Introduction to Spectroscopy

Spectroscopy Definition: The study of the interaction between electromagnetic radiation ( $\text{EMR}$ ) and matter (molecules, atoms, ions, or polymers).
Electromagnetic Radiation ( $\text{EMR}$ ): Energy propagating through space as oscillating electric and magnetic fields that are mutually perpendicular and perpendicular to the direction of wave propagation.
Key Wave Parameters & Equations:
- Frequency ( $\nu$ ): Number of vibrations per second, measured in Hertz ( $\text{Hz}$ ) or $\text{s}^{-1}$ .
- Wavelength ( $\lambda$ ): Distance between adjacent crests or troughs, measured in meters ( $\text{m}$ ) or nanometers ( $\text{nm}$ ).
- Wavenumber ( $\bar{\nu}$ ): Reciprocal of wavelength, $\bar{\nu} = \frac{1}{\lambda}$ , expressed in $\text{cm}^{-1}$ .
- Speed of Light ( $c$ ): $2.998 \times 10^8\,\text{m}\,\text{s}^{-1}$ .
- Planck's Constant ( $h$ ): $6.626 \times 10^{-34}\,\text{J}\,\text{s}$ ( $6.6260 \times 10^{-34}\,\text{kg}\,\text{m}^2\,\text{s}^{-1}$ ).
- Energy Transitions: $\Delta E = h\nu = \frac{hc}{\lambda} = hc\bar{\nu}$

Quantization: Molecular energy levels are quantized, meaning molecules can exist only in specific, discrete energy states.
Internal Energy Components:
- Electronic Energy ( $E_e$ ): Energy from electrons in fixed electronic orbitals.
- Vibrational Energy ( $E_v$ ): Energy from the oscillation of chemical bonds.
- Rotational Energy ( $E_r$ ): Energy from molecular rotation about an axis.
Energy Magnitude Order: $E_e > E_v > E_r$

X-ray Spectroscopy / Auger Effect:
- Energy Range: $10^4 - 10^6\,\text{kJ}\,\text{mol}^{-1}$ (or higher)
- Mechanism: Excitation of inner-shell electrons.
UV-Visible Spectroscopy:
- Energy Range: $100 - 1000\,\text{kJ}\,\text{mol}^{-1}$
- Mechanism: Excitation of outer valence electrons.
Infrared (IR) Spectroscopy:
- Energy Range: $1 - 50\,\text{kJ}\,\text{mol}^{-1}$
- Mechanism: Molecular vibrational transitions.
Microwave (Rotational) Spectroscopy:
- Energy Range: $0.01 - 1\,\text{kJ}\,\text{mol}^{-1}$
- Mechanism: Changes in rotational energy levels.
Nuclear Magnetic Resonance (NMR) Spectroscopy:
- Energy Range: $4 \times 10^{-5} - 2 \times 10^{-4}\,\text{kJ}\,\text{mol}^{-1}$
- Mechanism: Nuclear spin transitions induced by radio frequencies.

Microwave Spectroscopy:
- Frequency / Wavelength: $\nu \approx 300\,\text{MHz} - 300\,\text{GHz}$ ; $\lambda \approx 1\,\text{mm} - 1\,\text{m}$ .
- Transition Energy: $\approx 0.1 - 120\,\text{J}\,\text{mol}^{-1}$ .
- Selection Rule: Molecule must possess a permanent dipole moment (e.g., $\text{HCl}$ , $\text{CO}$ , $\text{NO}$ , $\text{H}_2\text{O}$ , $\text{CH}_3\text{Cl}$ ).
- Applications: Determination of bond lengths, isotopic masses, and dipole moments.
Vibrational (Infrared) Spectroscopy:
- Spectral Region: $4000 - 400\,\text{cm}^{-1}$ ( $2.5 - 25\,\mu\text{m}$ ).
- Transition Energy: $\approx 5 - 50\,\text{kJ}\,\text{mol}^{-1}$ .
- Mechanism: Occurs when IR frequency matches the natural vibrational frequency of oscillating bond masses.
UV-Visible Spectroscopy:
- Spectral Region: Wavelength $200 - 800\,\text{nm}$ ; energy $150 - 600\,\text{kJ}\,\text{mol}^{-1}$ .
- Mechanism: Promotes electrons from the highest occupied molecular orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO).
- Electronic Transitions: Includes $\sigma \rightarrow \sigma^*$ , $n \rightarrow \sigma^*$ , $\pi \rightarrow \pi^*$ , and $n \rightarrow \pi^*$ .
- Applications: Quantitative analysis via the Beer-Lambert law, reaction monitoring, and materials characterization.

Population Distribution: Describes the population ratio of excited state ( $N_1$ ) to ground state ( $N_0$ ) at thermal equilibrium: $\frac{N_1}{N_0} = \exp\left(-\frac{\Delta E}{k T}\right)$
Parameters:
- $k$ (Boltzmann constant): $1.381 \times 10^{-23}\,\text{J}\,\text{K}^{-1}$ .
- $T$ : Absolute temperature in Kelvin ( $\text{K}$ ).
- $\Delta E$ : Energy difference between levels ( $E_1 - E_0$ ).
Key Principles:
- Ground state population is higher than excited state population at thermal equilibrium.
- Excited state population decreases as $\Delta E$ increases.
- Temperature increases the population of the excited state.

Absorption Spectrum:
- Process: Electrons absorb photons and move from ground state to excited state.
- Appearance: Dark lines or bands superimposed on a continuous spectrum.
- Energy Change: Energy is absorbed.
- Applications: Compound identification and concentration analysis (UV-Vis).
Emission Spectrum:
- Process: Excited electrons relax to lower energy levels, emitting photons.
- Appearance: Bright colored lines or bands on a dark background.
- Energy Change: Energy is released.
- Applications: Fluorescence analysis, atomic emission, and plasma analysis.