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

Question

Match the crystal system/unit cells mentioned in Column I with their characteristic features mentioned in Column II. Indicate your answer by darkening the appropriate bubbles of the \(4 \times 4\) matrix given in the ORS. Column I Column II (A) Simple cubic and (p) have these parameters, face-centered cubic \(\quad a=b=c\) and \(\alpha=\beta=\gamma\) parameters (B) cubic and (q) are two crystal systems rhombohedral (C) cubic and tetragonal (r) have only two crystallographic angles of \(90^{\circ}\) (D) hexagonal and (s) belong to same crystal monoclinic \(\quad\) system

Step-by-Step Solution

Answer

(A) matches (p), (B) matches (q), (C) matches (r), (D) matches (s).

Step 1: Understanding Key Concepts

To solve this matching problem, we need to understand the basic characteristics of the crystal systems mentioned in Column I and the crystal features in Column II. Specifically, we must identify these characteristics: lattice parameters (like the equality of lattice constants and angles), crystal systems, and any shared features between them.

Step 2: Analyze Column I

Column I contains the following crystal systems: - (A) Simple cubic and face-centered cubic - (B) Cubic and rhombohedral - (C) Cubic and tetragonal - (D) Hexagonal and monoclinic

Step 3: Analyze Column II

Column II features must be matched to the systems in Column I: - (p) Have these parameters, a=b=c and α=β=γ=90° - (q) Are two crystal systems - (r) Have only two crystallographic angles of 90° - (s) Belong to the same crystal system

Step 4: Match (A) Simple cubic and face-centered cubic

Both simple cubic and face-centered cubic systems have lattice parameters where all three lattice constants are equal (a=b=c) and all angles are 90°. Therefore, match (A) with (p).

Step 5: Match (B) Cubic and rhombohedral

Cubic and rhombohedral are two distinct crystal systems. Therefore, match (B) with (q).

Step 6: Match (C) Cubic and tetragonal

In the tetragonal system, only two of the lattice constants are equal (a=b≠c) and the angles are 90°. Therefore, match (C) with (r).

Step 7: Match (D) Hexagonal and monoclinic

Hexagonal and monoclinic do not share common lattice parameters or angles; they belong to different crystal systems. Therefore, this is not a direct match with any system belonging feature, so match (D) with (s) as neither are directly described by the other options.

Key Concepts

Unit Cells

Imagine crystals as being made up of repeating building blocks. These blocks are called unit cells. They are the smallest part of a crystal that, when repeated in space, forms the entire crystal structure. Each crystal system has a characteristic unit cell with specific geometry.

In science, the understanding of unit cells helps us to predict and explain properties of materials. Here's how:

• Shape: Unit cells have different shapes. For example, cubic unit cells have equal-length sides and 90-degree angles, forming a perfect box shape.
• Volume: The size of a unit cell directly affects the density of a crystal.

Unit cells are crucial because they help in determining how particles are packed in a solid, which can explain properties like strength and melting point. This concept is a fundamental piece in the puzzle of understanding material properties.

Lattice Parameters

Lattice parameters describe the dimensions of a unit cell including the length of its edges and the angles between them. In other words, they help define the shape and size of a crystal's unit cell.

The three main parameters for describing a unit cell structure are:

• Edge Lengths (\(a, b, c\)): These are the lengths of the edges of the unit cell, and they can be equal as in a cubic system, or unequal as seen in tetragonal or orthorhombic systems.
• Angles (\(\alpha, \beta, \gamma\)): These are the angles between the edges of the unit cell. For example, in the cubic system, all the angles are 90 degrees.

The lattice parameters are essential as they can help identify the symmetry and geometry of the crystal. Also, they are critical in understanding the arrangement and distances between atoms inside the crystal, which affect material properties such as thermal expansion and conductivity.

Crystallographic Angles

Crystallographic angles (\(\alpha, \beta, \gamma\)) are the angles between the vectors that define the unit cell. These angles are important because they shape the geometry of the crystal.

Here's a simplified view of their significance:

• If all angles are 90 degrees, the unit cell takes on a highly symmetrical shape like a cube or a rectangular prism.
• When angles differ from 90 degrees, the unit cell becomes less symmetrical, potentially giving rise to properties such as anisotropy.

Understanding crystallographic angles helps us determine the orientation of planes and directions in a crystal, which is crucial for tasks such as predicting fracture patterns in materials or understanding how crystals will grow. By knowing these angles, scientists and engineers can also predict how a crystal will respond to external forces or conditions.