Question

(i) What is the formula of a compound based on a cubic closepacked structure in which one quarter of the tetrahedral holes are filled? What are the coordination numbers of both the atoms and the filled holes? (ii) What is the formula of a compound whose structure is based on hexagonal close packing when one quarter of the octahedral holes are filled? What are the coordination numbers of both the atoms and the filled holes in this case?

Short answer

(i) The formula of a compound based on a cubic close-packed structure in which one quarter of the tetrahedral holes are filled is AB\(_{4}\). The coordination number of the atoms is 12 and the filled tetrahedral holes have a coordination number of 4. (ii) The formula of a compound whose structure is based on hexagonal close packing when one quarter of the octahedral holes are filled is also AB\(_{4}\). In this case, the atoms have a coordination number of 12 and the filled octahedral holes have a coordination number of 6.

Step-by-step solution

Find the number of atoms and holes in close-packed structures.

In a close-packed structure, there are two types of holes: tetrahedral holes and octahedral holes. In a cubic close-packed (ccp) structure, the number of atoms is equal to the number of tetrahedral holes. It also has half the number of octahedral holes. In a hexagonal close-packed (hcp) structure, it has the same number of atoms and tetrahedral holes, and the same number of octahedral holes as there are atoms. (i)

Determine the number of atoms and filled tetrahedral holes in the ccp structure.

For a cubic close-packed structure, let's say there are A atoms. Since there are a total of A tetrahedral holes in ccp structure, and one quarter of them are filled, there will be \(\frac{1}{4}A\) filled tetrahedral holes.

Write the formula of the compound.

Since we have A atoms and \(\frac{1}{4}A\) filled tetrahedral holes, the formula of the compound will be A\(_{4}\)B, which can also be written as AB\(_{4}\).

Determine the coordination numbers.

In a close-packed structure, the coordination number of an atom is the same as that of a filled hole. For the cubic close-packed structure, the coordination number of a filled tetrahedral hole is 4 and for the surrounding atoms is 12. Thus, the formula of the compound in the ccp structure is AB\(_{4}\), the atoms have a coordination number of 12, and the filled tetrahedral holes have a coordination number of 4. (ii)

Determine the number of atoms and filled octahedral holes in the hcp structure.

For a hexagonal close-packed structure, let's say there are A atoms. Since there are a total of A octahedral holes in hcp structure, and one quarter of them are filled, there will be \(\frac{1}{4}A\) filled octahedral holes.

Write the formula of the compound.

Since we have A atoms and \(\frac{1}{4}A\) filled octahedral holes, the formula of the compound will be A\(_{4}\)B, which can also be written as AB\(_{4}\).

Determine the coordination numbers.

In a close-packed structure, the coordination number of an atom is the same as that of a filled hole. For the hexagonal close-packed structure, the coordination number of a filled octahedral hole is 6 and for the surrounding atoms is 12. Thus, the formula of the compound in the hcp structure is AB\(_{4}\), the atoms have a coordination number of 12, and the filled octahedral holes have a coordination number of 6.

Key concepts

Cubic Close-Packed (ccp) Structure

The cubic close-packed (ccp) structure, also known as face-centered cubic (fcc) structure, is one of the two main types of close-packing of equal spheres. In this structure, each layer of atoms is arranged in a way that the atoms in the next layer nestle in the crevices of the previous layer, leading to a highly efficient packing.

When visualizing this structure, imagine a layer of spheres with another layer stacking over it such that the second layer's spheres rest in the hollows formed by the first layer. This is then continued with more layers, following the same pattern. The ccp structure results in a unit cell with an atom at each corner and one at the center of each face.

The ccp structure is commonly observed in metals such as copper, aluminum, and silver. This arrangement is preferred due to its high packing efficiency and density, which enables stronger metallic bonding.

Hexagonal Close-Packed (hcp) Structure

In contrast to the ccp structure, the hexagonal close-packed (hcp) structure is the second primary form of packing equal spheres. Each atom in hcp still has 12 closest neighbors, maintaining a high level of packing efficiency, but it differs in the stacking sequence.

The hcp structure consists of layers where each sphere is surrounded by six others in the same layer, three in the layer above, and three in the layer below, resembling a hexagon. The unit cell of an hcp structure looks like a hexagonal prism with atoms at the corners and center of the two hexagonal bases, and three atoms arranged in a triangle in the middle of the cell.

This structure is prevalent in metals such as magnesium, zinc, and titanium. The choice between hcp and ccp structures in metals can affect their physical properties, including malleability and ductility.

Coordination Number

The coordination number in a crystal lattice refers to the number of adjacent atoms surrounding a central atom. It's a critical parameter indicating how tightly the atoms are packed together, which in turn influences a material's physical and chemical properties.

In a ccp or fcc lattice, the coordination number is 12, meaning every atom is immediately surrounded by 12 neighbors. This high coordination number is a result of the efficient packing in these structures and lends the material greater strength and metallic character. In the hcp lattice, the coordination number is likewise 12, despite the difference in packing arrangement from ccp.

The coordination number is also applicable to the holes within the structures, such as tetrahedral or octahedral holes, where it indicates the number of atoms that the holes touch.

Tetrahedral and Octahedral Holes

Understanding the concept of tetrahedral and octahedral holes is important in close-packed structures. These holes are the interstices or gaps left between atoms packed together.

A tetrahedral hole is the space where four atoms in close-packing touch. It resembles a tetrahedron, with the four atoms located at the corners. In both ccp and hcp structures, the coordination number for tetrahedral holes is 4 because each hole is surrounded by four surrounding atoms.

An octahedral hole, on the other hand, is surrounded by six atoms, forming an octahedron. The coordination number for octahedral holes in both ccp and hcp is 6. The ratio of the number of tetrahedral holes to the number of atoms in both ccp and hcp is 2:1, while the ratio of octahedral holes to atoms in ccp is 1:1 and in hcp it is also 1:1.

These holes are vital for understanding ionic compounds, where smaller ions (typically cations) occupy these interstitial sites amidst the larger ions (typically anions) in the close-packed arrangement.