Question

The radius of \(\mathrm{Ag}^{+}\)ion is \(126 \mathrm{pm}\) while that of \(\mathrm{I}\) ion is \(216 \mathrm{pm}\). The coordination number of \(\mathrm{Ag}\) in \(\mathrm{AgI}\) is (a) 8 (b) 6 (c) 4 (d) 2

Short answer

The coordination number of Ag in AgI is 4.

Step-by-step solution

Understand the Concept of Coordination Number

The coordination number of a central atom in a complex or crystal refers to the number of atoms (or ions) directly bonded to it. In minerals and ionic compounds, coordination number depends on the size of the ions and the geometry of the molecule.

Determining Factors for Coordination Number

The coordination number is affected by the relative sizes of the ions. Larger size differences can allow more ions to fit around each central ion, increasing the coordination number. Typically, the ratio between the radii of the cation and anion helps predict the coordination number.

Calculate the Radius Ratio

The radius ratio can be calculated using \[ \text{Radius Ratio} = \frac{\text{Radius of } \mathrm{Ag}^{+}}{\text{Radius of } \mathrm{I}^{-}} = \frac{126 \text{ pm}}{216 \text{ pm}} \approx 0.583 \]This ratio helps predict the coordination number based on standard geometrical considerations.

Interpret the Radius Ratio

Based on standard tables, a radius ratio of approximately 0.4 to 0.7 typically suggests a coordination number of 4, which corresponds to a tetrahedral geometry. This is common for an ionic compound like \( \mathrm{AgI} \).

Select the Correct Option

Given the calculated radius ratio and its interpretation, the coordination number in \( \mathrm{AgI} \) is determined to be 4. Hence, the correct answer is option (c) 4.

Key concepts

Radius Ratio

The radius ratio is an essential concept in determining the coordination number of an ionic compound. It is the ratio of the radius of the cation to the radius of the anion. In our example, for the compound \( \text{AgI} \), the radius of \( \text{Ag}^+ \) is \( 126 \text{ pm} \), and the radius of \( \text{I}^- \) is \( 216 \text{ pm} \). Therefore, the radius ratio can be calculated as follows:\[\text{Radius Ratio} = \frac{126 \text{ pm}}{216 \text{ pm}} \approx 0.583\]This numeric value indicates how well the cations fit into the spaces formed by the surrounding anions. A smaller radius ratio means the cations are much smaller compared to anions. This affects the possible ways the cations can be surrounded by anions, thereby influencing the coordination number.

• A radius ratio between 0.414 and 0.732 typically suggests a coordination number of 4, which relates to a tetrahedral coordination.
• More extreme radius ratios might suggest coordination numbers of different values, highlighting the importance of this simple calculation in creating chemical structures.

Cation and Anion Sizes

The sizes of the cations and anions in an ionic crystal directly impact the structure and properties of the compound. Cations are usually smaller than anions because they lose an electron shell upon ionization, whereas anions gain one. For \( \text{AgI} \), the \( \text{Ag}^+ \) ion has a size of \( 126 \text{ pm} \) while \( \text{I}^- \) is \( 216 \text{ pm} \). This difference in size influences how these ions arrange themselves in a crystal.

• Because \( \text{Ag}^+ \) is much smaller than \( \text{I}^- \), this allows iodine ions to form a four-sided frame (like a pyramid), where each \( \text{Ag}^+ \) ion fits into the central void.
• Smaller cations can fit into tetrahedral holes, which are smaller yet available within the regular close packed structures formed by anions.

The size of these ions has broader implications such as influencing the compound's melting point, solubility, and overall stability.

Tetrahedral Geometry

Tetrahedral geometry is a common type of spatial arrangement for atoms in various molecules and complex ions. This geometry arises when a central atom is surrounded by four other atoms placed at the corners of a tetrahedron. In the case of ionic compounds like \( \text{AgI} \), tetrahedral geometry is highly relevant.

• When we say that \( \text{Ag}^+ \) has a coordination number of 4 in \( \text{AgI} \), it means \( \text{Ag}^+ \) is surrounded by four \( \text{I}^- \) ions in a tetrahedral arrangement.
• This arrangement optimizes spatial distribution, balancing electrical charges by placing surrounding anions equidistantly to the central cation. This minimizes potential energy and maximizes stability.

Understanding this geometry is crucial since it affects the ionic bond's strength and the compound's physical properties. Moreover, the tetrahedral arrangement plays a vital role in dictating how \( \text{AgI} \) may interact with other chemicals or undergo phase changes.