Question

In the closest packing of atoms \(A\) (radius : \(r_{a}\) ), the radius of atom \(B\) that can be fitted into tetrahedral void is: (a) \(0.155 r_{a}\) (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 a tetrahedral void in the closest packing of atoms A is (c) \(0.414 r_{a}\).

Step-by-step solution

Understanding the closest packing and tetrahedral voids

In the closest packing of atoms, the atoms are arranged in a way that they occupy the maximum space possible. This arrangement leaves behind some voids, commonly known as interstitial sites. A tetrahedral void is the space between three atoms in the same layer and one atom in the adjacent layer forming a tetrahedron. The radius of an atom that can fit into a tetrahedral void can be calculated using geometric relationships.

Calculating the radius of the tetrahedral void

In a tetrahedral void, the center-to-center distance between the atoms forming the tetrahedron equals the edge of the tetrahedron. This can also be expressed as the sum of the radii of the large atom and the small atom fitted in the void, plus the radius of the large atom, as they touch each other: 2*rA + rB = edge of tetrahedron. For a regular tetrahedron, the edge length is related to the radius of the spheres in close packing by the equation edge = \(\sqrt{2} * 2 * r_{a}\).

Deriving the formula for the radius of atom B

Now, we will use the Pythagorean theorem. Since the center of the small atom (B) is at the center of the tetrahedron, we have a right-angled triangle with the edge of the tetrahedron as the hypotenuse and a radius of large atom A plus radius of small atom B as one of the sides. Thus, the relation is \(\sqrt{2} * r_{a}^2 = (r_{a} + r_{b})^2\). By solving the equation, we can express rB as a fraction of rA.

Solving the equation

The equation \(\sqrt{2} * r_{a}^2 = (r_{a} + r_{b})^2\) can be simplified to \(2 * r_{a}^2 = r_{a}^2 + 2 * r_{a} * r_{b} + r_{b}^2\). After rearranging terms and solving for \(r_{b}\), we get \(r_{b} = \frac{\sqrt{2} - 1}{2} * r_{a}\). Upon calculating, this gives us \(r_{b} = 0.414 * r_{a}\).

Key concepts

Closest Packing in Crystals

Closest packing in crystals is a way to efficiently organize atoms or molecules in a structure to occupy the maximum available space. This arrangement is found in many crystal structures where atoms are packed as closely together as possible, minimizing any wasted space. There are two primary types of closest packing: hexagonal closest packing (hcp) and cubic closest packing (ccp), which is also known as face-centered cubic (fcc) packing. In both of these arrangements, each atom is surrounded by 12 other atoms, creating a dense structure.

Understanding closest packing is fundamental when studying the properties of materials, as it dictates how atoms interact and bond with each other. The denser the packing, the stronger the resulting material tends to be due to closer atomic interactions. This concept also relates to how defects, such as interstitial sites, can occur within the otherwise densely packed structures.

Interstitial Sites

Interstitial sites are the small spaces or 'voids' that are left between atoms in a closely packed crystal structure. They are where atoms or ions smaller than the host atoms can fit into. These sites are crucial because they contribute to the properties of materials, such as hardness and elasticity. In the context of solving for the tetrahedral void, interstitial sites must be understood in order to know where additional atoms or ions can be accommodated in the crystal lattice.

There are two common types of interstitial sites: octahedral and tetrahedral voids. Tetrahedral voids are smaller and are formed by four atoms arranged in a tetrahedron. An atom occupying this void touches all four of these atoms. In contrast, octahedral voids are larger and are formed by six atoms. These geometric considerations are essential when calculating the size of an atom that can fit into these interstitial spaces.

Geometric Relationships in Crystal Structures

The geometric relationships in crystal structures determine how atoms are spatially related to each other and how they fit together. Understanding these geometric relationships helps us calculate various parameters of the crystal, such as the size of voids and the distance between atoms. For example, in the closest packed structures, the tetrahedral void is quantified through its geometric relationship to the surrounding atoms.

The solution to the tetrahedral void radius calculation employs the geometry of a regular tetrahedron. By using the Pythagorean theorem in conjunction with the known edge lengths of a tetrahedron related to atomic radii, we can calculate unknown quantities like the radius of a smaller atom that would fit into the interstitial tetrahedral void. Such calculations are not just academic; they're necessary for understanding the fundamental attributes of materials and designing new materials with specific characteristics.