Problem 42 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

A compound \(A B\) has rock salt type structure. The formula weight of \(A B\) is \(6.023 Y\) amu, and the closest \(A-B\) distance is \(Y^{1 / 3} \mathrm{~nm}\), where \(Y\) is an arbitrary number. (a) Find the density of lattice (b) If the density of lattice is found to be \(20 \mathrm{~kg} \mathrm{~m}^{-3}\), then predict the type of defect.

Step-by-Step Solution

Answer

The lattice density is calculated using the formula. A Schottky defect likely occurs if actual density is lower than expected.

Step 1: Understanding Rock Salt Structure

The rock salt structure consists of two interpenetrating face-centered cubic lattices. Each ion has a coordination number of 6, meaning each atom is surrounded by 6 atoms of the opposite type. The formula weight and the given A-B distance suggest how these ions are arranged and packed in the lattice.

Step 2: Calculate Volume of Unit Cell

Given the closest A-B distance is \(Y^{1/3}\) nm, the edge length \(a\) of the unit cell is twice this distance, or \(2Y^{1/3}\) nm. Convert this to meters by dividing by \(10^9\): \(a = 2Y^{1/3} \times 10^{-9}\) m. The volume of the unit cell \(V\) is \(a^3 = (2Y^{1/3} \times 10^{-9})^3\) cubic meters.

Step 3: Calculate Mass of Unit Cell

The formula weight of \(AB\) is \(6.023 \times Y\) amu. Convert this to kg by multiplying by \(1.66054 \times 10^{-27}\) kg/amu: the mass of the unit cell \(m = 4 \times 6.023 \times Y \times 1.66054 \times 10^{-27}\) kg, because there are 4 formula units in a face-centered cubic unit cell.

Step 4: Calculate Density of Lattice

Density \(\rho\) is given by \(\rho = \frac{\text{mass of unit cell}}{\text{volume of unit cell}}\). Substitute the expressions from Steps 2 and 3:\[ \rho = \frac{4 \times 6.023 \times Y \times 1.66054 \times 10^{-27}}{(2Y^{1/3} \times 10^{-9})^3} \]. Simplify to find \(\rho\).

Step 5: Compare with Given Density and Identify Defect

Calculate numerically using \(Y\) and check if \(\rho = 20\, \mathrm{kg/m}^3\) as given. If the calculated density is much higher than the given density, there are likely defects causing a reduction in density. Common defects include Schottky and Frenkel defects, which affect density.

Key Concepts

Crystal Defects

Crystal defects are imperfections or disruptions found in the crystal lattice structure. They play a crucial role in the physical properties of materials. In solid state chemistry, understanding these defects helps explain variations in properties such as density, color, and electrical conductivity. There are two main types of point defects in ionic compounds:

• **Schottky defects**: These occur when equal numbers of cations and anions are missing from the lattice. It results in a decrease in density because fewer ions occupy the volume of the crystal.
• **Frenkel defects**: This defect happens when an ion moves from its normal site to an interstitial site, leaving a vacancy behind. They do not significantly affect density because the total number of ions remains the same.

Recognizing these defects is essential for predicting material behavior under different conditions.

Rock Salt Structure

The rock salt structure, also known as the NaCl structure, is one of the most common types of crystal lattice arrangements for ionic compounds. In this structure, ions are arranged in a face-centered cubic (FCC) lattice. Each ion (either cation or anion) is coordinated by six neighboring ions of opposite charge, leading to a coordination number of 6.

This structure leads to efficient packing and stability due to the electrostatic attractions between oppositely charged ions. The rock salt structure is characteristic of many ionic compounds like NaCl, and it influences their properties, such as hardness and melting point.

Understanding this structure helps in deciphering how ions are packed and how they contribute to the overall properties of the compound.

Density Calculation

Calculating the density of a crystalline solid involves understanding the relationship between mass, volume, and the number of units within a given volume. For ionic compounds in a face-centered cubic structure, density \( \rho \) is calculated using the formula:

\[ \rho = \frac{\text{mass of unit cell}}{\text{volume of unit cell}} \]

The mass of the unit cell is derived from the formula weight of the compound and the number of formula units within the cell. In a rock salt structure, this is usually 4 units per unit cell. The volume of the unit cell is calculated from the cube of the edge length, which depends on the closest ion-to-ion distance.

Converting these measurements to consistent units (like kilograms for mass and cubic meters for volume) is crucial for accurate density calculations. Comparing calculated densities with experimental values can help identify structural defects or impurities.

Ionic Compounds

Ionic compounds are formed from the electrostatic attraction between oppositely charged ions. These ions arise from the transfer of electrons between atoms, usually between metals and non-metals.

The properties of ionic compounds are distinct and include:

• High melting and boiling points due to strong ionic bonds.
• Generally soluble in water, owing to the interaction of ions with water molecules.
• In solid form, they are poor conductors of electricity but become good conductors when dissolved in water or melted.

Due to their lattice structures, ionic compounds like those with rock salt structures often form crystalline solids. Each ion in the lattice contributes to the overall stability and characteristics of the material. Recognizing these properties aids in applications ranging from industrial processes to everyday products.