Problem 55 - Chemical Principles

Question

An oxide of rhenium crystallizes with a cubic unit cell that has a rhenium cation at each corner and an oxide ion at the center of each edge of the crystal. (a) Determine the coordination numbers of the two ions. (b) Write the formula of the oxide.

Step-by-Step Solution

Answer

The coordination number of rhenium is 6 and that of oxide is 4. The formula of the oxide is ReO3.

Step 1: Calculate the Coordination Number of Rhenium

The coordination number is the number of atoms of opposite charge that surround an ion in a crystal. Since rhenium cations are at each corner of the cubic unit cell, each rhenium ion is surrounded by 6 oxide ions (one on each face-center of the six faces that meet at that corner). Therefore, the coordination number of rhenium is 6.

Step 2: Calculate the Coordination Number of Oxide

Each oxide ion is located at the center of each edge of the cube and is thus in contact with four rhenium cations (one at each end of the edge and one at each end of the perpendicular edges). Therefore, the coordination number of oxide is 4.

Step 3: Calculate the Formula of the Oxide

In a cubic unit cell, 1/8 of each corner atom is within the cell and each edge atom is shared between 4 unit cells. There are 8 corners with 1/8 of a cation each and 12 edges with 1/4 of an anion each. This leads to: Cations = 8 * (1/8) = 1, Anions = 12 * (1/4) = 3. The simplest whole number ratio is 1:3, giving the formula ReO3.

Key Concepts

Coordination Number

In crystallography, the coordination number is an essential concept that refers to the number of nearest neighbors an atom or ion has within a crystal structure. This number is a critical factor as it determines the size and spatial arrangement of atoms within the crystal, which has direct implications on the properties of the material.

For instance, a rhenium ion in our oxide crystal has a coordination number of 6, indicating that it is surrounded by six oxygen ions. The oxide ion, in contrast, has a coordination number of 4, which means it's surrounded by four rhenium ions. These values can affect the ionic bonding strength and influence characteristics such as electrical conductivity, melting point, and structural stability of the material.

Understanding the coordination number not only aids in visualizing the three-dimensional layout of the crystal but also helps predict how ions will bind to each other in different types of crystal structures, like cubic or hexagonal systems.

Cubic Unit Cell

A cubic unit cell is a basic repeating unit that defines a crystal's shape and volume, constituting its smallest orderly arrangement. There are several types of cubic unit cells, but in this context, we're looking at one where cations are at the corners, and anions are centered on the edges. This arrangement effectively fills space and determines the unit cell's symmetry and dimensions.

Each corner of a cubic unit cell is shared with adjacent cells, meaning that only a fraction of the ion at the corner belongs to one particular cell. In the oxide example, the rhenium ion (a cation) at the cell's corner has only 1/8th of its volume within the cell. Such fractional accounting is crucial when determining the chemical formula of the crystalline compound.

Importance of Cubic Unit Cells

Understanding these cells is fundamental as they directly influence the physical and chemical properties of the material by providing a framework for how ions or atoms pack within a solid. The unit cell's geometry dictates how the crystal absorbs, transmits, and reflects light, among other attributes.

Chemical Formula Calculation

The chemical formula calculation in crystallography involves deducing the simplest whole-number ratio of ions within a unit cell, leading to the empirical formula of a compound. This task is crucial as it reveals the fundamental composition of crystalline structures, which can then be used to infer a variety of properties, and allows chemists and material scientists to engineer substances with specific characteristics.

In the given oxide of rhenium, by considering the contributions of rhenium cations and oxide anions based on their locations—corners and edges, respectively—you can calculate that there is one full rhenium ion and three oxide ions. Thus, by translating these fractions into a stoichiometric formula, the oxide is determined to be ReO3.

This calculation approach is widely applicable in designing and synthesizing new materials, like battery materials or catalysts, and also aids in characterizing and understanding the composition of naturally occurring minerals.