Problem 64 - A Complete Resource Book in Chemistry for JEE Main

Question

An electron is moving in Bohr's fourth orbit, its de-Broglie wavelength is \(X\). What is the circumference of the fourth orbit? (a) \(2 \lambda\) (b) \(2 / \lambda\) (c) \(3 \lambda\) (d) \(4 \lambda\)

Step-by-Step Solution

Answer

(d) \(4 \lambda\)

Step 1: Understanding de Broglie's Postulate

De Broglie's hypothesis states that particles, like electrons, have wave properties and the wavelength of an electron is given by its momentum. In Bohr's model of the atom, the circumference of an electron's orbit is an integral multiple of its de Broglie wavelength.

Step 2: Recognize the Formula for Circumference

For an electron moving in a Bohr orbit, the circumference of the nth orbit is given by the formula: \( C_n = n \lambda \), where \( n \) is the orbit number and \( \lambda \) is the de Broglie wavelength of the electron.

Step 3: Substitute the Values for the Fourth Orbit

Since the electron is moving in Bohr's fourth orbit, substitute \( n = 4 \) an and \( \lambda = X \) into the formula for circumference: \( C_4 = 4 \times X = 4X \).

Step 4: Identify the Correct Option

From the calculation, the circumference of the fourth orbit is \( 4X \), which corresponds to option (d) \(4 \lambda \).

Key Concepts

De Broglie Wavelength

The concept of de Broglie wavelength was introduced by Louis de Broglie in 1924. This was a groundbreaking addition to quantum mechanics. The central idea is that particles usually thought of as matter, such as electrons, also exhibit wave-like behavior. This behavior can be described using a specific wavelength known as the de Broglie wavelength.
The de Broglie wavelength () of a particle is \[ \lambda = \frac{h}{p} \]where:

• \( h \) is Planck's constant.
• \( p \) is the momentum of the particle.

This relationship is particularly useful in describing the behavior of particles at the quantum level, such as electrons in an atomic orbit.
Previously thought entities as purely particles are now better described by this dual wave-particle nature.

Electron Orbit

In the context of the Bohr model, an electron orbit refers to the path that an electron takes as it moves around the nucleus of an atom. Bohr proposed that electrons travel in specific, quantized orbits. Each orbit corresponds to a certain energy level.
These orbits are defined by their radius and quantified by the principal quantum number, denoted as \( n \). The principal quantum number helps determine the energy level of the orbit.
According to Bohr's theory, an electron can only have certain, allowed orbits, that are spaced at specific distances from the nucleus:

• The electron remains in a stable orbit without radiating energy.
• Electrons can jump between orbits, absorbing or emitting energy.

The Bohr model introduced the idea of quantization in orbits, which was a crucial step in the development of modern quantum mechanics.

Circumference Formula

The circumference formula in the context of Bohr's atomic model is essential to understanding the behavior of electrons in atoms. For electrons orbiting an atom in a stable path, the circumference of each orbit relates directly to their de Broglie wavelength. According to the Bohr model, the circumference of an electron's orbit must be an integral multiple of its de Broglie wavelength.
The formula for determining the circumference () of an electron moving in the nth Bohr orbit is:\[ C_n = n \lambda \]where:

• \( C_n \) is the circumference of the nth orbit
• \( n \) is the orbit number
• \(  \) is the de Broglie wavelength of the electron

This formula emerges from the fundamental principle that only specific conditions allow for stable electron paths, integrating the wave properties as described by de Broglie.

Quantum Mechanics

Quantum mechanics is a fundamental branch of physics that deals with the physical phenomena at the scale of atoms and subatomic particles. This field of study emerged in the early 20th century. It aims to explain the quirks and behaviors of matter on a small scale, including electrons, protons, and neutrons.
Quantum mechanics challenges classical physics through principles like:

• Wave-Particle Duality: Suggesting entities like electrons have properties of both particles and waves.
• Quantization: Physical properties, such as energy, take on discrete values.
• Uncertainty Principle: Introduced by Heisenberg, it states the impossibility of simultaneously knowing exact values of certain pairs of variables, like position and momentum.
• Superposition: Particles exist in multiple states or locations until measured.

These principles underpin much of our understanding of molecular, atomic, and subatomic behavior, leading to technological advances like the development of semiconductors and lasers.