Problem 110 - A Complete Resource Book in Chemistry for JEE Main

Question

A metallic element crystallizes into a lattice containing a sequence of layers of ABABAB... Any packing of spheres leaves out voids in the lattice. What percentage by volume of this lattice is empty space? (a) \(26 \%\) (b) \(74 \%\) (c) \(50 \%\) (d) \(85 \%\)

Step-by-Step Solution

Answer

The percentage of empty space is 26% (option a).

Step 1: Understand the structure

The lattice in question is a hexagonal close-packed (HCP) structure, commonly denoted by the sequence ABABAB. In HCP structures, atoms are packed together in a very efficient manner.

Step 2: Calculate packing efficiency

The packing efficiency of a hexagonal close-packed structure is a measure of how much volume is occupied by spheres in the total volume. For HCP, the packing efficiency is known to be approximately 74%.

Step 3: Determine empty space

To find the percentage of empty space, subtract the packing efficiency from 100%. So, if 74% is packed, the empty space is given by: \[100 ext{%} - 74 ext{%} = 26 ext{%} \]

Step 4: Identify the correct option

From the given options, we identify that (a) \(26\%\) corresponds to the calculated empty space in the HCP structure.

Key Concepts

Packing Efficiency in Hexagonal Close-Packed Structure

The hexagonal close-packed (HCP) structure exemplifies one of the most efficient packing arrangements known in crystalline materials. In this structure, atoms are organized in layers following an ABABAB sequence, which allows the maximum number of atoms to fit into a given volume. To quantify this efficiency, we use the term "packing efficiency," which represents the fraction of volume actually filled by the atoms themselves.
The packing efficiency is a crucial parameter when analyzing crystal structures, as it reveals how tightly the constituent spheres, or atoms, are packed. In the case of the HCP, the packing efficiency is about 74%. This value signifies that 74% of the space in the lattice is occupied by atoms, making HCP both dense and space-efficient. This efficiency is achieved by the specific geometric arrangement of atoms that minimizes empty space.
A high packing efficiency means fewer voids and higher density, making materials with HCP structures generally more robust and capable of withstanding various mechanical stresses. Understanding packing efficiency helps scientists and engineers in predicting material properties and optimizing materials for various applications.

The Metallic Lattice

A metallic lattice describes the unique way atoms are structured in metallic elements. Metals are known for their ability to conduct electricity and heat, which is largely due to the arrangement of atoms in the lattice and the freedom of electron movement. Unlike covalent or ionic bonds, metallic bonds involve a "sea" of delocalized electrons that are free to move throughout the lattice, allowing metallic properties.
In a metallic lattice, such as in the hexagonal close-packed (HCP) structure, the orderly arrangement of atoms or ions ensures that each atom is surrounded equally by others. This close proximity of atoms enables the metallic lattice to maintain its structural integrity while allowing flexibility, due to the electron cloud that cushions and adapts to pressure.
It's important to note that this highly organized pattern not only contributes to the physical and chemical properties but also influences the mechanical properties like malleability, ductility, and tensile strength. Engineers often exploit these features when designing metal-based components, selecting materials with suitable lattices to fit specific applications and environments.

Voids in Crystal Lattices

Voids in crystal lattices refer to the empty spaces left between tightly packed atoms or ions in a crystalline structure. These voids are significant as they affect the overall density and sometimes the stability of the material. In a hexagonal close-packed (HCP) structure, despite the high packing efficiency, about 26% of the lattice volume remains unoccupied, constituting these voids.
These voids can take on several forms, such as octahedral or tetrahedral, depending on their position relative to the surrounding atoms. Their formation is fundamentally due to geometric constraints when arranging spheres uniformly in space. While seemingly inconsequential, voids can have profound effects on material properties.
For instance, they can influence ionic conductivity and diffusion pathways in materials, affecting properties like catalysis and corrosion resistance. Additionally, voids can offer sites for impurity atoms or molecules, which is beneficial in alloy formation and doping processes. Understanding and manipulating voids can thus lead to the development of new materials with tailored properties to fit advanced technological needs.