps in materials science, chemistry, and physics, allowing for advancements in various applications and new material synthesis.