Problem 41 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

In face centred cubic \((f c)\) crystal lattice, edge length is \(400 \mathrm{pm}\). Find the diameter of greatest sphere which can be fit into the interstitial void without distortion of lattice.

Step-by-Step Solution

Answer

The diameter of the greatest sphere is approximately 282.84 pm.

Step 1: Understand the Face-Centered Cubic (FCC) Lattice

In an FCC lattice, atoms or ions are located at each of the corners and the centers of all the faces of the cube. This formation creates octahedral and tetrahedral voids within the lattice where additional atoms or ions can fit.

Step 2: Identify the Largest Interstitial Void

In an FCC lattice, the largest void is the octahedral void. It is larger than the tetrahedral void and can accommodate the largest sphere without distorting the lattice.

Step 3: Recall Octahedral Void Size in FCC

For an FCC lattice with edge length \(a\), the radius of an octahedral void is given by \(r = \frac{a}{2\sqrt{2}}\). This is because the octahedral voids are located at the center of the edges as well as at the body center of the cube.

Step 4: Calculate the Radius of the Largest Sphere

Given the edge length \(a = 400\ pm\), substitute into the formula for the octahedral void: \(r = \frac{400}{2\sqrt{2}}\).Calculate \(\sqrt{2} \approx 1.414\), so: \(r = \frac{400}{2 \times 1.414} = \frac{400}{2.828}\). Calculating further gives \(r \approx 141.42\ pm\).

Step 5: Determine the Diameter of the Sphere

The diameter of the sphere is twice the radius since \(\text{Diameter} = 2 \times r\). Therefore, the diameter of the largest sphere is \(2 \times 141.42 = 282.84\ pm\).

Key Concepts

Interstitial Voids

Interstitial voids are fascinating components of crystal structures, acting as the "gaps" in which smaller atoms or ions can reside. Imagine a crystal lattice as a structure made up of stacked spheres, like a pile of marbles. Between these marbles, spaces are left unfilled. These spaces are what we call interstitial voids.

These voids play a significant role in determining the properties of the crystal. They can accommodate foreign atoms, which sometimes significantly alter the material's characteristics. In a face-centered cubic (FCC) lattice, you can find two primary types of interstitial voids: octahedral and tetrahedral voids. Both have specific roles and differ in size, influencing how and where atoms fit into the lattice.

Octahedral Voids

Octahedral voids are a specific type of interstitial void found within crystal structures like the face-centered cubic (FCC) lattice. These voids are larger than tetrahedral voids, making them suitable for accommodating larger atoms or ions. In an FCC lattice, octahedral voids are located at the center of the cube's edges and at the body center of the cube itself.

These voids are formed when six spheres (atoms or ions) touch to create an octahedron. This arrangement leaves enough space at the center for additional atoms or ions to fit snugly. The radius of these octahedral voids can be calculated using the formula:

\[ r = \frac{a}{2\sqrt{2}} \]

where \(a\) represents the edge length of the cube. This formula helps us determine the maximum size of an atom that can occupy the octahedral void without disturbing the overall structure.

Sphere Diameter Calculation

Calculating the diameter of a sphere that can fit into an interstitial void, specifically an octahedral void, is an essential step in understanding how atoms fit within a crystal lattice. Given an edge length of 400 pm for a face-centered cubic lattice, you can calculate the radius of the largest sphere fitting into an octahedral void using the formula:

\[ r = \frac{400}{2\sqrt{2}} \]

By simplifying this, we approximate \(\sqrt{2} \approx 1.414\), resulting in:

\[ r \approx \frac{400}{2 \times 1.414} = \frac{400}{2.828} \approx 141.42 \text{ pm} \]

To find the diameter, simply double the radius:

\[ \text{Diameter} = 2 \times 141.42 = 282.84 \text{ pm} \]

This calculation shows the size of the largest sphere that can be accommodated in the void, essential for understanding crystal packing.

Edge Length in Crystals

Edge length is a fundamental measurement in the study of crystal structures. It defines the size of the repeating unit cell within the crystal lattice, a building block that, when repeated, forms the entire structure. In a face-centered cubic (FCC) lattice, this edge length forms the basis for calculating the dimensions of interstitial voids and the arrangement of atoms.

For FCC lattices, where atoms are located at each corner and the centers of each face of the cube, the edge length helps determine how tightly packed the atoms are. This affects not only the stability of the crystal but also its physical properties, such as density and melting point.

By knowing the edge length, you can use it in various equations to compute other parameters like the radius of an octahedral void (as mentioned earlier) or derive the lattice's geometric properties. Understanding edge lengths unlocks deeper insights into how crystals function and interact with external elements.