Problem 107 - A Complete Resource Book in Chemistry for JEE Main

Question

In a compound \(\mathrm{XY}\), the ionic radii \(\mathrm{X}^{+}\)and \(\mathrm{Y}\) are \(88 \mathrm{pm}\) and \(200 \mathrm{pm}\) respectively. What is the coordination number of \(\mathrm{X}^{+}\)? (a) 4 (b) 6 (c) 8 (d) 10

Step-by-Step Solution

Answer

The coordination number of \( \mathrm{X}^{+} \) is 6.

Step 1: Understanding Coordination Number

The coordination number of an ion in a compound refers to the number of opposite charged ions that are directly surrounding it. For a cation like \( \mathrm{X}^{+} \) in compound \( \mathrm{XY} \), we need to consider how many anions (\( \mathrm{Y}^{-} \)) can pack around it.

Step 2: Calculating Radius Ratio

The radius ratio can be calculated using the formula: \( \text{Radius Ratio} = \frac{r_c}{r_a} \), where \( r_c \) is the radius of the cation and \( r_a \) is the radius of the anion. Here, \( r_c = 88 \mathrm{pm} \) and \( r_a = 200 \mathrm{pm} \). Thus, \( \text{Radius Ratio} = \frac{88}{200} = 0.44 \).

Step 3: Interpreting Radius Ratio

The radius ratio helps us predict the possible coordination number. For cations, a radius ratio of different ranges corresponds to different coordination numbers. Generally, a radius ratio of 0.414 to 0.732 corresponds to a coordination number of 6.

Step 4: Conclusion from Radius Ratio

The calculated radius ratio for \( \mathrm{X}^{+} \) is 0.44, which falls in the range of 0.414 to 0.732. Hence, the coordination number of \( \mathrm{X}^{+} \) is 6 in the compound \( \mathrm{XY} \).

Key Concepts

Coordination Number

The coordination number is a term used to describe the number of ions of opposite charge that directly surround a given ion in a crystal lattice structure. It plays a critical role in understanding the arrangement of ions within a compound. For instance, when discussing a compound like \( \mathrm{XY} \), examining the coordination number helps reveal how many \( \mathrm{Y}^{-} \) anions can cluster around a single \( \mathrm{X}^{+} \) cation in the compound.
Certain coordination numbers are more common due to how efficiently these structures can pack in space. For example, common coordination numbers like 4, 6, or 8 are associated with tetrahedral, octahedral, and cubic arrangements respectively. These geometric arrangements minimize repulsion between ions, stabilizing the structure.
In the case of \( \mathrm{X}^{+} \), understanding its coordination number is essential to predicting the structure and properties of the compound \( \mathrm{XY} \). A number like 6 indicates an octahedral shape where each \( \mathrm{X}^{+} \) is surrounded symmetrically by 6 \( \mathrm{Y}^{-} \) ions.

Radius Ratio

Radius ratio is calculated as the ratio of the radius of a cation to that of an anion. This simple ratio offers significant insight into the potential structure of the compound. By using the formula:

• \( \text{Radius Ratio} = \frac{r_c}{r_a} \)

where \( r_c \) represents the radius of the cation and \( r_a \) represents the radius of the anion, we can deduce critical information about possible coordination numbers.
For instance, with the compound \( \mathrm{XY} \), if \( r_c = 88 \, \mathrm{pm} \) and \( r_a = 200 \, \mathrm{pm} \), then the calculated radius ratio is \( 0.44 \).
The value obtained from the radius ratio can then be compared against known ranges:

• A radius ratio between 0.414 and 0.732 often suggests a coordination number of 6, reflecting an octahedral arrangement.
• A smaller or larger ratio might suggest a different coordination geometry, like tetrahedral or cubic.

Understanding the radius ratio is key to predicting which geometric arrangement offers the most stable configuration for ions in a lattice.

Cations and Anions

Cations and anions are the positively and negatively charged ions in a compound, respectively. In ionic compounds like \( \mathrm{XY} \), these ions arrange themselves in a regular, repeating structure to balance the electrical charges.
Cations, like \( \mathrm{X}^{+} \), are positively charged due to a deficiency of electrons. Typically smaller than anions due to this electron loss, cations take up less space in the lattice and are surrounded by a number of anions depending on their charge and size, which contributes to calculating the coordination number.
Anions, in contrast, like \( \mathrm{Y}^{-} \), carry a negative charge because of an excess of electrons. They are generally larger than their corresponding cations, allowing multiple anions to surround a single cation in a structure. The balance between the size of cations and anions greatly influences the type and strength of ionic bonds, which in turn defines the physical properties of the compound.
Understanding the characteristics of cations and anions aids in predicting how they interact in compounds, which is essential for the arrangement and stability of the resulting ionic lattice.