Problem 37 - 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 volume of this \(h c p\) unit cell is - (a) \(24 \sqrt{2} r^{3}\) (b) \(16 \sqrt{2} r^{3}\) (c) \(12 \sqrt{2} r^{3}\) (d) \(\frac{64}{3 \sqrt{3}} r^{3}\)

Step-by-Step Solution

Answer

The volume of the HCP unit cell is 16√2 r³ (option b).

Step 1: Understanding the Structure

The hexagonal close-packed (HCP) structure consists of three layers of spheres. The top and bottom layers form regular hexagons, and the middle layer consists of three spheres that fit into the gaps of the bottom layer's hexagonal arrangement.

Step 2: Determine Dimensions of the HCP Unit Cell

Each sphere has a radius \(r\), so the side of the regular hexagon equals \(2r\). The height of the HCP unit cell is the distance between two identical layers, which equals \((2\sqrt{2}/3)r\). Using the relationship between layers, the height of the unit cell \((c)\) can also be derived from the patterns of the HCP and is equal to \(\sqrt{\frac{8}{3}}r\).

Step 3: Calculate Volume of the Hexagonal Prism

The area of the hexagonal base can be calculated using the formula for the area of a hexagon: \({Area} = \frac{3\sqrt{3}}{2}(2r)^2 = 6\sqrt{3}r^2 \). Next, multiply the area of the hexagonal base by the height of the prism (from the previous step): \ Volume = \text{Base Area} \times \text{Height} = 6\sqrt{3}r^2 \times \sqrt{\frac{8}{3}}r = 16\sqrt{2}r^3 \

Step 4: Final Answer Matching

Compare the calculated volume with the given options. Our calculation yields \(16 \sqrt{2} r^3\), which matches option (b).

Key Concepts

Volume calculation

Understanding the volume of a hexagonal close-packed (HCP) unit cell is critical for grasping the geometrical nature of hexagonal systems of crystals. To compute this volume, we begin by analyzing the structure composed of three layers of packed spheres. Each sphere in the pattern has a radius denoted by \(r\). The base of this unit cell, given the hexagonal nature, is a regular hexagon, where each side measures \(2r\).
To calculate the volume, you must determine the area of this hexagonal base. The formula for a hexagon’s area, given its side \(s\), is \(\frac{3\sqrt{3}}{2}s^2\). Here, substituting \(2r\) for \(s\), we have an area of \(6\sqrt{3}r^2\). This calculation establishes the foundation upon which the height of the hexagonal prism is built. The height of the HCP unit cell is derived from the arrangement pattern and is equivalent to \(\sqrt{\frac{8}{3}}r\).
Multiplying the base area by this height provides the volume of the hexagonal prism: \(6\sqrt{3}r^2 \times \sqrt{\frac{8}{3}}r = 16\sqrt{2}r^3\). This fuses the geometric prowess of hexagons with radial dimensions to provide a cohesive volume calculation for the structure.

Hexagonal systems of crystals

In hexagonal crystal systems, the structural formation is foundational in nature, involving unique geometrical arrangements that maximize space efficiency. The hexagonal close-packed (HCP) arrangement elucidates one such system where spheres are meticulously arranged to occupy minimal space while maintaining structural integrity. The HCP unit cell, characterized by hexagonal prisms, comprises symmetrical upper and lower hexagonal layers with companionably arranged atoms or ions sandwiched in between.
This crystalline form narrows focus on precise spatial orientation. Hexagonal systems allow for adherence to specific symmetry operations, dictated by the angles of interconnections and uniformity among atomic radii. Such a packing structure not only optimizes atomic arrangement but also elevates material properties that are crucial for various applications across engineering and material sciences.
In real-world applications, such structure translates into enhanced stability and aesthetic forms, influencing everything from architectural designs to advanced construction materials.

Sphere arrangement in crystals

The arrangement of spheres within the hexagonal close-packed (HCP) configuration is a prime example of efficient packing in crystalline solids. In this structure, spheres are stacked in three layers: starting with the bottom layer forming a hexagonal pattern with each sphere touching its neighbors. The next layer comprises spheres nestled in the depressions of the first layer, ensuring a tight fit that maximizes coverage without unnecessary gaps.
This intermediate layer typically has three spheres touching the base layer beneath it, reflecting a quasi-triangular arrangement. The top layer resumes the bottom layer's positioning, creating a repeating sequence that typifies the HCP methodology. This symmetry offers a delicate equilibrium of forces and minimizes potential void spaces within the lattice. Atoms or spheres align symmetrically to optimize space usage in a dense configuration for material strength and stability. Such precision in geometry underpins the scientific understanding and engineering ingenuity seen in today's crystalline systems.
This strategic arrangement ensures that each sphere occupies a position maximizing contact and stability, critical for the structural robustness and practical application of these materials.