Question

In a closed packed array of \(\mathrm{N}\) spheres, the number of tetrahedral holes are (a) \(\mathrm{N}\) (b) \(2 \mathrm{~N}\) (c) \(\frac{\mathrm{N}}{2}\) (d \(\frac{\mathrm{N}}{4}\)

Short answer

The number of tetrahedral holes is option (b) 2N.

Step-by-step solution

Understanding Closed Packed Arrangement

In a closed packed array of spheres, the spheres are arranged such that they touch each other. This arrangement creates different types of voids or spaces, including tetrahedral and octahedral holes. Understanding the relationship between spheres and these voids is essential for finding the correct answer.

Identifying Tetrahedral Holes

In a close-packed structure, each sphere is in contact with the maximum number of adjacent spheres. For each sphere added to the structure, two tetrahedral holes are formed. This is a crucial detail that connects the number of spheres and holes.

Calculating the Number of Tetrahedral Holes

Given that each sphere contributes two tetrahedral holes, the total number of tetrahedral holes in the structure is twice the number of spheres. Therefore, if there are N spheres, there will be 2N tetrahedral holes.

Confirm the Logical Conclusion

Using the information that two tetrahedral holes form for every sphere, we can confirm that option (b) 2N matches our calculation for the number of tetrahedral holes. Other options do not fit this logic.

Key concepts

Close-Packed Structure

The close-packed structure is a highly efficient way of arranging identical spheres in a way that maximizes their packing density. When spheres are stacked together in such a manner, they form a compact structure where each sphere is in direct contact with its neighbors. This type of arrangement is pivotal in materials science as it helps in understanding the properties of metals and crystalline solids.

There are two types of close-packed structures usually discussed: **Hexagonal Close-Packed (HCP)** and **Cubic Close-Packed (CCP)**, which is also known as Face-Centered Cubic (FCC). In both of these structures, the spheres are arranged layer by layer such that each sphere in a layer rests in a depression of the layer beneath it. Such close packing leads to different types of voids that are significant for material properties.

Spheres Arrangement

In understanding how spheres are arranged in a close-packed structure, it becomes essential to delve into the sequential stacking of these spheres. The arrangement can be visualized through layers of spheres, typically labeled as A, B, or C.

**Hexagonal Close-Packed (HCP)** arrangement follows an ABAB... pattern, where spheres from alternate layers align directly over one another. On the other hand, the **Cubic Close-Packed (CCP)** structure exhibits an ABCABC... pattern, in which no two layers share the same alignment consecutively.

Both arrangements result in a coordination number of 12, meaning each sphere touches 12 others. This tight packing accounts for the high density and typically low porosity of metals and other crystalline structures.

Void Spaces

The spaces that are left empty between spheres in a close-packed structure are known as void spaces. These are crucial in determining the structural and chemical properties of the material. Void spaces typically exist in two forms: **tetrahedral** and **octahedral** voids.

A **tetrahedral void** is formed when four spheres make a triangular base, creating a vacant space inside the arrangement. Conversely, an **octahedral void** is created when six spheres surround an empty space.

Understanding the distribution and number of these voids helps in explaining how atoms fit within a crystal lattice and affects critical properties like ionic conductivity and density.

Tetrahedral and Octahedral Voids

Among the void spaces, tetrahedral and octahedral voids offer remarkable insights into the atomic arrangement and potential interstitial sites in crystals. **Tetrahedral voids** appear smaller and are formed by four surrounding spheres. For every sphere added into the arrangement, two tetrahedral voids are generated, which accounts for twice the number of spheres present.

Meanwhile, **octahedral voids** form larger spaces compared to tetrahedral voids. Each of these voids corresponds to the space between three spheres on top and three spheres below in the arrangement. The number of octahedral voids is typically equal to the number of spheres in the structure.

This classification and presence of voids assist in determining how atoms or molecules might occupy spaces within materials, influencing concepts like alloying and solid-solution formation.