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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

Expert verified
(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

01

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, \( l \).
02

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: \( \bar{u} = \frac{1}{\lambda (cm)} \).
03

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 \( 2B \).
04

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 \( l \to l-1 \).
05

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 \( r \).
06

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

The gap between valence and conduction bands in diamond is 5.47 eV. (a) What is the maximum wavelength of a photon that can excite an electron from the top of the valence band into the conduction band? In what region of the electromagnetic spectrum does this photon lie? (b) Explain why pure diamond is transparent and colorless. (c) Most gem diamonds have a yellow color. Explain how impurities in the diamond can cause this color.

Our galaxy contains numerous \(molecular\) \(clouds\), regions many lightyears in extent in which the density is high enough and the temperature low enough for atoms to form into molecules. Most of the molecules are H\(_2\), but a small fraction of the molecules are carbon monoxide (CO). Such a molecular cloud in the constellation Orion is shown in Fig. P42.42. The upper image was made with an ordinary visiblelight telescope; the lower image shows the molecular cloud in Orion as imaged with a radio telescope tuned to a wavelength emitted by CO in a rotational transition. The different colors in the radio image indicate regions of the cloud that are moving either toward us (blue) or away from us (red) relative to the motion of the cloud as a whole, as determined by the Doppler shift of the radiation. (Since a molecular cloud has about 10,000 hydrogen molecules for each CO molecule, it might seem more reasonable to tune a radio telescope to emissions from H\(_2\) than to emissions from CO. Unfortunately, it turns out that the H\(_2\) molecules in molecular clouds do not radiate in either the radio or visible portions of the electromagnetic spectrum.) (a) Using the data in Example 42.2 (Section 42.2), calculate the energy and wavelength of the photon emitted by a CO molecule in an \(l\) \(=\) 1\(\rightarrow\) \(l\) \(=\) 0 rotational transition. (b) As a rule, molecules in a gas at temperature \(T\) will be found in a certain excited rotational energy level, provided the energy of that level is no higher than \(kT\) (see Problem 42.39). Use this rule to explain why astronomers can detect radiation from CO in molecular clouds even though the typical temperature of a molecular cloud is a very low 20 K.

For a solid metal having a Fermi energy of 8.500 eV, what is the probability, at room temperature, that a state having an energy of 8.520 eV is occupied by an electron?

Consider a system of \(N\) free electrons within a volume \(V\). Even at absolute zero, such a system exerts a pressure \(p\) on its surroundings due to the motion of the electrons. To calculate this pressure, imagine that the volume increases by a small amount \(dV\). The electrons will do an amount of work \(p\) \(dV\) on their surroundings, which means that the total energy \(E_{tot}\) of the electrons will change by an amount \(dE_{tot} = -p dV\). Hence \(p = -dE_{tot}/dV\). (a) Show that the pressure of the electrons at absolute zero is \(p = \frac{3^{2/3}\pi^{4/3}\hbar^{2}}{5m} \lgroup \frac{N}{V}\ \rgroup^{5/3}\) (b) Evaluate this pressure for copper, which has a freeelectron concentration of \(8.45 \times 10^{28} m^{-3}\). Express your result in pascals and in atmospheres. (c) The pressure you found in part (b) is extremely high. Why, then, don't the electrons in a piece of copper simply explode out of the metal?

When an OH molecule undergoes a transition from the \(n = 0\) to the \(n = 1\) vibrational level, its internal vibrational energy increases by 0.463 eV. Calculate the frequency of vibration and the force constant for the interatomic force. (The mass of an oxygen atom is \(2.66 \times 10^{-26}\) kg, and the mass of a hydrogen atom is \(1.67 \times 10^{-27} kg.)\)

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