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

The circumference of the fourth orbit is \(4X\), which is option (d).

Step 1: Understand the Given Information

The problem states that the electron is moving in Bohr's fourth orbit, and its de Broglie wavelength is given as \( X \). We need to find the circumference of this orbit.

Step 2: Apply Bohr's Model of the Atom

According to Bohr's model, the circumference of the \( n^{th} \) orbit is given by the formula: \( 2 \pi r_n = n \lambda \), where \( n \) is the principal quantum number, and \( \lambda \) is the de Broglie wavelength of the electron.

Step 3: Identify the Principal Quantum Number

Since the electron is in the fourth orbit, the principal quantum number \( n \) is 4.

Step 4: Substitute the Known Values

Plug in the values into the circumference formula: \( 2 \pi r = 4X \). However, we are more focused on the simple relationship \( n \lambda \), so the formula simplifies to \( 4X \).

Step 5: Match with Given Options

Compare the calculated circumference \( 4X \) with the provided options. - (a) \( 2X \) - (b) \( 2/X \) - (c) \( 3X \) - (d) \( 4X \)The correct answer based on our calculation is (d) \( 4X \).

Key Concepts

de Broglie wavelength

The de Broglie wavelength is a fundamental concept that links the wave and particle nature of matter. Introduced by Louis de Broglie in 1924, it suggests that every moving particle or object has an associated wave. This wave is invisible to the naked eye but has important implications in quantum physics.
To calculate the de Broglie wavelength, you use the formula:

• \( \lambda = \frac{h}{mv} \)

Here, \( \lambda \) represents the wavelength, \( h \) is Planck's constant (approximately \(6.626 \times 10^{-34} \text{Js}\)), \( m \) is the mass of the particle, and \( v \) is its velocity.
The concept of de Broglie wavelength is crucial for understanding phenomena at the quantum level, such as electron diffraction and interference patterns. It helps bridge the gap between classical and quantum physics, providing a deeper understanding of the wave-particle duality seen in electrons and other tiny particles.

principal quantum number

The principal quantum number, denoted as \( n \), is a fundamental part of quantum mechanics and atomic theory. It defines the energy level of an electron within an atom and directly correlates with the electron's probable distance from the nucleus.
In Bohr’s atomic model, the principal quantum number determines the electron shell, or orbit, the electron occupies:

• The higher the principal quantum number, the higher the energy level and the further the electron is from the nucleus.
• These values are positive integers: \( n = 1, 2, 3, \) and so on.
• In our exercise, the electron is in the fourth orbit, so the principal quantum number is 4.

Understanding the principal quantum number is essential for grasping the arrangement of electrons in atoms. It is like the address system in an atomic "city," telling us the street, or orbit, where the electron "lives." It’s a stepping stone for understanding more complex topics like electron configurations and spectral lines.

circumference of orbit

The circumference of an orbit within Bohr’s model provides critical insights into how electrons travel around the nucleus of an atom. Bohr's model was revolutionary at its time, incorporating quantum theory to explain atomic structure.
For an electron in a circular orbit, the circumference is directly related to the de Broglie wavelength. According to Bohr's theory:

• The relationship is given by \( 2 \pi r_n = n \lambda \).
• Here, \( 2 \pi r_n \) is the circumference of the orbit, \( n \) is the principal quantum number, and \( \lambda \) is the de Broglie wavelength.

For example, if an electron is in the fourth orbit using the de Broglie wavelength, as in our exercise, then the circumference can be calculated as \( 2 \pi r = 4 \lambda \), indicating that the total path length around the nucleus accommodates an integer number of wavelengths.
This principle not only aids in understanding Bohr's model of the atom but also demonstrates the harmonious dance of particles and waves in quantum mechanics, articulating why only specific orbits are allowed.