Problem 5 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

In a monoclinic unit cell, the relation of sides and angles are respectively: (a) \(\mathrm{a}=\mathrm{b} \neq \mathrm{c}\) and \(\alpha=\beta=\gamma=90^{\circ}\) (b) \(a \neq b \neq c\) and \(\alpha=\beta=\gamma=90^{\circ}\) (c) \(\mathrm{a} \neq \mathrm{b} \neq \mathrm{c}\) and \(\beta=\gamma=90^{\circ} \neq \alpha\) (d) \(\mathrm{a} \neq \mathrm{b} \neq \mathrm{c}\) and \(\alpha \neq \beta \neq \gamma \neq 90^{\circ}\)

Step-by-Step Solution

Answer

The correct option is (c).

Step 1: Identifying Characteristics of a Monoclinic Unit Cell

In a monoclinic unit cell, the relationship between the edges and angles is such that the lengths of all three axes are unequal. Generally, this can be expressed as \(a eq b eq c\). Furthermore, two angles are right angles (\(90^{\circ}\)), specifically \(\alpha = \gamma = 90^{\circ}\), and one angle, \(\beta\), is not \(90^{\circ}\) and can have a different value. These characteristics are essential to solving the problem.

Step 2: Analyzing Given Options

We are given four choices to match with the monoclinic characteristics:1. (a) \(a = b eq c\) and \(\alpha = \beta = \gamma = 90^{\circ}\)2. (b) \(a eq b eq c\) and \(\alpha = \beta = \gamma = 90^{\circ}\)3. (c) \(a eq b eq c\) and \(\beta = \gamma = 90^{\circ} eq \alpha\)4. (d) \(a eq b eq c\) and \(\alpha eq \beta eq \gamma eq 90^{\circ}\).

Step 3: Matching Characteristics

Comparing each option with the monoclinic characteristics, option (c) matches the monoclinic conditions correctly because it states all sides are unequal \(a eq b eq c\), \(\beta = \gamma = 90^{\circ}\) and \(\alpha\) being not \(90^{\circ}\), fits the condition of a monoclinic unit cell where only one angle is not \(90^{\circ}\).

Key Concepts

Crystallography

Crystallography is the scientific study of crystal structures and properties. It is a significant field in both physical and materials sciences, as it helps us understand the arrangement of atoms within solids. These atomic arrangements result in crystals, which can be categorized into various systems based on their unit cells. The unit cell is the smallest repeating structure that makes up the crystal, and it depicts the symmetry and dimensions of the entire crystalline structure. Each crystal system, such as monoclinic, orthorhombic, or cubic, has unique characteristics defining the relationships of its axes and angles. In crystallography, careful determination of these cell parameters is crucial for understanding material properties, including mechanical strength and growth behavior. Crystallography also uses techniques such as X-ray diffraction to explore the internal lattice structure of crystals. This technique can precisely measure unit cell parameters. Through these measurements, we derive the coordinates of atoms and gain insights into bonding, which is essential for applications like drug design, mineralogy, and material science.

Unit Cell Angles

The angles between the edges of a unit cell are a defining feature that distinguishes different types of crystal systems. These angles are categorized as

• \(\alpha\) (alpha): angle between axes \(b\) and \(c\)
• \(\beta\) (beta): angle between axes \(a\) and \(c\)
• \(\gamma\) (gamma): angle between axes \(a\) and \(b\)

In a monoclinic unit cell, the angles are distinctive in that \(\beta\) is not a right angle, making the monoclinic system unique.Understanding these angles in the context of monoclinic cells involves recognizing that two angles (

• \(\alpha\)
• \(\gamma\)

) equal \(90^{\circ}\), while \(\beta\) deviates from \(90^{\circ}\). This angular deviation affects how we visualize and calculate properties of monoclinic crystals. For example, an altered \(\beta\) angle can impact how light traverses the crystal, potentially influencing optical properties. Additionally, it can affect the packing efficiency and stability, essential for practical applications in materials science.

Crystal Lattice Structures

Crystal lattice structures describe the arrangement of atoms in a crystal. Each structure is composed of unit cells repeated in a 3D space, and these repetitions form the crystal lattice. The arrangement is dictated by symmetry, lattice parameters including side lengths and angles, and atomic positions. Monoclinic crystals are a part of this diverse lattice family known for their unique set of angles. However, they retain two right angles and only one non-right angle, setting them apart from other systems like triclinic, where all angles are non-right. The monoclinic lattice is stable and ideal for substances with unequal forces acting along different axes, leading to its oblique shape and less symmetrical form. A monoclinic structure is often encountered in nature. Minerals like gypsum and some crystalline drugs use this form. The practical importance of studying these structures lies in material properties such as mechanical strength and thermal expansion behavior. These properties can significantly vary based on the orientation of the monoclinic units, hence detailed studies using methods like X-ray crystallography are vital.