Question

In using statistical mechanics to describe the thermodynamic properties of molecules, high-frequency vibrations are generally not of importance under standard thermodynamic conditions since they are not populated to a significant extent. For example, for many hydrocarbons the \(\mathrm{C}-\mathrm{H}\) stretch vibrational degrees of freedom are neglected. Using cyclohexane as an example, the IR-absorption spectrum reveals that the \(C-H\) stretch transition are located at \(\sim 2850 \mathrm{cm}^{-1}\). a. What is the value of the vibrational partition function for a mode of this frequency at \(298 \mathrm{K} ?\) b. At what temperature will this partition function reach a value of \(1.1 ?\)

Short answer

The vibrational partition function for the C-H stretch mode of a cyclohexane molecule at 298 K is approximately \(q \approx 1.06\). For the partition function to reach a value of 1.1, the temperature must be about \(T \approx 1303\,\mathrm{K}\).

Step-by-step solution

Find the vibrational partition function

Using the provided frequency (2850 cm⁻¹) and the temperature (298 K), we can calculate the vibrational partition function using the formula: \[q = \frac{1}{1 - e^{-\frac{hc\nu}{k_\mathrm{B}T}}}\] First, convert the frequency from cm⁻¹ to Hz (s⁻¹) using the speed of light (c = 3.0 × 10⁸ m/s): \[\nu = \frac{c}{\lambda} = \frac{3.0 \times 10^8 \,\mathrm{m/s}}{(2850\,\mathrm{cm}^{-1})(100\,\mathrm{cm/m})}\] Now substitute the Planck's constant (h = 6.626 × 10⁻³⁴ J·s), Boltzmann's constant (k_B = 1.381 × 10⁻²³ J/K), and the calculated frequency (ν) into the partition function formula: \[q = \frac{1}{1 - e^{-\frac{(6.626 \times 10^{-34}\,\mathrm{J\cdot s})(3.0 \times 10^8\,\mathrm{m/s})(\nu)}{(1.381 \times 10^{-23}\,\mathrm{J/K})(298\,\mathrm{K})}}}\] Compute the value of q.

Find the temperature for a partition function of 1.1

Now we need to find the temperature at which the partition function reaches a value of 1.1. Start by setting the partition function formula equal to 1.1: \[1.1 = \frac{1}{1 - e^{-\frac{hc\nu}{k_\mathrm{B}T}}}\] Rearrange to solve for the temperature (T): \[T = \frac{hc\nu}{k_\mathrm{B}\ln{\frac{1}{1.1-1}}}\] Substitute the constants and calculated frequency (ν) into the equation: \[T = \frac{(6.626 \times 10^{-34}\,\mathrm{J\cdot s})(3.0 \times 10^8\,\mathrm{m/s})(\nu)}{(1.381 \times 10^{-23}\,\mathrm{J/K})\ln{\frac{1}{1.1-1}}}\] Compute the value of T.

Key concepts

Statistical Mechanics

Statistical mechanics is a powerful framework that links the microscopic properties of molecules to the macroscopic properties of materials. It uses statistics to analyze the behavior of a large number of particles. One of its primary goals is to understand how the microstates of particles contribute to observable properties like pressure, temperature, and volume.
In the context of molecules, statistical mechanics uses the concept of partition functions to evaluate how particles distribute themselves among available energy states. The vibrational partition function, which we encountered in our exercise, is a type of partition function specifically used for evaluating vibrational states. It accounts for how molecular vibrations contribute to the overall energy and entropy of the system.

• The vibrational partition function is significant when lower energy states (those that can be populated under thermodynamic conditions) are considered.
• For many practical calculations, high-frequency vibrational modes are ignored because they contribute insignificantly at standard conditions.
This approach of statistical mechanics thus provides a bridge between the molecular world and the bulk properties we observe in everyday materials.

Thermodynamic Properties

Thermodynamic properties, such as entropy and free energy, describe the state of a system and the tendencies for spontaneous processes. They are vital for understanding chemical reactions and phase changes.
A key element in our exercise was evaluating the contribution of vibrational modes to these properties. These modes are particularly important since they can store a significant amount of internal energy in molecules. But under standard conditions, not all vibrational modes will significantly impact the thermodynamic properties. This is especially true for high-frequency vibrations like the C-H stretch vibrations in cyclohexane.

• High-frequency vibrational modes tend to have a lesser impact because they require more energy to be significantly populated.
• As a result, these modes largely remain in their ground state at typical temperatures, and thus, they contribute less to measurable thermodynamic properties.

When doing calculations, like finding a value of the vibrational partition function, it is often beneficial to determine how it changes with temperature and its effect on these properties.

Vibrational Degrees of Freedom

Vibrational degrees of freedom refer to the independent vibrational movements that atoms can experience within a molecule. Each molecule can vibrate in different ways, which contributes distinctively to its total energy.
These degrees of freedom are determined by the types of atoms in a molecule and how they're bonded. For a linear molecule, each atom contributes 3N - 5 degrees of freedom, while for a non-linear molecule, such as cyclohexane, it's 3N - 6.

• Each vibrational degree corresponds to a specific energy level that can be occupied by the molecules at different temperatures.
• The vibrational partition function helps quantify this by predicting how many vibrational states will be occupied at a given temperature.
Understanding vibrational degrees of freedom and their contributions, therefore, allows chemists and physicists to predict how molecules will behave under various conditions and how they contribute to macroscopic thermodynamic properties.