Question

To determine the equilibrium separation of the atoms in the HCl molecule, you measure the rotational spectrum of HCl. You find that the spectrum contains these wavelengths (among others): \(60.4 \mu m\), \(69.0 \mu m\), \(80.4 \mu m\), \(96.4 \mu m\), and \(120.4 \mu m\). (a) Use your measured wavelengths to find the moment of inertia of the HCl molecule about an axis through the center of mass and perpendicular to the line joining the two nuclei. (b) The value of \(l\) changes by \(\pm 1\) in rotational transitions. What value of \(l\) for the upper level of the transition gives rise to each of these wavelengths? (c) Use your result of part (a) to calculate the equilibrium separation of the atoms in the HCl molecule. The mass of a chlorine atom is \(5.81 \times 10^{-26}\) kg, and the mass of a hydrogen atom is \(1.67 \times 10^{-27}\) kg. (d) What is the longest-wavelength line in the rotational spectrum of HCl?

Short answer

(a) Find moment of inertia using the spectral lines. (b) Assign quantum numbers. (c) Calculate separation using \( I \). (d) Longest wavelength: 120.4 \( \mu \)m.

Step-by-step solution

Understanding Rotational Spectroscopy

First, recognize that the rotational spectrum of a diatomic molecule like HCl is characterized by transitions between different rotational states. When these transitions occur, they emit photons of specific wavelengths. Each wavelength corresponds to a change in the rotational quantum number, \( l \).

Convert Wavelengths to Wavenumbers

The energy difference \( \Delta E \) between rotational levels is related to the wavenumber \( \bar{u} \) (inverse of the wavelength \( \lambda \)) by \( \Delta E = hc \bar{u} \). Calculate the wavenumber for each given wavelength in cm⁻¹: \( \bar{u} = \frac{1}{\lambda (cm)} \).

Calculate Moment of Inertia

The rotational energy levels are given by \( E_l = \frac{\hbar^2}{2I} l(l+1) \), where \( I \) is the moment of inertia. Using the wavenumber difference between transitions, calculate \( I \): \( \Delta \bar{u} = B\) and \( B = \frac{\hbar}{4\pi c I} \). Find \( I \) by matching the increments of wavenumbers to \( 2B \).

Identify Quantum Level Changes

For each wavelength, determine the change in the rotational quantum number, \( l \), associated with the wavelength by using the formula \( \Delta l = 1 \). These correspond to transitions \( l \to l-1 \).

Calculate Equilibrium Separation

Use the relationship \( I = \mu r^2 \) to find the bond length \( r \). \( \mu \), the reduced mass \( \mu = \frac{m_H m_{Cl}}{m_H + m_{Cl}} \), and the already calculated \( I \), allow you to solve for \( r \).

Determine Longest Wavelength

The longest wavelength corresponds to the smallest energy transition (lowest \( \Delta E \)), typically \( l = 1 \to 0 \). Calculate it using the smallest transition wavenumber.

Key concepts

Moment of Inertia

The moment of inertia, often denoted as \( I \), is a key concept in the study of rotational spectroscopy, especially when analyzing diatomic molecules like HCl. It tells us how the mass of a molecule is distributed in space, crucial for understanding its rotational transitions. Unlike linear motion, where mass alone is a factor, rotation depends on how far the mass is from the axis of rotation.

For a diatomic molecule, the moment of inertia around an axis through its center of mass can be calculated using the formula:

• \( I = \mu r^2 \)

Here, \( \mu \) is the reduced mass, calculated as \( \mu = \frac{m_1 m_2}{m_1 + m_2} \), and \( r \) is the equilibrium separation between the two atoms. This relationship provides the foundation for determining \( I \) from spectral lines.

The rotational energy levels, \( E_l = \frac{\hbar^2}{2I} l(l+1) \), show that the energy depends on both \( I \) and the rotational quantum number \( l \). In practice, when you have spectral data, you can use the difference in wavenumbers to back-calculate the moment of inertia, which helps in understanding molecular structure.

Equilibrium Separation

Equilibrium separation, often symbolized as \( r \), is the average distance between the nuclei of the atoms in a molecule, such as HCl. This measurement is crucial because it deeply influences the molecule's vibrational and rotational spectra.

To find the equilibrium separation, you first calculate the moment of inertia \( I \), as discussed earlier. Once \( I \) is known, you can use the relationship:

• \( I = \mu r^2 \)

By plugging the reduced mass \( \mu \) and the moment of inertia \( I \) into this equation, you can solve for \( r \). This calculation provides not only the bond length but also insights into the potential energy surface of the molecule.

Knowing the equilibrium separation helps scientists predict how a molecule will behave under various conditions and how it will interact with other molecules. It also plays a critical role in fields such as spectroscopy and quantum chemistry.

Diatomic Molecules

Diatomic molecules, such as HCl, consist of only two atoms bonded together, making them ideal candidates for studying rotational spectroscopy. These molecules can be homonuclear (same element) or heteronuclear (different elements), as is the case with HCl, composed of hydrogen and chlorine.

The simplicity of diatomic molecules offers a clear picture of molecular rotational and vibrational behaviors. In the basic sense, the molecule rotates as if it were a tiny dumbbell, with its rotational spectra giving information about the bond length and the mass distribution between the two atoms.

In the context of rotational spectroscopy, transitions occur between rotational energy levels defined by the quantum number \( l \). These transitions emit or absorb radiation at specific wavelengths, providing a fingerprint of the molecule's internal structure.

Understanding diatomic molecules is not only fundamental in physics and chemistry classes but also vital for real-world applications like atmospheric science and remote sensing, where analyzing molecular spectra can reveal important environmental data.