Problem 71 - A Complete Resource Book in Chemistry for JEE Main

Question

The structure of \(\mathrm{TlCl}\) is similar to \(\mathrm{CsCl}\). What would be the radius ratio in \(\mathrm{TlCl} ?\) (a) \(0.225-0.414\) (b) \(0.155-0.225\) (c) \(0.732-1.000\) (d) \(0.414-0.732\)

Step-by-Step Solution

Answer

The radius ratio for \\( \text{TlCl} \\) is \\( 0.732-1.000 \\), which matches option (c).

Step 1: Understanding the Structure

The \( \text{CsCl} \) structure corresponds to a body-centered cubic arrangement where one of the ions, usually the larger one (Cs), occupies the cube's center, while the smaller ions (Cl) are at the cube's corners.

Step 2: Determine the Coordination Number

In the \( \text{CsCl} \) structure, each Cs atom is coordinated with 8 Cl atoms, so its coordination number is 8.

Step 3: Applying the Radius Ratio Rule

For a coordination number of 8, the ideal radius ratio \( r_{\text{cation}}/r_{\text{anion}} \) should be between \( 0.732 \) and \( 1.000 \). This is because larger cations permit more anions surrounding them, and this ratio corresponds to the stable configuration of the \( \text{CsCl} \) type structure.

Key Concepts

Coordination Number

In crystal structures, the coordination number is an essential concept. It refers to the number of nearest neighboring atoms or ions surrounding a particular atom or ion. This is a key factor in understanding how atoms are arranged within a solid. In the context of the CsCl structure, the coordination number is particularly interesting. Each cesium ion ( Cs ) is surrounded symmetrically by eight chloride ions ( Cl ) forming a cubic arrangement. This implies that the coordination number for both Cs and Cl in CsCl structure is 8. Knowing the coordination number helps in predicting and explaining the physical properties of the crystal, such as its stability and how it might interact with different elements.

CsCl Structure

The CsCl structure is an example of a body-centered cubic (bcc) arrangement, but with a twist. Normally, in a bcc lattice, identical atoms occupy the lattice points at the corners of the cube and one at its center. However, in CsCl, two different types of ions, Cs and Cl, alternate between these positions.

• The larger Cs cation often sits at the cube's center.
• The smaller Cl anions are located at the corners of the cube.

This arrangement leads to each ion being surrounded by eight of the opposing type, maintaining the structure's stability. The CsCl structure is common in compounds where there is a significant difference in size between the cation and the anion. This configuration provides insight into the compound's potential interactions and its mechanical and thermal properties.

Radius Ratio Rule

The radius ratio rule is a guideline used to predict the stability of ionic structures. It compares the ionic radii of the cation and anion in a compound. The rule suggests that for a compound to be stable, the ratio of the cation's radius ( r_{ ext{cation}} ) to the anion's radius ( r_{ ext{anion}} ) should fall within certain ranges, depending largely on the coordination number.

• For a coordination number of 8, like in the CsCl structure, this ratio should be between 0.732 and 1.000.

Simply put, this means the cation must be of sufficient size to allow the anions to surround it efficiently, ensuring the structure does not collapse or distort. This range indicates that the cation is large enough to fit comfortably in the gaps created by the anions, maintaining the stability of the structure. Using the radius ratio is a powerful way to predict possible structures for new and hypothetical compounds.