Question

In the closest packing of atoms \(A\) (radius : \(r_{a}\) ), the radius of atom \(B\) that can be fitted octahedral void is: (a) \(1.155 r_{a}\) (b) \(0.225 r_{a}\) (c) \(0.414 r_{a}\) (d) \(0.732 r_{a}\)

Short answer

The radius of atom B that can be fitted into an octahedral void is about \(0.414 r_{a}\), which corresponds to option (c).

Step-by-step solution

Understanding the Octahedral Void

In closest packing, octahedral voids are the spaces or gaps between the atoms where another atom can fit. The geometry of an octahedral void is such that it is surrounded by six atoms, forming an octahedron.

Calculating the Radius of the Atom for the Octahedral Void

The distance between the centers of two atoms forming the edge of the octahedron is equal to two radii of atom A, or \(2r_{a}\). The body diagonal of this octahedron will be equal to the distance between the centers of opposite atoms plus the diameter of the atom fitting inside the void. So, \(\sqrt{2} \times 2r_{A} = 2r_{A} + 2r_{B}\).

Solving for the Radius of Atom B

Isolate the radius of atom B, \(r_{B}\), by moving terms and simplifying the equation: \(r_{B} = (\sqrt{2} - 1)r_{A} \approx 0.414r_{A}\).

Key concepts

Closest Packing of Atoms

The arrangement of atoms in a crystal structure is crucial to understanding various physical properties of materials. Closest packing of atoms refers to the efficient arrangement of atoms to occupy the minimum amount of space. There are two main types of closest packing: hexagonal close-packed (hcp) and face-centered cubic (fcc) structures. Both of these packings result in a high packing efficiency of about 74%, meaning that 74% of the space is filled with atoms.

In closest packing, each atom touches several others, creating a stable and dense structure. Imagine a layer of spheres (atoms) laid out so each sphere touches six others. This is the first layer. The second layer of atoms fits into the depressions of the first layer, and so on, creating a three-dimensional lattice. Understanding this concept helps students visualize how materials are structured at an atomic level and is foundational for studying more complex crystallographic topics.

Octahedral Voids

In the context of crystal packing, voids or interstices are as significant as the atoms themselves. Octahedral voids occur in the closest packing of atoms and are gaps where another atom could potentially fit. These voids are surrounded by six atoms arranged at the vertices of an octahedron — a polyhedron with eight faces.

An important point to remember is that not all voids in a crystal are octahedrally shaped; some might be tetrahedral or even larger voids known as interstitial sites. However, octahedral voids offer a balance between size and availability, making them particularly interesting for insertion of different atoms, which can significantly alter the properties of the material. For students aiming to understand more about material science, grasping the concept of octahedral voids is essential.

Geometrical Calculations in Crystallography

Crystallography is the study of the arrangement of atoms within solids. In this discipline, geometrical calculations are frequently used to describe the shapes, angles, and distances between atoms in crystal lattices. When determining the size of an atom that can fit into an octahedral void, for instance, we apply mathematical relationships that arise from the geometry of Platonic solids.

We calculate the size of the atom that can fit into a void based on the distances between the surrounding atoms in the crystal lattice. Using principles from geometry and trigonometry, we can derive an equation that links the radius of the octahedral void, the radii of surrounding atoms, and the radius of the atom that can fit in the void. Understanding the mathematical underpinnings is crucial for students as it allows them to predict and explain the structures and properties of different materials.

Radius Ratio Rule

The radius ratio rule is a guideline used in crystallography to help predict the arrangement of ions in a crystalline compound based on the ratio of their ionic radii. It states that the stability of a crystal is dependent on how well the ions fit into their respective voids — whether those are octahedral, tetrahedral, or another type of void.

The rule essentially provides a way to anticipate the type of void an ion can fit into by comparing the radius of the ion to that of the ions surrounding the void. For example, specific radius ratio ranges correspond to coordination numbers (the number of atoms an ion is in contact with) and the resulting geometrical arrangement. Students will find the radius ratio rule very handy when studying ionic crystals and can use it to predict crystal structures and the types of voids that will be occupied by the ions.