Problem 14 - A Complete Resource Book in Chemistry for JEE Main

Question

If \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) are unequal and \(\alpha, \beta, \gamma\) are unequal and not \(90^{\circ}\), it represents a (a) triclinic system (b) tetragonal (c) monoclinic (d) orthorhombic

Step-by-Step Solution

Answer

(a) triclinic system

Step 1: Understanding Crystal Systems

A crystal system describes the arrangement of atoms in a crystalline solid. There are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, cubic, and trigonal.

Step 2: Analyzing Given Parameters

The parameters given are that the angles \( \alpha, \beta, \gamma \) are unequal and none is \(90^\circ\). Additionally, the sides \( A, B, C \) are unequal.

Step 3: Identifying Matching Crystal System

The triclinic crystal system is characterized by all three side lengths being unequal and all three angles being unequal without being \(90^\circ\). This matches the given parameters.

Step 4: Verifying Options

Verify against the options: - Triclinic: \( A eq B eq C \), \( \alpha eq \beta eq \gamma eq 90^\circ \) - matches.- Tetragonal: not possible, \( \alpha = \beta = \gamma = 90^\circ \) by definition.- Monoclinic: requires two \(90^\circ\) angles.- Orthorhombic: all angles \(90^\circ\).

Step 5: Conclusion

The crystal system that matches the given conditions (unequal side lengths, unequal angles not \(90^\circ\)) is the triclinic system.

Key Concepts

Triclinic System

In crystallography, the triclinic system is one of the seven crystal systems. The term 'triclinic' refers to a system in which none of the crystal axes are perpendicular, and all of them differ in length. This creates a very flexible and less symmetric structure compared to other systems. The key features of a triclinic system can be summarized as follows:

• None of the angles between the axes are equal to 90 degrees.
• All three axes (A, B, C) are of different lengths.
• Due to its asymmetrical nature, it lacks rotational symmetry.

The triclinic system is the least symmetric among all crystal systems and is often used to describe minerals with complex internal structures. Examples of minerals that form in this system include kyanite and turquoise.

Crystallography

Crystallography is the scientific study of crystal forms and structures. This field focuses on understanding the way atoms are arranged in solid matter. Through crystallography, scientists can determine:

• The internal structure of crystals.
• The geometric arrangement of atoms.
• The relationship between the structural arrangement and the physical properties of the material.

Crystallography uses techniques like X-ray diffraction to analyze the crystal lattice, revealing the precise size, shape, and orientation of the unit cell. It is a crucial field for materials science, chemistry, and biology, providing insights into everything from mineralogy to the design of pharmaceuticals. The study of these structures forms the basis of many innovations, such as new materials with improved properties and functionalities.

Crystal Lattice

A crystal lattice is the ordered arrangement of atoms, ions, or molecules within a crystal. This arrangement repeats itself in a regular pattern throughout the material, defining the crystal's structure. A few important points about crystal lattices are:

• They consist of repeating units called unit cells.
• The unit cell is the smallest portion that, when repeated, creates the entire crystal structure.
• The symmetry and arrangement of the lattice influence the material's physical properties.

Each crystal system has its specific lattice structure, influencing how light and other forms of energy interact with the material. Understanding crystal lattices is vital for predicting and explaining the various physical and chemical properties of materials, as they determine the crystal's hardness, melting point, electrical conductivity, and optical characteristics.