Question

Given that the ionic radius of \(\mathrm{Fe}^{2+}\) is \(0.77 \AA\) and that of \(\mathrm{O}^{2-}\) is \(1.26 \AA\) A, predict a likely structure for \(\mathrm{FeO}\) using ionic radius ratios.

Short answer

The radius ratio for FeO is 0.61, which falls in the range of 0.414 to 0.732, indicating a coordination number of 4 with a tetrahedral arrangement. Therefore, the likely structure for FeO is anionic close-packed with tetrahedral holes filled by Fe²⁺ cations.

Step-by-step solution

Calculate the radius ratio

First, let's calculate the radius ratio. The radius ratio is the ratio of the radius of the cation to the radius of the anion. In this case, the cation is Fe²⁺, and the anion is O²⁻. \[ \text{Radius ratio} = \dfrac{r_{\text{cation}}}{r_{\text{anion}}} \] For FeO, the radius of Fe²⁺ is 0.77 Å, and the radius of O²⁻ is 1.26 Å. Therefore, the radius ratio is: \[ \text{Radius ratio} = \dfrac{0.77 \,\mathring{A}}{1.26 \,\mathring{A}} = 0.61 \]

Determine the coordination number using the radius ratio rule

Now that we have the radius ratio, we can use the radius ratio rule to determine the coordination number of the FeO crystal structure. The coordination number is the number of anions surrounding a cation in the crystal lattice. The radius ratio rule states that the coordination number depends on the radius ratio: - If the radius ratio is between 0.414 and 0.732, the coordination number is 4 (tetrahedral arrangement). - If the radius ratio is between 0.732 and 1, the coordination number is 6 (octahedral arrangement). - If the radius ratio is greater than 1, the coordination number is 8 (cubic arrangement). Our calculated radius ratio is 0.61, which falls in the range of 0.414 to 0.732. Therefore, the FeO crystal structure has a coordination number of 4 and a tetrahedral arrangement.

Predict the likely structure for FeO

Since the coordination number is 4 and the arrangement is tetrahedral, the likely structure for FeO is anionic close-packed with tetrahedral holes filled by cations. In this structure, the O²⁻ anions are close-packed, and the Fe²⁺ cations fill the tetrahedral holes. Thus, the likely structure for FeO is a close-packed structure with tetrahedral holes filled with Fe²⁺ cations.

Key concepts

Coordination Number

When we talk about the coordination number in crystal structures, we're referring to the number of nearest neighbor ions that surround a particular ion. In simpler terms, it's like counting how many friends are closely gathered around you in a group photo. The coordination number is a critical factor in determining the structural layout of a crystal.

Using this concept with the given example of iron oxide (FeO), we found that the coordination number is crucial for predicting its specific structure. For FeO, our calculations showed a coordination number of 4. Historically, this number has been linked with the visual pattern of a tetrahedron—a pyramid shape with a triangular base, which indicates that each iron ion is surrounded by four oxygen ions in a very specific geometric arrangement.

Deducing the structural organization from the coordination number is essential in understanding the properties of the material since the arrangement of ions can affect the crystal’s stability, how it interacts with other substances, and its overall strength.

Crystal Structure

Crystal structure refers to the orderly geometric spatial arrangement of atoms in the crystalline state. Think of it as how different parts of a Lego set come together to create a specific shape—each block has its place. Crystals have a beautifully regular and repeating pattern that extends in all three dimensions.

For our FeO example, the crystal structure is defined by the arrangement of Fe²⁺ cations and O²⁻ anions. Using the coordination number with respect to the radius ratios, it suggests that FeO has a regular pattern where the oxygen ions form a close-packed structure, and the iron ions snugly fit into the tetrahedral holes within this arrangement. This specific arrangement is just one example of how elements can come together to create the diverse world of crystallography, which can range from the simple to the complex, each with its unique attributes and physical characteristics.

Radius Ratio Rule

The radius ratio rule helps us predict the coordination number based on the size comparison between the cations and anions in a crystal. Simply put, it's a bit like deciding if a small, medium, or large-sized ball can fit snugly into the hollows of a sponge. Each size can only fit comfortably into a particular type of space. In the radius ratio of FeO, with the cation (iron) being smaller than the anion (oxygen), our ratio falls between 0.414 and 0.732, which leads us to a coordination number of 4. This means that the iron ions fit perfectly into the smaller 'pockets' offered by the oxygen ions in the structure.

The radius ratio rule offers a straightforward method to take the guesswork out of crystal structure prediction. It's a testament to the ingenuity of scientific rules that a simple calculation involving the ionic radii of elements can provide us a window into the microscopic, yet magnificently orderly world of crystals.