Question

A tetrahedral site in a closest-packed lattice is formed by four spheres at the corners of a regular tetrahedron. This is equivalent to placing the spheres at alternate corners of a cube. In such a closest-packed arrangement the spheres are in contact, and if the spheres have a radius \(r\), the diagonal of the face of the cube is \(2 r\). The tetrahedral hole is inside the middle of the cube. Find the length of the body diagonal of this cube and then find the radius of the tetrahedral hole.

Short answer

The edge length of the cube is \(2\sqrt{2}r\), the body diagonal is \(2\sqrt{6}r\), and the radius of the tetrahedral hole is \(\frac{2\sqrt{6}r}{4} = \frac{\sqrt{6}r}{2}\).

Step-by-step solution

Find the length of the cube's edge

Given that the face diagonal of the cube is formed by placing two spheres in contact, the length of the face diagonal is equivalent to twice the diameter of a sphere, which is 2 times the radius, or 4r. Since the diagonal of a square face is longer than the edge by a factor of the square root of 2, we can set up the equation: \text{face diagonal} = edge length * \sqrt{2}. So, \( 4r = a \sqrt{2} \), where \( a \) is the edge length. Now, solve for \( a \)

Calculate the body diagonal of the cube

The body diagonal of a cube can be found using the formula which is the edge length multiplied by the square root of 3, or \( a \sqrt{3} \). We already found the edge length \( a \) in Step 1; now we substitute that value into this new formula to find the length of the body diagonal.

Find radius of the tetrahedral hole

The radius of the tetrahedral hole would be one-quarter of the body diagonal of the cube. This is because the tetrahedral hole is centered in the cube, with its corners touching the center of each of the four surrounding spheres (vertices of the cube). Therefore, the radius of the hole cannot exceed one-quarter of the body diagonal without protruding out of the surrounding spheres.

Key concepts

Solid State Chemistry

Solid state chemistry is a pivotal area within the realm of physical chemistry, concerned with the study and characterization of solid materials. It explores the structure, properties, and synthesis of solid substances including metals, minerals, ceramics, and semiconductor materials. A fundamental concept within this field is the arrangement of atoms or ions within a solid, which forms what is known as a crystal lattice.

Crystal lattices can be envisioned as three-dimensional frameworks where points, known as lattice points, denote the positions of atoms or ions within the solid. These lattice points are arranged in a periodic pattern that repeats throughout the material, providing a blueprint for the structure of the atomic or ionic ensemble. In the specific case of the closest-packed lattice, atoms are packed together as tightly as possible, leading to a highly ordered and efficient arrangement.

The study of the tetrahedral holes within the closest-packed lattice is integral in understanding how different atoms or ions can occupy these spaces, leading to the formation of various crystal structures and thus affecting the properties of the material. This level of structural comprehension is essential for predicting material behaviors and for creating new materials with desired physical and chemical characteristics.

Crystal Lattices and Unit Cells

The concept of a crystal lattice is intrinsically linked to the notion of a unit cell. A unit cell is the smallest repeating unit of a crystal lattice that, when repeated in three dimensions, creates the entire crystal. It essentially serves as a building block for the solid material. There are different types of unit cells, such as cubic, tetragonal, orthorhombic, among others, with the cubic lattice being prevalent due to its simplicity and prevalence in nature, particularly in the closest-packed arrangements.

In a closest-packed lattice, atoms are organized in a manner that uses the available space most efficiently. This creates two types of voids—or holes—called octahedral and tetrahedral. The tetrahedral hole, which is the focus of our exercise, is a space where a smaller atom can fit in the midst of four other atoms arranged at the corners of a tetrahedron. This can be visualized as the occupying of alternate corners of a cube — a geometric shape that is the foundation for understanding several unit cells in crystallography.

The insights into the spatial arrangement of atoms provided by the study of unit cells and holes are essential for predicting and explaining the physical and chemical properties of solid materials, such as ion exchange capabilities, electrical conductivity, and density.

Geometric Calculations in Chemistry

Geometric calculations often bridge the gap between abstract chemical concepts and tangible physical properties. For chemists, understanding the precise geometrical relationships within crystal structures is critical, as they influence material behavior and can dictate the physical properties of the substance. In the context of our exercise, the geometric calculation involves deducing the dimensions of a tetrahedral hole within a crystal lattice.

The face diagonal of a cube is crucial for this exercise as it represents the distance between two opposite corners of a cube's face, or square. By knowing that this diagonal is twice the diameter of a sphere in the closest-packed arrangement (or four times the radius, given as 4r), we can discern the length of the cube's edge through the Pythagorean theorem as it applies to a right-angled triangle. Further, the cube's body diagonal — connecting two opposite corners of the cube — is necessary for finding the radius of the tetrahedral hole. This is another application of the theorem, expanded into three dimensions.

The calculations become a handy tool for predicting the sizes of different types of voids in materials, and thus are essential for material design, determining the possible interstitial position of ions or atoms, and for understanding the material's ability to host impurities.