Question

\(\mathrm{ReO}_{3}\) is a structure type. The structure can be described as a simple cubic array of Re atoms with \(\mathrm{O}\) atoms located in the middle of each of the cube edges. Construct a unit cell of \(\mathrm{ReO}_{3}\). What are the coordination numbers of Re and \(\mathrm{O} ?\) Confirm that your unit cell gives the correct stoichiometry for \(\mathrm{ReO}_{3}\).

Short answer

Re has a coordination number of 6, and O has a coordination number of 2. The unit cell confirms the stoichiometry of ReO$_3$.

Step-by-step solution

Identify the Structure

ReO$_3$ has a perovskite-like cubic structure. In this structure, the Re atoms form a simple cubic lattice, and the oxygen atoms are located at the center of each Re-Re edge.

Draw the ReO$_3$ Unit Cell

Visualize the unit cell as a cube. Place the Re atoms at each of the eight corners of the cube. The O atoms are positioned at the center of each cube edge, totaling 12 edges.

Calculate the Formula From the Unit Cell

In the simple cubic structure: - The Re atoms at the corners each belong to 8 unit cells, contributing 1/8 of an atom per Re, for a total of 1 Re atom per unit cell. - Each O atom belongs to 4 unit cells (each sitting on an edge and shared by four cubes), contributing 1/4 atom per edge. Thus, the 12 edges contribute 12 × 1/4 = 3 O atoms per unit cell.

Determine Coordination Numbers

The coordination number (CN) of an atom is the number of nearest neighbors it has: - Re is surrounded by 6 O atoms along the cube edges, so its CN is 6. - Each O atom is at the bridge between two Re atoms, leading to a CN of 2 for oxygen.

Verify Stoichiometry

The unit cell contains 1 Re and 3 O atoms, confirming the stoichiometry ReO$_3$ for the unit cell.

Key concepts

Perovskite Structure

ReO extsubscript{3} is known for its unique perovskite-like structure, a term that denotes a specific type of crystal arrangement. The term "perovskite structure" originally derives from the mineral perovskite, with the chemical formula CaTiO extsubscript{3}. However, the same general arrangement appears in a variety of materials like ReO extsubscript{3}. In the case of ReO extsubscript{3}, the structure is simpler because the Re atoms (Rhenium) form a three-dimensional network in a cubic arrangement.

In this cubic structure, Rhenium atoms occupy the corners, creating a simple cubic lattice. Oxygen atoms, found at the midpoint of each edge of the cube, link up to form continuous Re-O chains throughout the crystal. This arrangement ensures an optimal packing of ions and contributes to the stability and distinctive properties of the material.

Such a structure can be imagined as a three-dimensional grid, where the positioning of atoms maximizes the stability and tightness of the crystal. This is essential for materials used in electronics, optics, and other technology-driven fields because it influences their physical properties, such as conductivity and resistance.

Coordination Number

The coordination number is a concept in chemistry and crystallography used to describe the number of neighboring atoms around a central atom in a crystal or molecule. In a crystal like ReO extsubscript{3}, understanding the coordination number is crucial to comprehend how the atoms interact with each other.

For Rhenium in ReO extsubscript{3}, the coordination number is 6. This means that each Re atom is surrounded by 6 oxygen atoms. These oxygen atoms are positioned evenly around the Rhenium at the edges of the cube, forming a stable three-dimensional network.

Meanwhile, each oxygen atom in ReO extsubscript{3} has a coordination number of 2. This indicates that each oxygen atom is bonded between two Rhenium atoms. It's like the oxygen serves as the linker holding two Rhenium atoms together. This dual interaction helps stabilize the structure and influences the overall material properties.

Grasping the concept of coordination number is key when predicting how changes in structure might influence material performance.

Unit Cell

A unit cell is the smallest divisible unit of a crystal lattice that retains the overall symmetry and properties of the entire crystal. In simple terms, it's like the basic building block that, when repeated in space, forms the whole crystal.

For ReO extsubscript{3}, the unit cell is cubic and contains both Rhenium and oxygen atoms. The corner of the cube houses the Re atoms, with only 1/8th of each Re atom located within a single unit cell because they are shared among 8 neighboring unit cells. Thus, a single Re unit cell contains 1 full Rhenium atom when all contributions are summed.

Each unit cell also features 12 edges, where oxygen atoms reside. Each oxygen is shared among 4 unit cells, so the contribution per edge results in 1/4 of an oxygen atom per unit cell. Hence, the complete unit cell contains a total of 3 oxygen atoms, ensuring the stoichiometry matches the chemical formula ReO extsubscript{3}.

Recognizing how the unit cell is constructed provides insight into the material's macroscopic properties, helping us understand how substances with similar structures can behave differently under various conditions.