Question

The rutile and fluorite structures, shown here (anions are colored green), are two of the most common structure types of ionic compounds where the cation to anion ratio is 1: 2 . (a) For \(\mathrm{CaF}_{2}\) and \(\mathrm{ZnF}_{2}\) use ionic radii, \(\mathrm{Ca}^{2+}(r=114 \mathrm{pm})\), \(\mathrm{Zn}^{2+}(r=88 \mathrm{pm})\), and \(\mathrm{F}^{-}(r=119 \mathrm{pm})\), to predict which compound is more likely to crystallize with the fluorite structure and which with the rutile structure. (b) What are the coordination numbers of the cations and anions in each of these structures?

Short answer

CaF2 is more likely to crystallize in the fluorite structure, and ZnF2 is more likely to crystallize in the rutile structure. For the fluorite structure, the cation coordination number is 8, and the anion coordination number is 4. For the rutile structure, both cation and anion coordination numbers are 6.

Step-by-step solution

Calculate cation-anion radius ratio for each compound

For CaF2, we need to calculate the ratio of the Ca2+ and F- ion radii: Cation-Anion Radius Ratio (CaF2) = \( \frac{r_{Ca^{2+}}}{r_{F^{-}}} \) = \( \frac{114 \, pm}{119 \, pm} = 0.958 \) Similarly, for ZnF2: Cation-Anion Radius Ratio (ZnF2) = \( \frac{r_{Zn^{2+}}}{r_{F^{-}}} \) = \( \frac{88 \, pm}{119 \, pm} = 0.739 \)

Compare the calculated radius ratios with known ratios for fluorite and rutile structures

In general, the cation-anion radius ratio for the fluorite structure (CaF2-type) is around 0.75 to 1, while for the rutile structure (TiO2-type) it is around 0.7 to 0.85. Comparing the calculated cation-anion radius ratios for CaF2 (0.958) and ZnF2 (0.739) with the known radius ratios for the fluorite and rutile structures, we can predict that CaF2 is more likely to crystallize in the fluorite structure, and ZnF2 is more likely to crystallize in the rutile structure.

Determine the coordination numbers for cations and anions in each structure

Coordination number can be defined as the number of nearest neighbors of a species (cation or anion) in the crystal structure. In the fluorite structure, the cations are surrounded by eight anions forming a cubic coordination polyhedron, while the anions are surrounded by four cations forming a tetrahedral coordination polyhedron. Thus, for the fluorite structure: Cation Coordination Number: 8 Anion Coordination Number: 4 In the rutile structure, both cations and anions are surrounded by six nearest neighbors in an octahedral arrangement. Thus, for the rutile structure: Cation Coordination Number: 6 Anion Coordination Number: 6

Summary

CaF2 is more likely to crystallize in the fluorite structure, and ZnF2 is more likely to crystallize in the rutile structure. For the fluorite structure, the cation coordination number is 8, and the anion coordination number is 4. For the rutile structure, both cation and anion coordination numbers are 6.

Key concepts

Cation-Anion Radius Ratio

The cation-anion radius ratio is crucial in determining the structure of ionic compounds. This ratio is calculated by dividing the radius of the cation by the radius of the anion. It helps predict how ions will pack together in a crystal lattice.

For example, in calcium fluoride (\(\text{CaF}_2\)), the ratio is \(\frac{114 \, \text{pm}}{119 \, \text{pm}} = 0.958\). In zinc fluoride (\(\text{ZnF}_2\)), it is \(\frac{88 \, \text{pm}}{119 \, \text{pm}} = 0.739\).

• A ratio between 0.75 and 1 often leads to the fluorite structure.
• A ratio around 0.7 to 0.85 usually results in the rutile structure.

This means \(\text{CaF}_2\) is more likely to have a fluorite structure, while \(\text{ZnF}_2\) prefers a rutile structure.

Fluorite Structure

The fluorite structure is a type of crystal layout commonly seen in compounds like calcium fluoride (\(\text{CaF}_2\)). This structure is characterized by a specific arrangement of ions where:

• Cations (like \(\text{Ca}^{2+}\)) are surrounded by eight anions, forming a cubic coordination sphere.
• Anions (like \(\text{F}^{-}\)) are surrounded by four cations, resulting in a tetrahedral arrangement.

This coordination leads to what is known as a coordination number of 8 for cations and 4 for anions.

It's stable for compounds where the cation-anion radius ratio falls between 0.75 and 1. This broad range allows for flexibility in ionic sizes while maintaining stability in the crystal.

Rutile Structure

The rutile structure is another common architecture found in certain ionic compounds, such as zinc fluoride (\(\text{ZnF}_2\)). It is defined by:

• A coordination number of 6 for both cations and anions, indicating that each ion is surrounded by six nearest neighbors.
• A typical octahedral configuration that creates a highly symmetrical and strong lattice.

This structure is favored when the cation-anion radius ratio is approximately between 0.7 and 0.85.

It exemplifies stability in ionic packing when both the cation and anion sizes are more comparable, allowing for an equal distribution of ions around each other.

Coordination Number

The coordination number in a crystal structure refers to the number of nearest neighboring ions surrounding a given ion. It reflects the spatial arrangement and stability within the crystal lattice.

In the fluorite structure,

• The cation's coordination number is 8, due to its surrounding by eight anions.
• The anion's coordination number is 4, as it is surrounded by four cations.

In contrast, the rutile structure offers a coordination number of 6 for both cations and anions, arranged in an octahedral shape.

This number provides insight into the bonding and geometry of the ionic compound, influencing both its physical and chemical properties.