Question

If \(\mathrm{R}\) is the radius of the sphere in the close packed arrangement and \(\mathrm{r}\) is the radius of the tetrahedral void, then (a) \(\mathrm{R}=0.414 \mathrm{r}\) (b) \(\mathrm{r}=0.224 \mathrm{R}\) (c) \(\mathrm{R}=0.225 \mathrm{r}\) (d) \(\mathrm{r}=0.414 \mathrm{R}\)

Short answer

The correct answer is (b) \( r = 0.224 R \).

Step-by-step solution

Understanding Tetrahedral Voids

In a close packed arrangement, the radius of a tetrahedral void \( r \) is the smaller sphere's radius that can fit into the void formed by four spheres of radius \( R \). The key is to find the mathematical relationship between \( r \) and \( R \).

Geometry of Tetrahedral Voids

In a tetrahedral void, the center of the spheres forms a regular tetrahedron. The distance between the centers of two touching spheres, \( R \), is equal to \( 2R \). The height of the tetrahedron from the base to the apex, within which the void lies, is \( \sqrt{\frac{2}{3}}(2R/3) \).

Relating Radii of Sphere and Void

The radius of the tetrahedral void (\( r \)) and the sphere's radius (\( R \)) are geometrically related. The distance from a center of the void sphere to the center of the corner spheres in the tetrahedron follows the Pythagorean theorem in three dimensions:

Calculate \( r \) Using a Formula

Using three-dimensional Pythagorean relation specific to tetrahedral geometry: \( r = \left(\frac{\sqrt{3}}{2} - \frac{1}{2}\right) R = 0.224 R \).

Verify the Correct Answer

The relation derived shows \( r = 0.224 R \), matching option (b). This confirms that the radius of the void is approximately 22.4% of the larger sphere's radius. Thus, the correct answer to the problem is option (b): \( r = 0.224 R \).

Key concepts

Tetrahedral Voids

In solid state chemistry, tetrahedral voids are small empty spaces present in close-packed structures. These voids are named "tetrahedral" because the center of the void and the centers of the four surrounding spheres form the vertices of a tetrahedron. The void is the space available within this tetrahedral shape. To understand how a sphere fits within this void, imagine four larger spheres touching each other. The small sphere, or void sphere, can nestle into this small open space.

• The radius of the void sphere, denoted as \( r \), is significantly smaller than the radius of the larger spheres, \( R \).
• Theoretical calculations show that the size of the tetrahedral void is determined by three-dimensional geometrical relationships.
• Thus, the radius of the tetrahedral void is about 22.4% of the larger spheres' radius, formulated as \( r = 0.224 R \).

These voids are crucial in materials science as they determine how smaller atoms and ions fit and move within a crystal structure.

Close-Packed Structures

Close-packed structures in solid state chemistry refer to arrangements where spheres (atoms, ions, or molecules) are packed together as closely as possible. There are two primary types of close packing: hexagonal close packing (HCP) and cubic close packing (CCP), also known as face-centered cubic (FCC). Both of these arrangements allow for maximization of space usage and stability.

• In these arrangements, each sphere is surrounded by 12 other spheres, optimizing their packing efficiency.
• HCP has layers arranged in an ABAB pattern, while CCP/FCC follows an ABCABC pattern.
• These patterns directly influence the formation of voids, such as tetrahedral voids, which occur between certain layers.

The concept of close packing is fundamental in understanding how materials are structured at the atomic level and why certain materials have specific physical and mechanical properties.

Geometric Relations in Crystals

Geometric relations in crystals are crucial to comprehending how solid materials are constructed. In crystal structures, geometry dictates how particles are aligned and how voids and spaces are formed within the structure. The relationships rely on geometric formulas to express distances, angles, and areas.

• In a simply packed structure, the distance between two touching spheres is twice their radius, so \( 2R \) if \( R \) is the sphere radius.
• The voids, like tetrahedral voids, form based on these distances and specific arrangements, creating a three-dimensional space within the layers.
• The geometric relationship within a tetrahedral void can be derived using the Pythagorean theorem in three dimensions, relating radius \( R \) and the smaller void radius \( r \).

Understanding these geometric relations helps in predicting how a crystal might behave or interact under certain conditions, which is essential for technological applications like the creation of new materials and studying their properties.