Question

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

Short answer

The most probable type of structure exhibited by \(\mathrm{Rb}^{+}\) and \(\mathrm{I}^{-}\) is NaCl type.

Step-by-step solution

Understanding Ionic Radii in Crystal Structures

In crystal structures, the type of structure formed is influenced by the ratio of the ionic radii of the cation to the anion. This ratio helps predict the coordination number of the ions, which refers to the number of oppositely charged ions that surround a given ion in the crystal lattice.

Calculating the Cation-to-Anion Radii Ratio

Find the ratio of the ionic radii of \(\mathrm{Rb}^{+}\) to \(\mathrm{I}^{-}\) by dividing their radii: \[\text{Ratio} = \frac{\text{Radius of } \mathrm{Rb}^{+}}{\text{Radius of } \mathrm{I}^{-}} = \frac{1.46}{2.16}\]

Comparing the Ratio to Known Structure Types

Compare the calculated ratio to the typical ionic radii ratios for different crystal structures. For CsCl type, the ratio ranges around 0.73 to 1.00; for NaCl type, 0.41 to 0.73; for ZnS type, less than 0.41; and for CaF2 type, greater than 1.00.

Identifying the Most Probable Type of Structure

Based on the ratio calculated, determine the most probable crystal structure. If the ratio falls within the range of a particular structure type, that structure is the likely arrangement for the compound.

Conclusion

Since the calculated ratio lies within the range of 0.41 to 0.73, the most probable type of structure exhibited by \(\mathrm{Rb}^{+}\) and \(\mathrm{I}^{-}\) is the NaCl type.

Key concepts

Ionic Radii Ratio

The ionic radii ratio is a pivotal factor when considering the packing of ions within an ionic crystal. It is calculated by simply dividing the radius of the cation (positively charged ion) by the radius of the anion (negatively charged ion). This ratio plays an essential role in determining the stability and type of the crystal lattice that will form between ions.

In our exercise, the ionic radii ratio aids in predicting which crystal structure type \(\mathrm{Rb}^{+}\) and \(\mathrm{I}^{-}\) will likely form. Mathematically, it is represented as the ratio between the ionic radii of Rubidium and Iodide ions, which is \[\frac{1.46 \dot{\mathrm{A}}}{2.16 \dot{\mathrm{A}}}\]. This calculated ratio then can be used to match with known values for various types of crystal structures, leading us to the most probable type. Understanding these ratios is crucial for chemistry students as it links the physical size of ions to the geometrical structure they form, influencing properties like melting point, hardness, and solubility.

Coordination Number

The coordination number in ionic crystals is a term used to define the number of nearest-neighbor oppositely charged ions surrounding a particular ion within a crystal lattice. It indicates how an ion is geometrically arranged with its neighbours and thus, can significantly affect the stability and physical properties of the crystal.

The coordination number is primarily influenced by the ionic radii ratio, as the size compatibility between the cations and anions determines how many of each can closely pack around the other. For instance, in our exercise, by predicting that \(\mathrm{Rb}^{+}\) and \(\mathrm{I}^{-}\) will form a NaCl-type structure, we are implying a coordination number of 6, meaning each ion is surrounded by six of the opposite charge. This insight is useful as it shapes the understanding of how crystal structures are formed and allows scientists and engineers to predict and modify the characteristics of materials.

Crystal Lattice

The crystal lattice represents the 3D arrangement of ions in a crystal, a orderly, repeating pattern that extends in all spatial dimensions. This arrangement is crucial, as it provides the framework for understanding the material's characteristics, including its optical, electrical, and magnetic properties.

In the context of the given exercise, knowing that \(\mathrm{Rb}^{+}\) and \(\mathrm{I}^{-}\) form a NaCl-type structure gives us a clear image of its crystal lattice: each \(\mathrm{Rb}^{+}\) ion is at the center of a cube surrounded by eight \(\mathrm{I}^{-}\) ions at the corners, and vice versa, forming a face-centered cubic (FCC) lattice. This type of structure is highly efficient packing for ions, assuming ionic radii that accommodate such a layout. Hence, the interplay between ionic sizes and shapes directly leads to the formation of different crystal lattices, determining the identity of countless solid materials.