Problem 109 - A Complete Resource Book in Chemistry for JEE Main

Question

Sodium metal crystallizes as a body-centred cubic lattice with the cell edge \(4.29 \AA\). What is the radius of sodium atom? (a) \(2.371 \times 10^{-7} \mathrm{~cm}\) (b) \(1.857 \times 10^{-8} \mathrm{~cm}\) (c) \(3.817 \times 10^{-8} \mathrm{~cm}\) (d) \(7.312 \times 10^{-7} \mathrm{~cm}\)

Step-by-Step Solution

Answer

The radius of the sodium atom is approximately \( 1.857 \times 10^{-8} \, \text{cm} \).

Step 1: Understanding BCC Structure

A body-centered cubic (BCC) lattice has an atom at each corner of the cube and one atom at the center. Atoms at the corners touch the central atom along the body diagonal of the cube.

Step 2: Formula for Body Diagonal Length

The body diagonal of a cube with edge length \( a \) is given by the formula: \( \sqrt{3} \times a \). For sodium, \( a = 4.29 \AA = 4.29 \times 10^{-8} \text{ cm} \).

Step 3: Relationship Between Atomic Radius and Body Diagonal

In a BCC lattice, the body diagonal is equal to four times the atomic radius \( (2r) \), where \( r \) is the radius of one atom. Hence, the formula is \( 4r = \sqrt{3} \times a \).

Step 4: Solving for Atomic Radius

Substitute the edge length \( a \) into the formula: \[ 4r = \sqrt{3} \times 4.29 \times 10^{-8} \]. \[ r = \frac{\sqrt{3} \times 4.29 \times 10^{-8}}{4} \]. Calculate \( r \).

Step 5: Calculation

Calculate \( \sqrt{3} \times 4.29 \) and then divide by 4. \( \sqrt{3} \approx 1.732 \), so \( \sqrt{3} \times 4.29 \approx 7.4355 \). Dividing by 4, \( r = \frac{7.4355}{4} \times 10^{-8} \approx 1.8589 \times 10^{-8} \mathrm{~cm} \).

Step 6: Comparing to Given Options

The calculated atomic radius \( 1.8589 \times 10^{-8} \mathrm{~cm} \) is closest to option (b) \( 1.857 \times 10^{-8} \mathrm{~cm} \).

Key Concepts

Atomic Radius Calculation

Atomic radius is a key concept when discussing the structure of crystallized metals such as sodium. It provides us with an idea of the size of the atoms in the crystal lattice. The atomic radius can be calculated by understanding the geometric arrangement of the atoms in the crystal structure. For a body-centered cubic (BCC) lattice like that of sodium, the calculation involves the cube's geometry.

To find the atomic radius in a BCC lattice, we focus on the relationship between the atomic radius and the cube's body diagonal. In a BCC lattice, the body diagonal is four times the atomic radius. The formula for the body diagonal of a cube with edge length \( a \) is given by \( \sqrt{3} \times a \). Hence, the formula relating the body diagonal and the atomic radius \( r \) is \( 4r = \sqrt{3} \times a \). Solving this equation gives us \( r = \frac{\sqrt{3} \times a}{4} \).

This formula represents how we utilize the geometric properties of the BCC lattice to deduce the size of the sodium atom. Applying this to sodium crystallized as a BCC structure with an edge length of \( 4.29 \AA \) translates to an atomic radius approximately equal to \( 1.857 \times 10^{-8} \mathrm{~cm} \). Ensuring accurate unit conversions is crucial here, as the units must match across the calculation.

Crystallography in Chemistry

Crystallography is the study of the structures and properties of crystals. It's an essential field in chemistry, enabling us to understand the arrangement of atoms within a material. By studying crystallography, we can learn how atoms are positioned in space and how they interact with one another, affecting the properties of the material.

In crystallography, different lattice structures such as body-centered cubic (BCC), face-centered cubic (FCC), and hexagonal close-packed are key to understanding how materials are formed. Each type of arrangement affects a material's properties, such as density and strength. In BCC, for instance, atoms are located at each corner of a cube with an additional atom at its center. This leads to a moderate packing efficiency, which means there's some space between the atoms. Understanding these arrangements helps chemists design and use materials effectively.

• Lattice Structures: Determines how atoms are packed in a crystal.
• Properties Influence: Affects material qualities like hardness and conductivity.
• Application: Helps in the synthesis of new materials and the improvement of old ones.

Crystallography informs material science and genetic engineering by providing insights into how atomic and molecular structures affect the characteristics of metals and other substances.

Sodium Metal Structure

Sodium metal is a fascinating element due to its simple yet informative structure. When crystallized, sodium exhibits a body-centered cubic (BCC) structure. This structure is characterized by having one atom at each corner of the cube and another atom placed at the center of the cube. Sodium, being a metal, forms metallic bonds where electrons are shared among the atoms, contributing to its conductivity.

The BCC structure of sodium affects its physical properties. It has a relatively low density and high malleability. This is due to the atomic arrangement allowing layers to slide over one another easily. Sodium's BCC lattice includes an edge length of \( 4.29 \AA \), forming a basic, repeating unit that contributes to its metallic characteristics. Understanding sodium's structure offers insight into its reactivity and applications in various products like street lamps and in atomic research.

• Simple BCC Structure: Enables easy calculation and modeling.
• Metallic Characteristics: High electrical conductivity and reactivity.
• Applications: Essential in chemical industries and electronics.

Studying sodium's metallic structure provides valuable information not only for educational purposes but also for industrial applications.