Problem 38 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

In hexagonal systems of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are regular hexagons and three atoms are sandwiched in between them. A space- filling model of this structure, called hexagonal close-packed (HCP), is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of these three spheres touches three spheres of the bottom layer. Finally, the second layer is covered with third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be 'r'. The empty space in this \(h c p\) unit cell is (a) \(74 \%\) (b) \(47.6 \%\) (c) \(32 \%\) (d) \(26 \%\)

Step-by-Step Solution

Answer

The empty space in the HCP unit cell is 26% (option d).

Step 1: Understanding the Hexagonal Close-Packed (HCP) Geometry

In an HCP structure, the spheres are packed as closely as possible in a hexagonal arrangement. Each sphere touches others around it. In addition, the second layer of spheres fits into the voids of the first layer, and the third layer aligns with the first layer. The structure forms a unit cell, which is a hexagonal prism with one complete layer of spheres inside.

Step 2: Calculating the Volume of the Unit Cell

The volume of the hexagonal unit cell is calculated based on the dimensions of the hexagon and the height of the prism. The side length of the hexagon is the radius of the spheres, noted as 'r'. The volume of the hexagonal base is given by:\[ \text{Area of hexagonal base} = \frac{3\sqrt{3}}{2} a^2 = \frac{3\sqrt{3}}{2} r^2 \] where 'a' is the side length of the hexagon. The height or c-axis is \( c = 2\sqrt{\frac{2}{3}}a \). Thus, the volume of the cell becomes \[ V = \frac{3\sqrt{3}}{2} r^2 \times 2\sqrt{\frac{2}{3}}r \approx 8.48r^3 \] after simplification.

Step 3: Calculating the Volume of the Spheres in the Unit Cell

Within the hexagonal close-packed structure, there are 2 complete spheres per unit cell due to the arrangement of partial spheres in the unit cell. The volume of one sphere is given by \( V = \frac{4}{3}\pi r^3 \). Thus, the total volume of the spheres in the unit cell is:\[ V_{spheres} = 2 \times \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \approx 8.38r^3 \] after calculation.

Step 4: Calculating the Packing Efficiency

The packing efficiency is given by the ratio of the volume occupied by the spheres to the total volume of the unit cell. Hence, packing efficiency is calculated as follows:\[ \frac{V_{spheres}}{V_{cell}} = \frac{8.38r^3}{8.48r^3} \approx 0.740 \] or 74%. This indicates the percentage of space occupied by the spheres.

Step 5: Determining the Empty Space

To find the empty space percentage in the unit cell, we subtract the packing efficiency from 100%. Since the packing efficiency is 74%, the empty space is:\[ 100\% - 74\% = 26\% \]. This accounts for the space not occupied by the atoms in the HCP structure.

Key Concepts

Crystallography

Crystallography is a branch of science that explores the arrangement of atoms in crystalline solids. It aims to understand how atoms are ordered over various distance scales within materials possessing a periodic structure. Crystals are grouped into different systems based on their geometric shape and symmetry. Each system is defined by the relative lengths of the crystal axes and the angles between them.

The hexagonal system is one of these, characterized by its distinct six-sided prisms. Within this system, the hexagonal close-packed (HCP) structure is an important arrangement wherein atoms are tightly grouped. In HCP, atoms form hexagonal layers stacked in a repeating ABAB pattern. This arrangement minimizes empty space and is seen in metals like magnesium and titanium. Understanding crystallography involves analyzing these geometric arrangements, evaluating how they affect the material properties, and studying their applications across different fields.

Packing Efficiency

Packing efficiency is a measure that indicates how tightly atoms are packed within a unit cell. It is expressed as a percentage representing the ratio of the occupied volume by atoms to the total volume of the unit cell. In the hexagonal close-packed (HCP) structure, which is known for its dense atomic arrangement, the packing efficiency is calculated at 74%.

This efficiency is important for understanding material properties, as tightly packed structures often result in high strength and density. HCP, along with face-centered cubic (FCC) structures, offers the highest packing efficiency among all crystalline arrangements, due to the optimal use of space with minimal voids. This efficiency influences mechanical properties such as ductility and hardness, making it an important factor in material science and physics.

• High packing efficiency leads to strong materials.
• Efficient atomic packing minimizes space, maximizing density.
• Crucial for applications in metallurgy and manufacturing.

Unit Cell Volume

The concept of unit cell volume is integral to understanding how atoms are arranged in a solid. In the hexagonal close-packed (HCP) structure, the unit cell is a hexagonal prism containing atoms arranged in a tight, repeating pattern. The volume of the unit cell is calculated based on the dimensions of its hexagonal base and the height (c-axis).

The area of the hexagonal base is given by the formula \( \frac{3\sqrt{3}}{2}r^2 \), where \( r \) is the atomic radius. The height of the prism in an HCP system is \( c = 2\sqrt{\frac{2}{3}}r \), allowing the calculation of the unit cell volume to be approximately \( 8.48r^3 \).

Understanding unit cell volume is vital for:

• Predicting material density
• Analyzing structural stability
• Calculating the mass of the crystal in applications requiring precision such as quality control.

Atomic Arrangement

Atomic arrangement refers to how atoms are organised in a crystal lattice within a solid. In the hexagonal close-packed (HCP) structure, atoms form a specific geometric pattern that involves highly efficient use of space. This pattern consists of horizontal layers of atoms arranged in a hexagon, followed by a second layer fitting into the spaces of the first, and a third layer aligning back with the bottom layer.

This ABAB stacking pattern results in a repeating sequence that generates a dense atomic framework. The atomic arrangement:

• Maximizes surface contact among atoms.
• Minimizes space wastage.
• Increases material strength and stability.

In materials like cobalt or zinc, this arrangement determines important properties such as malleability, thermal conductivity, and resistance to deformation. It is key to understanding how materials behave under different conditions, influencing their suitability for various applications.