Problem 148 - A Complete Resource Book in Chemistry for JEE Main

Question

CsCl crystallizes in body centred cubic lattice. if ' \(\mathrm{a}\) ' is its edge length then which of the following expression is correct: (a) \(\mathrm{r}_{\mathrm{C}^{+}}+\mathrm{r}_{\mathrm{CT}}=\frac{\sqrt{3}}{2} \mathrm{a}\) (b) \(\mathrm{r}_{\mathrm{Cs}^{*}}+\mathrm{r}_{\mathrm{c}-}=\sqrt{3} \mathrm{a}\) (c) \(\mathrm{r}_{\mathrm{Cs}^{+}}+\mathrm{r}_{\mathrm{Cr}}=3 \mathrm{a}\) (d) \(\mathrm{r}_{\mathrm{cs}^{*}}+\mathrm{r}_{\mathrm{Cr}^{-}}=\frac{3 \mathrm{a}}{2}\)

Step-by-Step Solution

Answer

The correct expression is option (a): \( r_{C^+}+r_{Cl^-}=\frac{\sqrt{3}}{2} a \).

Step 1: Understanding the Crystal Structure

Cesium Chloride (CsCl) crystallizes in a body-centered cubic (bcc) lattice. In this structure, a cesium ion (Cs⁺) is located at the center of the cube, and a chloride ion (Cl⁻) is located at each corner of the cube.

Step 2: Identifying the Relation

In a body-centered cubic lattice for CsCl, each Cs⁺ ion is surrounded by Cl⁻ ions. The ions touch along the body diagonal of the cube. Therefore, the diagonal of the cube is equal to twice the sum of the radii of Cs⁺ and Cl⁻, i.e., \[ \text{Body Diagonal} = 2(r_{Cs^+} + r_{Cl^-}) \]

Step 3: Calculating the Body Diagonal

The body diagonal of a cube can be calculated using Pythagoras' theorem. For a cubic unit cell with edge length 'a', the body diagonal is given by:\[ \text{Body Diagonal} = \sqrt{3}a \]

Step 4: Setting Up the Equation

Equating the expression from the structure with the calculated body diagonal gives:\[ 2(r_{Cs^+} + r_{Cl^-}) = \sqrt{3}a \]

Step 5: Solving for the Radius Sum

Divide both sides of the equation by 2 to find the relationship for the sum of the radii:\[ r_{Cs^+} + r_{Cl^-} = \frac{\sqrt{3}a}{2} \]

Step 6: Matching the Correct Option

Compare the derived equation \[ r_{Cs^+} + r_{Cl^-} = \frac{\sqrt{3}a}{2} \] with the given options, and identify that option (a) \( r_{C^+}+r_{Cl^-}=\frac{\sqrt{3}}{2} a \) is correctly representing the relationship.

Key Concepts

Body-Centered Cubic Lattice

In solid state chemistry, the body-centered cubic (bcc) lattice is a common crystal structure that is crucial to understand. This structure consists of a cube with one atom at each corner and one atom in the center of the cube. The atom in the center touches the corner atoms along the body diagonal. This type of arrangement is frequently found in metals and some ionic compounds, like cesium chloride.

• The bcc unit cell has atoms at the 8 corners of the cube.
• The central positioning of one atom allows it to contact others along the diagonal.
• This setup results in a very efficient packing within the crystal lattice, optimizing space usage.

Understanding this arrangement is vital for analyzing how ions are packed in a solid and how it affects properties like electrical conductivity and density.

Ion Radii in Crystals

The concept of ion radii is instrumental in understanding crystal structures. In a crystal lattice, ions are considered to have nearly spherical shapes, which helps in simplifying their arrangements within the structure. The radii of ions determine how closely they can pack together within the crystal lattice.

• In ionic compounds, such as CsCl, each type of ion will have a specific radius depending on its charge and size.
• The radii are critical in predicting how ions will touch, especially along specific lines or planes in the crystal.
• In a bcc lattice for CsCl, the ions are in contact along the body diagonal, so their combined radii are used in these calculations.

Grasping the concept of ion radii is essential for solving problems related to the properties of crystalline materials, as it directly influences the packing and overall structure stability.

Cesium Chloride Structure

The cesium chloride (CsCl) structure offers a fascinating look into how different ions can arrange themselves in a crystal. This structure falls under the body-centered cubic category, but with a slight twist. Here, the Cs⁺ ions occupy the center of the cube, while Cl⁻ ions are positioned at each of the eight corners.

• In the CsCl structure, each Cs⁺ ion is surrounded by eight Cl⁻ ions, which form a cubic arrangement.
• This positioning allows Cs⁺ to effectively interact with multiple Cl⁻ ions via the body diagonal.
• The structure is determined by the need to balance electrostatic attractions between the positive and negative ions, maximizing stability.

The knowledge of how these ions form a bcc lattice helps in deciphering the chemical and physical attributes of the compound, such as its bonding characteristics and spatial configuration.