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Chapter 12: Problem 9

The corundum crystal structure, found for \(\mathrm{Al}_{2} \mathrm{O}_{3}\), consists of an HCP arrangement of \(\mathrm{O}^{2-}\) ions; the \(\mathrm{Al}^{3+}\) ions occupy octahedral positions. (a) What fraction of the available octahedral positions are filled with \(\mathrm{Al}^{3+}\) ions? (b) Sketch two close-packed \(\mathrm{O}^{2-}\) planes stacked in an \(A B\) sequence, and note octahedral positions

Short Answer

Expert verified
Answer: In the corundum crystal structure, the fraction of filled octahedral positions is 1/6. To represent the octahedral positions between two close-packed oxygen planes, we can sketch two layers of oxygen ions in a hexagonal pattern with octahedral positions in between, forming triangles between the oxygen ions. Each octahedral position is surrounded by three oxygen ions of one layer and three oxygen ions of the other layer, with two octahedral positions per oxygen ion in the HCP arrangement.

Step by step solution

01

Determine the number of octahedral positions per unit cell

In an HCP arrangement, there are two octahedral positions per oxygen ion. Since there are six oxygen ions per unit cell (two in each layer for a total of three layers), there are a total of 12 octahedral positions per unit cell.
02

Determine the number of Al ions per unit cell

The corundum crystal structure is formed by \(\mathrm{Al}_{2} \mathrm{O}_{3}\). In one formula unit of \(\mathrm{Al}_{2} \mathrm{O}_{3}\), there are two Al ions. Since we have one formula unit of \(\mathrm{Al}_{2} \mathrm{O}_{3}\) in one unit cell, there are 2 Al ions per unit cell.
03

Calculate the fraction of filled octahedral positions

Now we can use the number of Al ions and octahedral positions to find the fraction of filled octahedral positions. The fraction of filled octahedral positions is equal to the ratio of Al ions to octahedral positions: \(\frac{[\text{Al}^{3+}]}{[\text{octahedral}]_{\text{positions}}} = \frac{2}{12} = \frac{1}{6}\). Hence, the fraction of filled octahedral positions is \(\frac{1}{6}\).
04

Sketch two close-packed O planes and indicate octahedral positions

To indicate octahedral positions between two close-packed \(\mathrm{O}^{2-}\) planes, draw two layers of oxygen ions arranged in a hexagonal pattern. Label the first layer as layer A and the second layer as layer B. Then, draw the octahedral positions in between the layers. Each octahedral position is surrounded by three oxygen ions of layer A and three oxygen ions of layer B. The octahedral positions form a triangle between the oxygen ions in the layers, and there are two octahedral positions per oxygen ion in the HCP arrangement.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

HCP Arrangement
The Hexagonal Close-Packed (HCP) arrangement is a highly efficient way of stacking atoms or ions in a crystal structure. It is one of the two most dense forms of packing, the other being Face-Centered Cubic (FCC).
In HCP, each layer of atoms is arranged in a hexagonal pattern. When adding a new layer, atoms sit in the depressions of the previous layer, leading to an ABAB... stacking sequence.
  • Every atom in HCP is surrounded by 12 others, which forms the closely packed structure.
  • The structure has hexagonal symmetry, evident in the arrangement when viewed from above.
  • Each atom contacts 6 atoms in its own layer, 3 in the layer above, and 3 in the layer below.
This specific arrangement plays a crucial role in determining the properties of the material, including density, melting point, and how it interacts with light and electricity.
Octahedral Positions
In crystal structures, octahedral positions are locations where an atom or ion resides surrounded by six others, positioned at the corners of an octahedron. This is a common coordination environment in many mineral structures, including the corundum crystal structure of \(\text{Al}_2\text{O}_3\).
  • The octahedral positions in an HCP structure are not randomly scattered; they are located at specific points where atoms from two layers align, allowing an atom to fit snugly between them.
  • Atoms occupying octahedral positions are coordinated by six others, three from the layer above and three from the layer below.
  • In corundum, \(\text{Al}^{3+}\) ions occupy these octahedral positions between the HCP layers of \(\text{O}^{2-}\) ions.
This arrangement allows for the strong ionic bonds characteristic of corundum, contributing to its hardness and high melting point.
Unit Cell
The concept of the unit cell is central to understanding crystal structures. It is the smallest repeating unit of a crystal lattice that displays the full symmetry of the crystal's structure.
  • The unit cell contains all the information needed to construct the entire crystal by stacking it in three-dimensional space.
  • In the HCP arrangement of corundum, the unit cell is a hexagonal prism, reflecting the symmetry and pattern of the entire crystal.
  • Knowing the unit cell allows us to calculate important properties of the material, such as its density and the fraction of filled lattice positions, which in the case of corundum, is \(\frac{1}{6}\) for the octahedral positions filled by \(\text{Al}^{3+}\) ions.
The precise knowledge of the unit cell dimensions and contents is crucial for understanding the properties of the crystal and for calculating its theoretical density, which is essential for identifying materials and understanding their behaviors.
Crystallography
Crystallography is the scientific study of crystals and their structures. It's an interdisciplinary field straddling chemistry, physics, geology, and materials science.
  • Crystallographers use techniques like X-ray diffraction to determine the arrangement of atoms within a crystal.
  • The patterns obtained can be interpreted to reveal the positions of atoms, the type of lattice, and the symmetries present within the crystal structure.
  • Crystallography provides insights into how the positions of atoms relate to the properties of a material, like hardness, conductivity, and reactivity.
Understanding crystallography is essential when examining the corundum structure of \(\text{Al}_2\text{O}_3\), as it elucidates how the HCP arrangement and the filling of octahedral positions by \(\text{Al}^{3+}\) ions influence the material's hardness and other characteristics, making it suitable for applications such as abrasives and cutting tools.

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Most popular questions from this chapter

Express cach of the following expressions using a single positive index: (a) \(x^{4} x^{7}\) (b) \(x^{2}(-x)\) (c) \(\frac{x^{2}}{x}\). (d) \(\frac{x^{-2}}{x^{-1}}\) (c) \(\left(x^{-2}\right)^{4}\) (f) \(\left(x^{-25} x^{-3.5}\right)^{2}\)

Show that the minimum cation-to-anion radius ratio for a coordination number of 6 is 0.414. (Hint: Use the NaCl crystal structure in Figure 12.2, and assume that anions and cations are just touching along cube edges and across face diagonals.)

Evaluate (a) \(2^{3}\) (b) \(3^{2}\) (c) \(\frac{5^{13}}{5^{12}}\) (d) \(\frac{19^{-11}}{19^{-13}}\) (c) \(\left(2^{1 / 4}\right)^{8}\) (f) \((-4)^{-2}\) (g) \(4^{-1 / 2}\) (h) \(\left(9^{1 / 3}\right)^{3 / 2}\) (i) \(\sqrt{32} \sqrt{2}\) (j) \(\sqrt{0.01}\) (k) \(81^{3 / 4}\)

A circular specimen of \(\mathrm{MgO}\) is loaded using a three-point bending mode. Compute the minimum possible radius of the specimen without fracture, given that the applied load is \(5560 \mathrm{~N}\left(1250 \mathrm{lb}_{\mathrm{f}}\right)\), the flexural strength is \(105 \mathrm{MPa}(15,000 \mathrm{psi})\), and the separation between load points is \(45 \mathrm{~mm}\) (1.75 in.).

The zinc blende crystal structure is one that may be generated from close- packed planes of anions. (a) Will the stacking sequence for this structure be FCC or HCP? Why? (b) Will cations fill tetrahedral or octahedral poitions? Why? (c) What fraction of the positions will be bccupied?
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