Question

For a certain crystal, the unit cell axial lengths are found to be \(a=5.62 \AA, b=7.41\) \(\AA\) and \(c=10.13 \AA\). The three coordinate axes are mutually perpendicular. The crystal system to which the crystal belongs is (a) tetragonal (b) orthorhombic (c) monoclinic (d) cubic

Short answer

The crystal belongs to the orthorhombic crystal system.

Step-by-step solution

Analyze Axial Lengths

Examine the given axial lengths, which are denoted as: a = 5.62\(\AA\), b = 7.41\(\AA\), and c = 10.13\(\AA\). These represent the lengths of the sides of the unit cell.

Determine Crystal System

Based on crystallography, the crystal system is determined by the axial lengths and the angles between them. For this crystal, since the three axes are mutually perpendicular, we focus on the lengths: if all three lengths are equal, the system is cubic; if two are equal, it's tetragonal; and if all three are unequal, it is orthorhombic.

Match Axial Lengths to Crystal System

Compare the three axial lengths. Since a != b != c, and since the problem states that the three axes are mutually perpendicular, the crystal system must be orthorhombic.

Key concepts

Physical Chemistry and Crystal Systems

Physical chemistry is a branch of chemistry that deals with the physical properties and behaviors of materials at an atomic and molecular level. One important aspect of this field is understanding crystal systems, which are categories used to classify crystals based on their symmetry and lattice parameters such as axial lengths and interaxial angles.

Crystals are solid materials whose atoms are arranged in a highly ordered, repeating pattern that extends in all three spatial dimensions. These patterns are defined by the crystal lattice, which serves as the fundamental geometric structure for describing crystal symmetry. A variety of properties of materials, such as their physical strength, melting point, and electronic properties, are influenced by their crystal structure.

When analyzing crystal systems, a key concept to grasp is that the arrangement of atoms has a direct impact on the material’s properties. For instance, in the exercise provided, understanding that the axial lengths are all different and the axes are perpendicular helps to determine the crystal system, which in turn could lead to insights on the material's anisotropic behaviors—where properties vary based on directional orientation.

Unit Cell: The Building Block of Crystalline Solids

The concept of a 'unit cell' is central to crystallography and to understanding the structure of crystalline solids. A unit cell is the smallest repeating unit of a crystal lattice that, when stacked in three dimensions, reproduces the entirety of the crystal's structure.

Each unit cell is defined by its lattice parameters: the lengths of its sides, known as axial lengths (denoted by a, b, and c), and the angles between these sides. The exercise showcases a scenario with unique axial lengths, which are critical in determining the type of unit cell, hence the crystal system classification.

The complexities and variations in crystalline structures stem from how these unit cells are shaped and how they connect with one another. If all sides are equal and all angles are 90 degrees, we have the simplest form: the cubic system. As shown in the provided exercise, when no two sides are equal in length, but all angles remain at 90 degrees, we identify the system as orthorhombic—a category with distinct physical properties due to its unique geometric arrangements.

Crystallography: Interpreting the Geometry of Solids

Crystallography is the scientific study of crystal structures and their properties. It involves measuring and understanding the atomic arrangement within a crystalline material, interpreting how this arrangement affects the material's properties, and classifying the material into a crystal system.

In the context of the exercise, we see how crystallography's principles are applied to determine the crystal system of a given crystalline material. By looking at the axial lengths and the angles between these axes—which are all perpendicular in this case—scientists can classify the material's structure. This classification into one of the seven crystal systems, such as cubic, tetragonal, orthorhombic, etc., is crucial for understanding how the material will behave physically and chemically.

The detailed analysis and categorization process require precision, as each system has specific trait implications on the crystal’s reactivity, durability, and various other chemical and physical characteristics. Thus, correctly identifying the crystal system paves the way for potential applications of the material and for predicting how it will respond in different conditions.