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 \(\mathrm{ABABAB} \ldots\) 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 in the lattice is 26%, so the answer is (a) 26%.

Step 1: Identify the Packing Type

The sequence \( \text{ABABAB}\ldots \) indicates that the lattice is a hexagonal close-packed (hcp) structure. In such structures, spheres pack as closely as possible.

Step 2: Determine Sphere Packing Efficiency

For a hexagonal close-packed (hcp) lattice, the packing efficiency, which is the fraction of volume occupied by spheres, is approximately \( 74\% \). This is because hcp structures have the same packing efficiency as face-centered cubic (fcc) lattices.

Step 3: Calculate Empty Space Percentage

The empty space percentage is the volume that is not occupied by spheres. Since the packing efficiency is \( 74\% \), the percentage of empty space is \( 100\% - 74\% = 26\% \).

Step 4: Select the Correct Answer

Compare the empty space percentage to the given multiple-choice answers. The correct choice is (a) \( 26\% \).

Key Concepts

Packing Efficiency

Understanding packing efficiency is crucial when discussing lattice structures like the hexagonal close-packed (hcp) and face-centered cubic (fcc) lattices. Packing efficiency refers to the fraction of space that is filled by the spheres in a structure. In an ideal setting, this measures how tightly the atoms (considered as spheres) are packed together. Packing efficiency helps us identify how much actual "material" is present in a given volume. For both hcp and fcc lattices, the spheres occupy about 74% of the total available space. This results from the spheres being arranged in such a way that they cover the same amount of space. Knowing this value is key, as it defines the compactness and potentially influences the properties of the crystalline material, such as density and stability. To sum up, a high packing efficiency of 74% indicates a highly dense structure.

Empty Space Percentage

The concept of empty space percentage provides insight into the unoccupied part of a lattice structure. In any crystalline arrangement, gaps or voids are inevitable. These voids are dictated by the arrangement and packing efficiency of the spheres. The empty space percentage is calculated by subtracting the packing efficiency from 100%. This calculation reveals the portion of the structure that consists of these voids. In hexagonal close-packed (and similarly in face-centered cubic) structures, since the packing efficiency is approximately 74%, the empty space percentage is simply 100% - 74% = 26%. This means that 26% of the volume within this lattice is just voids. The presence of this empty space can influence the physical properties, such as the ability of the material to absorb other substances or how it behaves under pressure.

Face-Centered Cubic Lattice

The face-centered cubic (fcc) lattice is one of the most efficiently packed structures and shares its packing efficiency with the hexagonal close-packed structure. In an fcc lattice, the spheres are arranged such that each is touched by its neighbors, achieving maximum contact in all three dimensions. This arrangement occurs due to the placement of spheres at each cube corner and one at each cube face center. The result is a structure where four spheres form a single cubic unit cell, contributing to its high packing efficiency of 74%. Moreover, this arrangement optimally utilizes the available space, leading to minimal void creation within the lattice. The fcc lattice is prevalent in metallic elements and affects their mechanical and thermal properties, often resulting in superior malleability and excellent thermal conductivity. Understanding this arrangement helps in comprehending why certain materials exhibit particular characteristics related to their atomic packing.