Problem 128 - A Complete Resource Book in Chemistry for JEE Main

Question

Ice crystallizes in hexagonal lattice. At a given temperature, the density of ice is \(0.92 \mathrm{gcm}^{-3}\). The volume of unit cell is \(1.3 \times 10^{-22} \mathrm{~cm}^{3} .\) The number of \(\mathrm{H}_{2} \mathrm{O}\) molecules per unit cell is

Step-by-Step Solution

Answer

The number of \(\mathrm{H}_2\mathrm{O}\) molecules per unit cell is approximately 4.

Step 1: Determine Molar Mass of Water

The molar mass of a water molecule (\(\mathrm{H}_2\mathrm{O}\)) is determined by adding the atomic masses: \(2 \times \text{Hydrogen (1 g/mol)} + 1 \times \text{Oxygen (16 g/mol)} = 18 \text{ g/mol}\).

Step 2: Calculate the Molar Volume

Using the density of ice, \(0.92 \text{ g/cm}^3\), we can calculate the molar volume of ice using the formula: \( \text{Molar Volume} = \frac{\text{Molar Mass}}{\text{Density}} = \frac{18 \text{ g/mol}}{0.92 \text{ g/cm}^3} \approx 19.57 \text{ cm}^3/\text{mol}\).

Step 3: Find Avogadro's Number Per Unit Volume

The number of molecules per unit volume can be found using the relation: \( \frac{1}{\text{Molar Volume in Liters}} \times \text{Avogadro's Number} = \frac{1}{0.01957} \times 6.022 \times 10^{23} \approx 3.08 \times 10^{25} \text{ molecules/cm}^3\).

Step 4: Calculate the Number of Molecules Per Unit Cell

Multiply the number of molecules per volume by the volume of the unit cell: \( \text{Number of molecules/unit cell} = 3.08 \times 10^{25} \text{ molecules/cm}^3 \times 1.3 \times 10^{-22} \text{ cm}^3 \approx 4 \).

Key Concepts

Hexagonal lattice

In the world of crystallography, understanding lattice structures is key. A hexagonal lattice is one such structure where points are arranged in a repeating hexagon pattern. This is important in materials like ice, where the water molecules form a repetitive pattern extending in three dimensions. The hexagonal structure aids in determining how molecules pack together. For instance:

• Each unit in a hexagonal lattice contains a repetitive pattern of hexagons.
• This arrangement allows molecules to be consistently spaced, influencing the material's overall characteristics.
• Materials with hexagonal lattices often display interesting properties, like the unique floating characteristic of ice.

Understanding the hexagonal lattice of ice helps us comprehend why ice is less dense than liquid water, allowing it to float.

Density of ice

Density is an essential concept in both physics and chemistry. For ice, the density is particularly interesting at approximately 0.92 g/cm³. This value means that ice is less dense than liquid water and explains why icebergs float on oceans. Key points about density include:

• Density is derived from the ratio of mass to volume.
• With water, density decreases when it freezes, leading to ice's unique behavior.
• Understanding density helps in solving problems related to buoyancy and is critical in studying climatology and environmental science.

By calculating density, one can predict how substances will interact with their environment, much like how ice interacts with bodies of water on Earth.

Molar mass

In chemistry, the molar mass of a substance is the mass of one mole of its entities (atoms, molecules, etc.). For water (H₂O), this is calculated by adding the atomic masses of its elements: two hydrogen atoms and one oxygen atom. The formula becomes:

• Molar mass of Hydrogen = 1 g/mol, so two hydrogens = 2 g/mol.
• Molar mass of Oxygen = 16 g/mol.
• Total = 18 g/mol for a single water molecule.

Understanding molar mass is crucial for converting between grams and moles, which is a fundamental step in many chemical calculations. It serves as the bridge connecting an element's atomic weight to everyday measurements.

Unit cell volume

In crystallography, the unit cell volume is crucial for understanding the properties of crystalline structures. The unit cell is the smallest repeating pattern in a crystal lattice, and its volume can provide insights into the material's overall characteristics. For ice, samples are often measured in cubic centimeters (cm³) for practical convenience. Some important aspects include:

• Unit cell volume helps determine the number of molecules within it.
• A larger volume may indicate a loosely packed structure, explaining the lower density.
• This measure assists in material sciences to predict and explain phenomena like thermal expansion and conductivity.

By calculating the unit cell volume from given dimensions, scientists and engineers gain valuable data about a material's composition and potential applications.