Problem 102 - A Complete Resource Book in Chemistry for JEE Main

Question

A solid XY has NaCl structure. If radius of \(\mathrm{X}^{+}\)is 100 \(\mathrm{pm}\), the radius of \(\mathrm{Y}\)-ion is (a) \(136.6\) to \(241.6 \mathrm{pm}\) (b) \(341.6\) to \(134.6 \mathrm{pm}\) (c) \(128 \mathrm{pm}\) (d) \(136.8 \mathrm{pm}\)

Step-by-Step Solution

Answer

The radius of the Y-ion is between 136.6 pm and 241.6 pm, so option (a) is correct.

Step 1: Understanding the NaCl Structure

The NaCl structure, also known as the rock-salt structure, is a face-centered cubic lattice of ions. In this structure, each cation (X) is surrounded by six anions (Y), and each anion is surrounded by six cations, creating an octahedral coordination.

Step 2: Coordination Number and Radius Ratio

In a crystal structure with coordination number 6 (like NaCl), the ionic radius ratio rule determines the allowable size of ions. The radius ratio \(\frac{r}{R}\) for a stable octahedral coordination is approximately between 0.414 and 0.732, where \(r\) is the radius of the cation and \(R\) is the radius of the anion.

Step 3: Applying the Radius Ratio Formula

Given \(r (\mathrm{X}^{+}) = 100\ \mathrm{pm}\), calculate permissible \(R (\mathrm{Y}^{-})\) using the radius ratio range: \(\frac{100}{R} = 0.414\) gives \(R = \frac{100}{0.414} \approx 241.6\ \mathrm{pm}\) and \(\frac{100}{R} = 0.732\) gives \(R = \frac{100}{0.732} \approx 136.6\ \mathrm{pm}\).

Step 4: Deriving the Range for Y-ion Radius

Thus, the allowed range for \(R (\mathrm{Y}^{-})\) is from 136.6 pm to 241.6 pm, which means the possible size of \(\mathrm{Y}^{-}\) should fit within this range. By examining the given options, only option (a) corresponds to this calculated range.

Key Concepts

Ionic Radius Ratio

The ionic radius ratio is a crucial concept in understanding the structure of many ionic compounds, including NaCl which has a rock-salt structure. This ratio helps us determine the relative sizes of the cations and anions in the lattice structure. It is expressed as the ratio of the radius of the cation ( \(r\)) to the radius of the anion ( \(R\)). In the case of octahedral coordination, which is seen in the NaCl structure, the ionic radius ratio is quite specific. When an ion is surrounded by six opposite charges, stability is achieved if the radius ratio falls within the range of \(0.414 \, \text{to} \, 0.732\). This is critical because, at these values, the structure remains energetically favorable and stable. Many textbooks often list these ratios as guides for chemists to predict the types of coordination likely in different ionic compounds. If the ratio falls outside this range, the ions cannot maintain contact, disrupting the stable structure. Using this concept, we ensure that the ions stay in their correct positions, maintaining the integrity and geometry of the crystal lattice.

Octahedral Coordination

Octahedral coordination is a common arrangement in many crystal structures, including the NaCl structure. In this configuration, each ion is directly surrounded by six opposite charges, forming an octahedral shape. This happens because of electrostatic attraction. For instance, in the NaCl structure, each sodium ion ( \( ext{Na}^+\)) is surrounded by six chloride ions ( \( ext{Cl}^-\)), and vice versa. This ensures a balance of charge and stability within the crystal lattice. This arrangement maximizes the packing efficiency and minimizes repulsions between like charges. Each ion in the octahedral arrangement is equidistant from its neighbors, which provides the lattice with additional stability. The equal distance is important because if one ion is too close or too far, it can destabilize the whole structure. The octahedral coordination is not just limited to salts; it is commonly found in various other mineral structures and is one of the primary coordination modes seen in transition metal complexes.

Face-Centered Cubic Lattice

The face-centered cubic (FCC) lattice is a common type of crystal structure that is highly efficient in terms of space usage. In an FCC lattice, each corner and the center of each cube face within the crystal cell is occupied by one ion or atom. This arrangement is also found in the structure of NaCl, which exemplifies the rock-salt structure.In the NaCl face-centered cubic lattice, ions arrange themselves systematically in a manner that allows for dense packing. Here, chloride ions ( \( ext{Cl}^-\)) form the face-centered lattice, while sodium ions ( \( ext{Na}^+\)) fill in the octahedral holes, creating a perfectly balanced and stable crystal structure. The FCC arrangement allows for the maximum use of space, reducing voids and ensuring that the ions are as closely packed as possible. It’s crucial because any gaps or inconsistencies can cause structural weaknesses. This type of lattice is not exclusive to salts. It is also found in metals, like copper and aluminum, highlighting its versatility and widespread use in varying contexts across materials science.