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

Expert verified
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

01

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
02

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 = \(0.5 × 1 = 0.5\)
03

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
04

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.

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