Problem 103 - A Complete Resource Book in Chemistry for JEE Main

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

Step-by-Step Solution

Answer

The most probable structure is the NaCl type.

Step 1: Calculate the Radius Ratio

The radius ratio is calculated by dividing the radius of the cation by the radius of the anion. Therefore, the radius ratio \( \text{RR} \) can be found by: \[ \text{RR} = \frac{1.46}{2.16} \approx 0.676 \] where \( 1.46 \AA \) is the radius of \( \text{Rb}^+ \) and \( 2.16 \AA \) is the radius of \( \text{I}^- \).

Step 2: Compare with Known Values

The radius ratio 0.676 falls within the range typically associated with the octahedral (NaCl-type) structure. Specifically, a radius ratio between roughly 0.414 and 0.732 is indicative of octahedral coordination.

Step 3: Identify the Structure Type

Based on the radius ratio calculated and the known range of radius ratios for different crystal structures, the most probable structure type for \( \text{RbI} \) is \( \text{NaCl} \) type. This is because the calculated radius ratio aligns with that typical of the \( \text{NaCl} \) structure.

Key Concepts

Radius Ratio

Understanding the concept of radius ratio is essential when studying the structures of ionic compounds. The radius ratio is a straightforward calculation where you divide the ionic radius of the cation by that of the anion. Here, it is calculated using the given radii of \( \text{Rb}^+ \) and \( \text{I}^- \). To recap from the exercise:

• The radius of \( \text{Rb}^+ \) is \( 1.46 \ \mathrm{\AA} \).
• The radius of \( \text{I}^- \) is \( 2.16 \ \mathrm{\AA} \).

Therefore, the radius ratio (RR) is:\[ \text{RR} = \frac{1.46}{2.16} \approx 0.676 \]This calculation helps in determining which crystal structure a compound is likely to adopt. The radius ratio influences how ions pack together in a solid state, affecting the overall stability and arrangement of the crystal lattice.

Crystal Structure

The term "crystal structure" refers to the ordered arrangement of atoms, ions, or molecules in a crystalline material. The arrangement is defined by repeating patterns in three dimensions, which can influence the material's properties. Several types of crystal structures are common for ionic compounds, including:

• NaCl-type structure (or rock salt structure) - typically has a face-centered cubic lattice with octahedral coordination.
• CaF₂-type structure (or fluorite structure) - characterized by a simple cubic lattice.
• ZnS-type structure (or zinc blende structure) - also features tetrahedral coordination within its face-centered cubic structure.
• CsCl-type structure - known for its body-centered cubic lattice.

When identifying the crystal structure of a compound like RbI, experts use the radius ratio and other properties to compare against known structures.

NaCl Structure

The \( \text{NaCl} \) structure, also known as the rock salt structure, is a classic example of ionic crystals where each ion is surrounded by six oppositely charged ions, forming an octahedral coordination. This is a very stable arrangement, allowing strong ionic bonds to form, similar to how Lego bricks fit together.In a typical \( \text{NaCl} \) structure:

• The lattice is face-centered cubic, creating a repeating pattern of ions.
• Each \( \text{Na}^+ \) ion is surrounded by six \( \text{Cl}^- \) ions, and vice versa.
• This arrangement results in a highly stable and symmetrical crystal lattice.

For \( \text{RbI} \), the calculated radius ratio of approximately 0.676 aligns with the characteristics of an \( \text{NaCl} \) structure, confirming the likeliest crystal type for this compound.

Octahedral Coordination

Octahedral coordination is a specific arrangement in ionic compounds where each cation is surrounded symmetrically by six anions, or vice versa. This structure forms an octahedron around each central ion. It is common in many ionic crystals, like those following the \( \text{NaCl} \) structure.Features of octahedral coordination include:

• The ions are configured so that the cation sits at the center of an octahedron.
• Each anion is one of the six vertices of the octahedron, stabilizing the central ion.
• This configuration allows for efficient packing and typically occurs when the radius ratio is between approximately 0.414 and 0.732.

When the radius ratio was calculated to be 0.676 for \( \text{RbI} \), it indicated that the compound would likely adopt an octahedral coordination, which is consistent with the stable and efficient packing seen in the \( \text{NaCl} \) crystal structure.