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

Question

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

Step-by-Step Solution

Answer

Option (a) \( r_{Cs^+} + r_{Cl^-} = \frac{\sqrt{3}}{2}a \) is correct.

Step 1: Identify Crystal Structure Type

CsCl crystallizes in a body-centered cubic (BCC) structure. In this type of lattice, the cesium ion is at the center of the cube and chloride ions are at the corners. The central atom touches the corner atoms along the body diagonal of the cube.

Step 2: Determine Body Diagonal Length

In a BCC lattice, the body diagonal of the cube is equal to the sum of twice the radius of the cesium ion and twice the radius of the chloride ion. This relationship is due to the fact that the ions are touching along the body diagonal: \[ \mathrm{Body \ Diagonal} = a\sqrt{3} \]

Step 3: Relate Body Diagonal to Radii Sum

For the cesium and chloride ions touching along the body diagonal, the relationship can be expressed as:\[ 2r_{Cs^+} + 2r_{Cl^-} = a\sqrt{3} \]Rearranging, we get:\[ r_{Cs^+} + r_{Cl^-} = \frac{a\sqrt{3}}{2} \]

Step 4: Choose the Correct Option

Comparing the derived expression \[ r_{Cs^+} + r_{Cl^-} = \frac{a\sqrt{3}}{2} \]with the options given in the problem, it matches with option (a) which states:\[ r_{Cs^+} + r_{Cl^-} = \frac{\sqrt{3}}{2}a \]

Key Concepts

Body-Centered Cubic Lattice

A body-centered cubic (BCC) lattice is a type of crystal structure that is common in many metallic and ionic compounds. In this structure, there is an atom located at the center of the cube in addition to the atoms located at each of the eight corners of the cube.
In the case of cesium chloride (CsCl), the Cs⁺ ion occupies the center of the cube, while the Cl⁻ ions are positioned at the corners. This formation allows for a dense and stable configuration as the central atom interacts with all corner atoms.
The BCC lattice is pivotal in understanding how the spherical ions pack together in a three-dimensional space to form solids. This structure is different from a more straightforward cubic lattice, which only has atoms at its corners with none in the center. The presence of the central atom alters both packing density and symmetry within the crystal lattice.

Ionic Radii

In a crystal lattice, the concept of ionic radii is essential to determine how closely the ions pack together. The ionic radius refers to the size of an ion in a crystal lattice and influences how ions of different sizes interact in solid form.
For CsCl, cesium ions (Cs⁺) and chloride ions (Cl⁻) come together to form the crystal. The interplay of their ionic radii dictates the entire structure and stability of the compound. The Cs⁺ ions are relatively large compared to many other ions, and the Cl⁻ ions fit well with the Cs⁺ ions to maximize the stability of the structure.
Ionic radii are critical for predicting the behavior of the ions in the lattice, as the balance between the sizes of the cations and anions determines the overall packing efficiency. Larger ions take up more space and might result in less dense packing, whereas smaller ions might fit closely together, increasing the crystal's density.

Body Diagonal in Crystals

The body diagonal in a body-centered cubic (BCC) lattice plays a crucial role in determining the relationships between the lattice parameters and ionic radii. The body diagonal runs across the cube from one corner, passing through the central atom, and reaching the opposite corner.
In CsCl, the body diagonal is crucial because it is the alignment along which the central Cs⁺ ion and the corner Cl⁻ ions are in direct contact. This alignment is defined mathematically by the equation: \[ \text{Body Diagonal} = a\sqrt{3} \]where \(a\) is the edge length of the cube.
This equation is significant because it helps relate the macroscopic (observable) properties of the crystal to the microscopic (atomic-scale) distances. Understanding the length of the body diagonal allows scientists to calculate the sum of the ionic radii, ensuring that these ions sit snugly along the diagonal line, as seen in how the radii add up to match the length of the body diagonal.
Observing these relationships in the context of the CsCl lattice can help understand the detailed geometry of crystal structures and predict how changes in the atomic arrangement might affect material properties.