Problem 104 - Master Resource Book in JEE Main Physics

Question

A diatomic molecule has moment of inertia \(I\). By Bohr's quantization condition its rotational energy in the \(n\)th level \((n=0\) is not allowed) is (a) \(\frac{1}{n^{2}}\left(\frac{h^{2}}{8 \pi^{2} l}\right)\) (b) \(\frac{1}{n}\left(\frac{h^{2}}{8 \pi^{2} l}\right)\) (c) \(n\left(\frac{h^{2}}{8 \pi^{2} I}\right)\) (d) \(n^{2}\left(\frac{h^{2}}{8 \pi^{2} 1}\right)\)

Step-by-Step Solution

Answer

The correct answer is (d) \( n^2\left(\frac{h^2}{8 \pi^2 I}\right) \).

Step 1: Understand the Problem

We need to determine the energy associated with the rotational levels of a diatomic molecule using Bohr's quantization condition. The objective is to determine which of the given options correctly expresses this rotational energy in terms of the quantum number \( n \), Planck's constant \( h \), and the moment of inertia \( I \).

Step 2: Recall Bohr's Quantization Condition

Bohr's quantization condition for angular momentum is given by \( L = n\hbar \), where \( \hbar = \frac{h}{2\pi} \). For a rotating system, angular momentum \( L \) is also given by \( L = I \omega \). This can be equated to \( n\hbar \) for specific quantized energy levels.

Step 3: Relate Angular Momentum to Rotational Energy

The rotational energy \( E \) of the nth level is given by \( E = \frac{L^2}{2I} \). Replacing \( L \) with \( n\hbar \), we have: \( E = \frac{(n\hbar)^2}{2I} \). Substituting \( \hbar = \frac{h}{2\pi} \) results in \( E = \frac{n^2 h^2}{8\pi^2 I} \).

Step 4: Compare the Expression With Options

Given the expression \( E = \frac{n^2 h^2}{8 \pi^2 I} \), compare it with the provided options. It matches exactly with option \( (d) \), which is \( n^2\left(\frac{h^2}{8\pi^2 I}\right) \).

Key Concepts

Bohr's Quantization Condition

In quantum mechanics, Bohr's quantization condition plays a key role in determining the properties of microscopic systems. This principle arose from Niels Bohr's model of the atom, which introduced the idea that angular momentum is quantized. In simple terms, this means that not all values of angular momentum are allowed. Instead, angular momentum can only exist in specific, discrete amounts or 'quanta'.
Bohr's quantization condition for angular momentum is expressed as \( L = n\hbar \), where \( n \) is a positive integer known as the quantum number, and \( \hbar \) is the reduced Planck's constant, defined as \( \hbar = \frac{h}{2\pi} \).
To understand this better:

• \( h \) is Planck's constant, a fundamental constant used in quantum mechanics.
• \( n \) is a quantum number that determines the energy level of a system.
• This formula explains why only certain 'orbits' or states of a system, like an electron in an atom or a diatomic molecule, are stable and correspond to integer values of \( n \).

This concept is crucial in predicting the energy levels of rotating systems in molecular physics, helping us understand why molecules have specific rotational energies and why they transition between discrete states.

Moment of Inertia

Moment of inertia is a fundamental concept in rotational dynamics, akin to mass in linear motion. It measures an object's resistance to changes in its rotational motion. In simpler terms, it tells us how difficult it is to spin an object around a given axis. For diatomic molecules, which are two-atom systems, calculating the moment of inertia helps determine their rotational energy.
The moment of inertia \( I \) for a diatomic molecule depends on:

• The masses of the two atoms, usually denoted \( m_1 \) and \( m_2 \).
• The distance between the atoms, usually denoted \( r \).

It’s calculated using the formula: \[ I = \frac{m_1m_2}{m_1 + m_2} r^2 \].
This formula highlights how both mass and distance from the rotational axis influence the moment of inertia.

• Larger masses or greater distances lead to a higher moment of inertia, making it harder to change rotational motion.
• In the context of molecular physics, a higher moment of inertia usually results in lower rotational energy levels.

Understanding the moment of inertia allows us to comprehend how molecules behave when they rotate, offering insights into their energy states and how they interact with radiation.

Angular Momentum Quantization

Angular momentum quantization may sound complex, but it's a fundamental concept that explains many behaviors of microscopic systems. At the quantum level, angular momentum is not continuous but instead comes in discrete chunks or 'quanta'. This quantization affects how systems can rotate and what energy levels they can attain.
The quantization of angular momentum is described by the relation \( L = n\hbar \), reaffirming that angular momentum is always a multiple of \( \hbar \), the reduced Planck's constant.

• \( n \) signifies quantum numbers that start from 1, never zero or negative, indicating possible rotational states.
• This results in only specific rotational states being allowed for diatomic molecules.

To connect it with energy, we use the formula for rotational energy: \( E = \frac{L^2}{2I} \). By substituting \( L = n\hbar \) into this equation, we derive \( E = \frac{n^2 h^2}{8 \pi^2 I} \), showing that rotational energy also gets quantized.
Angular momentum quantization explains why rotational spectra appear as discrete lines and why transition between levels involves specific energy changes. This concept is foundational, offering insights into molecular behavior and interaction with electromagnetic fields.