Question

A certain metal fluoride crystallizes in such a way that the fluoride ions occupy simple cubic lattice sites, while the metal ions occupy the body centers of half the cubes. What is the formula of the metal fluoride?

Short answer

The formula of the metal fluoride is MF₂, based on the arrangement of fluoride ions in a simple cubic lattice and metal ions occupying the body centers of half the cubes.

Step-by-step solution

1. Determine the arrangement of fluoride ions

Fluoride ions are arranged in a simple cubic lattice. In a simple cubic lattice, there is one atom/ion at each corner of the cube. There are 8 corners in a cube, and each corner contributes 1/8 of an ion to a unit cell. Thus, the number of fluoride ions (F⁻) in a unit cell is: Number_of_F⁻ = Corner_ions × Contribution_per_corner = \(8 × \frac{1}{8}\) = 1

2. Determine the arrangement of metal ions

Metal ions occupy the body centers of half the cubes. Since there is only one body center per cube, half the cubes will have one metal ion (M) and half the cubes will have none. On average, the number of metal ions per cube is: Number_of_M = \(0.5 × 1 = 0.5\)

3. Simplify the ratio of metal and fluoride ions in the lattice

To determine the formula of the metal fluoride, we must create a ratio of metal to fluoride ions in the lattice: Ratio_of_M:F⁻ = Number_of_M : Number_of_F⁻ = 0.5 : 1 Since the ratio has a fractional value, we multiply both sides by 2 to obtain the simplest whole number ratio: 2 × (0.5 : 1) = 1 : 2

4. Determine the formula of the metal fluoride

We can now deduce the formula of the metal fluoride based on the simplified ratio: Formula: M₁F₂, where M represents the metal ion The formula of the metal fluoride is M₁F₂, or simply MF₂.

Key concepts

Simple Cubic Lattice

In crystallography, a simple cubic lattice is one of the most straightforward and fundamental structures. Imagine a cube where each corner hosts an atom or ion, creating a three-dimensional grid.
This lattice type has a repeating unit that forms the bulk of crystalline structures, an essential aspect in materials science and chemistry.
Within a simple cubic lattice:

• There are 8 corner positions.
• Each corner atom is shared by eight adjacent cubes.
• Therefore, each corner contributes 1/8 of an atom to a given unit cell.

So, if we talk about a simple cubic unit cell, it fundamentally contains only 1 full atom since 8 corners imes 1/8 contribution per corner = 1 atom per unit cell.
This basic understanding helps in calculating how many atoms exist in a compound's crystal lattice when they occupy certain positions, as seen in metal fluoride compounds.

Unit Cell

The concept of a unit cell is central to understanding crystal lattices. A unit cell is the simplest repeating unit in a space lattice that completely describes the symmetry and structure of the entire crystal.
Think of it as the building block of a crystal lattice. By repeating this block in three dimensions, you can recreate the entire crystal structure.
In the context of our metal fluoride problem:

• The fluoride ions form a simple cubic lattice.
• The metal ions occupy the body centers of half the cubes.

When considered together within a unit cell, each ion's location and contribution to the unit cell is calculated to simplify into a chemical formula.
Ultimately, understanding the unit cell aids in determining the ratio of different atoms or ions in the entire crystal, leading to the correct formula for the compound.

Metal Fluoride Formula

Determining the formula of a metal fluoride involves understanding how the metal and fluoride ions are arranged in the lattice. The ratio between the ions gives us the chemical formula of the compound.
In our example:

• Fluoride ions occupy the simple cubic lattice. Here, each unit cell contains 1 fluoride ion.
• The metal ions are found at the body centers of half the cubes, giving us 0.5 metal ions per cube.

To find the simplest ratio, we compare the number of metal ions to fluoride ions:

• Metal ions = 0.5 per unit cell.
• Fluoride ions = 1 per unit cell.
• The ratio is therefore 0.5:1.
• When multiplied by 2 to get whole numbers, we have 1:2.

Thus, the formula is determined to be MF₂, where M is the metal.
This step-by-step approach ensures we understand the chemistry behind metal fluoride formation and derive the correct stoichiometric relationships.