Question

Show how the fluorite structure accommodates a cation-to-anion ratio of \(1 : 2 .\)

Short answer

The fluorite structure has 4 cations and 8 anions per unit cell, resulting in a 1:2 cation-to-anion ratio when simplified.

Step-by-step solution

Understanding the Fluorite Structure

The fluorite structure is the crystal structure adopted by calcium fluoride, CaF2. It consists of a face-centered cubic (fcc) arrangement of cations, with anions occupying all the tetrahedral holes. Each cation is coordinated by eight anions, and each anion is coordinated by four cations.

Counting Cations and Anions in the Unit Cell

In a fluorite structure, the cations occupy all the eight corners and six face centers of the cubic unit cell. Each corner atom is shared among eight unit cells and each face-centered atom is shared with two cells. Therefore, the contribution of cations to one unit cell is: Corners: (1/8) * 8 = 1, Faces: (1/2) * 6 = 3, Total cations = 4. The anions fill all the tetrahedral sites, and there are eight tetrahedral holes in an fcc lattice. Since anions in the tetrahedral holes are not shared, all eight contribute to the unit cell. Hence, the total anions = 8.

Calculating the Cation-to-Anion Ratio

Having counted the cations and anions in the unit cell separately, we see that there are a total of 4 cations and 8 anions. This gives us a cation-to-anion ratio of 4:8. When simplified by dividing both numbers by 4, we get a ratio of 1:2. Thus, the fluorite structure accomplishes a 1 to 2 ratio of cation to anion.

Key concepts

Cation-to-Anion Ratio

Understanding the cation-to-anion ratio is crucial when studying crystal structures, especially for compounds like calcium fluoride (CaF2), which forms a fluorite structure. In a crystalline compound, the ratio of positively charged ions (cations) to negatively charged ions (anions) determines the stability and geometry of the structure.

In the fluorite structure, this ratio shapes the entire crystal lattice. Each cation requires two anions to balance the charge, creating a 1:2 ratio. This is reflected in the cube-shaped unit cell of the fluorite structure, where we find four cations and eight anions. The precise stoichiometry is paramount for maintaining the electrical neutrality of the compound, ensuring that there are enough anions to counterbalance the positive charge of the cations.

Unit Cell

A unit cell is the smallest representative volume of a crystal's structure, which when repeated in three dimensions forms the entire crystal. In other words, it's like a building block that, brick by brick, constructs the larger crystal edifice.

The fluorite structure's unit cell is a perfect example; it's a face-centered cubic (fcc) lattice, with a cation occupying each corner and the centers of each face. The clever arrangement allows the unit cell to pack maximum density of ions.

Cation Positioning in the Unit Cell

Each corner cation is shared among eight cells and each face-centered cation among two, contributing to a net count of four cations per cell. This is a beautiful synergy of geometry and chemistry that gives the fluorite its characteristic cation-to-anion ratio.

Tetrahedral Holes

The concept of tetrahedral holes is integral to understanding many crystal structures. These holes are spaces within the lattice where smaller ions can fit. They are surrounded by four lattice points, creating a tetrahedron-shaped space – hence the name.

In an fcc lattice like that of the fluorite structure, for each cation at the corners and centers of the faces, there are corresponding anions snugly fitting into these tetrahedral holes. In the case of CaF2, there are eight of these holes in the unit cell, all occupied by anions. The 'lock-and-key' fit ensures that each anion is tetrahedrally coordinated by four cations, which reinforces the structural integrity and determines many of the crystal's properties.