Question

The ionic radii of \(\mathrm{Rb}^{+}\)and \(\mathrm{I}^{-}\)are \(1.46 \AA\) and \(2.16 \AA\) respectively. The most probable type of structure exhibited by it is (a) \(\mathrm{CaF}_{2}\) type (b) \(\mathrm{ZnS}\) type (c) \(\mathrm{CsCl}\) type (d) \(\mathrm{NaCl}\) type

Short answer

The structure of \( \mathrm{RbI} \) is most likely NaCl type.

Step-by-step solution

Understand the Problem

We are given the ionic radii of \(\mathrm{Rb}^{+}\) and \(\mathrm{I}^{-}\) and asked to determine the most probable structure type of the compound \(\mathrm{RbI}\). The possible structures are \(\mathrm{CaF}_{2}\) type, \(\mathrm{ZnS}\) type, \(\mathrm{CsCl}\) type, and \(\mathrm{NaCl}\) type.

Evaluate the Ratio of Ionic Radii

The structure type is determined by the ratio of ionic radii, \(\frac{r_+}{r_-}\), where \(r_+\) is the radius of the cation and \(r_-\) is the radius of the anion. For \(\mathrm{Rb}^{+}\) and \(\mathrm{I}^{-}\), \(r_+ = 1.46 \ \text{Å}\) and \(r_- = 2.16 \ \text{Å}\). Calculate the ratio: \[\frac{r_+}{r_-} = \frac{1.46}{2.16} = 0.68\]

Determine the Suitable Structure

Compare the calculated ionic radius ratio to the typical ratios for known crystal structures:- \(\text{NaCl type (rock salt)}\): \(0.414 < \frac{r_+}{r_-} < 0.732\)- \(\text{CsCl type}\): \(\frac{r_+}{r_-} \approx 0.732\)- \(\text{ZnS type (sphalerite)}\): \(0.225 < \frac{r_+}{r_-} < 0.414\)- \(\text{CaF}_2\) type: not applicable here.The calculated ratio of 0.68 falls within the range for NaCl type structures.

Key concepts

Ionic Compounds

Ionic compounds are formed when atoms transfer electrons, resulting in positive and negative ions that attract each other. These electrostatic attractions hold the ions in a stable structure. One of the hallmark features of ionic compounds is their ability to form well-defined crystal lattices. This happens due to the regular and repeating arrangement of ions, which reflect the levels of attraction and types of interactions between each type of ion.

• Positive Ions (Cations): These are atoms that have lost electrons. In our exercise, \(\mathrm{Rb}^{+}\) is a typical cation.
• Negative Ions (Anions): These atoms gain electrons. An example from the exercise is \(\mathrm{I}^{-}\).
• Stability: The greater the attraction between ions, the more stable the compound. This attraction is directly related to the charges on the ions and their sizes.

When ions like \(\mathrm{Rb}^{+}\) and \(\mathrm{I}^{-}\) combine, they create ionic compounds such as \(\mathrm{RbI}\), which can exhibit different crystal structures based on their ionic radii.

Crystal Structures

Crystal structures describe how ions are arranged in a solid. The structure depends on sizes of ions, their charges, and the cation-anion radius ratio. Here, we'll look at a few typical structures:

• NaCl Type: Often referred to as the rock salt structure, this is one of the most common types. It occurs when the cation-anion radius ratio is between 0.414 and 0.732. In our exercise, \(\mathrm{RbI}\) fits this category with a ratio of 0.68, indicating a likely NaCl structure.
• CsCl Type: Another common structure, typically forming when the radius ratio is around 0.732. It is a simple cubic lattice where both the cation and anion occupy equivalent lattice points.
• ZnS and CaF₂ Types: Despite being mentioned, these aren't as relevant for \(\mathrm{RbI}\) given its radius ratio. These structures have more specific radius requirements that \(\mathrm{RbI}\) does not meet.

Understanding these structures is crucial, as they influence the compound's properties, such as melting points, solubility, and hardness.

Cation-Anion Radius Ratio

The cation-anion radius ratio is a key factor that determines the type of crystal structure an ionic compound will adopt. This ratio is calculated by dividing the radius of the cation by the radius of the anion, represented by the formula: \[ \frac{r_+}{r_-} \]

• Importance of Ratio: It helps predict the possible crystal structure based on the sizes of the involved ions. It essentially tells us the relative size difference between the cation and anion.
• Implications for Structure Types:
  • NaCl Type: A ratio between 0.414 and 0.732 suggests that the ions can pack efficiently in a face-centered cubic system.
  • CsCl Type: Ratios close to or around 0.732 are ideal for cubic structures with coordinated symmetry at each lattice point.
• Relevance to RbI: With a ratio of 0.68, \(\mathrm{RbI}\) aligns with the NaCl type structure, indicating a rock salt configuration which is capable of forming stable ionic bonds due to effective packing.

By understanding this ratio, predictions about the compound’s behavior and interaction become much clearer, aiding in the study of materials science.