Problem 116 - A Complete Resource Book in Chemistry for JEE Main

Question

Identify the correct statements. (a) The size of octahedral site is given by \(r=0.414 R\) where \(\mathrm{r}\) is the radius of octahedral hole and \(\mathrm{R}\) is the radius of spheres enclosing it. (b) In CCP or HCP arrangement, there are two octahedral sites per closely packed sphere. (c) The centres of octahedral sites lie at the mid point of the sides and at the body centre of face centred cubic unit cell. (d) The radius of tetrahedral hole (r) and that of spheres (R) forming it are related as \(\mathrm{r}=0.155 \mathrm{R}\).

Step-by-Step Solution

Answer

All the statements (a), (b), (c), and (d) are correct.

Step 1: Analyzing Statement (a)

The statement claims that the size of an octahedral site is given by \(r = 0.414 R\), where \(\mathrm{r}\) is the radius of the octahedral hole, and \(\mathrm{R}\) is the radius of the enclosing spheres. This is a known formula from crystallography, which is correct.

Step 2: Evaluating Statement (b)

This statement refers to the number of octahedral sites in a cubic close-packed (CCP) or hexagonal close-packed (HCP) structure. It is true that in these arrangements, there are two octahedral sites per sphere.

Step 3: Verifying Statement (c)

The statement mentions the locations of octahedral sites in a face-centered cubic (FCC) unit cell. It is accurate that the centers of octahedral sites are located at the midpoint of the edges and at the body center of the FCC unit cell.

Step 4: Confirming Statement (d)

The statement provides the relationship between the radius of a tetrahedral hole (\(r\)) and the radius of forming spheres (\(R\)) as \(\mathrm{r} = 0.155 \mathrm{R}\). This is a standard relationship and is correct.

Key Concepts

Octahedral sites

In crystallography, octahedral sites are cavities formed between the spheres that make up the lattice structures in crystals, like CCP and HCP. These sites are called "octahedral" because they are shaped like an octahedron, a polyhedron with eight faces. The octahedral sites are crucial because they can accommodate smaller atoms or ions, such as in metallic crystals. This affects the properties of the material, such as its density and stability.

The radius of an octahedral site is smaller than the radius of the spheres that make up the lattice. The formula for the size of an octahedral site is given by:

• the octahedral hole radius, \\(r = 0.414 R\), where \\(R\) is the radius of the spheres enclosing it.

This formula helps crystallographers predict how atoms can fit into these sites, ensuring that the crystal structure remains stable. Understanding these sites is fundamental in materials science, as they influence how a material behaves at the atomic level.

Cubic close-packed (CCP)

The cubic close-packed (CCP) arrangement is one of the most efficient ways to pack spheres to minimize empty space. In this structure, the spheres are packed together in a cubic lattice, with each sphere surrounded by twelve others, maximizing packing efficiency. CCP is also known as FCC (face-centered cubic) because of its structural similarity.

A significant aspect of CCP is the availability of interstitial sites:

• Octahedral sites: CCP contains octahedral sites which are often occupied by smaller atoms. There are two octahedral sites for each sphere.
• Tetrahedral sites: Each sphere has access to tetrahedral sites, which are smaller than octahedral sites due to their geometric arrangement.

This packing style is commonly found in metals like copper, lead, and nickel, providing them with unique properties like high density and strength.

Face-centered cubic (FCC)

Face-centered cubic (FCC) is a type of crystal lattice structure with high packing efficiency. It is synonymous with CCP because of its similar arrangement of spheres. In FCC structures, each face of the cube has an additional atom located at its center, on top of the atom at each corner of the cube. This configuration ensures that each sphere is in contact with twelve others.

The placement of octahedral sites in FCC is essential for understanding the distribution of atoms within the crystal:

• The centers of octahedral sites are located:
  • at the midpoint of each edge of the unit cell,
  • at the center of the unit cell itself.

This thorough integration of atoms makes FCC an ideal structure for many metals and compounds, enhancing their mechanical properties. The FCC arrangement is particularly significant in metallurgy, affecting how metals deform and conduct heat and electricity.

Hexagonal close-packed (HCP)

The hexagonal close-packed (HCP) structure represents another efficient way to pack atoms in a crystal, maximizing space usage. Unlike the cubic arrangements, the HCP structure forms a tessellated pattern, where the spheres pack in hexagonal layers. This design allows each sphere to touch twelve surrounding spheres just like in FCC. Yet, HCP differs by having a different type of symmetry and stacking pattern.

HCP structures feature:

• An A-B-A-B stacking sequence, where alternate layers align with each other.
• Two octahedral sites per sphere, providing spaces for different atoms or ions within the structure.

Metals like magnesium and zinc often crystallize in the HCP form. This is important in shaping the properties of the metal, including durability and density. The understanding of HCP helps in material design, aiding in the development of stronger and lighter metallic systems.