Problem 86 - Chemistry: An Atoms-Focused Approach

Question

What is the chemical formula of the compound that crystallizes in a simple cubic arrangement of fluoride ions with \(\mathrm{Ba}^{2+}\) ions occupying half of the cubic holes?

Step-by-Step Solution

Answer

Answer: The chemical formula of the compound is \(\mathrm{BaF}_8\).

Step 1: Understand the simple cubic arrangement of fluoride ions

In a simple cubic arrangement, each unit cell has one ion at each corner. Since there are eight corners in a cubic unit cell, there are eight fluoride ions in the simple cubic arrangement.

Step 2: Determine the number of available cubic holes

In a simple cubic arrangement, there is one cubic hole available at the center of the unit cell.

Step 3: Determine the number of occupied cubic holes by \(\mathrm{Ba}^{2+}\) ions

The problem states that \(\mathrm{Ba}^{2+}\) ions occupy half of the cubic holes. Since there is only one cubic hole available in the simple cubic arrangement, half of it means that there is only one occupied hole by a \(\mathrm{Ba}^{2+}\) ion.

Step 4: Calculate the ratio of fluoride ions to \(\mathrm{Ba}^{2+}\) ions

The unit cell has eight fluoride ions and one \(\mathrm{Ba}^{2+}\) ion. The ratio of fluoride ions to \(\mathrm{Ba}^{2+}\) ions is 8:1.

Step 5: Find the chemical formula of the compound

According to the ratio calculated in Step 4, there are eight fluoride ions for every one \(\mathrm{Ba}^{2+}\) ion in the unit cell. Thus, the chemical formula of the compound is \(\mathrm{BaF_8}\). However, we can simplify this formula, as a compound can contain ions in their lowest whole-number ratio. We can find the simplest whole-number ratio by dividing the number of ions by the greatest common divisor. The greatest common divisor of 8 and 1 is 1. Therefore, the chemical formula is \(\mathrm{BaF}_8\).