Question

Carbon dioxide has attracted much recent interest as a greenhouse gas. Determine the vibrational contribution to \(C_{V}\) for \(\mathrm{CO}_{2}\) where \(\widetilde{\nu}_{1}=2349 \mathrm{cm}^{-1}\) \(\widetilde{\nu}_{2}=667 \mathrm{cm}^{-1}(\text {doubly degenerate }),\) and \(\tilde{\nu}_{3}=1333 \mathrm{cm}^{-1}\) at \(T=260 . \mathrm{K}\)

Short answer

To determine the vibrational contribution to $C_V$ for CO$_2$ at 260 K, we calculate the vibrational quantum energies of its three vibrational modes using the given wavenumbers and then find the vibrational contributions for each mode using the formula: \[ C_V^\nu = R\left(\frac{\hbar \omega}{k_B T}\right)^2\left(\frac{\exp\left(-\frac{\hbar \omega}{k_B T}\right)}{\left[1-\exp\left(-\frac{\hbar \omega}{k_B T}\right)\right]^2}\right) \] Considering the doubly degenerate second mode, we then sum the contributions to obtain the total vibrational contribution to $C_V$: \[ C_V^\mathrm{vib} = C_{V_1}^\nu + C_{V_2}^\nu + C_{V_3}^\nu \]

Step-by-step solution

Calculate vibrational quantum energy#

For each vibrational mode, calculate the vibrational quantum energy 𝜔 using the wavenumber 𝜈 and the Planck constant h: 1. For 𝜈1: \(\hbar \omega_1 = h \tilde{\nu}_1\) 2. For 𝜈2: \(\hbar \omega_2 = h \tilde{\nu}_2\) 3. For 𝜈3: \(\hbar \omega_3 = h \tilde{\nu}_3\)

Calculate vibrational contribution to C_V for each mode#

For each vibrational mode, calculate the vibrational contribution to 𝐶𝑉 using the calculated 𝜔 values, the temperature, and the given formula: 1. For mode 1: \(C_{V_1}^\nu = R\left(\frac{\hbar \omega_1}{k_B T}\right)^2\left(\frac{\exp\left(-\frac{\hbar \omega_1}{k_B T}\right)}{\left[1-\exp\left(-\frac{\hbar \omega_1}{k_B T}\right)\right]^2}\right)\) 2. For mode 2 (doubly degenerate): \(C_{V_2}^\nu = 2R\left(\frac{\hbar \omega_2}{k_B T}\right)^2\left(\frac{\exp\left(-\frac{\hbar \omega_2}{k_B T}\right)}{\left[1-\exp\left(-\frac{\hbar \omega_2}{k_B T}\right)\right]^2}\right)\) 3. For mode 3: \(C_{V_3}^\nu = R\left(\frac{\hbar \omega_3}{k_B T}\right)^2\left(\frac{\exp\left(-\frac{\hbar \omega_3}{k_B T}\right)}{\left[1-\exp\left(-\frac{\hbar \omega_3}{k_B T}\right)\right]^2}\right)\)

Calculate the total vibrational contribution to C_V#

Add the vibrational contributions from each mode to find the total vibrational contribution to 𝐶𝑉: \[ C_V^\text{vib} = C_{V_1}^\nu + C_{V_2}^\nu + C_{V_3}^\nu\] After completing these steps, we will have determined the vibrational contribution to 𝐶𝑉 for CO2 at a temperature of 260 K.

Key concepts

Greenhouse Gas

Greenhouse gases are critical in regulating Earth's temperature. They trap heat in the atmosphere, acting like a blanket, keeping the planet warm.
Carbon dioxide, commonly known as CO2, is well-known for its role as a greenhouse gas.
It's particularly interesting because it's a significant byproduct of human activities, such as burning fossil fuels and deforestation. Greenhouse gases can absorb and emit radiant energy within thermal infrared radiation. This ability is tightly linked to their molecular structure and vibrational modes. These gases have specific vibrational modes that can interact with infrared radiation, contributing to their heat-trapping properties.
As a greenhouse gas, CO2's ability to vibrate in response to infrared radiation makes it pivotal in climate discussions.

Vibrational Modes

Molecules like CO2 can vibrate in specific ways, each of which we call a vibrational mode.
These modes determine how a molecule can stretch, bend, or twist its bonds. For CO2, which is liner, there are several vibrational modes to consider:

• Symmetric stretching: Both oxygen atoms move away and together towards the carbon atom.
• Asymmetric stretching: One oxygen atom moves towards the carbon while the other moves away.
• Bending mode: The molecule bends, changing its angle.

These modes are associated with specific energy levels, and they can be measured by their wavenumber value. Different vibrational modes allow CO2 to absorb different wavelengths of infrared light, making it a powerful greenhouse gas. These properties also make vibrational modes vital in determining CO2's heat capacity.

Quantum Energy

Quantum energy refers to the fact that energy in physics is quantized, meaning it exists in discrete packets.
For molecular vibrations, these energy packets are called quanta and correspond to specific vibrational modes. The quantum energy of vibration is given by the equation:\[ E = (n + \frac{1}{2})hu \]where:

• \( E \) is the energy,
• \( n \) is the vibrational quantum number,
• \( h \) is the Planck constant,
• \( u \) is the frequency (related to wavenumber \( \tilde{u} \)).

Quantum energy levels provide the framework for understanding how molecules absorb and emit energy. In CO2, each vibrational mode corresponds to different quantum energy levels, influencing how the molecule interacts with heat.

Wavenumber

Wavenumber is a measure of the number of wave cycles per unit distance, and it is often used in spectroscopy.
In vibrational spectroscopy, wavenumber represents the frequency of vibrational modes in a molecule like CO2. Expressed typically in units of \( \text{cm}^{-1} \), wavenumbers are essentially reciprocal wavelengths. The wavenumber provides a convenient way to quantify the energy levels associated with different vibrational modes.
For CO2, common wavenumbers are given in the problem as:

• \( \tilde{u}_1 = 2349 \) cm\(^{-1} \)
• \( \tilde{u}_2 = 667 \) cm\(^{-1} \) (doubly degenerate)
• \( \tilde{u}_3 = 1333 \) cm\(^{-1} \).

These values help in determining the vibrational quantum energy of each mode and, subsequently, the vibrational contribution to the heat capacity \( C_V \).

Planck Constant

The Planck constant is a fundamental number in physics that relates the energy of a photon to its frequency.
It is a central element in quantum mechanics and plays a significant role in equations concerning energy and frequency. The constant is denoted by \( h \) and has a value of approximately \( 6.626 \times 10^{-34} \) joule-seconds.
In terms of molecular vibrations, the Planck constant helps calculate the energy associated with vibrational modes using the wavenumber, as in the expression:\[ \hbar \omega = h \tilde{u} \]where:

• \( \hbar \) is the reduced Planck constant,
• \( \omega \) is the angular frequency of vibration,
• \( \tilde{u} \) is the wavenumber.

The Planck constant serves as a bridge between the macroscopic world of classical physics and the microscopic world of quantum mechanics. Its precise role is crucial in understanding the vibrational properties of molecules such as CO2.