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, \(\mathrm{Ca}^{2+}(r=1.14 \mathrm{~A}), \mathrm{Zn}^{2+}(r=0.88 \mathrm{~A})\), and \(\mathrm{F}(r=1.19 \mathrm{~A})\), 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

In summary, CaF2 is more likely to crystallize with the rutile structure, with coordination numbers of 6 for Ca and 3 for F, while ZnF2 is more likely to crystallize with the fluorite structure, with coordination numbers of 8 for Zn and 4 for F.

Step-by-step solution

Understanding the structures

In order to predict the structures, it is important to understand the coordination of cations and anions in these structures: - Fluorite structure: In this structure, cations are surrounded by 8 anions in a cubic arrangement, and anions are surrounded by 4 cations in a tetrahedral arrangement. - Rutile structure: In this structure, cations are surrounded by 6 anions in an octahedral arrangement, and anions are surrounded by 3 cations in a trigonal planar arrangement.

Calculate the cation-anion radius ratio for both compounds

The cation-anion radius ratio is an important factor in predicting the crystalline structures of ionic compounds. To calculate the radius ratio for both compounds, we need to divide the cation's ionic radius by the anion's ionic radius: For CaF2: \(radius\ ratio_{CaF_2} = \frac{1.14}{1.19}\) For ZnF2: \(radius\ ratio_{ZnF_2} = \frac{0.88}{1.19}\) Now, calculate the ratios: \(radius\ ratio_{CaF_2} \approx 0.958\) \(radius\ ratio_{ZnF_2} \approx 0.739\)

Predict which compound will likely form which structure

As a general rule, the larger the cation-anion radius ratio, the higher the coordination number of the cation. Let's compare the radius ratios with the general rules for coordination numbers: - A radius ratio between 0.732 and 1.000 corresponds to a coordination number of 6 (octahedral arrangement). - A radius ratio between 0.414 and 0.732 corresponds to a coordination number of 4 (tetrahedral arrangement). With these rules in mind: - CaF2 has a radius ratio of 0.958, which falls within the 6-coordination range, so it is more likely to form the rutile structure. - ZnF2 has a radius ratio of 0.739, which falls within the 4-coordination range, so it is more likely to form the fluorite structure.

Determine the coordination numbers in each structure

Since we predicted the structures for each compound, we can now find the coordination numbers in each case: - CaF2 (rutile structure): Each cation (Ca) is surrounded by 6 anions (F) in an octahedral arrangement (coordination number of Ca is 6), and each anion (F) is surrounded by 3 cations (Ca) in a trigonal planar arrangement (coordination number of F is 3). - ZnF2 (fluorite structure): Each cation (Zn) is surrounded by 8 anions (F) in a cubic arrangement (coordination number of Zn is 8), and each anion (F) is surrounded by 4 cations (Zn) in a tetrahedral arrangement (coordination number of F is 4). In summary, CaF2 is more likely to crystallize with the rutile structure, with coordination numbers of 6 for Ca and 3 for F, while ZnF2 is more likely to crystallize with the fluorite structure, with coordination numbers of 8 for Zn and 4 for F.

Key concepts

Fluorite Structure

The fluorite structure is a common arrangement found in ionic compounds where the general formula is AB2. It is best visualized as a cubic lattice. In this structure, each cation is surrounded by eight anions, forming a cubic arrangement. This is why we say the cations have a coordination number of 8. On the other hand, each anion in the fluorite structure is surrounded by four cations, forming a tetrahedral arrangement. The anions thus have a coordination number of 4.

One of the classic compounds that crystallize in the fluorite structure is calcium fluoride (CaF2). Here, calcium ions (Ca²⁺) are positioned within the cubic sites of the lattice, while fluoride ions (F⁻) occupy tetrahedral sites. The structure provides a good balance between the size and charge of different ions, making it energetically favorable for certain substances. This arrangement allows for strong ionic bonding with a fundamental balance between the forces of attraction and repulsion. Understanding this structure can be crucial in exploring the properties of solid ionic materials.

Rutile Structure

In contrast, the rutile structure is another common lattice type that is seen in certain ionic compounds, especially when the ratio of cations to anions is 1:2. This structure is characterized by an octahedral arrangement, where each cation is surrounded by six anions. In rutile, the coordination number of the cations is 6. Anions, however, are surrounded by just three cations, giving them a trigonal planar arrangement and a coordination number of 3.

An example of a rutile structure is titanium dioxide (TiO2), although other compounds like CaF2 can also form this configuration under specific conditions. The properties arising from this structural form include stability and a distinct anisotropy in physical characteristics. Rutile structures often show different properties in different spatial directions, which influences their electrical and optical behaviors. This knowledge is utilized in fields ranging from materials science to electronic components.

Coordination Numbers

Coordination numbers are a way to quantify the arrangement of ions around a central ion in a crystalline lattice. They represent how many ions of opposite charge surround a given ion. In the context of ionic solids, coordination numbers play a crucial role in predicting and understanding crystal structures because they reflect the balance needed to maintain structural stability.

In the fluorite structure, the coordination number is 8 for cations and 4 for anions. This means each cation sees eight anions wrapped around it, while each anion sees only four cations. In the case of the rutile structure, the coordination number of 6 applies to cations, where they are encircled by six anions, and anions have a coordination number of 3, with only three cations surrounding them. Recognizing these numbers helps us to determine which type of structure will most likely form based on the ratios of ionic radii and provides insight into the compound's potential chemical properties.

Cation-Anion Radius Ratio

The cation-anion radius ratio is a critical factor in determining the most stable configuration for a crystalline compound. This ratio is obtained by dividing the ionic radius of the cation by that of the anion. Knowing this ratio is helpful in predicting what type of structure will form because it aligns closely with the possible coordination numbers.

As a rule of thumb, if the radius ratio is large (close to 1), the compound is more likely to display a higher coordination number, such as that seen in the rutile structure. Conversely, a smaller radius ratio suggests a lower coordination number typical of structures like fluorite. For example, in CaF2, the radius ratio is approximately 0.958, favoring the six-coordination in a rutile structure. Meanwhile, ZnF2 with a radius ratio of about 0.739 leans towards the four-coordination arrangement of the fluorite structure. Understanding these ratios assists in predicting crystal forms and their related properties, making it a fundamental tool in materials chemistry.