Question

Cobalt fluoride crystallizes in a closest packed array of fluoride ions with the cobalt ions filling one-half of the octahedral holes. What is the formula of this compound?

Short answer

The formula of cobalt fluoride with cobalt ions filling half of the octahedral holes in a closest packed array of fluoride ions is \(CoF_2\).

Step-by-step solution

Identify the closest packed array structure

In a closest packed array of ions, the ions are arranged in a lattice, typically either hexagonal close-packed (hcp) or cubic close-packed (ccp, also known as face-centered cubic, fcc). Both hcp and ccp have the same number of octahedral holes (interstitial positions for other ions) relative to the number of ions in the lattice.

Determine the number of octahedral holes in the lattice

For both hcp and ccp structures, each ion in the lattice is associated with one octahedral hole. Therefore, the number of octahedral holes is equal to the number of fluoride ions in the lattice. Let's represent the number of fluoride ions as \(F\).

Determine the number of cobalt ions

We are given that cobalt ions fill half of the octahedral holes. Since the number of octahedral holes is equal to the number of fluoride ions (\(F\)), the number of cobalt ions in the lattice can be represented as \(\frac{1}{2}F\).

Find the formula of the compound

To find the formula of the compound, we need to determine the ratio between cobalt and fluoride ions. Since there are \(\frac{1}{2}F\) cobalt ions and \(F\) fluoride ions, the ratio is \(\frac{1}{2}\). To represent this ratio as a whole number, we can multiply both sides by 2. In this case, for every 1 cobalt ion, there are 2 fluoride ions in the compound. This means the formula of cobalt fluoride is \(CoF_2\).