Problem 96 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following statements is correct? (1) The coordination number of each type of ion in CsCl crystal is 8 (2) A metal that crystallizes in BCC structure has a coordination number of 12 (3) A unit cell of an ionic crystal shares some of its ion with other units cells (4) the length of unit cell in \(\mathrm{NaCl}\) is \(552 \mathrm{pm}\) (r \(\mathrm{Na}^{+}=\) \(\left.95 \mathrm{pm}, \mathrm{r} \mathrm{Cl}^{-}=181 \mathrm{pm}\right)\) (a) 1,2 (b) 1,3 (c) 2,3 (d) 2,4

Step-by-Step Solution

Answer

The correct statements are (1) and (3), so option (b) is correct.

Step 1: Understanding Coordination Number in CsCl

CsCl has a coordination number of 8 for both Cs and Cl ions because each Cs ion is surrounded by 8 Cl ions and vice versa, which defines its simple cubic or body-centered cubic structure.

Step 2: Coordination Number in BCC Structures

In a body-centered cubic (BCC) structure, each atom is in contact with 8 nearest neighbors. Hence, the coordination number for a metal in BCC is 8, not 12. Therefore, statement 2 is incorrect.

Step 3: Unit Cell Sharing in Ionic Crystals

In an ionic crystal, the unit cell shares some of its ions with adjacent cells. Hence, statement 3 is correct because each unit cell is a part of a larger lattice structure, and ions at the corners, edges, or faces of the cell are shared with adjacent unit cells.

Step 4: Determining Unit Cell Length in NaCl

The calculated length of the unit cell can be determined from the formula for the edge length in terms of the ionic radii. Generally, length \( a = 2(r_{Na^+} + r_{Cl^-}) \). Calculating gives \( a = 2(95 + 181) \) pm = 552 pm, making the fourth statement correct.

Step 5: Final Selection of Correct Statements

From the analysis of each statement: Statements 1 and 3 are correct. Statement 1 agrees with the coordination number of ions in CsCl, and Statement 3 accurately describes the sharing of ions in ionic crystal structures.

Key Concepts

CsCl coordination number

The concept of coordination number is crucial when analyzing crystal structures. In a CsCl crystal, each cesium (Cs) ion is surrounded by eight chlorine (Cl) ions, and vice versa. This mutual arrangement results in a coordination number of 8 for both types of ions. This high level of coordination is typical of CsCl because its structure allows each Cs ion to directly interact with 8 Cl ions, forming a cubic geometry. The symmetric arrangement maintains stability and ensures uniform distribution of electrostatic forces across the crystal lattice.

BCC structure

The Body-Centered Cubic (BCC) structure is a common crystal arrangement for metals. In this configuration, each atom is at the center of an imaginary cube, surrounded by eight corner atoms. The coordination number, which represents the number of nearest-neighbor atoms, is 8 in a BCC structure, contrary to the incorrect statement that it would be 12. This structure is quite efficient in tightly packing atoms, although it is less dense compared to some other structures, like the Face-Centered Cubic (FCC) structure. The BCC structure is important in materials with high strength and hardness.

ionic crystal unit cell

In ionic crystals, the unit cell represents the smallest repeating pattern in the crystal lattice. However, it does not operate in isolation. The ions at the corners, edges, or faces of each unit cell are shared among neighboring unit cells. This sharing reflects the continuous nature of the crystal structure, where atoms, ions, or molecules spanning multiple unit cells contribute to the entire solid's stability. This interconnectedness is the reason ionic crystals, such as NaCl, form extensive lattice networks. Thus, understanding ionic crystal unit cells helps explain how they maintain structural integrity and exhibit distinct physical properties.

NaCl unit cell length

The length of the unit cell in sodium chloride (NaCl) is calculated based on ionic radii. Each NaCl unit cell can be visualized as a cube, where

• Na⁺ ions touch four Cl^- ions at the unit cell corners.
• The total ionic radius is the sum of half the length of the cell.

Using this, the unit cell's edge length can be determined by the formula \[ a = 2(r_{Na^+} + r_{Cl^-}) \] where

• \( r_{Na^+} = 95 \, \text{pm} \)
• \( r_{Cl^-} = 181 \, \text{pm} \).

Thus, \[ a = 2(95 + 181) \, \text{pm} = 552 \, \text{pm} \].This confirms the correctness of the fourth statement regarding NaCl unit cell length, illustrating the precision and predictability inherent in crystallographic calculations.