Question

Evaluate the vibrational partition function for \(\mathrm{H}_{2} \mathrm{O}\) at \(2000 .\) K where the vibrational frequencies are 1615,3694 and \(3802 \mathrm{cm}^{-1}\)

Short answer

To evaluate the vibrational partition function for H2O at 2000 K with vibrational frequencies of 1615 cm⁻¹, 3694 cm⁻¹, and 3802 cm⁻¹, we converted the frequencies to energies and computed the individual partition functions for each vibrational mode using the formula \(q_i = \frac{1}{1 - e^{-\beta E_i}}\). We then calculated the total vibrational partition function as the product of the individual partition functions: \(q_{vib} = q_1 \times q_2 \times q_3\).

Step-by-step solution

Calculate the partition function for each vibrational frequency

First, we need to convert the vibrational frequencies from \(\mathrm{cm}^{-1}\) to the energy. We will use the relation: Energy = \(\nu \times h \times c\) Here, \(\nu\) is the frequency in \(\mathrm{cm}^{-1}\), h is the Planck's constant (6.626 x 10⁻³⁴ J s), and c is the speed of light in a vacuum (2.998 x 10¹⁰ cm/s). For each frequency, the individual vibrational partition function is given by: \(q_{i} = \frac{1}{1-e^{-\beta E_{i}}}\) Here, \(q_{i}\) is the partition function of the i-th vibrational mode, \(\beta\) is the inverse of the thermal energy (\(\frac{1}{k_{B}T}\)), \(k_{B}\) is the Boltzmann constant (1.38 x 10⁻²³ J K⁻¹), and \(T\) is the temperature in Kelvin. Let's calculate the partition functions using the given frequencies: Frequency 1: 1615 cm⁻¹ Frequency 2: 3694 cm⁻¹ Frequency 3: 3802 cm⁻¹

Compute the partition function for each vibrational frequency

Compute the partition functions using the equations provided in Step 1 for each frequency: For Frequency 1: Energy: \(E_{1} = 1615 \times 6.626 \times 10^{-34} \times 2.998 \times 10^{10}\) \(q_{1} = \frac{1}{1-e^{-\frac{1}{1.38 \times 10^{-23} \times 2000} E_{1}}}\) For Frequency 2: Energy: \(E_{2} = 3694 \times 6.626 \times 10^{-34} \times 2.998 \times 10^{10}\) \(q_{2} = \frac{1}{1-e^{-\frac{1}{1.38 \times 10^{-23} \times 2000} E_{2}}}\) For Frequency 3: Energy: \(E_{3} = 3802 \times 6.626 \times 10^{-34} \times 2.998 \times 10^{10}\) \(q_{3} = \frac{1}{1-e^{-\frac{1}{1.38 \times 10^{-23} \times 2000} E_{3}}}\)

Calculate the total vibrational partition function

The total vibrational partition function for H2O is given by the product of the partition functions of all the modes: \(q_{vib} = q_{1} \times q_{2} \times q_{3}\) Calculate the total vibrational partition function using the values obtained in Step 2: \(q_{vib} = q_{1} \times q_{2} \times q_{3}\) Thus, we have calculated the vibrational partition function for H2O at a temperature of 2000 K using the given vibrational frequencies.

Key concepts

Vibrational Frequencies

Vibrational frequencies are crucial in understanding how molecules behave. They represent the natural frequencies at which molecules vibrate internally, consisting of stretching and bending movements of atoms. The vibrational motion occurs along chemical bonds and is characterized by specific vibrational modes. These frequencies are often measured in units of \({\text{cm}}^{-1}\), also known as wavenumbers.

• Each mode of vibration corresponds to a specific energy level.
• The higher the frequency, the higher the energy associated with that vibrational mode.

For example, the molecule of water (\(\text{H}_2\text{O}\)) has unique vibrational frequencies like 1615, 3694, and 3802 \({\text{cm}}^{-1}\). These values help calculate the vibrational partition function, which relates to the energy levels populated at a given temperature. High precision in these frequencies is vital for accurate physical predictions about the behavior of molecules under various conditions.

Planck's Constant

The Planck's constant, denoted as \( h \), is a fundamental constant in quantum mechanics and plays a crucial role in the quantization of energy. It is approximately 6.626 x 10⁻³⁴ J s, representing the smallest discrete unit of energy exchange. In the context of vibrational frequencies:

• It helps translate the vibrational frequency (in wavenumbers) directly to energy.
• The energy associated with a vibrational mode can be calculated using the formula: \( E = u \times h \times c \).

Here, \( u \) is the frequency, \( c \) is the speed of light in a vacuum (2.998 x 10¹⁰ cm/s), and \( h \) is Planck’s constant. The knowledge of energy levels within a molecule allows us to predict how the molecule interacts with thermal energies and other external influences. Planck's constant thus bridges the gap between microscopic quantum actions and macroscopic observable phenomena.

Boltzmann Constant

The Boltzmann constant \( k_B \) is a key component in the statistical mechanics realm. It relates the average kinetic energy of particles in a gas with the temperature of the gas. Its value is about 1.38 x 10⁻²³ J K⁻¹. Understanding this constant is vital for calculating the vibrational partition function because:

• It appears in the exponent of the vibrational partition function formula.
• The exponent term \( \beta = \frac{1}{k_{B}T} \) characterizes the inverse of thermal energy available at temperature \( T \).

Using the vibrational partition function \( q_{i} = \frac{1}{1-e^{-\beta E_{i}}} \), where \( E_{i} \) is the vibrational energy, the Boltzmann constant helps determine how energy is distributed among various vibrational modes. It thus plays an instrumental role in predicting how molecules behave under thermal stress, influencing reactions and molecular configurations.

Thermal Energy

Thermal energy is a measure of the total energy of the molecular motion in a substance. It’s a crucial concept in thermodynamics and relates directly to the temperature of a system. In vibrational analysis:

• The term thermal energy \( k_B T \) is used, where \( T \) is the temperature in Kelvin.
• It indicates the energy available to the system to overcome potential energy barriers between vibrational energy levels.

In the context of the vibrational partition function, thermal energy determines how many vibrational modes will be excited at a given temperature. For instance, at 2000 K, the thermal energy is sufficient to populate higher vibrational states, affecting the molecular properties. The interplay between thermal energy and vibrational frequencies helps understand molecular interactions' dynamics, allowing scientists to predict reaction rates and mechanisms more accurately.