Problem 142 - A Complete Resource Book in Chemistry for JEE Main

Question

The edge length of a face centred cubic cell of an ionic substance is \(508 \mathrm{pm}\). If the radius of the cation is 110 \(\mathrm{pm}\), the radius of the anion is (a) \(288 \mathrm{pm}\) (b) \(398 \mathrm{pm}\) (c) \(618 \mathrm{pm}\) (d) \(144 \mathrm{pm}\)

Step-by-Step Solution

Answer

The radius of the anion is 144 pm.

Step 1: Understand the Problem

We are given a face-centered cubic (FCC) unit cell with an edge length of 508 pm. We know the radius of the cation is 110 pm, and we need to find the radius of the anion.

Step 2: Identify Relevant Formula

In a face-centered cubic lattice, the relationship between the radii of cation and anion (R) and the edge length (a) is given by: \( a = 2R + 2r_c \), where \( r_c \) is the radius of the cation and \( R \) is the radius of the anion.

Step 3: Set Up the Equation

Substitute the given values into the formula: \( 508 = 2R + 2 \times 110 \).

Step 4: Solve for Anion Radius

First, calculate \( 2 \times 110 = 220 \). Then, subtract 220 from 508 to find \( 2R \): \( 2R = 508 - 220 = 288 \). Finally, solve for \( R \) by dividing by 2: \( R = \frac{288}{2} = 144 \mathrm{pm} \).

Step 5: Verify Your Solution

Confirm that all calculations were correctly performed and the conditions of the problem are satisfied, leading us to conclude that the radius of the anion is 144 pm.

Key Concepts

Ionic Radius

The ionic radius is a measure of the size of an ion in a crystal lattice. It is crucial when understanding how ions pack together within a lattice structure.
The ionic radius differs from one ion to another due to the number of electrons they possess and their subsequent energy levels.

• Anions, which are negatively charged ions, typically have larger radii because their gaining of electrons can spread out the charge over a larger volume.
• Cations, which are positive, are generally smaller as they lose electrons, leading to less electron-electron repulsion.

In the context of a Face-Centered Cubic (FCC) lattice structure, the ionic radius is important for determining how closely packed the ions are. The given exercise shows how we can use known values of ionic radii in conjunction with lattice parameters to find unknown radii.
This highlights the relationship between structure and properties in solid-state chemistry.

Lattice Parameters

Lattice parameters are fundamental measurements that describe the size and shape of a unit cell in a crystal lattice. They include dimensions such as the edge length, angles between edges, and sometimes even the relationships between different dimensions of the structure.
In a face-centered cubic lattice, the key lattice parameter to consider is the edge length. This dimension directly influences the calculation of ionic radii when determining the arrangement of ions within the cell.

• For FCC structures, lattice parameters help relate the distances between ions to measurable quantities like edge lengths.
• They are used in equations that take into account the geometry of the crystal, such as the ones employed in the exercise.

In the provided exercise, the edge length of 508 pm was used as a crucial piece of information to find the anion's radius. Understanding lattice parameters helps students grasp how crystalline structures are quantitatively analyzed.

Cubic Unit Cell

The cubic unit cell is the simplest and most basic cell geometry within crystal structures, characterized by its cube-shaped geometry. Cubic unit cells come in several forms, including the simple cubic, body-centered cubic (BCC), and face-centered cubic (FCC) structures, all of which have different coordination numbers and packing efficiencies.
In the face-centered cubic unit cell mentioned in the exercise, each face of the cube has an ion at its center, in addition to the corners of the cube.

• This arrangement results in a coordination number of 12, meaning each ion is surrounded by 12 other ions.
• FCC structures are notable for their high packing efficiency, which is close to 74%.

Understanding the structure of a cubic unit cell, especially of an FCC, helps students visualize how ions fit together within a given space. This knowledge allows for the effective calculation of properties like ionic radii, crucial in many practical applications of solid-state chemistry.