⚛Molecular Physics Unit 6 – Molecular Vibrations and Rotations

Fundamentals of Molecular Motion

• Molecules possess various types of motion including vibrational, rotational, and translational
• Vibrational motion involves the periodic displacement of atoms within a molecule from their equilibrium positions
  • Occurs due to the stretching and compression of chemical bonds
  • Frequency of vibration depends on the mass of the atoms and the strength of the bonds
• Rotational motion refers to the rotation of a molecule about its center of mass
  • Influenced by the molecule's moment of inertia and the temperature of the system
• Translational motion describes the linear movement of the entire molecule through space
• The total energy of a molecule is the sum of its vibrational, rotational, and translational energies
• The study of molecular motion provides insights into the structure, bonding, and interactions of molecules

Vibrational Modes and Energy Levels

• Vibrational modes are the specific patterns of atomic motion within a molecule during vibration
• The number of vibrational modes depends on the number of atoms (N) in the molecule
  • Non-linear molecules have 3N-6 vibrational modes
  • Linear molecules have 3N-5 vibrational modes
• Each vibrational mode has a characteristic frequency and associated energy level
• The energy levels of a harmonic oscillator are quantized and given by $E_n = (n + \frac{1}{2})hν$, where $n$ is the vibrational quantum number, $h$ is Planck's constant, and $ν$ is the frequency of vibration
• The ground state ($n=0$) of a vibrational mode has a non-zero energy, known as the zero-point energy ($\frac{1}{2}hν$)
• Transitions between vibrational energy levels occur when a molecule absorbs or emits a photon with energy equal to the difference between the levels
• Selection rules govern the allowed transitions between vibrational energy levels
  • Transitions typically occur between adjacent levels ($Δn = ±1$) for harmonic oscillators

Rotational Motion in Molecules

• Rotational motion is described by the molecule's moment of inertia ($I$), which depends on the mass distribution and geometry of the molecule
• The rotational energy levels of a rigid rotor are given by $E_J = \frac{h^2}{8π^2I}J(J+1)$, where $J$ is the rotational quantum number
• The spacing between rotational energy levels decreases with increasing moment of inertia
• Molecules with higher symmetry (e.g., homonuclear diatomic molecules like $N_2$) have no permanent dipole moment and are not active in rotational spectroscopy
• Heteronuclear diatomic molecules (e.g., $CO$) and polyatomic molecules with asymmetric charge distributions have a permanent dipole moment and can undergo rotational transitions
• Rotational spectra appear in the microwave and far-infrared regions of the electromagnetic spectrum
• The analysis of rotational spectra provides information about bond lengths, molecular geometry, and dipole moments

Spectroscopy Techniques

• Spectroscopy is the study of the interaction between matter and electromagnetic radiation
• Vibrational spectroscopy techniques probe the vibrational modes and energy levels of molecules
  • Infrared (IR) spectroscopy measures the absorption of IR radiation by molecules
  • Raman spectroscopy detects the inelastic scattering of monochromatic light by molecules
• Rotational spectroscopy techniques investigate the rotational motion and energy levels of molecules
  • Microwave spectroscopy uses microwave radiation to induce rotational transitions
  • Rotational-vibrational spectroscopy combines the study of rotational and vibrational transitions
• Fourier-transform spectroscopy (FTIR, FT-Raman) improves the signal-to-noise ratio and resolution of spectra by using an interferometer and Fourier transform analysis
• Cavity ring-down spectroscopy (CRDS) is a highly sensitive technique that measures the decay of light intensity in an optical cavity containing the sample
• Spectroscopic techniques provide valuable information about molecular structure, bonding, and interactions

Quantum Mechanical Approach

• The quantum mechanical treatment of molecular vibrations and rotations provides a more accurate description of molecular motion
• The Schrödinger equation is used to determine the wavefunctions and energy levels of a molecule
  • The Hamiltonian operator includes terms for the kinetic and potential energy of the system
• The Born-Oppenheimer approximation separates the electronic and nuclear motions, allowing for the independent treatment of vibrational and rotational motion
• The harmonic oscillator model assumes a parabolic potential energy curve and equally spaced energy levels
  • Anharmonicity corrections account for deviations from the ideal harmonic behavior
• The rigid rotor model treats the molecule as a rigid body with a fixed bond length and moment of inertia
  • Centrifugal distortion corrections consider the stretching of bonds due to rotational motion
• Selection rules for vibrational and rotational transitions are derived from the transition dipole moment integral
• The intensity of spectral lines depends on the population of the initial state and the transition probability
• Quantum mechanical calculations provide insights into the structure, dynamics, and spectroscopic properties of molecules

Applications in Molecular Analysis

• Vibrational and rotational spectroscopy have numerous applications in molecular analysis and characterization
• Identification of functional groups and molecular structure
  • IR and Raman spectroscopy provide fingerprint regions for identifying specific functional groups (e.g., carbonyl, hydroxyl, amine)
  • Rotational spectroscopy can determine bond lengths and angles in molecules
• Quantitative analysis and concentration determination
  • The intensity of spectral lines is proportional to the concentration of the absorbing or emitting species (Beer-Lambert law)
• Monitoring of chemical reactions and kinetics
  • Time-resolved spectroscopy can track the formation and disappearance of intermediates and products during a reaction
• Environmental monitoring and trace gas detection
  • High-resolution spectroscopy can detect and quantify trace amounts of pollutants, greenhouse gases, and atmospheric species
• Astrochemistry and the study of interstellar molecules
  • Rotational and vibrational spectroscopy are used to identify and characterize molecules in interstellar space and planetary atmospheres
• Biomedical applications, such as breath analysis and disease diagnosis
  • Spectroscopic techniques can detect biomarkers and metabolites associated with specific diseases or physiological states

Key Equations and Formulas

• Number of vibrational modes:
  • Non-linear molecules: $3N-6$
  • Linear molecules: $3N-5$
• Vibrational energy levels (harmonic oscillator): $E_n = (n + \frac{1}{2})hν$
• Zero-point energy: $\frac{1}{2}hν$
• Rotational energy levels (rigid rotor): $E_J = \frac{h^2}{8π^2I}J(J+1)$
• Moment of inertia (diatomic molecule): $I = μr^2$, where $μ$ is the reduced mass and $r$ is the bond length
• Selection rules for vibrational transitions (harmonic oscillator): $Δn = ±1$
• Selection rules for rotational transitions: $ΔJ = ±1$
• Transition dipole moment integral: $μ_{fi} = \int ψ_f^* \hat{μ} ψ_i dτ$
• Beer-Lambert law: $A = εbc$, where $A$ is absorbance, $ε$ is molar absorptivity, $b$ is path length, and $c$ is concentration

Real-World Examples and Case Studies

• Greenhouse gas monitoring using IR spectroscopy
  • Measuring the concentration of $CO_2$, $CH_4$, and other greenhouse gases in the atmosphere
  • Tracking the sources and sinks of these gases to understand climate change
• Breath analysis for disease diagnosis
  • Detecting volatile organic compounds (VOCs) in exhaled breath as biomarkers for lung cancer, asthma, and other respiratory diseases
  • Developing non-invasive and early-stage diagnostic tools based on spectroscopic techniques
• Astrochemistry and the search for interstellar molecules
  • Identifying new molecules in interstellar clouds and circumstellar envelopes using radio and submillimeter spectroscopy
  • Understanding the chemical complexity and evolution of the universe
• Monitoring industrial processes and product quality control
  • Using vibrational spectroscopy to ensure the purity and consistency of pharmaceutical products, polymers, and materials
  • Detecting contaminants or impurities in real-time during manufacturing processes
• Studying the structure and dynamics of biomolecules
  • Investigating the secondary structure of proteins using IR and Raman spectroscopy
  • Probing the hydrogen bonding and hydration of DNA and RNA using vibrational spectroscopy techniques