Question

Determine the vibrational contribution to \(C_{V}\) for HCN where \(\tilde{\nu}_{1}=2041 \mathrm{cm}^{-1}, \widetilde{\nu}_{2}=712 \mathrm{cm}^{-1}\) (doubly degenerate) and \(\tilde{\nu}_{3}=3369 \mathrm{cm}^{-1}\) at \(T=298,500 .,\) and \(1000 . \mathrm{K}\)

Short answer

The vibrational contribution to \(C_V\) for HCN at \(T=298 K, 500 K,\) and \(1000 K\) can be calculated using the equation \(C_{V, i} = R \frac{(\frac{hν_i}{k_BT})^2e^{\frac{-hν_i}{k_BT}}}{(1-e^{\frac{-hν_i}{k_BT}})^2}\). Convert wavenumbers to frequencies (\(ν_i = \tilde{ν}_i \times c\)), then calculate the heat capacity for each vibrational mode and sum them up, considering the doubly degenerate mode: \(C_V = C_{V,1} + 2C_{V,2} + C_{V,3}\). Repeat this process for each temperature to determine the vibrational contribution to \(C_V\) at \(298 K,\) \(500 K,\) and \(1000 K\).

Step-by-step solution

Convert wavenumbers to frequencies

To use the given wavenumbers in the equation, we need to first convert them to frequencies (in Hz) by multiplying with the speed of light: \(ν_i = \tilde{ν}_i \times c\) Where c is the speed of light in m/s.

Calculate vibrational heat capacity for each mode

Now, we'll use the equation for vibrational heat capacity and the frequencies obtained for each vibrational mode and temperature: \(C_{V, i} = R \frac{(\frac{hν_i}{k_BT})^2e^{\frac{-hν_i}{k_BT}}}{(1-e^{\frac{-hν_i}{k_BT}})^2}\)

Sum up individual heat capacities

Because one of the vibrational modes is doubly degenerate, we'll sum up individual heat capacities with this consideration: \(C_V = C_{V,1} + 2C_{V,2} + C_{V,3}\)

Calculate the vibrational contribution to C_V at different temperatures

We'll now repeat steps 2 and 3 for each given temperature, 298 K, 500 K, and 1000 K, to obtain the vibrational contribution to C_V at each of these temperatures. In this way, we can find the vibrational contributions to C_V for HCN at different temperatures using the given vibrational frequencies.

Key concepts

Quantum Harmonic Oscillator

In quantum mechanics, the quantum harmonic oscillator is a model that describes systems with a potential energy that varies with the square of the displacement from equilibrium, such as a mass on a spring. This model is important because it can be solved exactly, providing a cornerstone for understanding more complex systems.

The energy levels of a quantum harmonic oscillator are quantized, meaning that they can only take on certain discrete values. The equation \( E_n = \hbar \omega \left(n + \frac{1}{2}\right) \) defines the energy of the nth level, where \( \hbar \) is the reduced Planck's constant, \( \omega \) is the angular frequency, and \( n \) is an integer representing the quantum number. This discreteness is crucial when we discuss vibrational modes in molecules, as it explains why only specific energy packets, or 'quanta,' can be absorbed or emitted.

Vibrational Modes

Molecules have various ways they can move: translating, rotating, and vibrating. Vibrational modes are specific patterns in which the atoms in a molecule can move relative to each other. Each mode has a characteristic frequency, and in quantum mechanics, these frequencies correspond to the quantized energy levels of the harmonic oscillator.

In the case of HCN (hydrogen cyanide), the exercise cites three vibrational frequencies. These are related to three different movements within the molecule: stretching along the H-C bond, bending of the molecule, and stretching along the C-N bond. The frequencies mentioned, such as \( \tilde{u}_{1} = 2041 \mathrm{cm}^{-1} \), define how often a molecule vibrates in a particular mode per second. When multiple atoms are moving together in a symmetrical fashion, as in the doubly degenerate \( \tilde{u}_{2} \) mode, it is counted twice because it represents two equivalent ways the molecule can vibrate.

Statistical Thermodynamics

Statistical thermodynamics is a branch of physics that combines quantum mechanics and classical thermodynamics. It allows us to understand and predict the properties of matter at the molecular level by considering the behavior of large ensembles of particles.

One key concept in statistical thermodynamics is the partition function, which provides a link between the microscopic properties of molecules, such as energies of vibrational modes and the macroscopic thermodynamic properties of the material. By using the partition function, we can calculate important thermodynamic quantities like entropy, free energy, and, relevant to this discussion, heat capacity. This field of study is essential for explaining the molecular origins of heat capacity and how it changes with temperature.

Boltzmann Distribution

The Boltzmann distribution is a probability distribution in statistical thermodynamics that provides the likelihood that a particle will have a certain energy at a given temperature. The formula given by \( e^{-E_i/k_BT} \) describes this distribution, where \( E_i \) is the energy level of a state \( i \) , \( k_B \) is Boltzmann's constant, and \( T \) is the temperature in Kelvin.

This distribution tells us that at higher temperatures, the probability of finding molecules in higher energy states increases. This relates directly to heat capacity since the ability of a substance to absorb heat without a large change in temperature depends on the distribution of molecular energy levels.

Heat Capacity

Heat capacity (\( C_V \) for constant volume) is a measure of the amount of heat required to raise the temperature of a substance by one degree Celsius. The steps outlined in the original exercise demonstrate how the heat capacity of a system with discrete energy levels, such as vibrational states in a molecule, can be calculated.

In the provided methods, we see the application of the Boltzmann distribution and quantum mechanics to determine \( C_{V,i} \) for individual vibrational modes and how they sum to give the total vibrational contribution to the heat capacity of a molecule. Higher vibrational modes tend to contribute more to the heat capacity at higher temperatures because the likelihood of these states being populated increases, as predicted by the Boltzmann distribution. This is especially significant for polyatomic molecules like HCN where multiple vibrational modes exist.