Question

An oxygen molecule has a vibrational mode that behaves approximately like a simple harmonic oscillator with frequency \(2.99 \cdot 10^{14} \mathrm{rad} / \mathrm{s} .\) Calculate the energy of the ground state and the first two excited states.

Short answer

Question: Calculate the energy of an oxygen molecule in its ground state and the first two excited states using a frequency value of \(2.99 \times 10^{14}\,\mathrm{rad/s}\). Answer: The energy of the oxygen molecule in its ground state is \(E_0 = 9.922 \times 10^{-20}\,\mathrm{J}\), in the first excited state is \(E_1 = 29.767 \times 10^{-20}\,\mathrm{J}\), and in the second excited state is \(E_2 = 49.611 \times 10^{-20}\,\mathrm{J}\).

Step-by-step solution

Identify Given Information

We are given the frequency of the oxygen molecule: \(f = 2.99 \times 10^{14}\,\mathrm{rad/s}\). Step 2: Recall the Energy Formula for a Harmonic Oscillator

Recall the Energy Formula for a Harmonic Oscillator

The energy of a simple harmonic oscillator can be calculated using the formula: \(E_n = (n+\frac{1}{2})hf\) Where - \(E_n\) is the energy of the \(n\)-th state, - \(n\) is the excited state number (0 for ground state, 1 for the first excited state, and so on), - \(h\) is the Planck constant (\(6.626 \times 10^{-34}\,\mathrm{J\,s}\)), - \(f\) is the frequency of the oscillator. Step 3: Calculate the Energy of the Ground State

Calculate the Energy of the Ground State

First, let's calculate the energy of the ground state, which corresponds to \(n=0\): \(E_0 = (\frac{1}{2})hf = \frac{1}{2} \times 6.626 \times 10^{-34}\,\mathrm{J\,s} \times 2.99 \times 10^{14}\,\mathrm{rad/s}\) \(E_0 = 9.922 \times 10^{-20}\,\mathrm{J}\) Step 4: Calculate the Energy of the First Excited State

Calculate the Energy of the First Excited State

Next, let's calculate the energy of the first excited state, which corresponds to \(n=1\): \(E_1 = (\frac{3}{2})hf = \frac{3}{2} \times 6.626 \times 10^{-34}\,\mathrm{J\,s} \times 2.99 \times 10^{14}\,\mathrm{rad/s}\) \(E_1 = 29.767 \times 10^{-20}\,\mathrm{J}\) Step 5: Calculate the Energy of the Second Excited State

Calculate the Energy of the Second Excited State

Finally, let's calculate the energy of the second excited state, which corresponds to \(n=2\): \(E_2 = (\frac{5}{2})hf = \frac{5}{2} \times 6.626 \times 10^{-34}\,\mathrm{J\,s} \times 2.99 \times 10^{14}\,\mathrm{rad/s}\) \(E_2 = 49.611 \times 10^{-20}\,\mathrm{J}\) Step 6: Present the Results

Present the Results

The energy of the oxygen molecule in its ground state is \(E_0 = 9.922 \times 10^{-20}\,\mathrm{J}\), in the first excited state is \(E_1 = 29.767 \times 10^{-20}\,\mathrm{J}\), and in the second excited state is \(E_2 = 49.611 \times 10^{-20}\,\mathrm{J}\).

Key concepts

Vibrational Mode

In the context of a harmonic oscillator, a vibrational mode refers to how a molecule like oxygen can oscillate at a specific frequency. This oscillation is an inherent property of the molecule and is dictated by its physical structure and the bonds connecting the atoms. The frequency of the vibration is crucial because it determines how the molecule behaves at different energy levels. The frequency for an oxygen molecule in the given scenario is approximately \(2.99 \times 10^{14} \, \text{rad/s}\).

Different vibrational modes exist, and each represents a unique oscillatory behavior determined by quantum mechanical principles. The simple harmonic oscillator model is a simplified version that allows us to understand these behaviors in a more accessible way.

• It assumes a perfect, symmetrical oscillation similar to a weight bouncing on a spring.
• The molecule can occupy numerous vibrational modes as it gains energy.
• Knowing the vibrational mode helps us calculate energy levels accurately.

Energy Levels

Energy levels in a harmonic oscillator are quantized, which means that the molecule can exist only at specific energy states. These states in the exercise are labeled as the ground state, the first excited state, and the second excited state.

Each energy level corresponds to a unique vibrational quantum number \(n\):
- Ground State (\(n = 0\))
- First Excited State (\(n = 1\))
- Second Excited State (\(n = 2\))

The energy of each state is calculated using the formula \(E_n = (n + \frac{1}{2})hf\), where \(h\) is the Planck constant and \(f\) is the frequency.

This model helps us:

• Predict how a molecule behaves as it absorbs energy
• Provide accurate calculations of the state energies using known constants
• Understand the molecular behavior of different materials under various conditions

Planck Constant

The Planck constant, denoted as \(h\), is a fundamental constant in quantum mechanics. It is essential in calculating energy levels for quantum systems, such as the energy levels of a harmonic oscillator. With a value of \(6.626 \times 10^{-34} \, \text{J\,s}\), it bridges the concepts of wave and particle in the quantum realm.

The Planck constant is vital because:

• It sets the scale for quantum effects - very small and often not noticeable in classical physics.
• It is used in the formula \(E_n = (n + \frac{1}{2})hf\) to determine the energy levels.
• Ensures that quantities like energy and frequency are related in a consistent way across all quantum systems.

Without the Planck constant, we wouldn't be able to accurately predict how molecules behave at a quantum level, making it an indispensable tool in physics and chemistry.